ReliaWiki - User contributions [en]

ReliaSoft API Reference

2024-04-12T20:00:59Z

Melinda Caroline:

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The ReliaSoft ''Application Programming Interface'' (API) enables you to programmatically read and write data to ReliaSoft repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the ReliaSoft API allows you to connect data in repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/xfracas-failure-reporting-analysis-corrective-action-system-fracas-software XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/blocksim-system-reliability-availability-maintainability-ram-analysis-software BlockSim].
*'''Read/write specific data elements'''
:Read or write data to resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from ReliaSoft repositories, as well as load data into the Reliability Data Warehouse (RDW).
*'''Obtain calculated results'''
:Read data from published models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [https://hbkworld.com/en/products/software/analysis-simulation/reliability/weibull-life-data-analysis-software Weibull++'s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a ReliaSoft API license.)

==Prerequisites==
*'''Getting the latest version'''
:The ReliaSoft API consists of a library file that contains the classes, methods and events that provide access to ReliaSoft applications. The file is located in the folder where the ReliaSoft desktop applications are installed. By default, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file. Prior to Version 2019, the files to use were SynthesisAPI.tlb (VBA) and SynthesisAPI.dll (.NET).

:To obtain the latest version of the library file and the ReliaSoft applications, choose '''File > Help > Check for Updates''' or download the latest version from the [https://support.hbmprenscia.com HBM Prenscia Support Portal].

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime. Starting in Version 2022, .NET Framework 4.8 developer pack is required.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the ReliaSoft applications that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls.

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. It is necessary to ensure your application bitness matches the bitness of the ReliaSoft desktop applications installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [https://help.reliasoft.com/basics/content/reliasoft_installation_and_licensing.htm locally hosted licensing] with Prenscia Access.

*'''Access to calculations functionality
:Access to Weibull++'s analysis and plotting engines requires a ReliaSoft API license. To obtain a license, please see [https://www.hbkworld.com/en/contact-us/reliasoft-worldwide-contacts contact ReliaSoft].

==Getting Started==

====Referencing the ReliaSoft API Library File====

<onlyinclude>To use the ReliaSoft API in your code, you must reference the ReliaSoft API library in your VBA or .NET projects. The file is located in the folder where the ReliaSoft desktop applications are installed. By default for Version 2023, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the ReliaSoftAPI23.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node of your project and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the ReliaSoftAPI23.dll file on your computer. Click '''OK'''.
:#Edit your project file, either directly using any text editor, or by right-clicking your project and choosing '''Unload Project''', then right-clicking your project again and choosing '''Edit Project File'''
:#Insert the following line just before the last </Project> clause (change the path if necessary):
:::
::::::<Import Project="C:\Program Files\ReliaSoft\ReliaSoft 2023\ReliaSoftAPI.targets" />
:::
:::This will ensure the required files are copied into your project’s destination folder during the build.
:#<li value="5">Save the project file. If you unloaded the project in step 3, right-click the project and choose '''Reload Project'''.</li>
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the ReliaSoft installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples23\API.

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the ReliaSoft API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the ReliaSoft API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]].

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the ReliaSoft repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the ReliaSoft resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Reliability Data Warehouse (RDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

ReliaSoft API Reference

2024-03-11T21:43:32Z

Melinda Caroline: /* Prerequisites */

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The ReliaSoft ''Application Programming Interface'' (API) enables you to programmatically read and write data to ReliaSoft repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the ReliaSoft API allows you to connect data in repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/xfracas-failure-reporting-analysis-corrective-action-system-fracas-software XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/blocksim-system-reliability-availability-maintainability-ram-analysis-software BlockSim].
*'''Read/write specific data elements'''
:Read or write data to resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from ReliaSoft repositories, as well as load data into the Reliability Data Warehouse (RDW).
*'''Obtain calculated results'''
:Read data from published models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [https://hbkworld.com/en/products/software/analysis-simulation/reliability/weibull-life-data-analysis-software Weibull++'s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a ReliaSoft API license.)

==Prerequisites==
*'''Getting the latest version'''
:The ReliaSoft API consists of a library file that contains the classes, methods and events that provide access to ReliaSoft applications. The file is located in the folder where the ReliaSoft desktop applications are installed. By default, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file. Prior to Version 2019, the files to use were SynthesisAPI.tlb (VBA) and SynthesisAPI.dll (.NET).

:To obtain the latest version of the library file and the ReliaSoft applications, choose '''File > Help > Check for Updates''' or download the latest version from the [https://support.hbmprenscia.com HBM Prenscia Support Portal].

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime. Starting in Version 2022, .NET Framework 4.8 developer pack is required.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the ReliaSoft applications that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls.

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. It is necessary to ensure your application bitness matches the bitness of the ReliaSoft desktop applications installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [https://help.reliasoft.com/basics/content/reliasoft_installation_and_licensing.htm locally hosted licensing] and you’ll need to install the ReliaSoft API Ante Service to scale the implementation (the service is not required when running the API on a Windows desktop environment). For instructions on how to install and configure the service, see https://support.hbmprenscia.com/s/article/ReliaSoft-API-Ante-Service.

*'''Access to calculations functionality
:Access to Weibull++'s analysis and plotting engines requires a ReliaSoft API license. To obtain a license, please see [https://www.hbkworld.com/en/contact-us/reliasoft-worldwide-contacts contact ReliaSoft].

==Getting Started==

====Referencing the ReliaSoft API Library File====

<onlyinclude>To use the ReliaSoft API in your code, you must reference the ReliaSoft API library in your VBA or .NET projects. The file is located in the folder where the ReliaSoft desktop applications are installed. By default for Version 2023, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the ReliaSoftAPI23.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node of your project and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the ReliaSoftAPI23.dll file on your computer. Click '''OK'''.
:#Edit your project file, either directly using any text editor, or by right-clicking your project and choosing '''Unload Project''', then right-clicking your project again and choosing '''Edit Project File'''
:#Insert the following line just before the last </Project> clause (change the path if necessary):
:::
::::::<Import Project="C:\Program Files\ReliaSoft\ReliaSoft 2023\ReliaSoftAPI.targets" />
:::
:::This will ensure the required files are copied into your project’s destination folder during the build.
:#<li value="5">Save the project file. If you unloaded the project in step 3, right-click the project and choose '''Reload Project'''.</li>
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the ReliaSoft installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples23\API.

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the ReliaSoft API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the ReliaSoft API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]].

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the ReliaSoft repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the ReliaSoft resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Reliability Data Warehouse (RDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

ReliaSoft API Reference

2024-03-11T21:42:21Z

Melinda Caroline: /* Prerequisites */

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The ReliaSoft ''Application Programming Interface'' (API) enables you to programmatically read and write data to ReliaSoft repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the ReliaSoft API allows you to connect data in repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/xfracas-failure-reporting-analysis-corrective-action-system-fracas-software XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/blocksim-system-reliability-availability-maintainability-ram-analysis-software BlockSim].
*'''Read/write specific data elements'''
:Read or write data to resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from ReliaSoft repositories, as well as load data into the Reliability Data Warehouse (RDW).
*'''Obtain calculated results'''
:Read data from published models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [https://hbkworld.com/en/products/software/analysis-simulation/reliability/weibull-life-data-analysis-software Weibull++'s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a ReliaSoft API license.)

==Prerequisites==
*'''Getting the latest version'''
:The ReliaSoft API consists of a library file that contains the classes, methods and events that provide access to ReliaSoft applications. The file is located in the folder where the ReliaSoft desktop applications are installed. By default, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file. Prior to Version 2019, the files to use were SynthesisAPI.tlb (VBA) and SynthesisAPI.dll (.NET).

:To obtain the latest version of the library file and the ReliaSoft applications, choose '''File > Help > Check for Updates''' or download the latest version from the [https://support.hbmprenscia.com HBM Prenscia Support Portal].

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime. Starting in Version 2022, .NET Framework 4.8 developer pack is required.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the ReliaSoft applications that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls.

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. It is necessary to ensure your application bitness matches the bitness of the ReliaSoft desktop applications installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [https://www.hbkworld.com/en/knowledge/resource-center/resources/2022/locally-hosted-licensing locally hosted licensing] and you’ll need to install the ReliaSoft API Ante Service to scale the implementation (the service is not required when running the API on a Windows desktop environment). For instructions on how to install and configure the service, see https://support.hbmprenscia.com/s/article/ReliaSoft-API-Ante-Service.

*'''Access to calculations functionality
:Access to Weibull++'s analysis and plotting engines requires a ReliaSoft API license. To obtain a license, please see [https://www.hbkworld.com/en/contact-us/reliasoft-worldwide-contacts contact ReliaSoft].

==Getting Started==

====Referencing the ReliaSoft API Library File====

<onlyinclude>To use the ReliaSoft API in your code, you must reference the ReliaSoft API library in your VBA or .NET projects. The file is located in the folder where the ReliaSoft desktop applications are installed. By default for Version 2023, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the ReliaSoftAPI23.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node of your project and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the ReliaSoftAPI23.dll file on your computer. Click '''OK'''.
:#Edit your project file, either directly using any text editor, or by right-clicking your project and choosing '''Unload Project''', then right-clicking your project again and choosing '''Edit Project File'''
:#Insert the following line just before the last </Project> clause (change the path if necessary):
:::
::::::<Import Project="C:\Program Files\ReliaSoft\ReliaSoft 2023\ReliaSoftAPI.targets" />
:::
:::This will ensure the required files are copied into your project’s destination folder during the build.
:#<li value="5">Save the project file. If you unloaded the project in step 3, right-click the project and choose '''Reload Project'''.</li>
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the ReliaSoft installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples23\API.

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the ReliaSoft API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the ReliaSoft API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]].

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the ReliaSoft repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the ReliaSoft resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Reliability Data Warehouse (RDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

ReliaSoft API Reference

2023-07-13T17:06:03Z

Melinda Caroline:

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The ReliaSoft ''Application Programming Interface'' (API) enables you to programmatically read and write data to ReliaSoft repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the ReliaSoft API allows you to connect data in repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/xfracas-failure-reporting-analysis-corrective-action-system-fracas-software XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/blocksim-system-reliability-availability-maintainability-ram-analysis-software BlockSim].
*'''Read/write specific data elements'''
:Read or write data to resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from ReliaSoft repositories, as well as load data into the Reliability Data Warehouse (RDW).
*'''Obtain calculated results'''
:Read data from published models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [https://hbkworld.com/en/products/software/analysis-simulation/reliability/weibull-life-data-analysis-software Weibull++'s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a ReliaSoft API license.)

==Prerequisites==
*'''Getting the latest version'''
:The ReliaSoft API consists of a library file that contains the classes, methods and events that provide access to ReliaSoft applications. The file is located in the folder where the ReliaSoft desktop applications are installed. By default, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file. Prior to Version 2019, the files to use were SynthesisAPI.tlb (VBA) and SynthesisAPI.dll (.NET).

:To obtain the latest version of the library file and the ReliaSoft applications, choose '''File > Help > Check for Updates''' or download the latest version from the [https://support.hbmprenscia.com HBM Prenscia Support Portal].

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime. Starting in Version 2022, .NET Framework 4.8 developer pack is required.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the ReliaSoft applications that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls.

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. It is necessary to ensure your application bitness matches the bitness of the ReliaSoft desktop applications installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [https://www.hbkworld.com/en/knowledge/resource-center/resources/2022/locally-hosted-licensing locally hosted licensing] and you’ll need to install the ReliaSoft API Ante Service to scale the implementation (the service is not required when running the API on a Windows desktop environment). For instructions on how to install and configure the service, see https://www.hbkworld.com/en/knowledge/resource-center/resources/2022/reliasoft-api-ante-service.

*'''Access to calculations functionality
:Access to Weibull++'s analysis and plotting engines requires a ReliaSoft API license. To obtain a license, please see [https://www.hbkworld.com/en/contact-us/reliasoft-worldwide-contacts contact ReliaSoft].

==Getting Started==

====Referencing the ReliaSoft API Library File====

<onlyinclude>To use the ReliaSoft API in your code, you must reference the ReliaSoft API library in your VBA or .NET projects. The file is located in the folder where the ReliaSoft desktop applications are installed. By default for Version 2023, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the ReliaSoftAPI23.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node of your project and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the ReliaSoftAPI23.dll file on your computer. Click '''OK'''.
:#Edit your project file, either directly using any text editor, or by right-clicking your project and choosing '''Unload Project''', then right-clicking your project again and choosing '''Edit Project File'''
:#Insert the following line just before the last </Project> clause (change the path if necessary):
:::
::::::<Import Project="C:\Program Files\ReliaSoft\ReliaSoft 2023\ReliaSoftAPI.targets" />
:::
:::This will ensure the required files are copied into your project’s destination folder during the build.
:#<li value="5">Save the project file. If you unloaded the project in step 3, right-click the project and choose '''Reload Project'''.</li>
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the ReliaSoft installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples23\API.

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the ReliaSoft API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the ReliaSoft API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]].

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the ReliaSoft repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the ReliaSoft resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Reliability Data Warehouse (RDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

ReliaSoft API Reference

2023-07-13T17:00:48Z

Melinda Caroline:

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The ReliaSoft ''Application Programming Interface'' (API) enables you to programmatically read and write data to ReliaSoft repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the ReliaSoft API allows you to connect data in repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/xfracas-failure-reporting-analysis-corrective-action-system-fracas-software XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/blocksim-system-reliability-availability-maintainability-ram-analysis-software BlockSim].
*'''Read/write specific data elements'''
:Read or write data to resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from ReliaSoft repositories, as well as load data into the Synthesis Data Warehouse (SDW).
*'''Obtain calculated results'''
:Read data from published models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [https://hbkworld.com/en/products/software/analysis-simulation/reliability/weibull-life-data-analysis-software Weibull++'s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a ReliaSoft API license.)

==Prerequisites==
*'''Getting the latest version'''
:The ReliaSoft API consists of a library file that contains the classes, methods and events that provide access to ReliaSoft applications. The file is located in the folder where the ReliaSoft desktop applications are installed. By default, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file. Prior to Version 2019, the files to use were SynthesisAPI.tlb (VBA) and SynthesisAPI.dll (.NET).

:To obtain the latest version of the library file and the ReliaSoft applications, choose '''File > Help > Check for Updates''' or download the latest version from the [https://support.hbmprenscia.com HBM Prenscia Support Portal].

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime. Starting in Version 2022, .NET Framework 4.8 developer pack is required.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the ReliaSoft applications that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls.

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. It is necessary to ensure your application bitness matches the bitness of the ReliaSoft desktop applications installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [https://www.hbkworld.com/en/knowledge/resource-center/resources/2022/locally-hosted-licensing locally hosted licensing] and you’ll need to install the ReliaSoft API Ante Service to scale the implementation (the service is not required when running the API on a Windows desktop environment). For instructions on how to install and configure the service, see https://www.hbkworld.com/en/knowledge/resource-center/resources/2022/reliasoft-api-ante-service.

*'''Access to calculations functionality
:Access to Weibull++'s analysis and plotting engines requires a ReliaSoft API license. To obtain a license, please see [https://www.hbkworld.com/en/contact-us/reliasoft-worldwide-contacts contact ReliaSoft].

==Getting Started==

====Referencing the ReliaSoft API Library File====

<onlyinclude>To use the ReliaSoft API in your code, you must reference the ReliaSoft API library in your VBA or .NET projects. The file is located in the folder where the ReliaSoft desktop applications are installed. By default for Version 2023, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the ReliaSoftAPI23.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node of your project and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the ReliaSoftAPI23.dll file on your computer. Click '''OK'''.
:#Edit your project file, either directly using any text editor, or by right-clicking your project and choosing '''Unload Project''', then right-clicking your project again and choosing '''Edit Project File'''
:#Insert the following line just before the last </Project> clause (change the path if necessary):
:::
::::::<Import Project="C:\Program Files\ReliaSoft\ReliaSoft 2023\ReliaSoftAPI.targets" />
:::
:::This will ensure the required files are copied into your project’s destination folder during the build.
:#<li value="5">Save the project file. If you unloaded the project in step 3, right-click the project and choose '''Reload Project'''.</li>
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the ReliaSoft installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples23\API.

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the ReliaSoft API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the ReliaSoft API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]].

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the ReliaSoft repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the ReliaSoft resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Synthesis Data Warehouse (SDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

ReliaSoft API Reference

2023-07-13T16:34:04Z

Melinda Caroline:

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The ReliaSoft ''Application Programming Interface'' (API) enables you to programmatically read and write data to ReliaSoft repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the ReliaSoft API allows you to connect data in repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/xfracas-failure-reporting-analysis-corrective-action-system-fracas-software XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/blocksim-system-reliability-availability-maintainability-ram-analysis-software BlockSim].
*'''Read/write specific data elements'''
:Read or write data to resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from ReliaSoft repositories, as well as load data into the Synthesis Data Warehouse (SDW).
*'''Obtain calculated results'''
:Read data from published models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [https://hbkworld.com/en/products/software/analysis-simulation/reliability/weibull-life-data-analysis-software Weibull++'s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a ReliaSoft API license.)

==Prerequisites==
*'''Getting the latest version'''
:The ReliaSoft API consists of a library file that contains the classes, methods and events that provide access to ReliaSoft applications. The file is located in the folder where the ReliaSoft desktop applications are installed. By default, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file. Prior to Version 2019, the files to use were SynthesisAPI.tlb (VBA) and SynthesisAPI.dll (.NET).

:To obtain the latest version of the library file and the ReliaSoft applications, choose '''File > Help > Check for Updates''' or download the latest version from the [https://support.hbmprenscia.com HBM Prenscia Support Portal].

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime. Starting in Version 2022, .NET Framework 4.8 developer pack is required.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the ReliaSoft applications that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls.)

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. It is necessary to ensure your application bitness matches the bitness of the ReliaSoft desktop applications installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [https://www.hbkworld.com/en/knowledge/resource-center/resources/2022/locally-hosted-licensing locally hosted licensing] and you’ll need to install the ReliaSoft API Ante Service to scale the implementation (the service is not required when running the API on a Windows desktop environment). For instructions on how to install and configure the service, see https://www.hbkworld.com/en/knowledge/resource-center/resources/2022/reliasoft-api-ante-service.

*'''Access to calculations functionality
:Access to Weibull++'s analysis and plotting engines requires a ReliaSoft API license. To obtain a license, please see [https://www.hbkworld.com/en/contact-us/reliasoft-worldwide-contacts contact ReliaSoft].

==Getting Started==

====Referencing the ReliaSoft API Library File====

<onlyinclude>To use the ReliaSoft API in your code, you must reference the ReliaSoft API library in your VBA or .NET projects. The file is located in the folder where the ReliaSoft desktop applications are installed. By default for Version 2023, this folder is at "C:\Program Files\ReliaSoft\ReliaSoft 2023." For VBA, you will use the '''ReliaSoftAPI23.tlb''' file; for .NET applications, you will use the '''ReliaSoftAPI23.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the ReliaSoftAPI23.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node of your project and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the ReliaSoftAPI23.dll file on your computer. Click '''OK'''.
:#Edit your project file, either directly using any text editor, or by right-clicking your project and choosing '''Unload Project''', then right-clicking your project again and choosing '''Edit Project File'''
:#Insert the following line just before the last </Project> clause (change the path if necessary):
:::
::::::<Import Project="C:\Program Files\ReliaSoft\ReliaSoft 2023\ReliaSoftAPI.targets" />
:::
:::This will ensure the required files are copied into your project’s destination folder during the build.
:#<li value="5">Save the project file. If you unloaded the project in step 3, right-click the project and choose '''Reload Project'''.</li>
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the ReliaSoft installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples23\API.

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the ReliaSoft API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the ReliaSoft API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]].

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the ReliaSoft repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the ReliaSoft resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Synthesis Data Warehouse (SDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

File:Fmra table 3.png

2017-03-06T15:25:39Z

Melinda Caroline: uploaded a new version of "File:Fmra table 3.png"

State Change Triggers

2017-02-06T18:18:29Z

Melinda Caroline: /* Assumptions */

<noinclude>{{Banner BlockSim Articles}}{{Navigation box}}
''This article, which discusses the use of State Change Triggers (SCT) in simulation diagrams, also appears in the [[Repairable_Systems_Analysis_Through_Simulation#State_Change_Triggers|System Analysis Reference]] book.''
</noinclude>
Consider a case where you have two generators, and one (A) is primary while the other (B) is standby. If A fails, you will turn B on. When A is repaired, it then becomes the standby. State change triggers (SCT) allow you to simulate this case. You can specify events that will activate and/or deactivate the block during simulation. The figure below shows the options for state change triggers in the Block Properties window.
[[Image:State Change Trigger Options.png‎|center|600px|link=]]

Once you have enabled state change triggers for a block, there are several options.

*'''Initial state''' allows you to specify the initial state for the block, either ON or OFF.
*'''State upon repair''' allows you to specify the state of the block after its repair. There are four choices: Always ON, Always OFF, Default ON unless SCT Overridden and Default OFF unless SCT Overridden. In the Assumptions sections, we will explain what these choices mean and illustrate them using an example.
*'''Add a state change trigger''' allows you to add a state change trigger to the block.

The state change trigger can either activate or deactivate the block when items in specified maintenance groups go down or are restored. To define the state change trigger, specify the triggering event (i.e., an item goes down or an item is restored), the state change (i.e., the block is activated or deactivated) and the maintenance group(s) in which the triggering event must happen in order to trigger the state change. Note that the current block does not need to be part of the specified maintenance group(s) to use this functionality.

The State Change Trigger window is shown in the figure below:
[[Image:State_change_trigger_window.png‎|center|500px|link=]]

== Assumptions ==
*A block cannot trigger events on itself. For example, if Block 1 is the only block that belongs to MG 1 and Block 1 is set to be turned ON or OFF based on MG 1, this trigger is ignored.
*OFF events cannot trigger other events. This means that things cannot be turned OFF in cascade. For example, if Block 1 going down turns OFF Block 2 and Block 2 going down turns OFF Block 3, a failure by Block 1 will not turn OFF Block 3. Block 3 would have to be directly associated with downing events of Block 1 for this to happen. The reason for this restriction is that allowing OFF events to trigger other events can cause circular reference problems. For example, four blocks A, B, C and D are in parallel. Block A belongs to MG A and initially it is ON. Block B belongs to MG B and its initial status is also ON. Block C belongs to MG C and its initial status is OFF. Block D belongs to MG D and its initial status is ON. A failure of Block A will turn OFF Block B. Then Block B will turn Block C ON and finally C will turn OFF Block D. However, if an OFF event for Block D will turn Block B ON, and an ON event for Block B will turn Block C OFF, and an OFF event for Block C will turn Block D ON, then there is a circular reference problem.
*Upon restoration states:
**Always ON: Upon restoration, the block will always be on.
**Always OFF: Upon restoration, the block will always be off.
**Default ON unless SCT overridden: Upon restoration, the block will be on unless a request is made to turn this block off while the block is down and the request is still applicable at the time of restoration. For example, assume Block A's state upon repair is ON unless SCT overridden. If a failure of Block B triggers a request to turn Block A off but Block A is down, when the maintenance for Block A is completed, Block A will be turned off if Block B is still down.
**Default off unless SCT overridden: Upon restoration, the block will be off unless a request is made to turn this block on while the block is down and the request is still applicable at the time of restoration
*Maintenance while block is off: Maintenance tasks will be performed. At the end of the maintenance, "upon restoration" rules will be checked to determine the state of the block.
*Assumptions for phases: In Versions 10 and earlier, the state of a block (on/off) was determined at the beginning of each phase based on the "Initial state" setting of the block for that phase. Starting in Version 11, the state of the block transfers across phases instead of resetting based on initial settings.
*If there are multiple triggering requests put on a block when it is down, only the latest one is considered. The latest request will cancel all requests before it. For example, Block A fails at 20 and is down until 70. Block B fails at 30 and Block C fails at 40. Block A has state change triggers enabled such that it will be activated when Block B fails and it will be deactivated when Block C fails. Thus from 20 to 70, at 30, Block B will put a request on Block A to turn it ON and at 40, Block C will put another request to turn it OFF. In this case, according to our assumption, the request from Block C at 40 will cancel the request from Block B at 30. In the end, only the request from Block C will be considered. Thus, Block A will be turned OFF at 70 when it is done with repair.

==Example: Using SCT for Standby Rotation==
{{:Example_Using_SCT_for_Standby_Rotation}}

<div class="noprint">
{{Examples Box|BlockSim Examples|More examples are available for using State Change Triggers (SCTs) in simulation diagrams. See also:{{Examples Link|Example_Using_SCT_to_Analyze_Tire_Maintenance|Using SCT to Analyze Tire Maintenance}}{{Examples Link|Example_Using_SCT_to_Analyze_Standby_with_Delay|Using SCT to Analyze Standby with Delay}}{{Examples Link|Example_Using_SCT_to_Model_Two_Standby_Blocks|Using SCT to Model Two Standby Blocks}}{{Examples Link|Example_Demonstrating_the_State_Upon_Repair_Option_for_SCT|SCT: The State Upon Repair Option}}{{Examples Link|BlockSim_Example:_Default_OFF_unless_SCT_Overridden|Default OFF Unless SCT Overridden}}{{Examples Link|BlockSim_Example:_Default_OFF_unless_SCT_Overridden|Default OFF Unless SCT Overridden}}}}
</div>

Markov Diagrams

2016-10-17T17:38:13Z

Melinda Caroline:

{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector <math>\underline{X}\,\!</math>:

::<math>\underline{X}(0)= (X_1 X_2 \cdot\cdot\cdot X_i) \,\!</math>

where the term <math>X_i\,\!</math> represents the probability of the system being in state <math>i\,\!</math> and with:

::<math>\sum_{i=1}^{n}X_{i}=1\,\!</math>

The transitions between the states can be represented by a matrix <math>\underline{P}\,\!</math>:

::<math> \underline{P} =
\begin{pmatrix}
P_{11} & P_{12} & \cdots & P_{1i} \\
P_{21} & P_{22} & \cdots & P_{2i} \\
\vdots & \vdots & \ddots & \vdots \\
P_{i1} & P_{i2} & \cdots & P_{ii}
\end{pmatrix}\,\!</math>

where, for example, the term <math>P_{12}\,\!</math> is the transition probability from state 1 to state 2, and for any row <math>m\,\!</math> with <math>i\,\!</math> states:

::<math>\sum_{j=1}^{i}P_{mj}=1\,\!</math>

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector <math>\underline{X}(0)\,\!</math> with the transition matrix <math>\underline{P}\,\!</math>:

::<math>\underline{X}(1)=\underline{X}(0) \cdot \underline{P}\,\!</math>

This will give us the state probability vector after the first step, <math>\underline{X}(1)\,\!</math>.

