Simulation with RGA Models - Revision history

David J. Groebel at 20:20, 2 June 2014

2014-06-02T20:20:09Z

← Older revision		Revision as of 20:20, 2 June 2014
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	When analyzing developmental systems for reliability growth, and conducting data analysis of fielded repairable systems, it is often useful to experiment with various ''what if'' scenarios or put together hypothetical analyses before data sets become available in order to plan for the best way to perform the analysis. With that in mind, the RGA software offers applications based on Monte Carlo simulation that can be used in order to:		When analyzing developmental systems for reliability growth, and conducting data analysis of fielded repairable systems, it is often useful to experiment with various ''what if'' scenarios or put together hypothetical analyses before data sets become available in order to plan for the best way to perform the analysis. With that in mind, the RGA software offers applications based on Monte Carlo simulation that can be used in order to:

	:~~a) Better understand reliability growth concepts.~~		<ol style="list-style-type:lower-alpha">
	~~:b) Experiment with the impact of sample size, test time and growth parameters on analysis results.~~
	~~:c) Construct simulation~~-~~based confidence intervals.~~
	~~:d) Better understand concepts behind confidence intervals.~~
	~~:e) Design reliability demonstration tests.~~

			<li>Better understand reliability growth concepts.
			<li>Experiment with the impact of sample size, test time and growth parameters on analysis results.
			<li>Construct simulation-based confidence intervals.
			<li>Better understand concepts behind confidence intervals.
			<li>Design reliability demonstration tests.
			</li>
			</ol>
	There are two applications of the Monte Carlo simulation in the RGA software. One is called Generate Monte Carlo Data and the other is called SimuMatic.		There are two applications of the Monte Carlo simulation in the RGA software. One is called Generate Monte Carlo Data and the other is called SimuMatic.

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	The first time to failure, <math>{{t}_{1}},\,\!</math> is assumed to follow a Weibull distribution. It is obtained by solving for <math>{{t}_{1}}\,\!</math> :		The first time to failure, <math>{{t}_{1}},\,\!</math> is assumed to follow a Weibull distribution. It is obtained by solving for <math>{{t}_{1}}\,\!</math> :

	::<math>R(t_1)=e^{(-\frac{t_1}{\eta})^\beta}= Uniform (0,1)\,\!</math>		:<math>R(t_1)=e^{(-\frac{t_1}{\eta})^\beta}= Uniform (0,1)\,\!</math>

	:where:		where:

	~~::<math>\eta ={{\left( \frac{1}{\lambda } \right)}^{\tfrac{1}{\beta }}}\,\!</math>~~

			:<math>\eta ={{\left( \frac{1}{\lambda } \right)}^{\tfrac{1}{\beta }}}\,\!</math>

	Solving for <math>{{t}_{1}}\,\!</math> yields:		Solving for <math>{{t}_{1}}\,\!</math> yields:

	::<math>t_{1} = \eta \left [ -ln(Uniform(0,1)) \right ]^{\frac {1}{\beta}}</math>		:<math>t_{1} = \eta \left [ -ln(Uniform(0,1)) \right ]^{\frac {1}{\beta}}</math>


	The failure times are then obtained based on the conditional unreliability equation that describes the non-homogeneous Poisson process (NHPP):		The failure times are then obtained based on the conditional unreliability equation that describes the non-homogeneous Poisson process (NHPP):

	::<math>F(t_{i}\ \| \ t_{i-1}) = 1 -e^{-\lambda\left[t_{i}^{\beta}-t_{i-1}^{\beta}\right]} = Uniform(0,1)</math>		:<math>F(t_{i}\ \| \ t_{i-1}) = 1 -e^{-\lambda\left[t_{i}^{\beta}-t_{i-1}^{\beta}\right]} = Uniform(0,1)</math>


	and then solving for <math>{{t}_{i}}\,\!</math> yields:		and then solving for <math>{{t}_{i}}\,\!</math> yields:

	::<math>t_{i}=\left[-\frac {ln(1-Uniform(0,1))}{\lambda}+t_{i-1}^{\beta} \right ]^{\frac{1}{\beta}}</math>		:<math>t_{i}=\left[-\frac {ln(1-Uniform(0,1))}{\lambda}+t_{i-1}^{\beta} \right ]^{\frac{1}{\beta}}</math>