If one wants to determine the probabilities of the system being in a particular state after <math>n\,\!</math> steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after <math>n\,\!</math> steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the <math>n\,\!</math>th power, or:

::<math>\underline{X}(n)=\underline{X}(0) \cdot \underline{P}^n\,\!</math>

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

Then, we can create the transition matrix to represent all the various transition probabilities between states:

::<math>\underline{P}=\begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}\,\!</math>

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

::<math>\underline{X}(10)=\underline{X}(0) \cdot \underline{P}^{10}\,\!</math>

or:

::<math>\underline{X}(10)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}^{10}\,\!</math>

resulting in :

::<math>\underline{X}(10)=\begin{pmatrix} 0.596 & 0.353 & 0.051 \end{pmatrix}\,\!</math>

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chains=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

::<math>\frac{dP_j}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j}\,\!</math>

where:

*<math>n\,\!</math> is the total number of states
*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_l\,\!</math> is the probability of being in state <math>l\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state <math>l\,\!</math> to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state <math>j\,\!</math> to state <math>l\,\!</math>

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

::<math>\frac{dP_{working}}{dt}=0.01\cdot P_{repair} - 0.0001 \cdot P_{working} \,\!</math>

::<math>\frac{dP_{repair}}{dt}=0.0001\cdot P_{working} - 0.01 \cdot P_{repair} \,\!</math>

with initial bounds of:

::<math>P_{working}(0)=1 \,\!</math>

::<math>P_{repair}(0)=0 \,\!</math>

Solving this system of equations yields the following results:

::<math>P_{working}=\frac{0.01}{0.0101}+\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

::<math>P_{repair}=\frac{0.0001}{0.0101}-\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

::<math>f_j=\frac{dP_{j}}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j} \,\!</math>

::<math>k_1=hf(t_i,P_{l,i})\,\!</math>

::<math>k_2=hf\Big( t_i+\frac{h}{4},P_{l,i}+\frac{k_1}{4}\Big)\,\!</math>

::<math>k_3=hf\Big( t_i+\frac{3h}{8},P_{l,i}+\frac{3}{32}k_2\Big)\,\!</math>

::<math>k_4=hf\Big( t_i+\frac{12h}{13},P_{l,i}+\frac{1932}{2197}k_1+\frac{7200}{2197}k_2+\frac{7296}{2197}k_3\Big)\,\!</math>

::<math>k_5=hf\Big( t_i+h,P_{l,i}+\frac{439}{216}k_1+8k_2+\frac{3680}{513}k_3-\frac{845}{4104}k_4\Big)\,\!</math>

::<math>k_6=hf\Big( t_i+\frac{h}{2},P_{l,i}-\frac{8}{27}k_1+2k_2-\frac{3544}{2565}k_3+\frac{1859}{4104}k_4-\frac{11}{40}k_5\Big)\,\!</math>

::<math>P_{j,i+1}=P_{j,i}+\frac{25}{216}k_1+\frac{1408}{2565}k_3+\frac{2197}{4104}k_4-\frac{1}{5}k_5\,\!</math>

::<math>\tilde{P}_{j,i+1}=P_{j,i}+\frac{16}{135}k_1+\frac{6656}{12825}k_3+\frac{28561}{56430}k_4-\frac{9}{50}k_5+\frac{2}{55}k_6\,\!</math>

::<math>R=\frac{1}{h}|\tilde{P}_{j,i+1}-P_{j,i+1}|\,\!</math>

::<math>\delta=0.84\Big(\frac{\epsilon}{R}\Big)^{\frac{1}{4}}\,\!</math>

If <math>R\leq\epsilon\,\!</math> then keep <math>P_{i+1}\,\!</math> as the current solution, and move to next step with step size <math>\delta{h}\,\!</math>; if <math>R>\epsilon\,\!</math> then recalculate the current step with step size <math>\delta{h}\,\!</math>

where:

*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_{l,i}\,\!</math> is the probability of being in state <math>l\,\!</math> at time <math>i\,\!</math>
*<math>\lambda_{lj}\,\!</math> is the transition rate from state <math>l\,\!</math> to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> is the transition rate from state <math>j\,\!</math> to state <math>l\,\!</math>
*<math>f_j\,\!</math> is the change in the probability of being in state <math>P_j\,\!</math> (note that <math>f_j\,\!</math> is not a function of time for constant transition rate Markov chains)
*<math>h\,\!</math> is the time step size
*<math>t_i\,\!</math> is the time at <math>i\,\!</math>
*<math>\varepsilon\,\!</math> is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Optimum Replacement Time Example

2016-08-18T22:54:50Z

Melinda Caroline:

<noinclude>{{Banner BlockSim Examples}}
''This example appears in the article [[Preventive Maintenance]]''.

This example demonstrates BlockSim's Optimum Replacement utility, which allows you to determine the most cost-effective time to replace one or more blocks in one or more systems, based on their planned and unplanned replacement costs.

'''Optimum Replacement Time'''
</noinclude>
The failure distribution of a component is described by a 2-parameter Weibull distribution with <math>\beta = 2.5\,\!</math> and <math>\eta = 1000\,\!</math> hours.

:* The cost for a corrective replacement is $5. 
:* The cost for a preventive replacement is $1. 

Estimate the optimum replacement age in order to minimize these costs.

'''Solution'''

Prior to obtaining an optimum replacement interval for this component, the assumptions of the following equation must be checked.

::<math>\begin{align}
CPUT\left( t \right)= & \frac{\text{Total Expected Replacement Cost per Cycle}}{\text{Expected Cycle Length}} \\
= & \frac{{{C}_{P}}\cdot R\left( t \right)+{{C}_{U}}\cdot \left[ 1-R\left( t \right) \right]}{\int_{0}^{t}R\left( s \right)ds}
\end{align}\,\!</math>

The component has an increasing failure rate because it follows a Weibull distribution with <math>\beta \,\!</math> greater than 1. Note that if <math>\beta =1\,\!</math>, then the component has a constant failure rate, but if <math>\beta <1\,\!</math>, then it has a decreasing failure rate. If either of these cases exist, then preventive replacement is unwise. Furthermore, the cost for preventive replacement is less than the corrective replacement cost. Thus, the conditions for the optimum age replacement policy have been met.

Using BlockSim, enter the parameters of the Weibull distribution in the component's Block Properties window. Next, open the Optimum Replacement window and enter the '''1''' in the '''Planned Replacement Cost''' column, and '''5''' in the '''Unplanned Replacement Cost''' column. Click '''Calculate'''. In the Optimum Replacement Calculations window that appears, select the '''Individual''' option and click '''OK'''. The optimum replacement time for the component is estimated to be 493.0470, as shown next.

[[Image:optimum replacement.png|center|BlockSim's Optimum Replacement Utility|link=]]

The figure below shows the Cost vs. Time plot of the component (with the scaling adjusted and the plot annotated to show the minimum cost).

[[Image:BS7.6.png|center|600px|Cost vs. Replacement Time|link=]]

If we enter different cost values in the '''Unplanned Replacement Cost''' column and obtain the optimum replacement time at each value, we can use the data points to create a plot that shows the effect of the corrective cost on the optimum replacement interval. The following plot shows an example. In this case, the optimum replacement interval decreases as the cost ratio increases. This is an expected result because the corrective replacement costs are much greater than the preventive replacement costs. Therefore, it is more cost-effective to replace the component more frequently before it fails.

[[Image:BS7.7.png|center|600px|Replacement interval as a function of the corrective/preventive cost ratio.|link=]]

File:Optimum replacement.png

2016-08-18T22:53:22Z

Melinda Caroline: uploaded a new version of "File:Optimum replacement.png"

Optimum Replacement Time Example

2016-08-18T22:47:12Z

Melinda Caroline:

<noinclude>{{Banner BlockSim Examples}}
''This example appears in the article [[Preventive Maintenance]]''.

This example demonstrates BlockSim's Optimum Replacement utility, which allows you to determine the most cost-effective time to replace one or more blocks in one or more systems, based on their planned and unplanned replacement costs.

'''Optimum Replacement Time'''
</noinclude>
The failure distribution of a component is described by a 2-parameter Weibull distribution with <math>\beta = 2.5\,\!</math> and <math>\eta = 1000\,\!</math> hours.

:* The cost for a corrective replacement is $5. 
:* The cost for a preventive replacement is $1. 

Estimate the optimum replacement age in order to minimize these costs.

'''Solution'''

Prior to obtaining an optimum replacement interval for this component, the assumptions of the following equation must be checked.

::<math>\begin{align}
CPUT\left( t \right)= & \frac{\text{Total Expected Replacement Cost per Cycle}}{\text{Expected Cycle Length}} \\
= & \frac{{{C}_{P}}\cdot R\left( t \right)+{{C}_{U}}\cdot \left[ 1-R\left( t \right) \right]}{\int_{0}^{t}R\left( s \right)ds}
\end{align}\,\!</math>

The component has an increasing failure rate because it follows a Weibull distribution with <math>\beta \,\!</math> greater than 1. Note that if <math>\beta =1\,\!</math>, then the component has a constant failure rate, but if <math>\beta <1\,\!</math>, then it has a decreasing failure rate. If either of these cases exist, then preventive replacement is unwise. Furthermore, the cost for preventive replacement is less than the corrective replacement cost. Thus, the conditions for the optimum age replacement policy have been met.

Using BlockSim, enter the parameters of the Weibull distribution in the component's Block Properties window. Next, open the Optimum Replacement window and enter the '''1''' in the '''Planned Replacement Cost''' column, and '''5''' in the '''Unplanned Replacement Cost''' column. Click '''Calculate'''. In the Optimum Replacement Calculations window that appears, select the '''Individual''' option and click '''OK'''. The optimum replacement time for the component is estimated to be 493.0470, as shown next.

[[Image:optimum replacement.png|center|850px|BlockSim's Optimum Replacement Utility|link=]]

The figure below shows the Cost vs. Time plot of the component (with the scaling adjusted and the plot annotated to show the minimum cost).

[[Image:BS7.6.png|center|600px|Cost vs. Replacement Time|link=]]

If we enter different cost values in the '''Unplanned Replacement Cost''' column and obtain the optimum replacement time at each value, we can use the data points to create a plot that shows the effect of the corrective cost on the optimum replacement interval. The following plot shows an example. In this case, the optimum replacement interval decreases as the cost ratio increases. This is an expected result because the corrective replacement costs are much greater than the preventive replacement costs. Therefore, it is more cost-effective to replace the component more frequently before it fails.

[[Image:BS7.7.png|center|600px|Replacement interval as a function of the corrective/preventive cost ratio.|link=]]

Optimum Replacement Time Example

2016-08-18T22:44:52Z

Melinda Caroline:

<noinclude>{{Banner BlockSim Examples}}
''This example appears in the article [[Preventive Maintenance]]''.

This example demonstrates BlockSim's Optimum Replacement utility, which allows you to determine the most cost-effective time to replace one or more blocks in one or more systems, based on their planned and unplanned replacement costs.

'''Optimum Replacement Time'''
</noinclude>
The failure distribution of a component is described by a 2-parameter Weibull distribution with <math>\beta = 2.5\,\!</math> and <math>\eta = 1000\,\!</math> hours.

:* The cost for a corrective replacement is $5. 
:* The cost for a preventive replacement is $1. 

Estimate the optimum replacement age in order to minimize these costs.

'''Solution'''

Prior to obtaining an optimum replacement interval for this component, the assumptions of the following equation must be checked.

::<math>\begin{align}
CPUT\left( t \right)= & \frac{\text{Total Expected Replacement Cost per Cycle}}{\text{Expected Cycle Length}} \\
= & \frac{{{C}_{P}}\cdot R\left( t \right)+{{C}_{U}}\cdot \left[ 1-R\left( t \right) \right]}{\int_{0}^{t}R\left( s \right)ds}
\end{align}\,\!</math>

The component has an increasing failure rate because it follows a Weibull distribution with <math>\beta \,\!</math> greater than 1. Note that if <math>\beta =1\,\!</math>, then the component has a constant failure rate, but if <math>\beta <1\,\!</math>, then it has a decreasing failure rate. If either of these cases exist, then preventive replacement is unwise. Furthermore, the cost for preventive replacement is less than the corrective replacement cost. Thus, the conditions for the optimum age replacement policy have been met.

Using BlockSim, enter the parameters of the Weibull distribution in the component's Block Properties window. Next, open the Optimum Replacement window and enter the '''1''' in the '''Planned Replacement Cost''' column, and '''5''' in the '''Unplanned Replacement Cost''' column. Click '''Calculate'''. In the Optimum Replacement Calculations window that appears, select the '''Individual''' option and click '''OK'''. The optimum replacement time for the component is estimated to be 493.0470, as shown next.

[[Image:optimum replacement.png|center|850px|BlockSim's Optimum Replacement Utility|link=]]

The figure below shows the Cost vs. Time plot of the component (with the scaling adjusted and the plot annotated to show the minimum cost).

[[Image:BS7.6.png|center|600px|Cost vs. Replacement Time|link=]]

If we enter different cost values in the '''Additional Unplanned''' column and obtain the optimum replacement time at each value, we can use the data points to create a plot that shows the effect of the corrective cost on the optimum replacement interval. The following plot shows an example. In this case, the optimum replacement interval decreases as the cost ratio increases. This is an expected result because the corrective replacement costs are much greater than the preventive replacement costs. Therefore, it is more cost-effective to replace the component more frequently before it fails.

[[Image:BS7.7.png|center|600px|Replacement interval as a function of the corrective/preventive cost ratio.|link=]]

Markov Diagrams

2016-02-12T16:20:34Z

Melinda Caroline: /* Continuous Markov Chains */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector <math>\underline{X}\,\!</math>:

::<math>\underline{X}(0)= (X_1 X_2 \cdot\cdot\cdot X_i) \,\!</math>

where the term <math>X_i\,\!</math> represents the probability of the system being in state <math>i\,\!</math> and with:

::<math>\sum_{i=1}^{n}X_{i}=1\,\!</math>

The transitions between the states can be represented by a matrix <math>\underline{P}\,\!</math>:

::<math> \underline{P} =
\begin{pmatrix}
P_{11} & P_{12} & \cdots & P_{1i} \\
P_{21} & P_{22} & \cdots & P_{2i} \\
\vdots & \vdots & \ddots & \vdots \\
P_{i1} & P_{i2} & \cdots & P_{ii}
\end{pmatrix}\,\!</math>

where, for example, the term <math>P_{12}\,\!</math> is the transition probability from state 1 to state 2, and for any row <math>m\,\!</math> with <math>i\,\!</math> states:

::<math>\sum_{j=1}^{i}P_{mj}=1\,\!</math>

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector <math>\underline{X}(0)\,\!</math> with the transition matrix <math>\underline{P}\,\!</math>:

::<math>\underline{X}(n)=\underline{X}(0) \cdot \underline{P^n}\,\!</math>

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

Then, we can create the transition matrix to represent all the various transition probabilities between states:

::<math>\underline{P}=\begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}\,\!</math>

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

::<math>\underline{X}(10)=\underline{X}(0) \cdot \underline{P}^{10}\,\!</math>

or:

::<math>\underline{X}(10)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}^{10}\,\!</math>

resulting in :

::<math>\underline{X}(10)=\begin{pmatrix} 0.596 & 0.353 & 0.051 \end{pmatrix}\,\!</math>

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chains=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

::<math>\frac{dP_j}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j}\,\!</math>

where:

*<math>n\,\!</math> is the total number of states
*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_l\,\!</math> is the probability of being in state <math>l\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state l to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state j to state <math>l\,\!</math>

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

::<math>\frac{dP_{working}}{dt}=0.01\cdot P_{repair} - 0.0001 \cdot P_{working} \,\!</math>

::<math>\frac{dP_{repair}}{dt}=0.0001\cdot P_{working} - 0.01 \cdot P_{repair} \,\!</math>

with initial bounds of:

::<math>P_{working}(0)=1 \,\!</math>

::<math>P_{repair}(0)=0 \,\!</math>

Solving this system of equations yields the following results:

::<math>P_{working}=\frac{0.01}{0.0101}+\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

::<math>P_{repair}=\frac{0.0001}{0.0101}-\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

::<math>f_j=\frac{dP_{j}}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j} \,\!</math>

::<math>k_1=hf(t_i,P_{l,i})\,\!</math>

::<math>k_2=hf\Big( t_i+\frac{h}{4},P_{l,i}+\frac{k_1}{4}\Big)\,\!</math>

::<math>k_3=hf\Big( t_i+\frac{3h}{8},P_{l,i}+\frac{3}{32}k_2\Big)\,\!</math>

::<math>k_4=hf\Big( t_i+\frac{12h}{13},P_{l,i}+\frac{1932}{2197}k_1+\frac{7200}{2197}k_2+\frac{7296}{2197}k_3\Big)\,\!</math>

::<math>k_5=hf\Big( t_i+h,P_{l,i}+\frac{439}{216}k_1+8k_2+\frac{3680}{513}k_3-\frac{845}{4104}k_4\Big)\,\!</math>

::<math>k_6=hf\Big( t_i+\frac{h}{2},P_{l,i}-\frac{8}{27}k_1+2k_2-\frac{3544}{2565}k_3+\frac{1859}{4104}k_4-\frac{11}{40}k_5\Big)\,\!</math>

::<math>P_{j,i+1}=P_{j,i}+\frac{25}{216}k_1+\frac{1408}{2565}k_3+\frac{2197}{4104}k_4-\frac{1}{5}k_5\,\!</math>

::<math>\tilde{P}_{j,i+1}=P_{j,i}+\frac{16}{135}k_1+\frac{6656}{12825}k_3+\frac{28561}{56430}k_4-\frac{9}{50}k_5+\frac{2}{55}k_6\,\!</math>

::<math>R=\frac{1}{h}|\tilde{P}_{j,i+1}-P_{j,i+1}|\,\!</math>

::<math>\delta=0.84\Big(\frac{\epsilon}{R}\Big)^{\frac{1}{4}}\,\!</math>

If <math>R\leq\epsilon\,\!</math> then keep <math>P_{i+1}\,\!</math> as the current solution, and move to next step with step size <math>\delta{h}\,\!</math>; if <math>R>\epsilon\,\!</math> then recalculate the current step with step size <math>\delta{h}\,\!</math>

where:

*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_{l,i}\,\!</math> is the probability of being in state <math>l\,\!</math> at time <math>i\,\!</math>
*<math>\lambda_{lj}\,\!</math> is the transition rate from state <math>l\,\!</math> to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> is the transition rate from state <math>j\,\!</math> to state <math>l\,\!</math>
*<math>f_j\,\!</math> is the change in the probability of being in state <math>P_j\,\!</math> (note that <math>f_j\,\!</math> is not a function of time for constant transition rate Markov chains)
*<math>h\,\!</math> is the time step size
*<math>t_i\,\!</math> is the time at <math>i\,\!</math>
*<math>\varepsilon\,\!</math> is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-12T16:19:47Z

Melinda Caroline: /* Continuous Markov Chains */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector <math>\underline{X}\,\!</math>:

::<math>\underline{X}(0)= (X_1 X_2 \cdot\cdot\cdot X_i) \,\!</math>

where the term <math>X_i\,\!</math> represents the probability of the system being in state <math>i\,\!</math> and with:

::<math>\sum_{i=1}^{n}X_{i}=1\,\!</math>

The transitions between the states can be represented by a matrix <math>\underline{P}\,\!</math>:

::<math> \underline{P} =
\begin{pmatrix}
P_{11} & P_{12} & \cdots & P_{1i} \\
P_{21} & P_{22} & \cdots & P_{2i} \\
\vdots & \vdots & \ddots & \vdots \\
P_{i1} & P_{i2} & \cdots & P_{ii}
\end{pmatrix}\,\!</math>

where, for example, the term <math>P_{12}\,\!</math> is the transition probability from state 1 to state 2, and for any row <math>m\,\!</math> with <math>i\,\!</math> states:

::<math>\sum_{j=1}^{i}P_{mj}=1\,\!</math>

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector <math>\underline{X}(0)\,\!</math> with the transition matrix <math>\underline{P}\,\!</math>:

::<math>\underline{X}(n)=\underline{X}(0) \cdot \underline{P^n}\,\!</math>

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

Then, we can create the transition matrix to represent all the various transition probabilities between states:

::<math>\underline{P}=\begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}\,\!</math>

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

::<math>\underline{X}(10)=\underline{X}(0) \cdot \underline{P}^{10}\,\!</math>

or:

::<math>\underline{X}(10)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}^{10}\,\!</math>

resulting in :

::<math>\underline{X}(10)=\begin{pmatrix} 0.596 & 0.353 & 0.051 \end{pmatrix}\,\!</math>

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chains=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

::<math>\frac{dP_j}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j}\,\!</math>

where:

*<math>n\,\!</math> is the total number of states
*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_l\,\!</math> is the probability of being in state <math>l\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state l to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state j to state <math>l\,\!</math>

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

::<math>\frac{dP_{working}}{dt}=0.01\cdot P_{repair} - 0.0001 \cdot P_{working} \,\!</math>

::<math>\frac{dP_{repair}}{dt}=0.0001\cdot P_{working} - 0.01 \cdot P_{repair} \,\!</math>

with initial bounds of:

::<math>P_{working}(0)=1 \,\!</math>

::<math>P_{repair}(0)=0 \,\!</math>

Solving this system of equations yields the following results:

::<math>P_{working}=\frac{0.01}{0.0101}+\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

::<math>P_{repair}=\frac{0.0001}{0.0101}-\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

::<math>f_j=\frac{dP_{j}}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j} \,\!</math>

::<math>k_1=hf(t_i,P_{l,i})\,\!</math>

::<math>k_2=hf\Big( t_i+\frac{h}{4},P_{l,i}+\frac{k_1}{4}\Big)\,\!</math>

::<math>k_3=hf\Big( t_i+\frac{3h}{8},P_{l,i}+\frac{3}{32}k_2\Big)\,\!</math>

::<math>k_4=hf\Big( t_i+\frac{12h}{13},P_{l,i}+\frac{1932}{2197}k_1+\frac{7200}{2197}k_2+\frac{7296}{2197}k_3\Big)\,\!</math>

::<math>k_5=hf\Big( t_i+h,P_{l,i}+\frac{439}{216}k_1+8k_2+\frac{3680}{513}k_3-\frac{845}{4104}k_4\Big)\,\!</math>

::<math>k_6=hf\Big( t_i+\frac{h}{2},P_{l,i}-\frac{8}{27}k_1+2k_2-\frac{3544}{2565}k_3+\frac{1859}{4104}k_4-\frac{11}{40}k_5\Big)\,\!</math>

::<math>P_{j,i+1}=P_{j,i}+\frac{25}{216}k_1+\frac{1408}{2565}k_3+\frac{2197}{4104}k_4-\frac{1}{5}k_5\,\!</math>

::<math>\tilde{P}_{j,i+1}=P_{j,i}+\frac{16}{135}k_1+\frac{6656}{12825}k_3+\frac{28561}{56430}k_4-\frac{9}{50}k_5+\frac{2}{55}k_6\,\!</math>

::<math>R=\frac{1}{h}|\tilde{P}_{j,i+1}-P_{j,i+1}|\,\!</math>

::<math>\delta=0.84\Big(\frac{\epsilon}{R}\Big)^{\frac{1}{4}}\,\!</math>

If <math>R\leq\epsilon\,\!</math> then keep <math>P_{i+1}\,\!</math> as the current solution, and move to next step with step size <math>\delta{h}\,\!</math>; if <math>R>\epsilon\,\!</math> then recalculate the current step with step size <math>\delta{h}\,\!</math>

where:

*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_{l,i}\,\!</math> is the probability of being in state <math>l\,\!</math> at time <math>i\,\!</math>
*<math>\lambda_{lj}\,\!</math> is the transition rate from state <math>l\,\!</math> to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> is the transition rate from state <math>j\,\!</math> to state <math>l\,\!</math>
*<math>f_j\,\!</math> is the change in the probability of being in state <math>P_j\,\!</math> (Note that <math>f_j\,\!</math> is not a function of time for constant transition rate Markov chains)
*<math>h\,\!</math> is the time step size
*<math>t_i\,\!</math> is the time at <math>i\,\!</math>
*<math>\varepsilon\,\!</math> is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-12T16:19:07Z

Melinda Caroline: /* Continuous Markov Chains */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector <math>\underline{X}\,\!</math>:

::<math>\underline{X}(0)= (X_1 X_2 \cdot\cdot\cdot X_i) \,\!</math>

where the term <math>X_i\,\!</math> represents the probability of the system being in state <math>i\,\!</math> and with:

::<math>\sum_{i=1}^{n}X_{i}=1\,\!</math>

The transitions between the states can be represented by a matrix <math>\underline{P}\,\!</math>:

::<math> \underline{P} =
\begin{pmatrix}
P_{11} & P_{12} & \cdots & P_{1i} \\
P_{21} & P_{22} & \cdots & P_{2i} \\
\vdots & \vdots & \ddots & \vdots \\
P_{i1} & P_{i2} & \cdots & P_{ii}
\end{pmatrix}\,\!</math>

where, for example, the term <math>P_{12}\,\!</math> is the transition probability from state 1 to state 2, and for any row <math>m\,\!</math> with <math>i\,\!</math> states:

::<math>\sum_{j=1}^{i}P_{mj}=1\,\!</math>

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector <math>\underline{X}(0)\,\!</math> with the transition matrix <math>\underline{P}\,\!</math>:

::<math>\underline{X}(n)=\underline{X}(0) \cdot \underline{P^n}\,\!</math>

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

Then, we can create the transition matrix to represent all the various transition probabilities between states:

::<math>\underline{P}=\begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}\,\!</math>

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

::<math>\underline{X}(10)=\underline{X}(0) \cdot \underline{P}^{10}\,\!</math>

or:

::<math>\underline{X}(10)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}^{10}\,\!</math>

resulting in :

::<math>\underline{X}(10)=\begin{pmatrix} 0.596 & 0.353 & 0.051 \end{pmatrix}\,\!</math>

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chains=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

::<math>\frac{dP_j}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j}\,\!</math>

where:

*<math>n\,\!</math> is the total number of states
*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_l\,\!</math> is the probability of being in state <math>l\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state l to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state j to state <math>l\,\!</math>

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

::<math>\frac{dP_{working}}{dt}=0.01\cdot P_{repair} - 0.0001 \cdot P_{working} \,\!</math>

::<math>\frac{dP_{repair}}{dt}=0.0001\cdot P_{working} - 0.01 \cdot P_{repair} \,\!</math>

with initial bounds of:

::<math>P_{working}(0)=1 \,\!</math>

::<math>P_{repair}(0)=0 \,\!</math>

Solving this system of equations yields the following results:

::<math>P_{working}=\frac{0.01}{0.0101}+\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

::<math>P_{repair}=\frac{0.0001}{0.0101}-\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

::<math>f_j=\frac{dP_{j}}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j} \,\!</math>

::<math>k_1=hf(t_i,P_{l,i})\,\!</math>

::<math>k_2=hf\Big( t_i+\frac{h}{4},P_{l,i}+\frac{k_1}{4}\Big)\,\!</math>

::<math>k_3=hf\Big( t_i+\frac{3h}{8},P_{l,i}+\frac{3}{32}k_2\Big)\,\!</math>

::<math>k_4=hf\Big( t_i+\frac{12h}{13},P_{l,i}+\frac{1932}{2197}k_1+\frac{7200}{2197}k_2+\frac{7296}{2197}k_3\Big)\,\!</math>

::<math>k_5=hf\Big( t_i+h,P_{l,i}+\frac{439}{216}k_1+8k_2+\frac{3680}{513}k_3-\frac{845}{4104}k_4\Big)\,\!</math>

::<math>k_6=hf\Big( t_i+\frac{h}{2},P_{l,i}-\frac{8}{27}k_1+2k_2-\frac{3544}{2565}k_3+\frac{1859}{4104}k_4-\frac{11}{40}k_5\Big)\,\!</math>

::<math>P_{j,i+1}=P_{j,i}+\frac{25}{216}k_1+\frac{1408}{2565}k_3+\frac{2197}{4104}k_4-\frac{1}{5}k_5\,\!</math>

::<math>\tilde{P}_{j,i+1}=P_{j,i}+\frac{16}{135}k_1+\frac{6656}{12825}k_3+\frac{28561}{56430}k_4-\frac{9}{50}k_5+\frac{2}{55}k_6\,\!</math>

::<math>R=\frac{1}{h}|\tilde{P}_{j,i+1}-P_{j,i+1}|\,\!</math>

::<math>\delta=0.84\Big(\frac{\epsilon}{R}\Big)^{\frac{1}{4}}\,\!</math>

If <math>R\leq\epsilon\,\!</math> then keep <math>P_{i+1}\,\!</math> as the current solution, and move to next step with step size <math>\delta{h}\,\!</math>; if <math>R>\epsilon\,\!</math> then recalculate the current step with step size <math>\delta{h}\,\!</math>

where:

*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_{l,i}\,\!</math> is the probability of being in state <math>l\,\!</math> at time <math>i\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state <math>l\,\!</math> to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state <math>j\,\!</math> to state <math>l\,\!</math>
*<math>f_j\,\!</math> is the change in the probability of being in state <math>P_j\,\!</math> (Note that <math>f_j\,\!</math> is not a function of time for constant transition rate Markov chains)
*<math>h\,\!</math> is the time step size
*<math>t_i\,\!</math> is the time at <math>i\,\!</math>
*<math>\varepsilon\,\!</math> is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-12T16:18:08Z

Melinda Caroline: /* Continuous Markov Chains */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector <math>\underline{X}\,\!</math>:

::<math>\underline{X}(0)= (X_1 X_2 \cdot\cdot\cdot X_i) \,\!</math>

where the term <math>X_i\,\!</math> represents the probability of the system being in state <math>i\,\!</math> and with:

::<math>\sum_{i=1}^{n}X_{i}=1\,\!</math>

The transitions between the states can be represented by a matrix <math>\underline{P}\,\!</math>:

::<math> \underline{P} =
\begin{pmatrix}
P_{11} & P_{12} & \cdots & P_{1i} \\
P_{21} & P_{22} & \cdots & P_{2i} \\
\vdots & \vdots & \ddots & \vdots \\
P_{i1} & P_{i2} & \cdots & P_{ii}
\end{pmatrix}\,\!</math>

where, for example, the term <math>P_{12}\,\!</math> is the transition probability from state 1 to state 2, and for any row <math>m\,\!</math> with <math>i\,\!</math> states:

::<math>\sum_{j=1}^{i}P_{mj}=1\,\!</math>

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector <math>\underline{X}(0)\,\!</math> with the transition matrix <math>\underline{P}\,\!</math>:

::<math>\underline{X}(n)=\underline{X}(0) \cdot \underline{P^n}\,\!</math>

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

Then, we can create the transition matrix to represent all the various transition probabilities between states:

::<math>\underline{P}=\begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}\,\!</math>

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

::<math>\underline{X}(10)=\underline{X}(0) \cdot \underline{P}^{10}\,\!</math>

or:

::<math>\underline{X}(10)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}^{10}\,\!</math>

resulting in :

::<math>\underline{X}(10)=\begin{pmatrix} 0.596 & 0.353 & 0.051 \end{pmatrix}\,\!</math>

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chains=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

::<math>\frac{dP_j}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j}\,\!</math>

where:

*<math>n\,\!</math> is the total number of states
*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_l\,\!</math> is the probability of being in state <math>l\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state l to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state j to state <math>l\,\!</math>

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

::<math>\frac{dP_{working}}{dt}=0.01\cdot P_{repair} - 0.0001 \cdot P_{working} \,\!</math>

::<math>\frac{dP_{repair}}{dt}=0.0001\cdot P_{working} - 0.01 \cdot P_{repair} \,\!</math>

with initial bounds of:

::<math>P_{working}(0)=1 \,\!</math>

::<math>P_{repair}(0)=0 \,\!</math>

Solving this system of equations yields the following results:

::<math>P_{working}=\frac{0.01}{0.0101}+\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

::<math>P_{repair}=\frac{0.0001}{0.0101}-\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

::<math>f_j=\frac{dP_{j}}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j} \,\!</math>

::<math>k_1=hf(t_i,P_{l,i})\,\!</math>

::<math>k_2=hf\Big( t_i+\frac{h}{4},P_{l,i}+\frac{k_1}{4}\Big)\,\!</math>

::<math>k_3=hf\Big( t_i+\frac{3h}{8},P_{l,i}+\frac{3}{32}k_2\Big)\,\!</math>

::<math>k_4=hf\Big( t_i+\frac{12h}{13},P_{l,i}+\frac{1932}{2197}k_1+\frac{7200}{2197}k_2+\frac{7296}{2197}k_3\Big)\,\!</math>

::<math>k_5=hf\Big( t_i+h,P_{l,i}+\frac{439}{216}k_1+8k_2+\frac{3680}{513}k_3-\frac{845}{4104}k_4\Big)\,\!</math>

::<math>k_6=hf\Big( t_i+\frac{h}{2},P_{l,i}-\frac{8}{27}k_1+2k_2-\frac{3544}{2565}k_3+\frac{1859}{4104}k_4-\frac{11}{40}k_5\Big)\,\!</math>

::<math>P_{j,i+1}=P_{j,i}+\frac{25}{216}k_1+\frac{1408}{2565}k_3+\frac{2197}{4104}k_4-\frac{1}{5}k_5\,\!</math>

::<math>\tilde{P}_{j,i+1}=P_{j,i}+\frac{16}{135}k_1+\frac{6656}{12825}k_3+\frac{28561}{56430}k_4-\frac{9}{50}k_5+\frac{2}{55}k_6\,\!</math>

::<math>R=\frac{1}{h}|\tilde{P}_{j,i+1}-P_{j,i+1}|\,\!</math>

::<math>\delta=0.84\Big(\frac{\epsilon}{R}\Big)^{\frac{1}{4}}\,\!</math>

If <math>R\leq\epsilon\,\!</math> then keep <math>P_{i+1}\,\!</math> as the current solution, and move to next step with step size <math>\delta{h}\,\!</math>.

If <math>R>\epsilon\,\!</math> then recalculate the current step with step size <math>\delta{h}\,\!</math>

where:

*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_{l,i}\,\!</math> is the probability of being in state <math>l\,\!</math> at time <math>i\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state <math>l\,\!</math> to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state <math>j\,\!</math> to state <math>l\,\!</math>
*<math>f_j\,\!</math> is the change in the probability of being in state <math>P_j\,\!</math> (Note that <math>f_j\,\!</math> is not a function of time for constant transition rate Markov chains)
*<math>h\,\!</math> is the time step size
*<math>t_i\,\!</math> is the time at <math>i\,\!</math>
*<math>\varepsilon\,\!</math> is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-12T16:08:34Z

Melinda Caroline: /* Discrete Markov Chains */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector <math>\underline{X}\,\!</math>:

::<math>\underline{X}(0)= (X_1 X_2 \cdot\cdot\cdot X_i) \,\!</math>

where the term <math>X_i\,\!</math> represents the probability of the system being in state <math>i\,\!</math> and with:

::<math>\sum_{i=1}^{n}X_{i}=1\,\!</math>

The transitions between the states can be represented by a matrix <math>\underline{P}\,\!</math>:

::<math> \underline{P} =
\begin{pmatrix}
P_{11} & P_{12} & \cdots & P_{1i} \\
P_{21} & P_{22} & \cdots & P_{2i} \\
\vdots & \vdots & \ddots & \vdots \\
P_{i1} & P_{i2} & \cdots & P_{ii}
\end{pmatrix}\,\!</math>

where, for example, the term <math>P_{12}\,\!</math> is the transition probability from state 1 to state 2, and for any row <math>m\,\!</math> with <math>i\,\!</math> states:

::<math>\sum_{j=1}^{i}P_{mj}=1\,\!</math>

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector <math>\underline{X}(0)\,\!</math> with the transition matrix <math>\underline{P}\,\!</math>:

::<math>\underline{X}(n)=\underline{X}(0) \cdot \underline{P^n}\,\!</math>

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

Then, we can create the transition matrix to represent all the various transition probabilities between states:

::<math>\underline{P}=\begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}\,\!</math>

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

::<math>\underline{X}(10)=\underline{X}(0) \cdot \underline{P}^{10}\,\!</math>

or:

::<math>\underline{X}(10)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}^{10}\,\!</math>

resulting in :

::<math>\underline{X}(10)=\begin{pmatrix} 0.596 & 0.353 & 0.051 \end{pmatrix}\,\!</math>

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chains=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

::<math>\frac{dP_j}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j}\,\!</math>

where:

*<math>n\,\!</math> is the total number of states
*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_l\,\!</math> is the probability of being in state <math>l\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state l to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state j to state <math>l\,\!</math>

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

::<math>\frac{dP_{working}}{dt}=0.01\cdot P_{repair} - 0.0001 \cdot P_{working} \,\!</math>

::<math>\frac{dP_{repair}}{dt}=0.0001\cdot P_{working} - 0.01 \cdot P_{repair} \,\!</math>

with initial bounds of:

::<math>P_{working}(0)=1 \,\!</math>

::<math>P_{repair}(0)=0 \,\!</math>

Solving this system of equations yields the following results:

::<math>P_{working}=\frac{0.01}{0.0101}+\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

::<math>P_{repair}=\frac{0.0001}{0.0101}-\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

::<math>f_j=\frac{dP_{j}}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j} \,\!</math>

::<math>k_1=hf(t_i,P_{l,i})\,\!</math>

::<math>k_2=hf\Big( t_i+\frac{h}{4},P_{l,i}+\frac{k_1}{4}\Big)\,\!</math>

::<math>k_3=hf\Big( t_i+\frac{3h}{8},P_{l,i}+\frac{3}{32}k_2\Big)\,\!</math>

::<math>k_4=hf\Big( t_i+\frac{12h}{13},P_{l,i}+\frac{1932}{2197}k_1+\frac{7200}{2197}k_2+\frac{7296}{2197}k_3\Big)\,\!</math>

::<math>k_5=hf\Big( t_i+h,P_{l,i}+\frac{439}{216}k_1+8k_2+\frac{3680}{513}k_3-\frac{845}{4104}k_4\Big)\,\!</math>

::<math>k_6=hf\Big( t_i+\frac{h}{2},P_{l,i}-\frac{8}{27}k_1+2k_2-\frac{3544}{2565}k_3+\frac{1859}{4104}k_4-\frac{11}{40}k_5\Big)\,\!</math>

::<math>P_{j,i+1}=P_{j,i}+\frac{25}{216}k_1+\frac{1408}{2565}k_3+\frac{2197}{4104}k_4-\frac{1}{5}k_5\,\!</math>

::<math>\tilde{P}_{j,i+1}=P_{j,i}+\frac{16}{135}k_1+\frac{6656}{12825}k_3+\frac{28561}{56430}k_4-\frac{9}{50}k_5+\frac{2}{55}k_6\,\!</math>

::<math>R=\frac{1}{h}|\tilde{P}_{j,i+1}-P_{j,i+1}|\,\!</math>

::<math>\delta=0.84\Big(\frac{\epsilon}{R}\Big)^{\frac{1}{4}}\,\!</math>

if <math>R\leq\epsilon\,\!</math> keep <math>P_{i+1}\,\!</math> as the current solution, and move to next step with step size <math>\delta{h}\,\!</math>.

if <math>R>\epsilon\,\!</math> recalculate the current step with step size <math>\delta{h}\,\!</math>.

where:

*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_{l,i}\,\!</math> is the probability of being in state <math>l\,\!</math> at time <math>i\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state <math>l\,\!</math> to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state <math>j\,\!</math> to state <math>l\,\!</math>
*<math>f_j\,\!</math> is the change in the probability of being in state <math>P_j\,\!</math> (Note that <math>f_j\,\!</math> is not a function of time for constant transition rate Markov chains)
*<math>h\,\!</math> is the time step size
*<math>t_i\,\!</math> is the time at <math>i\,\!</math>
*<math>\varepsilon\,\!</math> is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-12T16:07:18Z

Melinda Caroline: /* Discrete Markov Chains */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector <math>\underline{X}\,\!</math>:

::<math>\underline{X}(0)= (X_1 X_2 \cdot\cdot\cdot X_i) \,\!</math>

where the term <math>X_i\,\!</math> represents the probability of the system being in state <math>i\,\!</math> and with:

::<math>\sum_{i=1}^{n}X_{i}=1\,\!</math>

The transitions between the states can be represented by a matrix <math>\underline{P}\,\!</math>:

::<math> \underline{P} =
\begin{pmatrix}
P_{11} & P_{12} & \cdots & P_{1i} \\
P_{21} & P_{22} & \cdots & P_{2i} \\
\vdots & \vdots & \ddots & \vdots \\
P_{i1} & P_{i2} & \cdots & P_{ii}
\end{pmatrix}\,\!</math>

where, for example, the term <math>P_{12}\,\!</math> is the transition probability from state 1 to state 2, and for any row <math>m\,\!</math> with <math>i\,\!</math> states:

::<math>\sum_{j=1}^{i}P_{mj}=1\,\!</math>

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector <math>\underline{X}(0)\,\!</math> with the transition matrix <math>\underline{P}\,\!</math>:

::<math>\underline{X}(n)=\underline{X}(0) \cdot \underline{P^n}\,\!</math>

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

::<math>\underline{X}(0)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\,\!</math>

Then, we can create the transition matrix to represent all the various transition probabilities between states:

::<math>\underline{P}=\begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}\,\!</math>

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

::<math>\underline{X}(10)=\underline{X}(0) \cdot \underline{P}^{10}\,\!</math>

or:

::<math>\underline{X}(10)=\begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix}
0.75 & 0.20 & 0.05 \\
0.40 & 0.59 & 0.01 \\
0.15 & 0.50 & 0.35
\end{pmatrix}^{10}\,\!</math>

resulting in :

::<math>\underline{X}(10)=\begin{pmatrix} 0.596 & 0.353 & 0.051 \end{pmatrix}\,\!</math>

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chains=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

::<math>\frac{dP_j}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j}\,\!</math>

where:

*<math>n\,\!</math> is the total number of states
*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_l\,\!</math> is the probability of being in state <math>l\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state l to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state j to state <math>l\,\!</math>

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

::<math>\frac{dP_{working}}{dt}=0.01\cdot P_{repair} - 0.0001 \cdot P_{working} \,\!</math>

::<math>\frac{dP_{repair}}{dt}=0.0001\cdot P_{working} - 0.01 \cdot P_{repair} \,\!</math>

with initial bounds of:

::<math>P_{working}(0)=1 \,\!</math>

::<math>P_{repair}(0)=0 \,\!</math>

Solving this system of equations yields the following results:

::<math>P_{working}=\frac{0.01}{0.0101}+\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

::<math>P_{repair}=\frac{0.0001}{0.0101}-\frac{0.0001}{0.0101}e^{-(0.0101)t} \,\!</math>

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

::<math>f_j=\frac{dP_{j}}{dt}= \displaystyle\sum_{l=1}^{n} \lambda_{lj}P_{l} - \displaystyle\sum_{l=1}^{n} \lambda_{jl}P_{j} \,\!</math>

::<math>k_1=hf(t_i,P_{l,i})\,\!</math>

::<math>k_2=hf\Big( t_i+\frac{h}{4},P_{l,i}+\frac{k_1}{4}\Big)\,\!</math>

::<math>k_3=hf\Big( t_i+\frac{3h}{8},P_{l,i}+\frac{3}{32}k_2\Big)\,\!</math>

::<math>k_4=hf\Big( t_i+\frac{12h}{13},P_{l,i}+\frac{1932}{2197}k_1+\frac{7200}{2197}k_2+\frac{7296}{2197}k_3\Big)\,\!</math>

::<math>k_5=hf\Big( t_i+h,P_{l,i}+\frac{439}{216}k_1+8k_2+\frac{3680}{513}k_3-\frac{845}{4104}k_4\Big)\,\!</math>

::<math>k_6=hf\Big( t_i+\frac{h}{2},P_{l,i}-\frac{8}{27}k_1+2k_2-\frac{3544}{2565}k_3+\frac{1859}{4104}k_4-\frac{11}{40}k_5\Big)\,\!</math>

::<math>P_{j,i+1}=P_{j,i}+\frac{25}{216}k_1+\frac{1408}{2565}k_3+\frac{2197}{4104}k_4-\frac{1}{5}k_5\,\!</math>

::<math>\tilde{P}_{j,i+1}=P_{j,i}+\frac{16}{135}k_1+\frac{6656}{12825}k_3+\frac{28561}{56430}k_4-\frac{9}{50}k_5+\frac{2}{55}k_6\,\!</math>

::<math>R=\frac{1}{h}|\tilde{P}_{j,i+1}-P_{j,i+1}|\,\!</math>

::<math>\delta=0.84\Big(\frac{\epsilon}{R}\Big)^{\frac{1}{4}}\,\!</math>

if <math>R\leq\epsilon\,\!</math> keep <math>P_{i+1}\,\!</math> as the current solution, and move to next step with step size <math>\delta{h}\,\!</math>.

if <math>R>\epsilon\,\!</math> recalculate the current step with step size <math>\delta{h}\,\!</math>.

where:

*<math>P_j\,\!</math> is the probability of being in state <math>j\,\!</math>
*<math>P_{l,i}\,\!</math> is the probability of being in state <math>l\,\!</math> at time <math>i\,\!</math>
*<math>\lambda_{lj}\,\!</math> transition rate from state <math>l\,\!</math> to state <math>j\,\!</math>
*<math>\lambda_{jl}\,\!</math> transition rate from state <math>j\,\!</math> to state <math>l\,\!</math>
*<math>f_j\,\!</math> is the change in the probability of being in state <math>P_j\,\!</math> (Note that <math>f_j\,\!</math> is not a function of time for constant transition rate Markov chains)
*<math>h\,\!</math> is the time step size
*<math>t_i\,\!</math> is the time at <math>i\,\!</math>
*<math>\varepsilon\,\!</math> is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-08T21:10:25Z

Melinda Caroline: /* Continuous Markov Chain */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector <math>\underline{X}\,\!</math>:

::<math>\underline{X}(0)= (X_1 X_2 \cdot\cdot\cdot X_i) \,\!</math>

where the term <math>X_i\,\!</math> represents the probability of the system being in state <math>i\,\!</math> and with:

::<math>\sum_{i=1}^{n}X_{i}=1\,\!</math>

The transitions between the states can be represented by a matrix <math>\underline{P}\,\!</math>:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term <math>P_{12}\,\!</math> is the transition probability from state 1 to state 2, and for any row <math>m\,\!</math> with <math>i\,\!</math> states:

::<math>\sum_{j=1}^{i}P_{mj}=1\,\!</math>

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector <math>\underline{X}(0)\,\!</math> with the transition matrix <math>\underline{P}\,\!</math>:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chains=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-05T19:10:29Z

Melinda Caroline: /* Example */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-05T19:09:57Z

Melinda Caroline: /* Example */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states — operational, standby or offline — at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-05T19:08:18Z

Melinda Caroline: /* Example */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states-- operational, standby or offline-- at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-05T19:07:15Z

Melinda Caroline: /* Example */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states-- operational, standby or offline-- at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-05T18:58:15Z

Melinda Caroline:

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states-- operational, standby or offline-- at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1,000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2,000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the "non-repair" assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

=Phases=
It is possible to do analyses with multiple phases for both discrete and continuous Markov chain calculations. When using phases, the software keeps track of state correspondence across phases using the name of the state (e.g., "State B" is one phase is considered to correspond to "State B" in all other phases). The starting state probability comes from the ending state probability from the previous phase. If a state is absent from a phase, then its state probability does not change during the phase.

Also, it is possible to perform an analysis where the number of steps, for discrete analyses, or the operational time, for continuous analyses, are greater/longer than the sum length of all the phases. In this case, the analysis will return to the first phase, without resetting the state probabilities, and continue from there.

Markov Diagrams

2016-02-05T18:56:10Z

Melinda Caroline: /* Example */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states-- operational, standby or offline-- at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|700px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the “non-repair” assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

Markov Diagrams

2016-02-05T18:55:57Z

Melinda Caroline: /* Example */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states-- operational, standby or offline-- at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|500px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|700px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the “non-repair” assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

File:Continuous markov results.png

2016-02-05T18:55:20Z

Melinda Caroline:

File:Continuous markov state mean probability plot.png

2016-02-05T18:55:09Z

Melinda Caroline:

File:Continuous markov diagram.png

2016-02-05T18:54:51Z

Melinda Caroline:

Markov Diagrams

2016-02-05T18:54:36Z

Melinda Caroline: /* Example */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states-- operational, standby or offline-- at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|500px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

[[Image:continuous_markov_diagram.png|center|628px|link=]]

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

[[Image:continuous_markov_state_mean_probability_plot.png|center|500px|link=]]

[[Image:continuous_markov_results.png|center|798px|link=]]

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the “non-repair” assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

Markov Diagrams

2016-02-05T18:52:46Z

Melinda Caroline: /* Continuous Markov Chain */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states-- operational, standby or offline-- at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|500px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

=Continuous Markov Chain=
A continuous Markov chain can be viewed as a Markov chain where the transitions between states are defined by (constant) transition rates, as opposed to transition probabilities at fixed steps. It is common to use continuous Markov chains when analyzing system reliability/availability problems.

Because we are no longer performing analysis using fixed probabilities and a fixed step, we are no longer able to simply multiply a state probability vector with a transition matrix in order to obtain new state probabilities after a given step.

Instead, our problem can be written as a system of ordinary differential equations, where each differential equation represents the change in the probability of being in a particular state:

(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

where:

*n is the total number of states
*P_j is the probability of being in state j
*P_l is the probability of being in state l
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l

Our initial conditions to solve the differential equations are our initial probabilities for each state.

As a quick example, let us take a system that can be in one of two states, either working or under repair, and is initially in the working state. The transition rate from working to repair is 0.0001/hour (MTTF of 10,000 hours) and the transition rate from repair to working is 0.01/hour (MTTR of 100 hours). In this case, the two differential equations would look like this:

(dP_working)/dt=0.01∙P_repair-0.0001∙P_working

(dP_repair)/dt=0.0001∙P_working-0.01∙P_repair

with initial bounds of:

P_working (0)=1

P_repair (0)=0

Solving this system of equations yields the following results:

P_working=0.01/0.0101+0.0001/0.0101 e^(-(0.0101)t)

P_repair=0.0001/0.0101-0.0001/0.0101 e^(-(0.0101)t)

For more complex analyses, numerical methodologies are preferred. There are many methods, both analytical and numerical, to solve systems of ordinary differential equations. One of these is the Runge-Kutta-Fehlberg method, also known as the RKF45 method, which is a numerical algorithm. This algorithm is practical because one extra calculation allows for the error to be estimated and controlled with the use of an automatic adaptive step size methodology.