	To access the data generation utility, choose '''Home > Tools > Generate Monte Carlo Data'''. There are different data types that can be generated with the Monte Carlo utility. For all of them, the basic parameters that are always specified are the beta <math>(\beta )\,\!</math> and lambda <math>(\lambda )\,\!</math> parameters of the Crow-AMSAA (NHPP) model. That does not mean that the generated data can be analyzed only with the Crow-AMSAA (NHPP) model. Depending on the data type, the Duane, Crow extended and power law models can also be used. They share the same growth patterns, which are based on the <math>\beta \,\!</math> and <math>\lambda \,\!</math> parameters. In the case of the Duane model, <math>\beta =1-\alpha \,\!</math>, where <math>\alpha \,\!</math> is the growth parameter for the Duane model. Below we present the available data types that can generated with the Monte Carlo utility.		To access the data generation utility, choose '''Home > Tools > Generate Monte Carlo Data'''. There are different data types that can be generated with the Monte Carlo utility. For all of them, the basic parameters that are always specified are the beta <math>(\beta )\,\!</math> and lambda <math>(\lambda )\,\!</math> parameters of the Crow-AMSAA (NHPP) model. That does not mean that the generated data can be analyzed only with the Crow-AMSAA (NHPP) model. Depending on the data type, the Duane, Crow extended and power law models can also be used. They share the same growth patterns, which are based on the <math>\beta \,\!</math> and <math>\lambda \,\!</math> parameters. In the case of the Duane model, <math>\beta =1-\alpha \,\!</math>, where <math>\alpha \,\!</math> is the growth parameter for the Duane model. Below we present the available data types that can generated with the Monte Carlo utility.
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	The next figure shows the Monte Carlo utility and all the necessary user inputs.		The next figure shows the Monte Carlo utility and all the necessary user inputs.

	[[Image:rga14.1.png~~\|thumb~~\|center\|~~400px\|Monte Carlo data generation utility.~~]]		[[Image:rga14.1.png\|center\|450px]]


	The seed determines the starting point from which the random numbers will be generated. The use of a seed forces the software to use the same sequence of random numbers, resulting in repeatability. In other words, the same failure times can be generated if the same seed, data type, parameters and number of points/systems are used. If no seed is provided, the computer's clock is used to initialize the random number generator and a different set of failure times will be generated at each new request.		The seed determines the starting point from which the random numbers will be generated. The use of a seed forces the software to use the same sequence of random numbers, resulting in repeatability. In other words, the same failure times can be generated if the same seed, data type, parameters and number of points/systems are used. If no seed is provided, the computer's clock is used to initialize the random number generator and a different set of failure times will be generated at each new request.

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	The next figure shows the Main tab of the SimuMatic window where all the necessary user inputs for a multiple systems - concurrent data set have been entered. The Analysis tab allows you to specify the confidence level for simulation-generated confidence bounds, while the Results tab gives you the option to compute for additional results, such as the instantaneous MTBF given a specific test time.		The next figure shows the Main tab of the SimuMatic window where all the necessary user inputs for a multiple systems - concurrent data set have been entered. The Analysis tab allows you to specify the confidence level for simulation-generated confidence bounds, while the Results tab gives you the option to compute for additional results, such as the instantaneous MTBF given a specific test time.

	[[Image:rga14.5.png~~\|thumb~~\|center\|~~400px\|The SimuMatic input window.~~]]		[[Image:rga14.5.png\|center\|450px]]


	The next figure shows the generated results based on the inputs shown above. The data sheet called "Sorted" allows us to extract conclusions about the simulation-generated confidence bounds because the lambda and beta parameters and any other additional output are sorted by percentage.		The next figure shows the generated results based on the inputs shown above. The data sheet called "Sorted" allows us to extract conclusions about the simulation-generated confidence bounds because the lambda and beta parameters and any other additional output are sorted by percentage.

	[[Image:rga14.6.png~~\|thumb~~\|center\|450px~~\|SimuMatic output sorted by percentile.~~]]		[[Image:rga14.6.png\|center\|450px]]


	The following plot shows the simulation-confidence bounds for the cumulative number of failures based on the input parameters specified.		The following plot shows the simulation-confidence bounds for the cumulative number of failures based on the input parameters specified.

	[[Image:rga14.7.png~~\|thumb~~\|center\|450px~~\|Simulation generated confidence bounds on cumulative number of failures.~~]]		[[Image:rga14.7.png\|center\|450px]]

	==SimuMatic Example==		==SimuMatic Example==
	{{:RGA_SimuMatic_Example}}		{{:RGA_SimuMatic_Example}}

Chris Kahn at 00:20, 24 April 2014

2014-04-24T00:20:18Z

← Older revision		Revision as of 00:20, 24 April 2014
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	The next figure shows the Main tab of the SimuMatic window where all the necessary user inputs for a multiple systems - concurrent data set have been entered. The Analysis tab allows you to specify the confidence level for simulation-generated confidence bounds, while the Results tab gives you the option to compute for additional results, such as the instantaneous MTBF given a specific test time.		The next figure shows the Main tab of the SimuMatic window where all the necessary user inputs for a multiple systems - concurrent data set have been entered. The Analysis tab allows you to specify the confidence level for simulation-generated confidence bounds, while the Results tab gives you the option to compute for additional results, such as the instantaneous MTBF given a specific test time.