The methodology uses a 4th order Runge-Kutta and a 5th order Runge-Kutta, where the former is used for the calculation and the latter is used for the error estimation. The formulas for the method are as follows:

f_j=(dP_j)/dt=∑_(l=1)^n▒〖λ_lj P_l 〗-∑_(l=1)^n▒〖λ_jl P_j 〗

k_1=hf(t_i,P_(l,i) )

k_2=hf(t_i+h/4,P_(l,i)+k_1/4)

k_3=hf(t_i+3h/8,P_(l,i)+3/32 k_1+9/32 k_2 )

k_4=hf(t_i+12h/13,P_(l,i)+1932/2197 k_1-7200/2197 k_2+7296/2197 k_3 )

k_5=hf(t_i+h,P_(l,i)+439/216 k_1-8k_2+3680/513 k_3-845/4104 k_4 )

k_6=hf(t_i+h/2,P_(l,i)-8/27 k_1+2k_2-3544/2565 k_3+1859/4104 k_4-11/40 k_5 )

P_(j,i+1)=P_(j,i)+25/216 k_1+1408/2565 k_3+2197/4104 k_4-1/5 k_5

P ̃_(j,i+1)=P_(j,i)+16/135 k_1+6656/12825 k_3+28561/56430 k_4-9/50 k_5+2/55 k_6

R=1/h |P ̃_(j,i+1)-P_(j,i+1) |

δ=0.84(ϵ/R)^(1/4)

if R≤ϵ keep P_(i+1) as the current solution, and move to next step with step size δh

if R>ϵ recalculate the current step with step size δh

where:

*P_j is the probability of being in state j
*P_(l,i) is the probability of being in state l at time i
*λ_lj transition rate from state l to state j
*λ_jl transition rate from state j to state l
*f_j is the change in the probability of being in state P_j (Note that f_j is not a function of time for constant transition rate Markov chains)
*h is the time step size
*t_i is the time at i
*ε is the chosen acceptable error

This methodology can then be used for each state at each time step, where a time step is accepted only if the time step size is acceptable for each state in the system.

Since continuous Markov chains are often used for system availability/reliability analyses, the continuous Markov chain diagram in BlockSim allows the user the ability to designate one or more states as unavailable states. This allows for the calculation of both availability and reliability of the system.

Availability is calculated as the mean probability that the system is in a state that is not an unavailable state.

Reliability is calculated in the same manner as availability, with the additional restriction that all transitions leaving any unavailable state are considered to have a transition rate of zero.

==Example==
Assume you have a system composed of two generators. The system can be in one of three states:

*Both generators are operational
*One generator is operational and the other is under repair
*Both generators are under repair. This is an unavailable state.

The system starts in the state in which both generators are operational.

We know that the failure rate of a generator is 1 per 2,000 hours, and the repair rate is 1 per 200 hours.

Therefore:

*The transition rate from the state in which both generators are operational to the state where only one is operational is 1 per 1000 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are operational is 1 per 200 hours.
*The transition rate from the state in which one generator is operational to the state where both generators are under repair is 1 per 2000 hours.
*The transition rate from the state in which both generators are under repair to the state where one generator is operational is 1 per 100 hours.

We would like to know the mean availability of our system after 20,000 hours for all three states so that we can estimate our output based on time spent at full, half and zero generator capacity.

Solving this system by hand using the RKF45 method would be very time consuming, so we will turn to BlockSim to help us solve this problem using a continuous Markov diagram. After creating the diagram, adding the states and the transition rates, the final diagram looks like this:

xxx

Once the diagram is complete, the analysis is set for 20,000 hours. The comparison between the mean probabilities of the states can be done using a bar graph, or by looking at the diagram result summary.

xxx

xxx

From the Mean Probability column, we can see that the system is expected to be fully operational 82.8% of the time, half operational 16.4% of the time, and non-operational 0.8% of the time.

From the Point Probability (Av) column, we can get the point probability of being in a state when all transitions are considered. From the Point Probability (Rel) column, we can get the point probability of being in a state if we assume that there is no return from unavailable states, or in other words we are assuming no repair once the system has entered an unavailable (failed) state. Using the “non-repair” assumption, there is only an 18.0% chance that the system would still be fully operational, a 3.3% chance that it would be half operational and a 78.7% chance that it would be non-operational.

Markov Diagrams

2016-02-05T18:43:59Z

Melinda Caroline: /* Continuous Markov Chains: Applications to Non-Repairable Systems */

Markov Diagrams

2016-02-05T18:43:37Z

Melinda Caroline: /* Discrete Markov Chains */

File:Discrete markov state point probability plot.png

2016-02-05T18:41:56Z

Melinda Caroline:

Markov Diagrams

2016-02-05T18:41:29Z

Melinda Caroline: /* Discrete Markov Chains: Limiting Probabilities */

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains=
A discrete Markov chain can be viewed as a Markov chain where at the end of a step, the system will transition to another state (or remain in the current state), based on fixed probabilities. It is common to use discrete Markov chains when analyzing problems involving general probabilities, genetics, physics, etc.
To represent all the states that the system can occupy, we can use a vector ▁X:

▁X(0)=(■(X_1&X_2&⋯&X_i ))

where the term X_i represents the probability of the system being in state i and with:

∑_(i=1)^n▒X_i =1

The transitions between the states can be represented by a matrix ▁P:

▁P=(■(P_11&P_12&⋯&P_1i@P_21&P_22&⋯&P_2i@⋮&⋮&⋱&⋮@P_i1&P_i2&⋯&P_ii ))

where, for example, the term P_12 is the transition probability from state 1 to state 2, and for any row m with i states:

∑_(j=1)^i▒P_mj =1

To determine the probability of the system being in a particular state after the first step, we have to multiply the initial state probability vector ▁X(0) with the transition matrix ▁P:

▁X (1)=▁X(0)∙▁P

This will give us the state probability vector after the first step, ▁X(1).

If one wants to determine the probabilities of the system being in a particular state after n steps, the Chapman-Kolmogorov equation can be used. This equation states that the probabilities of being in a state after n steps can be calculated by taking the initial state vector and multiplying by the transition matrix to the nth power, or:

▁X (n)=▁X (0)∙▁P^n

==Example==
Take a system that can be in any one of three states-- operational, standby or offline-- at a given time, and starts in the standby state.

After each step:

*If the system is in the operational state, there is a 20% chance that it moves to the standby state, and a 5% chance that it goes offline.
*If it is in the standby state, there is a 40% chance that it becomes operational, and a 1% chance that it goes offline.
*If it is in the offline state, there is a 15% chance that it becomes operational, and a 50% chance that it moves to the standby state.

We want to know the probability that it is offline after 10 steps.

First, we must create the state probability vector at time equal to zero, which in this case is:

▁X(0)=(■(0&1&0))

Then, we can create the transition matrix to represent all the various transition probabilities between states:

▁P=(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))

Lastly, we can calculate the state probabilities after 10 steps using the Chapman-Kolmogorov equation:

▁X (10)=▁X (0)∙▁P^10

or:

▁X (10)=(■(0&1&0))∙(■(0.75&0.20&0.05@0.40&0.59&0.01@0.15&0.50&0.35))^10

resulting in :

▁X (10)=(■(0.596&0.353&0.051))

We can also plot the point probabilities of each state at each step if we calculate the state probabilities after each step:

[[Image:discrete_markov_state_point_probability_plot.png|center|500px|link=]]

From the plot, we can also determine that the probabilities of being in a state reach steady-state after about 6 steps.

==Chapman-Kolmogorov Equation==
The Chapman-Kolmogorov equation was realized and defined independently by British mathematician Sydney Chapman and Russian mathematician Andrew Kolmogorov. It can be used to provide the transitional densities of a Markov sequence.

Let pi(n)=P(Xn=i)

then:

::P(X(n+1)=j) = <math>\sum_{i \mathop =0}^{N}P</math> (Xn+1 = j|Xn = i)

so:

::Pj(n+1) = <math>\sum_{i \mathop =0}^{N}P</math> (Xi(n) Pij

With vector notation <math>\underline{p}</math>(n) = (p0(n),p1(n), ... ,pN(n)) (row vector)

<math>\underline{p}</math>(n+1) = p(n)<math>\underline{P}</math> = (<math>\underline{p}</math>(n-1)<math>\underline{P}</math>2 = p(0)p(n+1)

Let Pij(m) = P (Xn+m = j| Xn = i) and <math>\underline{p}</math>(m) = Pij(m)

then:

::<math>\underline{P}</math>n+m = <math>\underline{P}</math>(n) * <math>\underline{P}</math>(m) and <math>\underline{P}</math>(n)= <math>\underline{P}</math>n

==Accessible and Communicating States==
State j is accessible from state i, if for some m:

::Pij(m) > 0

State i communicates with state j, if j is accessible from i and also state i is accessible from j:

::<math>\sum_{m \mathop =1}^{\infty}P</math>ij(m) and <math>\sum_{m \mathop =1}^{\infty}P</math>ji(m)

The Markov chain is irreducible if every state i communicates with all other states and with itself.
==Recurrent and Transient States==
Let fi = P (starting at state i, system will return to state i)

If fi = 1, then state i is recurrent, repeated infinitely often

If fi < 1, then state i is transient, and repeated returns have smaller and smaller probabilities.

::fi = <math>\sum_{m \mathop =1}^{\infty}P</math>ii(m)

The Markov chain is ergodic if all states are recurrent and not periodic (there is no d>0 such that Pii(m) > 0 if and only if m is multiple of d).

==Limiting Probabilities==
* Theorem:
For an irreducible, ergodic Markov chain
<math> \lim_{m \to \infty}P</math>ij(m) = πj for all j for all j (10.4)
and limit is independent of i ( steady state probabilities):

::0 ≦ πj≦ 1

[[File:LimitingProbab.Theorem.PNG]]

* Method:

[[File:LimitingProbab.Method.PNG]]

==Mean Time Spent in States==
Mean time spent in recurrent states = ∞

Mean time spent in transient states:

::Sij = Starting at state i , expected number of time periods that state is j

::Sij = [[File:MeanTimeSpentInStates.1.PNG]]

where P * contains rows and columns of transient states of matrix ▁P:
S = I + P* S
<math>\underline{S}</math> = (I-P *)-1

=Continuous Markov Chains: Applications to Non-Repairable Systems=

* Non-repairable component with failure rate λ
** P0(t) = P ( at time t component works)
** P1(t) = P ( at time t component is broken)

P0 (t+ ∆ t) = (1- λ ∆ t) P0 (t) +0 P1 (t) Does not fail during ∆ t times

P1(t+ ∆ t) = λ ∆ t P0 (t) + 1 P1 (t) since P (Fails in ∆ Time) =1- e- λ∆t ≈ 1- (1- <math>\tfrac{\lambda\Delta t}{1!}+\tfrac{(\lambda^2(\Delta t)^2)}{2!}</math> - …) ≈ λ∆t if ∆t is small

==Method==
The method employed to solve a continuous Markov Chain problem is a modified RK45 Runga-Kutta-Fehlberg, which is an adaptive step size Runga-Kutta method.

==User Inputs==
The user must provide an initial probability for each state and a transition probability between each state. The initial probabilites of all states must add up to exactly 1.0. If a transition probability between states is not given, it is assumed to be zero.

===Symbol Definitions===
* αj,0 is the initial probability of being in state j (given by the user).
* ε is the user defined tolerance (accuracy). Default should be 1e-5 and can only get smaller.
* λl,j is the transitional failure rate into state wj from state wl
* wl is the probability of being in the state associated with the λl,j’s
* λj,k is the transitional failure rate leaving state wj to state wk
* fj is the change in state probability function (for a given state wj):
::[[File:ChgInStateProbFunt.Jpg]]
fj is not a function of time, as only constant failure rates are used. This means that the various k values calculated during the RK45 method are only functions of all the w values and the constant failure rates, λ.

The formula for the Runge-Kutta-Fehlberg method (RK45) is given next.

w0 = α

k1 = hf(ti, wi)

k2 = hf(ti+<math>\tfrac{h}{4}</math>, wi+<math>\tfrac{k_1}{4}</math>)

k3 = hf(ti+<math>\tfrac{3h}{8}</math>, wi+<math>\tfrac{3}{32}</math>k1+<math>\tfrac{9}{32}</math>k2)

k4 = hf(ti+<math>\tfrac{12h}{13}</math>, wi+<math>\tfrac{1932}{2197}</math>k1-<math>\tfrac{7200}{2197}</math>k2+<math>\tfrac{7296}{2197}</math>k3)

k5 = hf(ti+h wi+<math>\tfrac{439}{216}</math>k1-8k2+<math>\tfrac{3680}{513}</math>k3-<math>\tfrac{845}{4104}</math>k4)

k6 = hf(ti+<math>\tfrac{h}{2}</math> wi+<math>\tfrac{8}{27}</math>k1-2k2+<math>\tfrac{3544}{2565}</math>k3-<math>\tfrac{1859}{4104}</math>k4-<math>\tfrac{11}{40}</math>k5)

wi+1 = wi+<math>\tfrac{25}{216}</math>k1+<math>\tfrac{1408}{2565}</math>k3-<math>\tfrac{2197}{4104}</math>k4-<math>\tfrac{1}{5}</math>k5)

w'i+1 = wi+<math>\tfrac{16}{135}</math>k1+<math>\tfrac{6656}{12825}</math>k1+<math>\tfrac{28561}{56430}</math>k4-<math>\tfrac{9}{50}</math>k5-<math>\tfrac{2}{55}</math>k6)

R = <math>\tfrac{1}{h}|</math>w'i+1-wi+1|

δ = 0.84 * <math>\tfrac{\varepsilon}{R}^\tfrac{1}{4}</math>

If R≤ε then keep w as the current step solution and move to the next step with step size δh

If R>ε then recalculate the current step with step size δh. The above method is for each individual state, and not for the system as a whole. The w is the equivalent of the probability of being in a particular state, where the subscript i represents the time based variation. This still has to be done for all the states in the system, which later on is represented by the subscript j (for each state).

==Detailed Methodology==
#Generate an initial step size h from the available failure rates (1% of the smallest MTTF).
#Use the RK45 method on all states simultaneously using the given h. (This means that all states must have their k1 values calculated/used together, then k2, then k3, etc).
#If all calculations are within tolerance, keep results (RK4, so the w without the hat) and increase h to the smallest of the increases generated by the method. If some of the calculations are not within tolerance, decrease the step size to the smallest of the decreases generated by the method and recalculate with the new h. h should not be increased to more than double, so s should have a stipulation on it that forbids from that occurring. Be aware that s may become infinite if the difference between the RK4 and the RK5 is zero. This should be addressed as well with a catch of some sort to make s = 2 in that case. (So basically if s is calculated to be greater than 2 for any state, make it equal to 2 for that state).
#Repeat steps 2 & 3 as necessary.

If there are multiple phases, then steps 1-4 need to be performed for each phase where the initial probability of being in a state is equal to the final value from the previous phase.

This methodology provides the ability to give availability and unavailability metrics for the system, as well as point probabilities of being in a certain state.
For the reliability metrics, the methodology differs in that each unavailable state is considered a "sink." In other words, all transitions from unavailable to available states are ignored. This could be calculated simultaneously with the availability using the same step sizes as generated there.

The two results that need to be stored are the time and the corresponding probability of being in the state for each state.

Markov Diagrams

2016-02-05T17:59:31Z

Melinda Caroline:

{{Template:InProgress}}
{{Template:bsbook|11}}
The term "Markov Chain," invented by Russian mathematician Andrey Markov, is used across many applications to represent a stochastic process made up of a sequence of random variables representing the evolution of a system. Events are "chained" or "linked" serially together though memoryless transitions from one state to another. The term "memoryless" is used because past events are forgotten, as they are irrelevant; an event or state is dependent only on the state or event that immediately preceded it.

=Concept and Methodology=
The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a whole.

Depending on the transitions between the states, the Markov chain can be considered to be a discrete Markov chain, which has a constant probability of transition per unit step, or a continuous Markov chain, which has a constant rate of transition per unit time.

=Discrete Markov Chains: Limiting Probabilities=
==Transition Matrix==
A system has a finite number of states {0, 1, 2…,N} and transition from state to state is random. The matrix shows the potential inputs and outputs from one state to another to describe transitions of a Markov chain.
P(X(n+1)=j│X,n=i)=Pij where 0≦Pij≦1

[[File:TransitionMatrixExp.jpg]]

===Markov Chain Diagram===
Markov chain diagrams can be used to label events and transitions based upon a transition matrix.

==Chapman-Kolmogorov Equation==
The Chapman-Kolmogorov equation was realized and defined independently by British mathematician Sydney Chapman and Russian mathematician Andrew Kolmogorov. It can be used to provide the transitional densities of a Markov sequence.

Let pi(n)=P(Xn=i)

then:

::P(X(n+1)=j) = <math>\sum_{i \mathop =0}^{N}P</math> (Xn+1 = j|Xn = i)

so:

::Pj(n+1) = <math>\sum_{i \mathop =0}^{N}P</math> (Xi(n) Pij

With vector notation <math>\underline{p}</math>(n) = (p0(n),p1(n), ... ,pN(n)) (row vector)

<math>\underline{p}</math>(n+1) = p(n)<math>\underline{P}</math> = (<math>\underline{p}</math>(n-1)<math>\underline{P}</math>2 = p(0)p(n+1)

Let Pij(m) = P (Xn+m = j| Xn = i) and <math>\underline{p}</math>(m) = Pij(m)

then:

::<math>\underline{P}</math>n+m = <math>\underline{P}</math>(n) * <math>\underline{P}</math>(m) and <math>\underline{P}</math>(n)= <math>\underline{P}</math>n

==Accessible and Communicating States==
State j is accessible from state i, if for some m:

::Pij(m) > 0

State i communicates with state j, if j is accessible from i and also state i is accessible from j:

::<math>\sum_{m \mathop =1}^{\infty}P</math>ij(m) and <math>\sum_{m \mathop =1}^{\infty}P</math>ji(m)

The Markov chain is irreducible if every state i communicates with all other states and with itself.
==Recurrent and Transient States==
Let fi = P (starting at state i, system will return to state i)

If fi = 1, then state i is recurrent, repeated infinitely often

If fi < 1, then state i is transient, and repeated returns have smaller and smaller probabilities.

::fi = <math>\sum_{m \mathop =1}^{\infty}P</math>ii(m)

The Markov chain is ergodic if all states are recurrent and not periodic (there is no d>0 such that Pii(m) > 0 if and only if m is multiple of d).

==Limiting Probabilities==
* Theorem:
For an irreducible, ergodic Markov chain
<math> \lim_{m \to \infty}P</math>ij(m) = πj for all j for all j (10.4)
and limit is independent of i ( steady state probabilities):

::0 ≦ πj≦ 1

[[File:LimitingProbab.Theorem.PNG]]

* Method:

[[File:LimitingProbab.Method.PNG]]

==Mean Time Spent in States==
Mean time spent in recurrent states = ∞

Mean time spent in transient states:

::Sij = Starting at state i , expected number of time periods that state is j

::Sij = [[File:MeanTimeSpentInStates.1.PNG]]

where P * contains rows and columns of transient states of matrix ▁P:
S = I + P* S
<math>\underline{S}</math> = (I-P *)-1

=Continuous Markov Chains: Applications to Non-Repairable Systems=

* Non-repairable component with failure rate λ
** P0(t) = P ( at time t component works)
** P1(t) = P ( at time t component is broken)

P0 (t+ ∆ t) = (1- λ ∆ t) P0 (t) +0 P1 (t) Does not fail during ∆ t times

P1(t+ ∆ t) = λ ∆ t P0 (t) + 1 P1 (t) since P (Fails in ∆ Time) =1- e- λ∆t ≈ 1- (1- <math>\tfrac{\lambda\Delta t}{1!}+\tfrac{(\lambda^2(\Delta t)^2)}{2!}</math> - …) ≈ λ∆t if ∆t is small

==Method==
The method employed to solve a continuous Markov Chain problem is a modified RK45 Runga-Kutta-Fehlberg, which is an adaptive step size Runga-Kutta method.

==User Inputs==
The user must provide an initial probability for each state and a transition probability between each state. The initial probabilites of all states must add up to exactly 1.0. If a transition probability between states is not given, it is assumed to be zero.

===Symbol Definitions===
* αj,0 is the initial probability of being in state j (given by the user).
* ε is the user defined tolerance (accuracy). Default should be 1e-5 and can only get smaller.
* λl,j is the transitional failure rate into state wj from state wl
* wl is the probability of being in the state associated with the λl,j’s
* λj,k is the transitional failure rate leaving state wj to state wk
* fj is the change in state probability function (for a given state wj):
::[[File:ChgInStateProbFunt.Jpg]]
fj is not a function of time, as only constant failure rates are used. This means that the various k values calculated during the RK45 method are only functions of all the w values and the constant failure rates, λ.

The formula for the Runge-Kutta-Fehlberg method (RK45) is given next.

w0 = α

k1 = hf(ti, wi)

k2 = hf(ti+<math>\tfrac{h}{4}</math>, wi+<math>\tfrac{k_1}{4}</math>)

k3 = hf(ti+<math>\tfrac{3h}{8}</math>, wi+<math>\tfrac{3}{32}</math>k1+<math>\tfrac{9}{32}</math>k2)

k4 = hf(ti+<math>\tfrac{12h}{13}</math>, wi+<math>\tfrac{1932}{2197}</math>k1-<math>\tfrac{7200}{2197}</math>k2+<math>\tfrac{7296}{2197}</math>k3)

k5 = hf(ti+h wi+<math>\tfrac{439}{216}</math>k1-8k2+<math>\tfrac{3680}{513}</math>k3-<math>\tfrac{845}{4104}</math>k4)

k6 = hf(ti+<math>\tfrac{h}{2}</math> wi+<math>\tfrac{8}{27}</math>k1-2k2+<math>\tfrac{3544}{2565}</math>k3-<math>\tfrac{1859}{4104}</math>k4-<math>\tfrac{11}{40}</math>k5)

wi+1 = wi+<math>\tfrac{25}{216}</math>k1+<math>\tfrac{1408}{2565}</math>k3-<math>\tfrac{2197}{4104}</math>k4-<math>\tfrac{1}{5}</math>k5)

w'i+1 = wi+<math>\tfrac{16}{135}</math>k1+<math>\tfrac{6656}{12825}</math>k1+<math>\tfrac{28561}{56430}</math>k4-<math>\tfrac{9}{50}</math>k5-<math>\tfrac{2}{55}</math>k6)

R = <math>\tfrac{1}{h}|</math>w'i+1-wi+1|

δ = 0.84 * <math>\tfrac{\varepsilon}{R}^\tfrac{1}{4}</math>

If R≤ε then keep w as the current step solution and move to the next step with step size δh

If R>ε then recalculate the current step with step size δh. The above method is for each individual state, and not for the system as a whole. The w is the equivalent of the probability of being in a particular state, where the subscript i represents the time based variation. This still has to be done for all the states in the system, which later on is represented by the subscript j (for each state).

==Detailed Methodology==
#Generate an initial step size h from the available failure rates (1% of the smallest MTTF).
#Use the RK45 method on all states simultaneously using the given h. (This means that all states must have their k1 values calculated/used together, then k2, then k3, etc).
#If all calculations are within tolerance, keep results (RK4, so the w without the hat) and increase h to the smallest of the increases generated by the method. If some of the calculations are not within tolerance, decrease the step size to the smallest of the decreases generated by the method and recalculate with the new h. h should not be increased to more than double, so s should have a stipulation on it that forbids from that occurring. Be aware that s may become infinite if the difference between the RK4 and the RK5 is zero. This should be addressed as well with a catch of some sort to make s = 2 in that case. (So basically if s is calculated to be greater than 2 for any state, make it equal to 2 for that state).
#Repeat steps 2 & 3 as necessary.

If there are multiple phases, then steps 1-4 need to be performed for each phase where the initial probability of being in a state is equal to the final value from the previous phase.

This methodology provides the ability to give availability and unavailability metrics for the system, as well as point probabilities of being in a certain state.
For the reliability metrics, the methodology differs in that each unavailable state is considered a "sink." In other words, all transitions from unavailable to available states are ignored. This could be calculated simultaneously with the availability using the same step sizes as generated there.

The two results that need to be stored are the time and the corresponding probability of being in the state for each state.

Mixture Design

2016-01-12T21:15:19Z

Melinda Caroline: /* Example */

{{Template:Doebook|14}}

==Introduction==

When a product is formed by mixing together two or more ingredients, the product is called a mixture, and the ingredients are called mixture components. In a general mixture problem, the measured response is assumed to depend only on the proportions of the ingredients in the mixture, not the amount of the mixture. For example, the taste of a fruit punch recipe (i.e., the response) might depend on the proportions of watermelon, pineapple and orange juice in the mixture. The taste of a small cup of fruit punch will obviously be the same as a big cup.

Sometimes the responses of a mixture experiment depend not only on the proportions of ingredients, but also on the settings of variables in the process of making the mixture. For example, the tensile strength of stainless steel is not only affected by the proportions of iron, copper, nickel and chromium in the alloy; it is also affected by process variables such as temperature, pressure and curing time used in the experiment.

One of the purposes of conducting a mixture experiment is to find the best proportion of each component and the best value of each process variable, in order to optimize a single response or multiple responses simultaneously. In this chapter, we will discuss how to design effective mixture designs and how to analyze data from mixture experiments with and without process variables.

==Mixture Design Types==

There are several different types of mixture designs. The most common ones are simplex lattice, simplex centroid, simplex axial and simplex vertex designs, each of which is used for a different purpose.

*If there are many components in a mixture, the first choice is to screen out the most important ones. Simplex axial and Simplex centroid designs are used for this purpose.
*If the number of components is not large, but a high order polynomial equation is needed in order to accurately describe the response surface, then a simplex lattice design can be used.
*Simplex vertex designs are used for the cases when there are constraints on one or more components (e.g., if the proportion of watermelon juice in a fruit punch recipe is required to be less than 30%, and the combined proportion of watermelon and orange juice should always be between 40% and 70%).

===Simplex Plot===

Since the sum of all the mixture components is always 100%, the experiment space usually is given by a plot. The experiment space for the fruit punch experiment is given in the following triangle or simplex plot.

[[Image:doe_14.1.png|500 px|center]]

The triangle area in the above plot is defined by the fact that the sum of the three ingredients is 1 (100%). For the points that are on the vertices, the punch only has one ingredient. For instance, point 1 only has watermelon. The line opposite of point 1 represents a mixture with no watermelon .

The coordinate system used for the value of each ingredient
<math>{{x}_{i}}\,\!</math>,<math>i=1,2,...,q\,\!</math> is called a simplex coordinate system. q is the number of ingredients. The simplex plot can only visually display three ingredients. If there are more than three ingredients, the values for other ingredients must be provided. For the fruit punch example, the coordinate for point 1 is (1, 0, 0). The interior points of the triangle represent mixtures in which none of the three components is absent. It means all
<math>{{x}_{i}}>0\,\!</math>, <math>i=1,2,3\,\!</math>. Point 0 in the middle of the triangle is called the center point. In this case, it is the centroid of a face/plane. The coordinate for point 0 is (1/3, 1/3, 1/3). Points 2, 4 and 6 are each called a centroid of edge. Their coordinates are (0.5, 0.5, 0), (0, 0.5, 0.5), and (0.5, 0, 0.5).

===Simplex Lattice Design===

The response in a mixture experiment usually is described by a polynomial function. This function represents how the components affect the response. To better study the shape of the response surface, the natural choice for a design would be the one whose points are spread evenly over the whole simplex. An ordered arrangement consisting of a uniformly spaced distribution of points on a simplex is known as a lattice.

A {q, m} simplex lattice design for q components consists of points defined by the following coordinate settings: the proportions assumed by each component take the m+1 equally spaced values from 0 to 1,

::<math>{{x}_{i}}=0,\frac{1}{m},\frac{2}{m},....,1\text{ }i=1,2,....,q\,\!</math>

and the design space consists of all the reasonable combinations of all the values for each factor. m is usually called the degree of the lattice. For example, for a {3, 2} design, <math>{{x}_{i}}=0,\frac{1}{2},1\,\!</math> and its design space has 6 points. They are:

[[Image:doe_14.2.png|500 px|center]]

For a {3, 3} design, <math>{{x}_{i}}=0,\frac{1}{3},\frac{2}{3},1\,\!</math>, and its design space has 10 points. They are:

[[Image:doe_14.3.png|500 px|center]]

For a simplex design with degree of m, each component has m + 1 different values, therefore, the experiment results can be used to fit a polynomial equation up to an order of m. A {3, 3} simplex lattice design can be used to fit the following model.

::<math>\begin{align}
& y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+{{\beta }_{13}}{{x}_{1}}{{x}_{3}}+{{\beta }_{23}}{{x}_{2}}{{x}_{3}} \\
& +{{\delta }_{12}}{{x}_{1}}{{x}_{2}}\left( {{x}_{1}}-{{x}_{2}} \right)+{{\delta }_{13}}{{x}_{1}}{{x}_{3}}\left( {{x}_{1}}-{{x}_{3}} \right)+{{\delta }_{23}}{{x}_{2}}{{x}_{3}}\left( {{x}_{2}}-{{x}_{3}} \right) \\
& +{{\beta }_{123}}{{x}_{1}}{{x}_{2}}{{x}_{3}}
\end{align}\,\!</math>

The above model is called the full cubic model. Note that the intercept term is not included in the model due to the correlation between all the components (their sum is 100%).

Simplex lattice design includes all the component combinations. For a {q, m} design, the total number of runs is <math>\left( \begin{align}
& q+m-1 \\
& m \\
\end{align} \right)\,\!</math>. Therefore to reduce the number of runs and still be able to fit a high order polynomial model, sometimes we can use simplex centroid design which is explained next.

===Simplex Centroid Design===

A simplex centroid design only includes the centroid points. For the components that appear in a run in a simplex centroid design, they have the same values.

[[Image:doe_14.4.png|500 px|center]]

In the above simplex plot, points 2, 4 and 6 are 2nd degree centroids. Each of them has two non-zero components with equal values. Point 0 is a 3rd degree centroid and all three components have the same value. For a design with q components, the highest degree of centroid is q. It is called the overall centroid, or the center point of the design.

For a q component simplex centroid design with a degree of centroid of q, the total number of runs is
<math>{{2}^{q}}-1\,\!</math>. The runs correspond to the q permutations of (1, 0, 0,…, 0),
<math>\left( \begin{align}
& q \\
& 2 \\
\end{align} \right)\,\!</math> permutations of (1/2, 1/2, 0, 0, 0, 0, …,0), the
<math>\left( \begin{align}
& q \\
& 3 \\
\end{align} \right)\,\!</math> permutations of (1/3, 1/3, 1/3, 0, 0, 0, 0,…, 0)…., and the overall centroid (1/q, 1/q, …, 1/q). If the degree of centroid is defined as <math>m\,\!</math> (m < q), then the total number of runs will be
<math>\left( \begin{align}
& q \\
& 1 \\
\end{align} \right)+\left( \begin{align}
& q \\
& 2 \\
\end{align} \right)+...+\left( \begin{align}
& q \\
& m \\
\end{align} \right)\,\!</math>.

Since a simplex centroid design usually has fewer runs than a simplex lattice design with the same degree, a polynomial model with fewer terms should be used. A {3, 3} simplex centroid design can be used to fit the following model.

::<math>y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+{{\beta }_{13}}{{x}_{1}}{{x}_{3}}+{{\beta }_{23}}{{x}_{2}}{{x}_{3}}+{{\beta }_{123}}{{x}_{1}}{{x}_{2}}{{x}_{3}}\,\!</math>

The above model is called the special cubic model. Note that the intercept term is not included due to the correlation between all the components (their sum is 100%).

===Simplex Axial Design===

The simplex lattice and simplex centroid designs are boundary designs since the points of these designs are positioned on boundaries (vertices, edges, faces, etc.) of the simplex factor space, with the exception of the overall centroid. Axial designs, on the other hand, are designs consisting mainly of the points positioned inside the simplex. Axial designs have been recommended for use when component effects are to be measured in a screening experiment, particularly when first degree models are to be fitted.

Definition of Axial: The axial of a component <math>i\,\!</math> is defined as the imaginary line extending from the base point <math>{{x}_{i}}=0\,\!</math>, <math>{{x}_{j}}=1/\left( q-1 \right)\,\!</math> for all <math>j\ne i\,\!</math>, to the vertex where <math>{{x}_{i}}=1,{{x}_{j}}=0\,\!</math> all <math>j\ne i\,\!</math>. [John Cornell]

In a simplex axial design, all the points are on the axial. The simplest form of axial design is one whose points are positioned equidistant from the overall centroid <math>\left( {1}/{q,{1}/{q,}\;{1}/{q,}\;...}\; \right)\,\!</math>. Traditionally, points located at the half distance from the overall centroid to the vertex are called axial points/blends. This is illustrated in the following plot.

[[Image:doe_14.5.png|500 px|center]]

Points 4, 5 and 6 are the axial blends.

By default, a simple axial design in DOE++ only has vertices, axial blends, centroid of the constraint planes and the overall centroid. For a design with q components, constraint plane centroids are the center points of dimension of q-1 space. One component is 0, and the remaining components have the same values for the center points of constraint planes. The number of the constraint plane centroids is the number of components q. The total number of runs in a simple axial design will be 3q+1. They are q vertex runs, q centroids of constraint planes, q axial blends and 1 overall centroid.

A simplex axial design for 3 components has 10 points as given below.

[[Image:doe_14.6.png|500 px|center]]

Points 1, 2 and 3 are the three vertices; points 4, 5, 6 are the axial blends; points 7, 8 and 9 are the centroids of constraint planes, and point 0 is the overall center point.

===Extreme Vertex Design===

Extreme vertex designs are used when both lower and upper bound constraints on the components are presented, or when linear constraints are added to several components. For example, if a mixture design with 3 components has the following constraints:

:*<math>{{x}_{2}}\le 0.7\,\!</math>
:*<math>-2{{x}_{1}}+2{{x}_{2}}+3{{x}_{3}}\ge 0\,\!</math>
:*<math>48{{x}_{1}}+13{{x}_{2}}-{{x}_{3}}\ge 0\,\!</math>

Then the feasible region is defined by the six points in the following simplex plot. To meet the above constraints, all the runs conducted in the experiment should be in the feasible region or on its boundary.

[[Image:doe_14.7.png|500 px|center]]

The CONSIM method described in [Snee 1979] is used in DOE++ to check the consistency of all the constraints and to get the vertices defined by them.

Extreme vertex designs by default use the vertices at the boundary. Additional points such as the centroid of spaces of different dimensions, axial points and the overall center point can be added. In extreme vertex designs, axial points are between the overall center point and the vertices. For the above example, if the axial points and the overall center point are added, then all the runs in the experiment will be:

[[Image:doe_14.8.png|500 px|center]]

Point 0 in the center of the feasible region is the overall centroid. The other red points are the axial points. They are at the middle of the lines connecting the center point with the vertices.

==Mixture Design Data Analysis==

In the following section, we will discuss the most popular regression models in mixture design data analysis. Due to the correlation between all the components in mixture designs, the intercept term usually is not included in the regression model.

===Models Used in Mixture Design===

For a design with three components, the following models are commonly used.

*Linear model:

::<math>y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}\,\!</math>

If the intercept were included in the model, then the linear model would be

::<math>y=\beta _{0}^{'}+\beta _{1}^{'}{{x}_{1}}+\beta _{2}^{'}{{x}_{2}}+\beta _{3}^{'}{{x}_{3}}\,\!</math>

However, since <math>{{x}_{1}}+{{x}_{2}}+{{x}_{3}}=1\,\!</math> (can be other constants as well), the above equation can be written as

::<math>\begin{align}
& y=\beta _{0}^{'}\left( {{x}_{1}}+{{x}_{2}}+{{x}_{3}} \right)+\beta _{1}^{'}{{x}_{1}}+\beta _{2}^{'}{{x}_{2}}+\beta _{3}^{'}{{x}_{3}} \\
& =\left( \beta _{0}^{'}+\beta _{1}^{'} \right){{x}_{1}}+\left( \beta _{0}^{'}+\beta _{2}^{'} \right){{x}_{2}}+\left( \beta _{0}^{'}+\beta _{3}^{'} \right){{x}_{3}} \\
& ={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}
\end{align}\,\!</math>

The equation has thus been reformatted to omit the intercept.

*Quadratic model:

::<math>y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+{{\beta }_{13}}{{x}_{1}}{{x}_{3}}+{{\beta }_{23}}{{x}_{2}}{{x}_{3}}\,\!</math>

There are no classic quadratic terms such as <math>x_{1}^{2}\,\!</math>. This is because

::<math>x_{1}^{2}={{x}_{1}}\left( 1-{{x}_{2}}-{{x}_{3}} \right)={{x}_{1}}-{{x}_{1}}{{x}_{2}}-{{x}_{1}}{{x}_{3}}\,\!</math>

*Full cubic model:

::<math>\begin{align}
& y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+{{\beta }_{13}}{{x}_{1}}{{x}_{3}}+{{\beta }_{23}}{{x}_{2}}{{x}_{3}} \\
& +{{\delta }_{12}}{{x}_{1}}{{x}_{2}}\left( {{x}_{1}}-{{x}_{2}} \right)+{{\delta }_{13}}{{x}_{1}}{{x}_{3}}\left( {{x}_{1}}-{{x}_{3}} \right)+{{\delta }_{23}}{{x}_{2}}{{x}_{3}}\left( {{x}_{2}}-{{x}_{3}} \right) \\
& +{{\beta }_{123}}{{x}_{1}}{{x}_{2}}{{x}_{3}}
\end{align}\,\!</math>

*Special cubic model: <math>{{\delta }_{ij}}{{x}_{i}}{{x}_{j}}\left( {{x}_{i}}-{{x}_{j}} \right)\,\!</math> are removed from the full cubic model.

::<math>\begin{align}
& y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+{{\beta }_{13}}{{x}_{1}}{{x}_{3}}+{{\beta }_{23}}{{x}_{2}}{{x}_{3}} \\
& +{{\beta }_{123}}{{x}_{1}}{{x}_{2}}{{x}_{3}}
\end{align}\,\!</math>

The above types of models are called Scheffe type models. They can be extended to designs with more than three components.

In regular regression analysis, the effect of an exploratory variable or factor is represented by the value of the coefficient. The ratio of the estimated coefficient and its standard error is used for the t-test. The t-test can tell us if a coefficient is 0 or not. If a coefficient is statistically 0, then the corresponding factor has no significant effect on the response. However, for Scheffe type models, since the intercept term is not included in the model, we cannot use the regular t-test to test each individual main effect. In other words, we cannot test if the coefficient for each component is 0 or not.

Similarly, in the ANOVA analysis, the linear effects of all the components are tested together as a single group. The main effect test for each individual component is not conducted. To perform ANOVA analysis, the Scheffe type model needs to be reformatted to include the hidden intercept. For example, the linear model

::<math>y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}\,\!</math>

can be rewritten as

::<math>\begin{align}
& y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}} \\
& ={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}\left( 1-{{x}_{1}}-{{x}_{2}} \right) \\
& ={{\beta }_{3}}+\left( {{\beta }_{1}}-{{\beta }_{3}} \right){{x}_{1}}+\left( {{\beta }_{2}}-{{\beta }_{3}} \right){{x}_{2}} \\
& ={{\beta }_{0}}+\beta _{1}^{'}{{x}_{1}}+\beta _{2}^{'}{{x}_{2}}
\end{align}\,\!</math>

where <math>{{\beta }_{0}}={{\beta }_{3}}\,\!</math>, <math>\beta _{1}^{'}={{\beta }_{1}}-{{\beta }_{3}}\,\!</math>, <math>\beta _{2}^{'}={{\beta }_{2}}-{{\beta }_{3}}\,\!</math>. All other models such as the quadratic, cubic and special cubic model can be reformatted using the same procedure. By including the intercept in the model, the correct sum of squares can be calculated in the ANOVA table. If ANOVA analysis is conducted directly using the Scheffe type models, the result will be incorrect.

===L-Pseudocomponent, Proportion, and Actual Values===

In mixture designs, the total amount of the mixture is usually given. For example, we can make either a one-pound or a two-pound cake. Regardless of whether the cake is one or two pounds, the proportion of each ingredient is the same. When the total amount is given, the upper and lower limits for each ingredient are usually given in amounts, which is easier for the experimenter to understand. Of course, if the limits or other constraints are given in terms of proportions, these proportions need be converted to the real amount values when conducting the experiment. To keep everything consistent, all the constraints in DOE++ are treated as amounts.

In regular factorial design and response surface methods, the regression model is calculated using coded values. Coded values scale all the factors to the same magnitude, which makes the analysis much easier and reduces convergence error. Similarly, the analysis in mixture design is conducted using the so-called L-pseudocomponent value. L-pseudocomponent values scale all the components' values within 0 and 1. In DOE++ all the designs and calculations for mixture factors are based on L-pseudocomponent values. The relationship between L-pseudocomponent values, proportions and actual amounts are explained next.

====Example for L-Pseudocomponent Value====

We are going to make one gallon (about 3.8 liters) of fruit punch. Three ingredients will be in the punch with the following constraints.

::<math>1.2\le A\le 3.8\,\!</math>, <math>1.5\le B\le 3\,\!</math>, <math>0\le C\le 3.8\,\!</math>

Let <math>x_{i}^{A}\,\!</math> (i = 1, 2, 3) be the actual amount value, <math>x_{i}^{{}}\,\!</math> be the L-pseudocomponent value and <math>x_{i}^{R}\,\!</math> be the proportion value. Then the equations for the conversion between them are:

::<math>{{x}_{i}}=\frac{x_{i}^{A}-{{l}_{i}}}{\left( T-\sum\limits_{j=1}^{p}{{{l}_{j}}} \right)}\,\!</math>, <math>x_{i}^{A}={{l}_{i}}+\left( T-\sum\limits_{j=1}^{p}{{{l}_{i}}} \right){{x}_{i}}\,\!</math>, <math>x_{i}^{R}=\frac{x_{i}^{A}}{T}\,\!</math>

where <math>{{x}_{1}}\,\!</math>, <math>x_{1}^{A}\,\!</math> and
<math>x_{1}^{R}\,\!</math> are for component A, <math>{{x}_{2}}\,\!</math>, <math>x_{2}^{A}\,\!</math> and <math>x_{2}^{R}\,\!</math> are for component B, and <math>{{x}_{3}}\,\!</math>, <math>x_{3}^{A}\,\!</math> and <math>x_{3}^{R}\,\!</math> are for component C.

Since components in this example have both lower and upper limit constraints, an extreme vertex design is used. The design settings are given below.

[[Image:doe_14.9.png|500 px|center]]

The created design in terms of L-pseudocomponent values is:

[[Image:doe_14.10.png|500 px|center]]

Displayed in amount values, it is:

[[Image:doe_14.11.png|500 px|center]]

Displayed in proportion values, it is:

[[Image:doe_14.12.png|500 px|center]]

====Check Constraint Consistency====

In the above example, all the constraints are consistent. However, if we set the constraints to

::<math>1.2\le A\le 3.8\,\!</math>, <math>1.5\le B\le 3\,\!</math>, <math>2\le C\le 3.8,\,\!</math>

then they are not consistent. This is because the total is only 3.8, but the sum of all the lower limits is 4.7. Therefore, not all the lower limits can be satisfied at the same time. If only lower limits and upper limits are presented for all the components, then we can adjust the lower bounds to make the constraints consistent. The method given by [Pieple 1983] is used and summarized below.

Defined the range of a component to be <math>{{R}_{i}}={{U}_{i}}-{{L}_{i}}\,\!</math>. <math>{{U}_{i}}\,\!</math> and <math>{{L}_{i}}\,\!</math> are the upper and lower limit for component i. The implied range of component i is
<math>R_{i}^{*}=U_{i}^{*}-L_{i}^{*}\,\!</math>, where <math>L_{i}^{*}=T-\sum\limits_{j\ne i}^{q}{{{U}_{i}}}\,\!</math>, and
<math>U_{i}^{*}=T-\sum\limits_{j\ne i}^{q}{{{L}_{i}}}\,\!</math>. T is the total amount. The steps for checking and adjusting bounds are given below.

Step 1: Check if <math>L_{i}^{*}\,\!</math> and <math>U_{i}^{*}\,\!</math> are greater than 0, if they are, then these constraints meet the basic requirement to be consistent. We can move forward to step 2. If not, these constraints cannot be adjusted to be consistent. We should stop.

Step 2: For each component, check if <math>{{L}_{i}}\ge L_{i}^{*}\,\!</math> and <math>{{U}_{i}}\le U_{i}^{*}\,\!</math>. If they are, then this component’s constraints are consistent. Otherwise, if <math>{{L}_{i}}<L_{i}^{*}\,\!</math>, then set <math>{{L}_{i}}=L_{i}^{*}\,\!</math>, and if <math>{{U}_{i}}>U_{i}^{*}\,\!</math>, then set <math>{{U}_{i}}=U_{i}^{*}\,\!</math>.

Step 3: Whenever a bound is changed, restart from Step 1 to use the new bound to check if all the constraints are consistent. Repeat this until all the limits are consistent.

For extreme vertex design where linear constraints are allowed, DOE++ will give a warning and stop creating the design if inconsistent linear combination constraints are found. No adjustment will be conducted for linear constraints.

===Response Trace Plot===

Due to the correlation between all the components, the regular t-test is not used to test the significance of each component. A special plot called the Response Trace Plot can be used to see how the response changes when each component changes from its reference point [John Cornell].

A reference point can be any point inside the experiment space. An imaginary line can be drawn from this reference point to each vertex <math>{{x}_{i}}=1\,\!</math>, and <math>{{x}_{j}}=0\,\!</math> (<math>i\ne j\,\!</math>). This line is the direction for component i to change. Component i can either increase or decrease its value along this line, while the ratio of other components <math>{{{x}_{j}}}/{{{x}_{k}}}\,\!</math> (<math>j,k\ne i\,\!</math>) will keep constant. If the simplex plot is defined in terms of proportion, then the direction is called Cox’s direction, and <math>{{{x}_{j}}}/{{{x}_{k}}}\,\!</math> is the ratio of proportion. If the simplex plot is defined in terms of pseduocomponent value, then the direction is called Pieple’s direction, and <math>{{{x}_{j}}}/{{{x}_{k}}}\,\!</math> will be the ratio of pseduocomponent values.

Assume the reference point in terms of proportion is <math>s=\left( {{s}_{1}},{{s}_{2}},...,{{s}_{q}} \right)\,\!</math> where <math>{{s}_{1}}+{{s}_{2}}+...+{{s}_{q}}=1\,\!</math>. Suppose the proportion of component <math>i\,\!</math> at <math>{{s}_{i}}\,\!</math> is now changed by <math>{{\Delta }_{i}}\,\!</math> (<math>{{\Delta }_{i}}\,\!</math> could be greater than or less than 0) in Cox’s direction, so that the new proportion becomes <math>{{x}_{i}}={{s}_{i}}+{{\Delta }_{i}}\,\!</math>

Then the proportions of the remaining <math>q-1\,\!</math> components resulting from the change from <math>{{s}_{i}}\,\!</math> will be

::<math>{{x}_{j}}={{s}_{j}}-\frac{{{\Delta }_{i}}{{s}_{j}}}{1-{{s}_{i}}}\,\!</math>

After the change, the ratio of component j and k is unchanged. This is because

::<math>\frac{{{x}_{j}}}{{{x}_{k}}}=\frac{{{s}_{j}}-\frac{{{\Delta }_{i}}{{s}_{j}}}{1-{{s}_{i}}}}{{{s}_{k}}-\frac{{{\Delta }_{i}}{{s}_{k}}}{1-{{s}_{i}}}}=\frac{{{s}_{j}}}{{{s}_{k}}}\frac{\frac{{{\Delta }_{i}}}{1-{{s}_{i}}}}{\frac{{{\Delta }_{i}}}{1-{{s}_{i}}}}=\frac{{{s}_{j}}}{{{s}_{k}}}\,\!</math>

While <math>{{x}_{i}}\,\!</math> is changed along Cox’s direction, we can use a fitted regression model to get the response value y. A response trace plot for a mixture design with three components will look like

[[Image:doe_14.13.png|500 px|center]]

The x-axis is the deviation amount from the reference point, and the y-value is the fitted response. Each component has one curve. Since the red curve for component A changes significantly, this means it has a significant effect along its axial. The blue curve for component C is almost flat; this means when C changes along Cox’s direction and other components keep the same ratio, the response Y does not change very much. The effect of component B is between component A and C.

===Example===

Watermelon (A), pineapple (B) and orange juice (C) are used for making 3.8 liters of fruit punch. At least 30% of the fruit punch must be watermelon. Therefore the constraints are

::<math>1.14\le A\le 3.8\,\!</math>, <math>0\le B\le 3.8\,\!</math>, <math>0\le C\le 3.8,\,\!</math>

Different blends of the three-juice recipe were evaluated by a panel. A value from 1 (extremely poor) to 9 (very good) is used for the response [John Cornell, page 74]. A {3, 2} simplex lattice design is used with one center point and three axial points. Three replicates were conducted for each ingredient combination. The settings for creating this design in DOE++ is

[[Image:doe_14.14.png|500 px|center]]

The generated design in L-pseudocomponent values and the response values from the experiment are

[[Image:doe_14.15.png|500 px|center]]

The simplex design point plot is

[[Image:doe_14.16.png|500 px|center]]

Main effect and 2-way interactions are included in the regression model. The result for the regression model in terms of L-pseudocomponents is

::<math>y=4.81{{x}_{1}}+6.03{{x}_{2}}+6.16{{x}_{3}}+1.13{{x}_{1}}{{x}_{2}}+2.45{{x}_{1}}{{x}_{3}}+1.69{{x}_{2}}{{x}_{3}}\,\!</math>

The regression information table is

{| border="1" border="1" cellpadding="5" cellspacing="0" align="center"
! colspan="8" style="text-align: center; font-weight: bold;" style="background:#CCCCCC"| Regression Information
|-
| style="text-align: center; font-weight: bold;" | Term
| style="text-align: center; font-weight: bold;" | Coefficient
| style="text-align: center; font-weight: bold;" | Standard Error
| style="text-align: center; font-weight: bold;" | Low Confidence
| style="text-align: center; font-weight: bold;" | High Confidence
| style="text-align: center; font-weight: bold;" | T Value
| style="text-align: center; font-weight: bold;" | P Value
| style="text-align: center; font-weight: bold;" | Variance Inflation Factor
|-
| style="text-align: center;" | A: Watermelon
| style="text-align: center;" | 4.8093
| style="text-align: center;" | 0.3067
| style="text-align: center;" | 4.2845
| style="text-align: center;" | 5.3340
| style="text-align: center;" |
| style="text-align: center;" |
| style="text-align: center;" | 1.9636
|-
| style="text-align: center;" | B: Pineapple
| style="text-align: center;" | 6.0274
| style="text-align: center;" | 0.3067
| style="text-align: center;" | 5.5027
| style="text-align: center;" | 6.5522
| style="text-align: center;" |
| style="text-align: center;" |
| style="text-align: center;" | 1.9636
|-
| style="text-align: center;" | C: Orange
| style="text-align: center;" | 6.1577
| style="text-align: center;" | 0.3067
| style="text-align: center;" | 5.6330
| style="text-align: center;" | 6.6825
| style="text-align: center;" |
| style="text-align: center;" |
| style="text-align: center;" | 1.9636
|-
| style="text-align: center;" | A • B
| style="text-align: center;" | 1.1253
| style="text-align: center;" | 1.4137
| style="text-align: center;" | -1.2934
| style="text-align: center;" | 3.5439
| style="text-align: center;" | 0.7960
| style="text-align: center;" | 0.4339
| style="text-align: center;" | 1.9819
|-
| style="text-align: center;" | A • C 
| style="text-align: center;" | 2.4525
| style="text-align: center;" | 1.4137
| style="text-align: center;" | 0.0339
| style="text-align: center;" | 4.8712
| style="text-align: center;" | 1.7348
| style="text-align: center;" | 0.0956 
| style="text-align: center;" | 1.9819
|-
| style="text-align: center;" | B • C
| style="text-align: center;" | 1.6889
| style="text-align: center;" | 1.4137
| style="text-align: center;" | -0.7298
| style="text-align: center;" | 4.1075
| style="text-align: center;" | 1.1947
| style="text-align: center;" | 0.2439
| style="text-align: center;" | 1.9819
|}

The result shows that the taste of the fruit punch is significantly affected by the interaction between watermelon and orange.