	[[Image:rga14.5.png\|thumb\|center\|~~450px~~\|The SimuMatic input window.]]		[[Image:rga14.5.png\|thumb\|center\|400px\|The SimuMatic input window.]]

Chris Kahn at 00:18, 24 April 2014

2014-04-24T00:18:30Z

← Older revision		Revision as of 00:18, 24 April 2014
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	The next figure shows the Monte Carlo utility and all the necessary user inputs.		The next figure shows the Monte Carlo utility and all the necessary user inputs.

	[[Image:rga14.1.png\|thumb\|center\|~~300px~~\|Monte Carlo data generation utility.]]		[[Image:rga14.1.png\|thumb\|center\|400px\|Monte Carlo data generation utility.]]

Kate Racaza: /* SimuMatic Example */

2014-01-29T19:22:13Z

SimuMatic Example

← Older revision		Revision as of 19:22, 29 January 2014
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	==SimuMatic Example==		==SimuMatic Example==
	A manufacturer wants to design a reliability growth test for a redesigned product, in order to achieve an MTBF of 1,000 hours. Simulation is chosen to estimate the 1-sided 90% confidence bound on the required time to achieve the goal MTBF of 1,000 hours and the 1-sided 90% lower confidence bound on the MTBF at the end of the test time. The total test time is expected to be 15,000 hours. Based on historical data for the previous version, the expected beta and lambda parameters of the test are 0.5 and 0.3, respectively. Do the following:		{{:RGA_SimuMatic_Example}}

	~~:1) Generate 1,000 data sets using SimuMatic along with the required output.~~
	~~:2) Plot the instantaneous MTBF vs. time with the 90% confidence bounds.~~
	~~:3) Estimate the 1-sided 90% lower confidence bound on time for an MTBF of 1,000 hours.~~
	~~:4) Estimate the 1-sided 90% lower confidence bound on the instantaneous MTBF at the end of the test.~~


	~~'''Solution'''~~

	~~1) The next figure shows the SimuMatic window with all the appropriate inputs for creating the data sets.~~

	~~[[Image:rga14.8.png\|thumb\|center\|450px\|The Main tab of the SimuMatic window.]]~~


	~~:The next three figures show the settings in the Analysis, Test Design and Results tab of the SimuMatic window in order to obtain the desired outputs.~~

	~~[[Image:rga14.9.png\|thumb\|center\|450px\|The '''Analysis''' tab of the SimuMatic window.]]~~

	~~[[Image:rga14.9.5.png\|thumb\|center\|450px\|The '''Test Design''' tab of the SimuMatic window.]]~~

	~~[[Image:rga14.10.png\|thumb\|center\|450px\|The '''Results''' tab of the SimuMatic window.]]~~


	:The following figure displays the results of the simulation. The columns labeled "Beta" and "Lambda" contain the different parameters obtained by calculating each data set generated via simulation for the 1,000 data sets. The "DMTBF" column contains the instantaneous MTBF at 15,000 hours (the end of test time), given the parameters obtained by calculating each data set generated via simulation. The "T(IMTBF=1000 Hr)" column contains the time required for the MTBF to reach 1,000 hours, given the parameters obtained from the simulation.

	~~[[Image:rga14.11.png\|thumb\|center\|450px\|Simulation results using SimuMatic.]]~~


	~~2) The next figure shows the plot of the instantaneous MTBF with the 90% confidence bounds.~~

	~~[[Image:rga14.12.png\|thumb\|center\|450px\|Instantaneous MTBF with 90% confidence bounds.]]~~


	3) The 1-sided 90% lower confidence bound on time, assuming MTBF = 1,000 hours, can be obtained from the results of the simulation. In the "Sorted" data sheet, this is the target DMTBF value that corresponds to 10.00%, as shown in the next figure. Therefore the 1-sided 90% lower confidence bound on time is 12,642.21 hours.

	~~[[Image:rga14.13.png\|thumb\|center\|450px\|The 1-sided 90% lower confidence bound on time assuming 1,000 hours MTBF.]]~~


	~~4) The next figure shows the 1-sided 90% lower confidence bound on time in the instantaneous MTBF plot. This is indicated by the target lines on the plot.~~

	~~[[Image:rga14.14.png\|thumb\|center\|450px\|Instantaneous MTBF with the target.]]~~


	5) The 1-sided 90% lower confidence bound on the instantaneous MTBF at the end of the test is again obtained from the "Sorted" data sheet by looking at the value in the "IMTBF(15,000)" column that corresponds to 10.00%. As seen in the simulation results shown above, the 1-sided 90% lower confidence bound on the instantaneous MTBF at the end of the test is 605.93 hours.