The ANOVA table is

{| border="1" border="1" cellpadding="5" cellspacing="0" align="center"
! colspan="6" style="text-align: center; font-weight: bold;" style="background:#CCCCCC"| Anova Table
|-
| style="text-align: center; font-weight: bold;" | Source of Variation
| style="text-align: center; font-weight: bold;" | Degrees of Freedom
| style="text-align: center; font-weight: bold;" | Standard ErrorSum of Squares [Partial]
| style="text-align: center; font-weight: bold;" | Mean Squares [Partial]
| style="text-align: center; font-weight: bold;" | F Ratio
| style="text-align: center; font-weight: bold;" | P Value
|-
| Model 
| style="text-align: center;" | 5
| style="text-align: center;" | 6.5517
| style="text-align: center;" | 1.3103
| style="text-align: center;" | 4.3181
| style="text-align: center;" | 0.0061 
|-
| style="text-align: center;" | Linear 
| style="text-align: center;" | 2
| style="text-align: center;" | 3.6513
| style="text-align: center;" | 1.8256
| style="text-align: center;" | 6.0162
| style="text-align: center;" | 0.0076 
|-
| style="text-align: center;" | A • B
| style="text-align: center;" | 1
| style="text-align: center;" | 0.1923
| style="text-align: center;" | 0.1923
| style="text-align: center;" | 0.6336
| style="text-align: center;" | 0.4339
|-
| style="text-align: center;" | A • C 
| style="text-align: center;" | 1
| style="text-align: center;" | 0.9133
| style="text-align: center;" | 0.9133
| style="text-align: center;" | 3.0097
| style="text-align: center;" | 0.0956 
|-
| style="text-align: center;" | B • C
| style="text-align: center;" | 1
| style="text-align: center;" | 0.4331
| style="text-align: center;" | 0.4331
| style="text-align: center;" | 1.4272
| style="text-align: center;" | 0.2439
|-
| Residual
| style="text-align: center;" | 24
| style="text-align: center;" | 7.2829
| style="text-align: center;" | 0.3035
| style="text-align: center;" |
| style="text-align: center;" |
|-
| style="text-align: center;" | Lack of Fit 
| style="text-align: center;" | 4
| style="text-align: center;" | 4.4563
| style="text-align: center;" | 1.1141
| style="text-align: center;" | 7.8825
| style="text-align: center;" | 0.0006 
|-
| style="text-align: center;" | Pure Error
| style="text-align: center;" | 20
| style="text-align: center;" | 2.8267
| style="text-align: center;" | 0.1413
|
|
|-
| Total
| style="text-align: center;" | 29
| style="text-align: center;" | 13.8347
| style="text-align: center;" |
|
|
|}

The simplex contour plot in L-pseudocomponent values is

[[Image:doe_14.17.png|500 px|center]]

From this plot we can see that as the amount of watermelon is reduced, the taste of the fruit punch becomes better.

In order to find the best proportion of each ingredient, the optimization tool in DOE++ can be utilized. Set the settings as

[[Image:doe_14.18.png|500 px|center]]

The resulting optimal plot is

[[Image:doe_14.19.png|500 px|center]]

This plot shows that when the amounts for watermelon, pineapple and orange juice are 1.141, 1.299 and 1.359, respectively, the rated taste of the fruit punch is highest.

==Mixture Design with Process Variables==

Process variables often play very important roles in mixture experiments. A simple example is baking a cake. Even with the same ingredients, different baking temperatures and baking times can produce completely different results. In order to study the effect of process variables and find their best settings, we need to consider them when conducting a mixture experiment.

An easy way to do this is to make mixtures with the same ingredients in different combinations of process variables. If all the process variables are independent, then we can plan a regular factorial design for these process variables. By combining these designs with a separated mixture design, the effect of mixture components and effect of process variables can be studied.

For example, a {3, 2} simplex lattice design is used for a mixture with 3 components. Together with the center point, it has total of 7 runs or 7 different ingredient combinations. Assume 2 process variables are potentially important and a two level factorial design is used for them. It has a total of 4 combinations for these 2 process variables. If the 7 different mixtures are made under each of the 4 process variable combinations, then the experiment has a total of 28 runs. This is illustrated in the figure below.

[[Image:doe_14.20.png|500 px|center]]

Of course, if it is possible, all the 28 experiments should be conducted in a random order.

===Model with Process Variables===

In DOE++, regression models including both mixture components and process variables are available. For mixture components, we use L-pseudocomponent values, and for process variables coded values are used.

Assume a design has 3 mixture components and 2 process variables, as illustrated in the above figure. We can use the following models for them.

*For the 3 mixture components, the following special cubic model is used.

::<math>y={{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{3}}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+{{\beta }_{13}}{{x}_{1}}{{x}_{3}}+{{\beta }_{23}}{{x}_{2}}{{x}_{3}}+{{\beta }_{123}}{{x}_{1}}{{x}_{2}}{{x}_{3}}\,\!</math>

*For the 2 process variables the following model is used.

::<math>y={{\alpha }_{0}}+{{\alpha }_{1}}{{z}_{1}}+{{\alpha }_{2}}{{z}_{2}}+{{\alpha }_{12}}{{z}_{1}}{{z}_{2}}\,\!</math>

*The combined model with both mixture components and process variables is

::<math>\begin{align}
& y=\sum\limits_{i=1}^{3}{\gamma _{i}^{0}{{x}_{i}}}+\sum{\sum\limits_{i<j}^{3}{\gamma _{ij}^{0}{{x}_{i}}{{x}_{j}}}+}\gamma _{123}^{0}{{x}_{1}}{{x}_{2}}{{x}_{3}} \\
& +\left( \sum\limits_{i=1}^{3}{\gamma _{i}^{1}{{x}_{i}}}+\sum{\sum\limits_{i<j}^{3}{\gamma _{ij}^{1}{{x}_{i}}{{x}_{j}}}+}\gamma _{123}^{1}{{x}_{1}}{{x}_{2}}{{x}_{3}} \right){{z}_{1}} \\
& +\left( \sum\limits_{i=1}^{3}{\gamma _{i}^{2}{{x}_{i}}}+\sum{\sum\limits_{i<j}^{3}{\gamma _{ij}^{2}{{x}_{i}}{{x}_{j}}}+}\gamma _{123}^{2}{{x}_{1}}{{x}_{2}}{{x}_{3}} \right){{z}_{2}} \\
& +\left( \sum\limits_{i=1}^{3}{\gamma _{i}^{12}{{x}_{i}}}+\sum{\sum\limits_{i<j}^{3}{\gamma _{ij}^{12}{{x}_{i}}{{x}_{j}}}+}\gamma _{123}^{12}{{x}_{1}}{{x}_{2}}{{x}_{3}} \right){{z}_{1}}{{z}_{2}}
\end{align}\,\!</math>

The above combined model has total of 7x4=28 terms. By expanding it, we get the following model:

::<math>\begin{align}
& y=\gamma _{1}^{0}{{x}_{1}}+\gamma _{2}^{0}{{x}_{2}}+\gamma _{3}^{0}{{x}_{3}}+\gamma _{12}^{0}{{x}_{1}}{{x}_{2}}+\gamma _{13}^{0}{{x}_{1}}{{x}_{3}}+\gamma _{23}^{0}{{x}_{2}}{{x}_{3}}+\gamma _{123}^{0}{{x}_{1}}{{x}_{2}}{{x}_{3}} \\
& +\gamma _{1}^{1}{{x}_{1}}{{z}_{1}}+\gamma _{2}^{1}{{x}_{2}}{{z}_{1}}+\gamma _{3}^{1}{{x}_{3}}{{z}_{1}}+\gamma _{12}^{1}{{x}_{1}}{{x}_{2}}{{z}_{1}}+\gamma _{13}^{1}{{x}_{1}}{{x}_{3}}{{z}_{1}}+\gamma _{23}^{1}{{x}_{2}}{{x}_{3}}{{z}_{1}}+\gamma _{123}^{1}{{x}_{1}}{{x}_{2}}{{x}_{3}}{{z}_{1}} \\
& +\gamma _{1}^{2}{{x}_{1}}{{z}_{2}}+\gamma _{2}^{2}{{x}_{2}}{{z}_{2}}+\gamma _{3}^{2}{{x}_{3}}{{z}_{2}}+\gamma _{12}^{2}{{x}_{1}}{{x}_{2}}{{z}_{2}}+\gamma _{13}^{2}{{x}_{1}}{{x}_{3}}{{z}_{2}}+\gamma _{23}^{2}{{x}_{2}}{{x}_{3}}{{z}_{2}}+\gamma _{123}^{2}{{x}_{1}}{{x}_{2}}{{x}_{3}}{{z}_{2}} \\
& +\gamma _{1}^{12}{{x}_{1}}{{z}_{1}}{{z}_{2}}+\gamma _{2}^{12}{{x}_{2}}{{z}_{1}}{{z}_{2}}+\gamma _{3}^{12}{{x}_{3}}{{z}_{1}}{{z}_{2}}+\gamma _{12}^{12}{{x}_{1}}{{x}_{2}}{{z}_{1}}{{z}_{2}}+\gamma _{13}^{12}{{x}_{1}}{{x}_{3}}{{z}_{1}}{{z}_{2}}+\gamma _{23}^{12}{{x}_{2}}{{x}_{3}}{{z}_{1}}{{z}_{2}}+\gamma _{123}^{12}{{x}_{1}}{{x}_{2}}{{x}_{3}}{{z}_{1}}{{z}_{2}}
\end{align}\,\!</math>

The combined model basically crosses every term in the mixture components model with every term in the process variables model. From a mathematical point of view, this model is just a regular regression model. Therefore, the traditional regression analysis method can still be used for obtaining the model coefficients and calculating the ANOVA table.

===Example===

Three kinds of meats (beef, pork and lamb) are mixed together to form burger patties. The meat comprises 90% of the total mixture, with the remaining 10% reserved for flavoring ingredients. A {3, 2} simplex design with the center point is used for the experiment. The design has 7 meat combinations, which are given below using L-pseudocomponent values.

{| border="1" cellpadding="5" cellspacing="0" align="center"
!A: Beef
!B: Pork
!C: Lamb
|-
|style="text-align: center"|1||style="text-align: center"| 0||style="text-align: center"| 0
|-
|style="text-align: center"|0.5||style="text-align: center"| 0.5||style="text-align: center"| 0
|-
|style="text-align: center"|0.5||style="text-align: center"| 0||style="text-align: center"| 0.5
|-
|style="text-align: center"|0||style="text-align: center"| 1||style="text-align: center"| 0
|-
|style="text-align: center"|0||style="text-align: center"| 0.5||style="text-align: center"| 0.5
|-
|style="text-align: center"|0||style="text-align: center"| 0||style="text-align: center"| 1
|-
|style="text-align: center"|0.333333||style="text-align: center"| 0.333333||style="text-align: center"| 0.333333
|}

Two process variables on making the patties are also studied: cooking temperature and cooking time. The low and high temperature values are 375°F and 425°F, and the low and high time values are 25 and 40 minutes. A two level full factorial design is used and displayed below with coded values.

{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
!Temperature
!Time
|-
| -1|| -1
|-
| -1|| 1
|-
| 1|| -1
|-
| 1|| 1
|}

One of the properties of the burger patties is texture. The texture is measured by a compression test that measures the grams of force required to puncture the surface of the patty.

Combining the simplex design and the factorial design together, we get the following 28 runs. The corresponding texture reading for each blend is also provided.

{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
!Standard Order
!A: Beef
!B: Pork
!C: Lamb
!Z1: Temperature
!Z2: Time
!Texture (<math>10^3\,\!</math> gram)
|-
|1|| 1|| 0|| 0|| -1|| -1|| 1.84
|-
|2|| 0.5|| 0.5|| 0|| -1|| -1|| 0.67
|-
|3|| 0.5|| 0|| 0.5|| -1|| -1|| 1.51
|-
|4|| 0|| 1|| 0|| -1|| -1|| 1.29
|-
|5|| 0|| 0.5|| 0.5|| -1|| -1|| 1.42
|-
|6|| 0|| 0|| 1|| -1|| -1|| 1.16
|-
|7|| 0.333|| 0.333|| 0.333|| -1|| -1|| 1.59
|-
|8|| 1|| 0|| 0|| 1|| -1|| 2.86
|-
|9|| 0.5|| 0.5|| 0|| 1|| -1|| 1.1
|-
|10|| 0.5|| 0|| 0.5|| 1|| -1|| 1.6
|-
|11|| 0|| 1|| 0|| 1|| -1|| 1.53
|-
|12|| 0|| 0.5|| 0.5|| 1|| -1|| 1.81
|-
|13|| 0|| 0|| 1|| 1|| -1|| 1.5
|-
|14|| 0.333|| 0.333|| 0.333|| 1|| -1|| 1.68
|-
|15|| 1|| 0|| 0|| -1|| 1|| 3.01
|-
|16|| 0.5|| 0.5|| 0|| -1|| 1|| 1.21
|-
|17|| 0.5|| 0|| 0.5|| -1|| 1|| 2.32
|-
|18|| 0|| 1|| 0|| -1|| 1|| 1.93
|-
|19|| 0|| 0.5|| 0.5|| -1|| 1|| 2.57
|-
|20|| 0|| 0|| 1|| -1|| 1|| 1.83
|-
|21|| 0.333|| 0.3333|| 0.333|| -1|| 1|| 1.94
|-
|22|| 1|| 0|| 0|| 1|| 1|| 4.13
|-
|23|| 0.5|| 0.5|| 0|| 1|| 1|| 1.67
|-
|24|| 0.5|| 0|| 0.5|| 1|| 1|| 2.57
|-
|25|| 0|| 1|| 0|| 1|| 1|| 2.26
|-
|26|| 0|| 0.5|| 0.5|| 1|| 1|| 3.15
|-
|27|| 0|| 0|| 1|| 1|| 1|| 2.22
|-
|28|| 0.333|| 0.333|| 0.333|| 1|| 1|| 2.6
|}

Using a quadratic model for the mixture component and a 2-way interaction model for the process variables, we get the following results.

{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
!Term
!Coefficient
!Standard Error
!T Value
!P Value
!Variance Inflation Factor
|-
| A:Beef || 2.9421|| 0.1236|| *|| *|| 1.5989
|-
| B:Pork || 1.7346|| 0.1236|| *|| *|| 1.5989
|-
| C:Lamb || 1.6596|| 0.1236|| *|| *|| 1.5989
|-
| A • B || -4.4170|| 0.5680|| -7.7766|| 0.0015 || 1.5695
|-
|A • C|| -0.9170|| 0.5680|| -1.6146|| 0.1817|| 1.5695
|-
| B • C || 2.4480|| 0.5680|| 4.3099|| 0.0125 || 1.5695
|-
| Z1 • A || 0.5324|| 0.1236|| 4.3084|| 0.0126 || 1.5989
|-
|Z1 • B|| 0.1399|| 0.1236|| 1.1319|| 0.3209|| 1.5989
|-
|Z1 • C|| 0.1799|| 0.1236|| 1.4557|| 0.2192|| 1.5989
|-
|Z1 • A • B|| -0.4123|| 0.5680|| -0.7260|| 0.5081|| 1.5695
|-
|Z1 • A • C|| -1.0423|| 0.5680|| -1.8352|| 0.1404|| 1.5695
|-
|Z1 • B • C|| 0.3727|| 0.5680|| 0.6561|| 0.5476|| 1.5695
|-
| Z2 • A || 0.6193|| 0.1236|| 5.0117|| 0.0074 || 1.5989
|-
| Z2 • B || 0.3518|| 0.1236|| 2.8468|| 0.0465 || 1.5989
|-
| Z2 • C || 0.3568|| 0.1236|| 2.8873|| 0.0447 || 1.5989
|-
|Z2 • A • B|| -0.9802|| 0.5680|| -1.7258|| 0.1595|| 1.5695
|-
|Z2 • A • C|| -0.3202|| 0.5680|| -0.5638|| 0.6030|| 1.5695
|-
|Z2 • B • C|| 0.9248|| 0.5680|| 1.6282|| 0.1788|| 1.5695
|-
|Z1 • Z2 • A|| 0.0177|| 0.1236|| 0.1433|| 0.8930|| 1.5989
|-
|Z1 • Z2 • B|| 0.0152|| 0.1236|| 0.1231|| 0.9080|| 1.5989
|-
|Z1 • Z2 • C|| 0.0052|| 0.1236|| 0.0422|| 0.9684|| 1.5989
|-
|Z1 • Z2 • A • B|| 0.0808|| 0.5680|| 0.1423|| 0.8937|| 1.5695
|-
|Z1 • Z2 • A • C|| 0.2308|| 0.5680|| 0.4064|| 0.7052|| 1.5695
|-
|Z1 • Z2 • B • C|| 0.2658|| 0.5680|| 0.4680|| 0.6641|| 1.5695
|}

The above table shows that all the terms with <math>{{z}_{1}}\times {{z}_{2}}\,\!</math> have very large P values, therefore, we can remove these terms from the model. We can also remove other terms with P values larger than 0.5. After recalculating with the desired terms, the final results are

{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
!Term
!Coefficient
!Standard Error
!T Value
!P Value
!Variance Inflation Factor
|-
| A:Beef || 2.9421|| 0.0875|| *|| *|| 1.5989
|-
| B:Pork || 1.7346|| 0.0875|| *|| *|| 1.5989
|-
| C:Lamb || 1.6596|| 0.0875|| *|| *|| 1.5989
|-
| A • B || -4.4170|| 0.4023|| -10.9782|| 6.0305E-08 || 1.5695
|-
| A • C || -0.9170|| 0.4023|| -2.2792|| 0.0402 || 1.5695
|-
| B • C || 2.4480|| 0.4023|| 6.0842|| 3.8782E-05 || 1.5695
|-
| Z1 • A || 0.4916|| 0.0799|| 6.1531|| 3.4705E-05 || 1.3321
|-
| Z1 • B || 0.1365|| 0.0725|| 1.8830|| 0.0823 || 1.0971
|-
| Z1 • C || 0.2176|| 0.0799|| 2.7235|| 0.0174 || 1.3321
|-
| Z1 • A • C || -1.0406|| 0.4015|| -2.5916|| 0.0224 || 1.5631
|-
| Z2 • A || 0.5910|| 0.0800|| 7.3859|| 5.3010E-06 || 1.3364
|-
| Z2 • B || 0.3541|| 0.0875|| 4.0475|| 0.0014 || 1.5971
|-
| Z2 • C || 0.3285|| 0.0800|| 4.1056|| 0.0012 || 1.3364
|-
| Z2 • A • B || -0.9654|| 0.4019|| -2.4020|| 0.0320 || 1.5661
|-
| Z2 • B • C || 0.9396|| 0.4019|| 2.3378|| 0.0360 || 1.5661
|}

The regression model is

::<math>\begin{align}
& y=2.9421{{x}_{1}}+1.7346{{x}_{2}}+1.6596{{x}_{3}}-4.4170{{x}_{1}}{{x}_{2}}-0.9170{{x}_{1}}{{x}_{3}}+2.4480{{x}_{2}}{{x}_{3}} \\
& +0.4916{{x}_{1}}{{z}_{1}}+0.1365{{x}_{2}}{{z}_{1}}+0.2176{{x}_{3}}{{z}_{1}}-1.0406{{x}_{1}}{{x}_{3}}{{z}_{1}}+0.5910{{x}_{1}}{{z}_{2}} \\
& +0.3541{{x}_{2}}{{z}_{2}}+0.3285{{x}_{3}}{{z}_{2}}-0.9654{{x}_{1}}{{x}_{2}}{{z}_{2}}+0.9396{{x}_{2}}{{x}_{3}}{{z}_{2}}
\end{align}\,\!</math>

The ANOVA table for this model is

{| border="1" cellpadding="5" cellspacing="0" align="center"
! colspan="6" style="background: #CCCCCC;"|ANOVA Table
|-
!Source of Variation||Degrees of Freedom||Sum of Squares [Partial]|| Mean Squares [Partial]|| F Ratio|| P Value
|-
| Model||style="text-align: center"| 14||style="text-align: center"| 14.5066||style="text-align: center"| 1.0362|| style="text-align: center"|33.5558||style="text-align: center"| 6.8938E-08
|-
|style="text-align: center"|Component Only || || || || ||
|-
|style="text-align: right"| Linear ||style="text-align: center"| 2||style="text-align: center"| 4.1446||style="text-align: center"| 2.0723||style="text-align: center"| 67.1102||style="text-align: center"| 1.4088E-07
|-
|style="text-align: right"| A • B ||style="text-align: center"| 1||style="text-align: center"| 3.7216||style="text-align: center"| 3.7216||style="text-align: center"| 120.5208||style="text-align: center"| 6.0305E-08
|-
|style="text-align: right"| A • C ||style="text-align: center"| 1||style="text-align: center"| 0.1604||style="text-align: center"| 0.1604||style="text-align: center"| 5.1949||style="text-align: center"| 0.0402
|-
|style="text-align: right"| B • C ||style="text-align: center"| 1||style="text-align: center"| 1.1431||style="text-align: center"| 1.1431||style="text-align: center"| 37.0173 ||style="text-align: center"|3.8782E-05
|-
|style="text-align: center"|Component • Z1|| || || || ||
|-
|style="text-align: right"| Z1 • A ||style="text-align: center"|1||style="text-align: center"| 1.1691||style="text-align: center"| 1.1691||style="text-align: center"| 37.8604||style="text-align: center"| 3.4705E-05
|-
|style="text-align: right"| Z1 • B ||style="text-align: center"| 1||style="text-align: center"| 0.1095||style="text-align: center"| 0.1095||style="text-align: center"| 3.5456||style="text-align: center"| 0.0823
|-
|style="text-align: right"| Z1 • C ||style="text-align: center"| 1||style="text-align: center"| 0.2290||style="text-align: center"| 0.2290||style="text-align: center"| 7.4172||style="text-align: center"| 0.0174
|-
|style="text-align: right"| Z1 • A • C ||style="text-align: center"| 1||style="text-align: center"| 0.2074||style="text-align: center"| 0.2074||style="text-align: center"| 6.7165||style="text-align: center"| 0.0224
|-
|style="text-align: center"|Component • Z2|| || || || ||
|-
|style="text-align: right"| Z2 • A ||style="text-align: center"| 1||style="text-align: center"| 1.6845||style="text-align: center"| 1.6845|| style="text-align: center"|54.5517||style="text-align: center"| 5.3010E-06
|-
|style="text-align: right"| Z2 • B ||style="text-align: center"| 1||style="text-align: center"| 0.5059||style="text-align: center"| 0.5059||style="text-align: center"| 16.3819||style="text-align: center"| 0.0014
|-
|style="text-align: right"| Z2 • C ||style="text-align: center"| 1||style="text-align: center"| 0.5205||style="text-align: center"| 0.5205||style="text-align: center"| 16.8556||style="text-align: center"| 0.0012
|-
|style="text-align: right"| Z2 • A • B ||style="text-align: center"| 1||style="text-align: center"| 0.1782||style="text-align: center"| 0.1782||style="text-align: center"| 5.7698||style="text-align: center"| 0.0320
|-
|style="text-align: right"| Z2 • B • C ||style="text-align: center"| 1||style="text-align: center"| 0.1688||style="text-align: center"| 0.1688||style="text-align: center"| 5.4651||style="text-align: center"| 0.0360
|-
|style="text-align: left"|Residual ||style="text-align: center"| 13||style="text-align: center"| 0.4014||style="text-align: center"| 0.0309|| ||
|-
|style="text-align: center"|Lack of Fit||style="text-align: center"| 13||style="text-align: center"| 0.4014||style="text-align: center"| 0.0309 || ||
|-
|style="text-align: left"|Total||style="text-align: center"| 27||style="text-align: center"| 14.9080|| || ||
|}

The above table shows both process factors have significant effects on the texture of the patties. Since the model is pretty complicate, the best settings for the process variables and for components cannot be easily identified.

The optimization tool in DOE++ is used for the above model. The target texture value is <math>3\times {{10}^{3}}\,\!</math> grams with an acceptable range of <math>2.5-3.5\times {{10}^{3}}\,\!</math> grams.

[[Image:doe_14.21.png|600 px|center]]

The optimal solution is Beef = 98.5%, Pork = 0.7%, Lamb = 0.7%, Temperature = 375.7, and Time = 40.

==References==
 
1. Cornell, John (2002), Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, John Wiley & Sons, Inc. New York.
 
2. Piepel, G. F. (1983), “Defining consistent constraint regions in mixture experiments,” Technometrics, Vol. 25, pp. 97-101.
 
3. Snee, R. D. (1979), “Experimental designs for mixture systems with multiple component constraints,” Communications in Statistics, Theory and Methods, Bol. A8, pp. 303-326.

Two Level Factorial Experiments

2015-12-15T16:07:57Z

Melinda Caroline: /* Design Resolution */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In DOE++, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|thumb|center|487px|Selection of full factorial experiments with two levels in DOE++.]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|thumb|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|thumb|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|thumb|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in DOE++, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in DOE++ as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|thumb|center|887px|Design properties for the experiment in the example.]]

[[Image:doe7_6.png|thumb|center|859px|Design summary for the experiment in the example.]]

[[Image:doe7_7.png|thumb|center|850px|Experiment design for the example to investigate the surface finish of automobile brake drums.]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|thumb|center|633px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|thumb|center|650px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from DOE++ to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In DOE++, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in DOE++, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|thumb|center|619px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into DOE++ as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|thumb|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of DOE++ plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. DOE++ then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|thumb|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|thumb|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|thumb|center|650px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].]]

[[Image:doe7_15.png|thumb|center|580px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].]]

[[Image:doe7_16.png|thumb|center|587px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png|thumb|center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|thumb|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|thumb|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|thumb|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of DOE++. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|thumb|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|thumb|center|400px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by DOE++ is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|thumb|center|700px| Adding block properties for the experiment in the example.]]

[[Image:doe7_24.png|thumb|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].]]

[[Image:doe7_25.png|thumb|center|650px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|thumb|center|650px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|thumb|center|400px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|thumb|center|631px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].]]

[[Image:doe7_29.png|thumb|center|635px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|thumb|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|thumb|center|567px|Variability analysis in DOE++.]]
 

[[Image:doe7_32.png|thumb|center|592px|Normal probability plot of effects for the variability analysis example.]]
 

[[Image:doe7_33.png|thumb|center|646px|Effect coefficients for the variability analysis example.]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|thumb|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|thumb|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|thumb|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|thumb|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|thumb|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|thumb|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|thumb|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|thumb|center|538px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].]]

[[Image:doe7_40.png|thumb|center|623px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|thumb|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|thumb|center|599px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|thumb|center|629px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

File:Doe7.38.png

2015-12-15T15:57:48Z

Melinda Caroline: uploaded a new version of "File:Doe7.38.png"

Highly Fractional Factorial Designs

2015-11-24T18:30:07Z

Melinda Caroline: /* Example */

{{Template:Doebook|9}}
This chapter discusses factorial designs that are commonly used in designed experiments, but are not necessarily limited to two level factors. These designs are the [[Highly_Fractional_Factorial_Designs#Plackett-Burman_Designs|Plackett-Burman designs]] and [[Highly_Fractional_Factorial_Designs#Taguchi.27s_Orthogonal_Arrays|Taguchi's orthogonal arrays]].

==Plackett-Burman Designs==

It was mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] that resolution III designs can be used as highly fractional designs to investigate <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs (provided that three factor and higher order interaction effects are not important to the experimenter). A limitation with these designs is that all runs in these designs have to be a power of 2. The valid runs for these designs are 4, 8, 16, 32, etc. Therefore, the next design after the 2 <math>_{\text{III}}^{3-1}\,\!</math> design with 4 runs is the 2 <math>_{\text{III}}^{7-4}\,\!</math> design with 8 runs, and the design after this is the 2 <math>_{\text{III}}^{15-11}\,\!</math> design with 32 runs and so on, as shown in the next table.

[[Image:doet8.1.png|thumb|center|487px|Highly fractional designs to investigate main effects.]]

Plackett-Burman designs solve this problem. These designs were proposed by R. L. Plackett and J.P. Burman (1946). They allow the estimation of <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs. In these designs, runs are a multiple of 4 (i.e., 4, 8, 12, 16, 20 and so on). When the runs are a power of 2, the designs correspond to the resolution III two factor fractional factorial designs. Although Plackett-Burman designs are all two level orthogonal designs, the alias structure for these designs is complicated when runs are not a power of 2.

As an example, consider the 12-run Plackett-Burman design shown in the figure below.

[[Image:doe8.1.png|thumb|center|400px|12-run Plackett-Burman design.]]

If 11 main effects are to be estimated using this design, then each of these main effects is partially aliased with all other two factor interactions not containing that main effect. For example, the main effect <math>A\,\!</math> is partially aliased with all two factor interactions except <math>AB\,\!</math>, <math>AC\,\!</math>, <math>AD\,\!</math>, <math>AE\,\!</math>, <math>AF\,\!</math>, <math>AG\,\!</math>, <math>AH\,\!</math>, <math>AJ\,\!</math>, <math>AK\,\!</math> and <math>AL\,\!</math>. There are 45 such two factor interactions that are aliased with <math>A\,\!</math>.

::<math>\begin{align}
& A= & A-\frac{1}{3}BC-\frac{1}{3}BD+\frac{1}{3}CD-\frac{1}{3}BE-\frac{1}{3}CE+\frac{1}{3}DE+... \\
& & ...+\frac{1}{3}EL-\frac{1}{3}FL-\frac{1}{3}GL+\frac{1}{3}HL+\frac{1}{3}JL-\frac{1}{3}KL
\end{align}\,\!</math>

Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs|Appendix C]].

==Taguchi's Orthogonal Arrays==

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays|Appendix D]].

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in DOE++. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

[[Image:doe8.3.png|thumb|center|400px|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]

Similarly, consider the L8(2^7) array shown in figure (a) below. This design can be used to estimate up to seven main effects using eight runs. This array is again similar to the 2 <math>_{\text{III}}^{7-4}\,\!</math> design shown in figure (b) below, except that the aliasing between the columns of the two designs differs in sign for some of the columns (see figure (c)).

[[Image:doe8.4.png|thumb|center|400px|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]

The L8 array can also be used as a full factorial three factor experiment design in the same way as a <math>2^{3}\,\!</math> design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, figure (c) above shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 <math>^{3}\,\!</math> design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the ''preferred columns'' for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in the next table.

[[Image:doet8.2.png|thumb|center|400px|Alias relations for the L8 array.]]

The cell value in any (<math>i,j\,\!</math>) cell of the table gives the column number of the two factor interaction for the <math>i\,\!</math>th row and <math>j\,\!</math>th column. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the ( <math>1,2\,\!</math> ) cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in [[Alias_Relations_for_Taguchi_Orthogonal_Arrays|Appendix E]].

===Example===

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. The three factors investigated in the experiment were honing pressure (factor A), number of strokes (factor B) and cycle time (factor C). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the <math>{2}^{3}\,\!</math> design that was used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on the figure (c) above, so that you can easily compare results obtained from the L8 array to the ones included in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]]. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in the third table. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

[[Image:doet8.3.png|thumb|center|639px|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]

The experiment design for this example can be set using the properties shown in the figure below.

[[Image:doe8_5.png|thumb|center|842px|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

Note that for this design, the factor properties are set up as shown in the design summary.

[[Image:doe8_6.png|thumb|center|828px|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

The resulting design along with the response values is shown in the figure below.

[[Image:doe8_7.png|thumb|center|650px|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

And the results from DOE++ for the design are shown in the next figure.

[[Image:doe8_8_1.png|thumb|center|613px|Results for the experiment in the example.]]

The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the <math>{2}^{3}\,\!</math> design used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]].

=== Preferred Columns in Taguchi OA===
One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

[[Image:DOEtableChapter9.png|center]]

This is a fractional factorial design for 7 factors. For any fractional factorial design, the first thing we need to do is check its alias structure. In general, the alias structures for Taguchi OAs are very complicated. People usually use the following table to represent the alias relations between each factor. For the above orthogonal array, the alias table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7
|-
|2x3|| 1x3|| 1x2|| 1x5|| 1x4|| 1x7|| 1x6
|-
|4x5|| 4x6|| 4x7|| 2x6|| 2x7|| 2x4|| 2x5
|-
|6x7|| 5x7|| 5x6|| 3x7|| 3x6|| 3x5|| 3x4
|}

In the above table, an Arabic number is used to represent a factor. For instance, “1” represents the factor assigned to the 1st column in the array. “2x3” represents the interaction effect of the two factors assigned to column 2 and 3. Each column in the above alias table lists all the 2-way interaction effects that are aliased with the main effect of the factor assigned to this column. For example, for the 1st column, the main effect of the factor assigned to it is aliased with interaction effects of 2x3, 4x5 and 6x7.
If an experiment has only 3 factors and these 3 factors A, B and C are assigned to the first 3 columns of the above L8(2^7) array, then the design table will be:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 3)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

The alias structure for the above table is:

{| style="text-align:center;" cellpadding="2" align="center"
|-
|[I] = I – ABC
|-
|[A] = A – BC
|-
|[B] = B – AC
|-
|[C] = C – AB
|}

This is a resolution 3 design. All the main effects are aliased with 2-way interactions. There are many ways to choose 3 columns from the 7 columns of L8(2^7). If the 3 factors are assigned to column 1, 2, and 4, then the design table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 4)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

====Example====
Design an experiment with 3 qualitative factors. Factors A and B have 2 levels; factor C has 4 levels. The experimenters are interested in all the main effects and the interaction effect AC.

Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.

[[Image:doe8_9.png|thumb|center|842px|Selecting the Taguchi OA design type.]]

Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.

[[Image:doe8_10.png|thumb|center|420px|Specifying the interaction terms of interest.]]

Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

[[Image:doe8_11.png|thumb|center|842px|The interaction terms of interest have been specified.]]

However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

====Example====
Design an experiment for a test with 4 qualitative factors. Factors A and B have 2 levels; C and D have 4 levels. We are interested in all the main effects and the interaction effects AC and BD.

Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.

[[Image:doe8_12.png|thumb|center|420px|Specifying the interaction terms of interest.]]

When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.

This can be explained by checking the alias table of an L16(2^6*4^3) design as given below.

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
|2x7|| 1x7|| 1x9|| 1x5|| 1x4|| 1x8|| 1x2|| 1x6|| 1x3
|-
|3x9|| 3x6|| 2x6|| 2x8|| 2x9|| 2x3|| 1x8|| 1x7|| 1x7
|-
|4x5|| 4x8|| 4x7|| 3x7|| 3x8|| 4x9|| 1x9|| 1x9|| 1x8
|-
|6x8|| 5x9|| 5x8|| 6x9|| 6x7|| 5x7|| 2x8|| 2x4|| 2x5
|-
|7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 2x9|| 2x7|| 2x7
|-
|7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 3x8|| 2x9|| 2x8
|-
|8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 3x9|| 3x5|| 3x7
|-
| || || || || || ||4x8|| 3x7|| 3x8
|-
| || || || || || ||4x9|| 3x9|| 4x6
|-
| || || || || || ||5x6|| 4x7|| 4x7
|-
| || || || || || ||5x8|| 4x9|| 4x8
|-
| || || || || || ||5x9|| 5x7|| 5x7
|-
| || || || || || ||6x8|| 5x9|| 5x8
|-
| || || || || || ||6x9|| 6x7|| 6x7
|-
| || || || || || ||8x9|| 6x9|| 6x8
|-
| || || || || || || || 7x9|| 7x8
|}

From this table, we can see that it is impossible to clearly estimate both AC and BD. Factors C and D can only be assigned to the last three columns since they are 4 level factors. Assume we assign factor C to column 7 and factor D to column 8. Factor B (2 level) will be in one of the columns from 1 to 6. Therefore, effect BD will be one of the effects highlighted in the table, where the first term of the interaction is between 1 and 6 and the last term is 8 (i.e., factor D).

The above alias table shows that factor C is aliased with one of those highlighted effects. Thus, no matter which of the first six columns is assigned to factor B, the main effect C will be aliased with interaction effect BD. This is also true if C is assigned to column 8 or 9. Therefore, if L16(2^6*4^3) is used, there is no way to clearly estimate all the main effects and the interaction effects AC and BD. Another Taguchi OA or other design types such as a general full factorial design should be used. A more efficient way might be to create an optimal custom design that can clearly estimate all the specified terms. For more detail, please refer to the chapter on [[Optimal Custom Designs]].

Highly Fractional Factorial Designs

2015-11-24T18:29:22Z

Melinda Caroline: /* Example */

{{Template:Doebook|9}}
This chapter discusses factorial designs that are commonly used in designed experiments, but are not necessarily limited to two level factors. These designs are the [[Highly_Fractional_Factorial_Designs#Plackett-Burman_Designs|Plackett-Burman designs]] and [[Highly_Fractional_Factorial_Designs#Taguchi.27s_Orthogonal_Arrays|Taguchi's orthogonal arrays]].

==Plackett-Burman Designs==

It was mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] that resolution III designs can be used as highly fractional designs to investigate <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs (provided that three factor and higher order interaction effects are not important to the experimenter). A limitation with these designs is that all runs in these designs have to be a power of 2. The valid runs for these designs are 4, 8, 16, 32, etc. Therefore, the next design after the 2 <math>_{\text{III}}^{3-1}\,\!</math> design with 4 runs is the 2 <math>_{\text{III}}^{7-4}\,\!</math> design with 8 runs, and the design after this is the 2 <math>_{\text{III}}^{15-11}\,\!</math> design with 32 runs and so on, as shown in the next table.

[[Image:doet8.1.png|thumb|center|487px|Highly fractional designs to investigate main effects.]]

Plackett-Burman designs solve this problem. These designs were proposed by R. L. Plackett and J.P. Burman (1946). They allow the estimation of <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs. In these designs, runs are a multiple of 4 (i.e., 4, 8, 12, 16, 20 and so on). When the runs are a power of 2, the designs correspond to the resolution III two factor fractional factorial designs. Although Plackett-Burman designs are all two level orthogonal designs, the alias structure for these designs is complicated when runs are not a power of 2.

As an example, consider the 12-run Plackett-Burman design shown in the figure below.

[[Image:doe8.1.png|thumb|center|400px|12-run Plackett-Burman design.]]

If 11 main effects are to be estimated using this design, then each of these main effects is partially aliased with all other two factor interactions not containing that main effect. For example, the main effect <math>A\,\!</math> is partially aliased with all two factor interactions except <math>AB\,\!</math>, <math>AC\,\!</math>, <math>AD\,\!</math>, <math>AE\,\!</math>, <math>AF\,\!</math>, <math>AG\,\!</math>, <math>AH\,\!</math>, <math>AJ\,\!</math>, <math>AK\,\!</math> and <math>AL\,\!</math>. There are 45 such two factor interactions that are aliased with <math>A\,\!</math>.

::<math>\begin{align}
& A= & A-\frac{1}{3}BC-\frac{1}{3}BD+\frac{1}{3}CD-\frac{1}{3}BE-\frac{1}{3}CE+\frac{1}{3}DE+... \\
& & ...+\frac{1}{3}EL-\frac{1}{3}FL-\frac{1}{3}GL+\frac{1}{3}HL+\frac{1}{3}JL-\frac{1}{3}KL
\end{align}\,\!</math>

Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs|Appendix C]].

==Taguchi's Orthogonal Arrays==

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays|Appendix D]].

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in DOE++. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

[[Image:doe8.3.png|thumb|center|400px|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]

Similarly, consider the L8(2^7) array shown in figure (a) below. This design can be used to estimate up to seven main effects using eight runs. This array is again similar to the 2 <math>_{\text{III}}^{7-4}\,\!</math> design shown in figure (b) below, except that the aliasing between the columns of the two designs differs in sign for some of the columns (see figure (c)).

[[Image:doe8.4.png|thumb|center|400px|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]

The L8 array can also be used as a full factorial three factor experiment design in the same way as a <math>2^{3}\,\!</math> design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, figure (c) above shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 <math>^{3}\,\!</math> design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the ''preferred columns'' for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in the next table.

[[Image:doet8.2.png|thumb|center|400px|Alias relations for the L8 array.]]

The cell value in any (<math>i,j\,\!</math>) cell of the table gives the column number of the two factor interaction for the <math>i\,\!</math>th row and <math>j\,\!</math>th column. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the ( <math>1,2\,\!</math> ) cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in [[Alias_Relations_for_Taguchi_Orthogonal_Arrays|Appendix E]].

===Example===

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. The three factors investigated in the experiment were honing pressure (factor A), number of strokes (factor B) and cycle time (factor C). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the <math>{2}^{3}\,\!</math> design that was used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on the figure (c) above, so that you can easily compare results obtained from the L8 array to the ones included in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]]. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in the third table. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

[[Image:doet8.3.png|thumb|center|639px|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]

The experiment design for this example can be set using the properties shown in the figure below.

[[Image:doe8_5.png|thumb|center|842px|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

Note that for this design, the factor properties are set up as shown in the design summary.

[[Image:doe8_6.png|thumb|center|828px|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

The resulting design along with the response values is shown in the figure below.

[[Image:doe8_7.png|thumb|center|650px|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

And the results from DOE++ for the design are shown in the next figure.

[[Image:doe8_8_1.png|thumb|center|613px|Results for the experiment in the example.]]

The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the <math>{2}^{3}\,\!</math> design used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]].

=== Preferred Columns in Taguchi OA===
One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

[[Image:DOEtableChapter9.png|center]]

This is a fractional factorial design for 7 factors. For any fractional factorial design, the first thing we need to do is check its alias structure. In general, the alias structures for Taguchi OAs are very complicated. People usually use the following table to represent the alias relations between each factor. For the above orthogonal array, the alias table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7
|-
|2x3|| 1x3|| 1x2|| 1x5|| 1x4|| 1x7|| 1x6
|-
|4x5|| 4x6|| 4x7|| 2x6|| 2x7|| 2x4|| 2x5
|-
|6x7|| 5x7|| 5x6|| 3x7|| 3x6|| 3x5|| 3x4
|}

In the above table, an Arabic number is used to represent a factor. For instance, “1” represents the factor assigned to the 1st column in the array. “2x3” represents the interaction effect of the two factors assigned to column 2 and 3. Each column in the above alias table lists all the 2-way interaction effects that are aliased with the main effect of the factor assigned to this column. For example, for the 1st column, the main effect of the factor assigned to it is aliased with interaction effects of 2x3, 4x5 and 6x7.
If an experiment has only 3 factors and these 3 factors A, B and C are assigned to the first 3 columns of the above L8(2^7) array, then the design table will be:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 3)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

The alias structure for the above table is:

{| style="text-align:center;" cellpadding="2" align="center"
|-
|[I] = I – ABC
|-
|[A] = A – BC
|-
|[B] = B – AC
|-
|[C] = C – AB
|}

This is a resolution 3 design. All the main effects are aliased with 2-way interactions. There are many ways to choose 3 columns from the 7 columns of L8(2^7). If the 3 factors are assigned to column 1, 2, and 4, then the design table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 4)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

====Example====
Design an experiment with 3 qualitative factors. Factors A and B have 2 levels; factor C has 4 levels. The experimenters are interested in all the main effects and the interaction effect AC.

Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.

[[Image:doe8_9.png|thumb|center|842px|Selecting the Taguchi OA design type.]]

Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.

[[Image:doe8_10.png|thumb|center|420px|Specifying the interaction terms of interest.]]

Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

[[Image:doe8_11.png|thumb|center|842px|The interaction terms of interest have been specified.]]

However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

====Example====
Design an experiment for a test with 4 qualitative factors. Factors A and B have 2 levels; C and D have 4 levels. We are interested in all the main effects and the interaction effects AC and BD.

Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.

[[Image:doe8_12.png|thumb|center|420px|Specifying the interaction terms of interest.]]

When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.

This can be explained by checking the alias table of a L16(2^6*4^3) design as given below.

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
|2x7|| 1x7|| 1x9|| 1x5|| 1x4|| 1x8|| 1x2|| 1x6|| 1x3
|-
|3x9|| 3x6|| 2x6|| 2x8|| 2x9|| 2x3|| 1x8|| 1x7|| 1x7
|-
|4x5|| 4x8|| 4x7|| 3x7|| 3x8|| 4x9|| 1x9|| 1x9|| 1x8
|-
|6x8|| 5x9|| 5x8|| 6x9|| 6x7|| 5x7|| 2x8|| 2x4|| 2x5
|-
|7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 2x9|| 2x7|| 2x7
|-
|7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 3x8|| 2x9|| 2x8
|-
|8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 3x9|| 3x5|| 3x7
|-
| || || || || || ||4x8|| 3x7|| 3x8
|-
| || || || || || ||4x9|| 3x9|| 4x6
|-
| || || || || || ||5x6|| 4x7|| 4x7
|-
| || || || || || ||5x8|| 4x9|| 4x8
|-
| || || || || || ||5x9|| 5x7|| 5x7
|-
| || || || || || ||6x8|| 5x9|| 5x8
|-
| || || || || || ||6x9|| 6x7|| 6x7
|-
| || || || || || ||8x9|| 6x9|| 6x8
|-
| || || || || || || || 7x9|| 7x8
|}

From this table, we can see that it is impossible to clearly estimate both AC and BD. Factors C and D can only be assigned to the last three columns since they are 4 level factors. Assume we assign factor C to column 7 and factor D to column 8. Factor B (2 level) will be in one of the columns from 1 to 6. Therefore, effect BD will be one of the effects highlighted in the table, where the first term of the interaction is between 1 and 6 and the last term is 8 (i.e., factor D).

The above alias table shows that factor C is aliased with one of those highlighted effects. Thus, no matter which of the first six columns is assigned to factor B, the main effect C will be aliased with interaction effect BD. This is also true if C is assigned to column 8 or 9. Therefore, if L16(2^6*4^3) is used, there is no way to clearly estimate all the main effects and the interaction effects AC and BD. Another Taguchi OA or other design types such as a general full factorial design should be used. A more efficient way might be to create an optimal custom design that can clearly estimate all the specified terms. For more detail, please refer to the chapter on [[Optimal Custom Designs]].

File:Doe8 12.png

2015-11-24T18:24:06Z

Melinda Caroline:

Highly Fractional Factorial Designs

2015-11-24T18:23:50Z

Melinda Caroline: /* Example */

{{Template:Doebook|9}}
This chapter discusses factorial designs that are commonly used in designed experiments, but are not necessarily limited to two level factors. These designs are the [[Highly_Fractional_Factorial_Designs#Plackett-Burman_Designs|Plackett-Burman designs]] and [[Highly_Fractional_Factorial_Designs#Taguchi.27s_Orthogonal_Arrays|Taguchi's orthogonal arrays]].

==Plackett-Burman Designs==

It was mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] that resolution III designs can be used as highly fractional designs to investigate <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs (provided that three factor and higher order interaction effects are not important to the experimenter). A limitation with these designs is that all runs in these designs have to be a power of 2. The valid runs for these designs are 4, 8, 16, 32, etc. Therefore, the next design after the 2 <math>_{\text{III}}^{3-1}\,\!</math> design with 4 runs is the 2 <math>_{\text{III}}^{7-4}\,\!</math> design with 8 runs, and the design after this is the 2 <math>_{\text{III}}^{15-11}\,\!</math> design with 32 runs and so on, as shown in the next table.

[[Image:doet8.1.png|thumb|center|487px|Highly fractional designs to investigate main effects.]]

Plackett-Burman designs solve this problem. These designs were proposed by R. L. Plackett and J.P. Burman (1946). They allow the estimation of <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs. In these designs, runs are a multiple of 4 (i.e., 4, 8, 12, 16, 20 and so on). When the runs are a power of 2, the designs correspond to the resolution III two factor fractional factorial designs. Although Plackett-Burman designs are all two level orthogonal designs, the alias structure for these designs is complicated when runs are not a power of 2.

As an example, consider the 12-run Plackett-Burman design shown in the figure below.

[[Image:doe8.1.png|thumb|center|400px|12-run Plackett-Burman design.]]

If 11 main effects are to be estimated using this design, then each of these main effects is partially aliased with all other two factor interactions not containing that main effect. For example, the main effect <math>A\,\!</math> is partially aliased with all two factor interactions except <math>AB\,\!</math>, <math>AC\,\!</math>, <math>AD\,\!</math>, <math>AE\,\!</math>, <math>AF\,\!</math>, <math>AG\,\!</math>, <math>AH\,\!</math>, <math>AJ\,\!</math>, <math>AK\,\!</math> and <math>AL\,\!</math>. There are 45 such two factor interactions that are aliased with <math>A\,\!</math>.

::<math>\begin{align}
& A= & A-\frac{1}{3}BC-\frac{1}{3}BD+\frac{1}{3}CD-\frac{1}{3}BE-\frac{1}{3}CE+\frac{1}{3}DE+... \\
& & ...+\frac{1}{3}EL-\frac{1}{3}FL-\frac{1}{3}GL+\frac{1}{3}HL+\frac{1}{3}JL-\frac{1}{3}KL
\end{align}\,\!</math>

Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs|Appendix C]].

==Taguchi's Orthogonal Arrays==

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays|Appendix D]].

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in DOE++. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

[[Image:doe8.3.png|thumb|center|400px|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]

Similarly, consider the L8(2^7) array shown in figure (a) below. This design can be used to estimate up to seven main effects using eight runs. This array is again similar to the 2 <math>_{\text{III}}^{7-4}\,\!</math> design shown in figure (b) below, except that the aliasing between the columns of the two designs differs in sign for some of the columns (see figure (c)).

[[Image:doe8.4.png|thumb|center|400px|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]

The L8 array can also be used as a full factorial three factor experiment design in the same way as a <math>2^{3}\,\!</math> design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, figure (c) above shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 <math>^{3}\,\!</math> design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the ''preferred columns'' for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in the next table.

[[Image:doet8.2.png|thumb|center|400px|Alias relations for the L8 array.]]

The cell value in any (<math>i,j\,\!</math>) cell of the table gives the column number of the two factor interaction for the <math>i\,\!</math>th row and <math>j\,\!</math>th column. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the ( <math>1,2\,\!</math> ) cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in [[Alias_Relations_for_Taguchi_Orthogonal_Arrays|Appendix E]].