Kate Racaza: /* Monte Carlo Data Example */

2014-01-29T19:20:26Z

Monte Carlo Data Example

@@ Line 55: / Line 55: @@
 ==Monte Carlo Data Example==
-A reliability engineer wants to experiment with different testing scenarios as the reliability growth test of the company's new product is being prepared. From the reliability growth test data of a similar product that was developed previously, the beta and lambda parameters are <math>\beta =0.5\,\!</math> and <math>\lambda =0.75.\,\!</math> Three systems are to be used to generate a representative data set of expected times-to-failure for the upcoming test. The purpose is to explore different test durations in order to demonstrate an MTBF of 200 hours.
+{{:RGA_Monte_Carlo_Simulation_Example}}
 =SimuMatic=

David J. Groebel at 00:57, 16 January 2014

2014-01-16T00:57:40Z

← Older revision		Revision as of 00:57, 16 January 2014
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	{{template:RGA BOOK\|7\|~~Additional Tools~~}}		{{template:RGA BOOK\|7\|Simulation with RGA Models}}
	When analyzing developmental systems for reliability growth, and conducting data analysis of fielded repairable systems, it is often useful to experiment with various ''what if'' scenarios or put together hypothetical analyses before data sets become available in order to plan for the best way to perform the analysis. With that in mind, the RGA software offers applications based on Monte Carlo simulation that can be used in order to:		When analyzing developmental systems for reliability growth, and conducting data analysis of fielded repairable systems, it is often useful to experiment with various ''what if'' scenarios or put together hypothetical analyses before data sets become available in order to plan for the best way to perform the analysis. With that in mind, the RGA software offers applications based on Monte Carlo simulation that can be used in order to:

David J. Groebel at 00:55, 16 January 2014

2014-01-16T00:55:20Z

← Older revision		Revision as of 00:55, 16 January 2014
Line 1:		Line 1:
	{{template:RGA BOOK\|14\|Additional Tools}}		{{template:RGA BOOK\|7\|Additional Tools}}
	When analyzing developmental systems for reliability growth, and conducting data analysis of fielded repairable systems, it is often useful to experiment with various ''what if'' scenarios or put together hypothetical analyses before data sets become available in order to plan for the best way to perform the analysis. With that in mind, the RGA software offers applications based on Monte Carlo simulation that can be used in order to:		When analyzing developmental systems for reliability growth, and conducting data analysis of fielded repairable systems, it is often useful to experiment with various ''what if'' scenarios or put together hypothetical analyses before data sets become available in order to plan for the best way to perform the analysis. With that in mind, the RGA software offers applications based on Monte Carlo simulation that can be used in order to:

Athanasios Gerokostopoulos: /* SimuMatic Example */

2013-11-27T19:38:32Z

SimuMatic Example

← Older revision		Revision as of 19:38, 27 November 2013
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	:The next ~~two~~ figures show the settings in the Analysis and Results tab of the SimuMatic window in order to obtain the desired outputs.		:The next three figures show the settings in the Analysis, Test Design and Results tab of the SimuMatic window in order to obtain the desired outputs.

	[[Image:rga14.9.png\|thumb\|center\|450px\|The '''Analysis''' tab of the SimuMatic window.]]		[[Image:rga14.9.png\|thumb\|center\|450px\|The '''Analysis''' tab of the SimuMatic window.]]
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	:The following figure displays the results of the simulation. The columns labeled "Beta" and "Lambda" contain the different parameters obtained by calculating each data set generated via simulation for the 1,000 data sets. The "~~IMTBF(15,000)~~" column contains the instantaneous MTBF at 15,000 hours (the end of test time), given the parameters obtained by calculating each data set generated via simulation. The "~~Target DMTBF~~" column contains the time required for the MTBF to reach 1,000 hours, given the parameters obtained from the simulation.		:The following figure displays the results of the simulation. The columns labeled "Beta" and "Lambda" contain the different parameters obtained by calculating each data set generated via simulation for the 1,000 data sets. The "DMTBF" column contains the instantaneous MTBF at 15,000 hours (the end of test time), given the parameters obtained by calculating each data set generated via simulation. The "T(IMTBF=1000 Hr)" column contains the time required for the MTBF to reach 1,000 hours, given the parameters obtained from the simulation.

	[[Image:rga14.11.png\|thumb\|center\|450px\|Simulation results using SimuMatic.]]		[[Image:rga14.11.png\|thumb\|center\|450px\|Simulation results using SimuMatic.]]

Athanasios Gerokostopoulos: /* SimuMatic Example */

2013-11-27T19:33:48Z

SimuMatic Example

← Older revision		Revision as of 19:33, 27 November 2013
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	[[Image:rga14.9.png\|thumb\|center