===Example===

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. The three factors investigated in the experiment were honing pressure (factor A), number of strokes (factor B) and cycle time (factor C). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the <math>{2}^{3}\,\!</math> design that was used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on the figure (c) above, so that you can easily compare results obtained from the L8 array to the ones included in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]]. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in the third table. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

[[Image:doet8.3.png|thumb|center|639px|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]

The experiment design for this example can be set using the properties shown in the figure below.

[[Image:doe8_5.png|thumb|center|842px|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

Note that for this design, the factor properties are set up as shown in the design summary.

[[Image:doe8_6.png|thumb|center|828px|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

The resulting design along with the response values is shown in the figure below.

[[Image:doe8_7.png|thumb|center|650px|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

And the results from DOE++ for the design are shown in the next figure.

[[Image:doe8_8_1.png|thumb|center|613px|Results for the experiment in the example.]]

The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the <math>{2}^{3}\,\!</math> design used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]].

=== Preferred Columns in Taguchi OA===
One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

[[Image:DOEtableChapter9.png|center]]

This is a fractional factorial design for 7 factors. For any fractional factorial design, the first thing we need to do is check its alias structure. In general, the alias structures for Taguchi OAs are very complicated. People usually use the following table to represent the alias relations between each factor. For the above orthogonal array, the alias table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7
|-
|2x3|| 1x3|| 1x2|| 1x5|| 1x4|| 1x7|| 1x6
|-
|4x5|| 4x6|| 4x7|| 2x6|| 2x7|| 2x4|| 2x5
|-
|6x7|| 5x7|| 5x6|| 3x7|| 3x6|| 3x5|| 3x4
|}

In the above table, an Arabic number is used to represent a factor. For instance, “1” represents the factor assigned to the 1st column in the array. “2x3” represents the interaction effect of the two factors assigned to column 2 and 3. Each column in the above alias table lists all the 2-way interaction effects that are aliased with the main effect of the factor assigned to this column. For example, for the 1st column, the main effect of the factor assigned to it is aliased with interaction effects of 2x3, 4x5 and 6x7.
If an experiment has only 3 factors and these 3 factors A, B and C are assigned to the first 3 columns of the above L8(2^7) array, then the design table will be:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 3)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

The alias structure for the above table is:

{| style="text-align:center;" cellpadding="2" align="center"
|-
|[I] = I – ABC
|-
|[A] = A – BC
|-
|[B] = B – AC
|-
|[C] = C – AB
|}

This is a resolution 3 design. All the main effects are aliased with 2-way interactions. There are many ways to choose 3 columns from the 7 columns of L8(2^7). If the 3 factors are assigned to column 1, 2, and 4, then the design table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 4)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

====Example====
Design an experiment with 3 qualitative factors. Factors A and B have 2 levels; factor C has 4 levels. The experimenters are interested in all the main effects and the interaction effect AC.

Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.

[[Image:doe8_9.png|thumb|center|842px|Selecting the Taguchi OA design type.]]

Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.

[[Image:doe8_10.png|thumb|center|420px|Specifying the interaction terms of interest.]]

Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

[[Image:doe8_11.png|thumb|center|842px|The interaction terms of interest have been specified.]]

However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

====Example====
Design an experiment for a test with 4 qualitative factors. Factors A and B have 2 levels; C and D have 4 levels. We are interested in all the main effects and the interaction effects AC and BD.

Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.

[[Image:doe8_12.png|thumb|center|420px|Specifying the interaction terms of interest.]]

When you click '''OK''', you will see a message saying that it is impossible to clearly estimate all the main effects and the specified interaction effect AC and BD.

This can be explained by checking the alias table of a L16(2^6*4^3) design as given below.

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
|2x7|| 1x7|| 1x9|| 1x5|| 1x4|| 1x8|| 1x2|| 1x6|| 1x3
|-
|3x9|| 3x6|| 2x6|| 2x8|| 2x9|| 2x3|| 1x8|| 1x7|| 1x7
|-
|4x5|| 4x8|| 4x7|| 3x7|| 3x8|| 4x9|| 1x9|| 1x9|| 1x8
|-
|6x8|| 5x9|| 5x8|| 6x9|| 6x7|| 5x7|| 2x8|| 2x4|| 2x5
|-
|7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 2x9|| 2x7|| 2x7
|-
|7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 3x8|| 2x9|| 2x8
|-
|8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 3x9|| 3x5|| 3x7
|-
| || || || || || ||4x8|| 3x7|| 3x8
|-
| || || || || || ||4x9|| 3x9|| 4x6
|-
| || || || || || ||5x6|| 4x7|| 4x7
|-
| || || || || || ||5x8|| 4x9|| 4x8
|-
| || || || || || ||5x9|| 5x7|| 5x7
|-
| || || || || || ||6x8|| 5x9|| 5x8
|-
| || || || || || ||6x9|| 6x7|| 6x7
|-
| || || || || || ||8x9|| 6x9|| 6x8
|-
| || || || || || || || 7x9|| 7x8
|}

From this table, we can see that it is impossible to clearly estimate both AC and BD. Factors C and D can only be assigned to the last three columns since they are 4 level factors. Assume we assign factor C to column 7 and factor D to column 8. Factor B (2 level) will be in one of the columns from 1 to 6. Therefore, effect BD will be one of the effects highlighted in the table, where the first term of the interaction is between 1 and 6 and the last term is 8 (i.e., factor D).

The above alias table shows factor C is aliased with one of those highlighted effects. Thus, no matter which of the first six columns is assigned to factor B, the main effect C will be aliased with interaction effect BD. This is also true if C is assigned to column 8 or 9. Therefore, if L16(2^6*4^3) is used, there is no way to clearly estimate all the main effects and the interaction effects AC and BD. Another Taguchi OA or other design types such as a general level full factorial design should be used. A more efficient way probably is to create an optimal custom design that can clearly estimate all the specified terms. For more detail, please refer to the chapter on Optimal Custom Designs.

Highly Fractional Factorial Designs

2015-11-23T23:35:06Z

Melinda Caroline: /* Example */

{{Template:Doebook|9}}
This chapter discusses factorial designs that are commonly used in designed experiments, but are not necessarily limited to two level factors. These designs are the [[Highly_Fractional_Factorial_Designs#Plackett-Burman_Designs|Plackett-Burman designs]] and [[Highly_Fractional_Factorial_Designs#Taguchi.27s_Orthogonal_Arrays|Taguchi's orthogonal arrays]].

==Plackett-Burman Designs==

It was mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] that resolution III designs can be used as highly fractional designs to investigate <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs (provided that three factor and higher order interaction effects are not important to the experimenter). A limitation with these designs is that all runs in these designs have to be a power of 2. The valid runs for these designs are 4, 8, 16, 32, etc. Therefore, the next design after the 2 <math>_{\text{III}}^{3-1}\,\!</math> design with 4 runs is the 2 <math>_{\text{III}}^{7-4}\,\!</math> design with 8 runs, and the design after this is the 2 <math>_{\text{III}}^{15-11}\,\!</math> design with 32 runs and so on, as shown in the next table.

[[Image:doet8.1.png|thumb|center|487px|Highly fractional designs to investigate main effects.]]

Plackett-Burman designs solve this problem. These designs were proposed by R. L. Plackett and J.P. Burman (1946). They allow the estimation of <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs. In these designs, runs are a multiple of 4 (i.e., 4, 8, 12, 16, 20 and so on). When the runs are a power of 2, the designs correspond to the resolution III two factor fractional factorial designs. Although Plackett-Burman designs are all two level orthogonal designs, the alias structure for these designs is complicated when runs are not a power of 2.

As an example, consider the 12-run Plackett-Burman design shown in the figure below.

[[Image:doe8.1.png|thumb|center|400px|12-run Plackett-Burman design.]]

If 11 main effects are to be estimated using this design, then each of these main effects is partially aliased with all other two factor interactions not containing that main effect. For example, the main effect <math>A\,\!</math> is partially aliased with all two factor interactions except <math>AB\,\!</math>, <math>AC\,\!</math>, <math>AD\,\!</math>, <math>AE\,\!</math>, <math>AF\,\!</math>, <math>AG\,\!</math>, <math>AH\,\!</math>, <math>AJ\,\!</math>, <math>AK\,\!</math> and <math>AL\,\!</math>. There are 45 such two factor interactions that are aliased with <math>A\,\!</math>.

::<math>\begin{align}
& A= & A-\frac{1}{3}BC-\frac{1}{3}BD+\frac{1}{3}CD-\frac{1}{3}BE-\frac{1}{3}CE+\frac{1}{3}DE+... \\
& & ...+\frac{1}{3}EL-\frac{1}{3}FL-\frac{1}{3}GL+\frac{1}{3}HL+\frac{1}{3}JL-\frac{1}{3}KL
\end{align}\,\!</math>

Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs|Appendix C]].

==Taguchi's Orthogonal Arrays==

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays|Appendix D]].

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in DOE++. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

[[Image:doe8.3.png|thumb|center|400px|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]

Similarly, consider the L8(2^7) array shown in figure (a) below. This design can be used to estimate up to seven main effects using eight runs. This array is again similar to the 2 <math>_{\text{III}}^{7-4}\,\!</math> design shown in figure (b) below, except that the aliasing between the columns of the two designs differs in sign for some of the columns (see figure (c)).

[[Image:doe8.4.png|thumb|center|400px|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]

The L8 array can also be used as a full factorial three factor experiment design in the same way as a <math>2^{3}\,\!</math> design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, figure (c) above shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 <math>^{3}\,\!</math> design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the ''preferred columns'' for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in the next table.

[[Image:doet8.2.png|thumb|center|400px|Alias relations for the L8 array.]]

The cell value in any (<math>i,j\,\!</math>) cell of the table gives the column number of the two factor interaction for the <math>i\,\!</math>th row and <math>j\,\!</math>th column. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the ( <math>1,2\,\!</math> ) cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in [[Alias_Relations_for_Taguchi_Orthogonal_Arrays|Appendix E]].

===Example===

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. The three factors investigated in the experiment were honing pressure (factor A), number of strokes (factor B) and cycle time (factor C). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the <math>{2}^{3}\,\!</math> design that was used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on the figure (c) above, so that you can easily compare results obtained from the L8 array to the ones included in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]]. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in the third table. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

[[Image:doet8.3.png|thumb|center|639px|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]

The experiment design for this example can be set using the properties shown in the figure below.

[[Image:doe8_5.png|thumb|center|842px|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

Note that for this design, the factor properties are set up as shown in the design summary.

[[Image:doe8_6.png|thumb|center|828px|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

The resulting design along with the response values is shown in the figure below.

[[Image:doe8_7.png|thumb|center|650px|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

And the results from DOE++ for the design are shown in the next figure.

[[Image:doe8_8_1.png|thumb|center|613px|Results for the experiment in the example.]]

The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the <math>{2}^{3}\,\!</math> design used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]].

=== Preferred Columns in Taguchi OA===
One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

[[Image:DOEtableChapter9.png|center]]

This is a fractional factorial design for 7 factors. For any fractional factorial design, the first thing we need to do is check its alias structure. In general, the alias structures for Taguchi OAs are very complicated. People usually use the following table to represent the alias relations between each factor. For the above orthogonal array, the alias table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7
|-
|2x3|| 1x3|| 1x2|| 1x5|| 1x4|| 1x7|| 1x6
|-
|4x5|| 4x6|| 4x7|| 2x6|| 2x7|| 2x4|| 2x5
|-
|6x7|| 5x7|| 5x6|| 3x7|| 3x6|| 3x5|| 3x4
|}

In the above table, an Arabic number is used to represent a factor. For instance, “1” represents the factor assigned to the 1st column in the array. “2x3” represents the interaction effect of the two factors assigned to column 2 and 3. Each column in the above alias table lists all the 2-way interaction effects that are aliased with the main effect of the factor assigned to this column. For example, for the 1st column, the main effect of the factor assigned to it is aliased with interaction effects of 2x3, 4x5 and 6x7.
If an experiment has only 3 factors and these 3 factors A, B and C are assigned to the first 3 columns of the above L8(2^7) array, then the design table will be:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 3)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

The alias structure for the above table is:

{| style="text-align:center;" cellpadding="2" align="center"
|-
|[I] = I – ABC
|-
|[A] = A – BC
|-
|[B] = B – AC
|-
|[C] = C – AB
|}

This is a resolution 3 design. All the main effects are aliased with 2-way interactions. There are many ways to choose 3 columns from the 7 columns of L8(2^7). If the 3 factors are assigned to column 1, 2, and 4, then the design table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 4)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

====Example====
Design an experiment with 3 qualitative factors. Factors A and B have 2 levels; factor C has 4 levels. The experimenters are interested in all the main effects and the interaction effect AC.

Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.

[[Image:doe8_9.png|thumb|center|842px|Selecting the Taguchi OA design type.]]

Click Specify Interaction Terms to specify the interaction terms of interest by the experimenters.

[[Image:doe8_10.png|thumb|center|420px|Specifying the interaction terms of interest.]]

Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

[[Image:doe8_11.png|thumb|center|842px|The interaction terms of interest have been specified.]]

However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

Example: Design an experiment for a test with 4 qualitative factors. Factors A and B have 2 levels; C and D have 4 levels. We are interested in all the main effects and the interaction effects AC and BD.

Assume again we want to use Taguchi OA L16(2^6*4^3). Click Taguchi Preferred Columns in the following screen.

Specify the interaction effects that you want to estimate in the experiment.

When click on OK, you will get a warning message saying that it is impossible to clearly estimate all the main effects and the specified interaction effect AC and BD.

This can be explained by checking the alias table of a L16(2^6*4^3) design as given below.

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
|2x7|| 1x7|| 1x9|| 1x5|| 1x4|| 1x8|| 1x2|| 1x6|| 1x3
|-
|3x9|| 3x6|| 2x6|| 2x8|| 2x9|| 2x3|| 1x8|| 1x7|| 1x7
|-
|4x5|| 4x8|| 4x7|| 3x7|| 3x8|| 4x9|| 1x9|| 1x9|| 1x8
|-
|6x8|| 5x9|| 5x8|| 6x9|| 6x7|| 5x7|| 2x8|| 2x4|| 2x5
|-
|7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 2x9|| 2x7|| 2x7
|-
|7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 3x8|| 2x9|| 2x8
|-
|8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 3x9|| 3x5|| 3x7
|-
| || || || || || ||4x8|| 3x7|| 3x8
|-
| || || || || || ||4x9|| 3x9|| 4x6
|-
| || || || || || ||5x6|| 4x7|| 4x7
|-
| || || || || || ||5x8|| 4x9|| 4x8
|-
| || || || || || ||5x9|| 5x7|| 5x7
|-
| || || || || || ||6x8|| 5x9|| 5x8
|-
| || || || || || ||6x9|| 6x7|| 6x7
|-
| || || || || || ||8x9|| 6x9|| 6x8
|-
| || || || || || || || 7x9|| 7x8
|}

From this table, we can see that it is impossible to clearly estimate both AC and BD. Factors C and D can only be assigned to the last three columns since they are 4 level factors. Assume we assign factor C to column 7 and factor D to column 8. Factor B (2 level) will be in one of the columns from 1 to 6. Therefore, effect BD will be one of the effects highlighted in the table, where the first term of the interaction is between 1 and 6 and the last term is 8 (i.e., factor D).

The above alias table shows factor C is aliased with one of those highlighted effects. Thus, no matter which of the first six columns is assigned to factor B, the main effect C will be aliased with interaction effect BD. This is also true if C is assigned to column 8 or 9. Therefore, if L16(2^6*4^3) is used, there is no way to clearly estimate all the main effects and the interaction effects AC and BD. Another Taguchi OA or other design types such as a general level full factorial design should be used. A more efficient way probably is to create an optimal custom design that can clearly estimate all the specified terms. For more detail, please refer to the chapter on Optimal Custom Designs.

File:Doe8 11.png

2015-11-23T23:34:55Z

Melinda Caroline:

File:Doe8 10.png

2015-11-23T23:34:43Z

Melinda Caroline:

File:Doe8 9.png

2015-11-23T23:07:08Z

Melinda Caroline: uploaded a new version of "File:Doe8 9.png"

Highly Fractional Factorial Designs

2015-11-23T23:05:45Z

Melinda Caroline: /* Preferred Columns in Taguchi OA */

{{Template:Doebook|9}}
This chapter discusses factorial designs that are commonly used in designed experiments, but are not necessarily limited to two level factors. These designs are the [[Highly_Fractional_Factorial_Designs#Plackett-Burman_Designs|Plackett-Burman designs]] and [[Highly_Fractional_Factorial_Designs#Taguchi.27s_Orthogonal_Arrays|Taguchi's orthogonal arrays]].

==Plackett-Burman Designs==

It was mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] that resolution III designs can be used as highly fractional designs to investigate <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs (provided that three factor and higher order interaction effects are not important to the experimenter). A limitation with these designs is that all runs in these designs have to be a power of 2. The valid runs for these designs are 4, 8, 16, 32, etc. Therefore, the next design after the 2 <math>_{\text{III}}^{3-1}\,\!</math> design with 4 runs is the 2 <math>_{\text{III}}^{7-4}\,\!</math> design with 8 runs, and the design after this is the 2 <math>_{\text{III}}^{15-11}\,\!</math> design with 32 runs and so on, as shown in the next table.

[[Image:doet8.1.png|thumb|center|487px|Highly fractional designs to investigate main effects.]]

Plackett-Burman designs solve this problem. These designs were proposed by R. L. Plackett and J.P. Burman (1946). They allow the estimation of <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs. In these designs, runs are a multiple of 4 (i.e., 4, 8, 12, 16, 20 and so on). When the runs are a power of 2, the designs correspond to the resolution III two factor fractional factorial designs. Although Plackett-Burman designs are all two level orthogonal designs, the alias structure for these designs is complicated when runs are not a power of 2.

As an example, consider the 12-run Plackett-Burman design shown in the figure below.

[[Image:doe8.1.png|thumb|center|400px|12-run Plackett-Burman design.]]

If 11 main effects are to be estimated using this design, then each of these main effects is partially aliased with all other two factor interactions not containing that main effect. For example, the main effect <math>A\,\!</math> is partially aliased with all two factor interactions except <math>AB\,\!</math>, <math>AC\,\!</math>, <math>AD\,\!</math>, <math>AE\,\!</math>, <math>AF\,\!</math>, <math>AG\,\!</math>, <math>AH\,\!</math>, <math>AJ\,\!</math>, <math>AK\,\!</math> and <math>AL\,\!</math>. There are 45 such two factor interactions that are aliased with <math>A\,\!</math>.

::<math>\begin{align}
& A= & A-\frac{1}{3}BC-\frac{1}{3}BD+\frac{1}{3}CD-\frac{1}{3}BE-\frac{1}{3}CE+\frac{1}{3}DE+... \\
& & ...+\frac{1}{3}EL-\frac{1}{3}FL-\frac{1}{3}GL+\frac{1}{3}HL+\frac{1}{3}JL-\frac{1}{3}KL
\end{align}\,\!</math>

Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs|Appendix C]].

==Taguchi's Orthogonal Arrays==

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays|Appendix D]].

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in DOE++. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

[[Image:doe8.3.png|thumb|center|400px|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]

Similarly, consider the L8(2^7) array shown in figure (a) below. This design can be used to estimate up to seven main effects using eight runs. This array is again similar to the 2 <math>_{\text{III}}^{7-4}\,\!</math> design shown in figure (b) below, except that the aliasing between the columns of the two designs differs in sign for some of the columns (see figure (c)).

[[Image:doe8.4.png|thumb|center|400px|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]

The L8 array can also be used as a full factorial three factor experiment design in the same way as a <math>2^{3}\,\!</math> design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, figure (c) above shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 <math>^{3}\,\!</math> design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the ''preferred columns'' for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in the next table.

[[Image:doet8.2.png|thumb|center|400px|Alias relations for the L8 array.]]

The cell value in any (<math>i,j\,\!</math>) cell of the table gives the column number of the two factor interaction for the <math>i\,\!</math>th row and <math>j\,\!</math>th column. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the ( <math>1,2\,\!</math> ) cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in [[Alias_Relations_for_Taguchi_Orthogonal_Arrays|Appendix E]].

===Example===

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. The three factors investigated in the experiment were honing pressure (factor A), number of strokes (factor B) and cycle time (factor C). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the <math>{2}^{3}\,\!</math> design that was used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on the figure (c) above, so that you can easily compare results obtained from the L8 array to the ones included in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]]. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in the third table. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

[[Image:doet8.3.png|thumb|center|639px|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]

The experiment design for this example can be set using the properties shown in the figure below.

[[Image:doe8_5.png|thumb|center|842px|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

Note that for this design, the factor properties are set up as shown in the design summary.

[[Image:doe8_6.png|thumb|center|828px|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

The resulting design along with the response values is shown in the figure below.

[[Image:doe8_7.png|thumb|center|650px|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example| example]].]]

And the results from DOE++ for the design are shown in the next figure.

[[Image:doe8_8_1.png|thumb|center|613px|Results for the experiment in the example.]]

The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the <math>{2}^{3}\,\!</math> design used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]].

=== Preferred Columns in Taguchi OA===
One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

[[Image:DOEtableChapter9.png|center]]

This is a fractional factorial design for 7 factors. For any fractional factorial design, the first thing we need to do is check its alias structure. In general, the alias structures for Taguchi OAs are very complicated. People usually use the following table to represent the alias relations between each factor. For the above orthogonal array, the alias table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7
|-
|2x3|| 1x3|| 1x2|| 1x5|| 1x4|| 1x7|| 1x6
|-
|4x5|| 4x6|| 4x7|| 2x6|| 2x7|| 2x4|| 2x5
|-
|6x7|| 5x7|| 5x6|| 3x7|| 3x6|| 3x5|| 3x4
|}

In the above table, an Arabic number is used to represent a factor. For instance, “1” represents the factor assigned to the 1st column in the array. “2x3” represents the interaction effect of the two factors assigned to column 2 and 3. Each column in the above alias table lists all the 2-way interaction effects that are aliased with the main effect of the factor assigned to this column. For example, for the 1st column, the main effect of the factor assigned to it is aliased with interaction effects of 2x3, 4x5 and 6x7.
If an experiment has only 3 factors and these 3 factors A, B and C are assigned to the first 3 columns of the above L8(2^7) array, then the design table will be:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 3)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

The alias structure for the above table is:

{| style="text-align:center;" cellpadding="2" align="center"
|-
|[I] = I – ABC
|-
|[A] = A – BC
|-
|[B] = B – AC
|-
|[C] = C – AB
|}

This is a resolution 3 design. All the main effects are aliased with 2-way interactions. There are many ways to choose 3 columns from the 7 columns of L8(2^7). If the 3 factors are assigned to column 1, 2, and 4, then the design table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 4)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

====Example====
Design an experiment with 3 qualitative factors. Factors A and B have 2 levels; factor C has 4 levels. The experimenters are interested in all the main effects and the interaction effect AC.

Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

In DOE++, we can choose L16(2^6*4^3) in the following window.

[[Image:doe8_9.png|thumb|center|842px|Selecting the Taguchi OA design type.]]

Click Taguchi Preferred Columns to specify the interaction terms of interest by the experimenters.

Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

Example: Design an experiment for a test with 4 qualitative factors. Factors A and B have 2 levels; C and D have 4 levels. We are interested in all the main effects and the interaction effects AC and BD.

Assume again we want to use Taguchi OA L16(2^6*4^3). Click Taguchi Preferred Columns in the following screen.

Specify the interaction effects that you want to estimate in the experiment.

When click on OK, you will get a warning message saying that it is impossible to clearly estimate all the main effects and the specified interaction effect AC and BD.

This can be explained by checking the alias table of a L16(2^6*4^3) design as given below.

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
|2x7|| 1x7|| 1x9|| 1x5|| 1x4|| 1x8|| 1x2|| 1x6|| 1x3
|-
|3x9|| 3x6|| 2x6|| 2x8|| 2x9|| 2x3|| 1x8|| 1x7|| 1x7
|-
|4x5|| 4x8|| 4x7|| 3x7|| 3x8|| 4x9|| 1x9|| 1x9|| 1x8
|-
|6x8|| 5x9|| 5x8|| 6x9|| 6x7|| 5x7|| 2x8|| 2x4|| 2x5
|-
|7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 2x9|| 2x7|| 2x7
|-
|7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 3x8|| 2x9|| 2x8
|-
|8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 3x9|| 3x5|| 3x7
|-
| || || || || || ||4x8|| 3x7|| 3x8
|-
| || || || || || ||4x9|| 3x9|| 4x6
|-
| || || || || || ||5x6|| 4x7|| 4x7
|-
| || || || || || ||5x8|| 4x9|| 4x8
|-
| || || || || || ||5x9|| 5x7|| 5x7
|-
| || || || || || ||6x8|| 5x9|| 5x8
|-
| || || || || || ||6x9|| 6x7|| 6x7
|-
| || || || || || ||8x9|| 6x9|| 6x8
|-
| || || || || || || || 7x9|| 7x8
|}

From this table, we can see that it is impossible to clearly estimate both AC and BD. Factors C and D can only be assigned to the last three columns since they are 4 level factors. Assume we assign factor C to column 7 and factor D to column 8. Factor B (2 level) will be in one of the columns from 1 to 6. Therefore, effect BD will be one of the effects highlighted in the table, where the first term of the interaction is between 1 and 6 and the last term is 8 (i.e., factor D).

The above alias table shows factor C is aliased with one of those highlighted effects. Thus, no matter which of the first six columns is assigned to factor B, the main effect C will be aliased with interaction effect BD. This is also true if C is assigned to column 8 or 9. Therefore, if L16(2^6*4^3) is used, there is no way to clearly estimate all the main effects and the interaction effects AC and BD. Another Taguchi OA or other design types such as a general level full factorial design should be used. A more efficient way probably is to create an optimal custom design that can clearly estimate all the specified terms. For more detail, please refer to the chapter on Optimal Custom Designs.

File:Doe8 9.png

2015-11-23T23:05:26Z

Melinda Caroline: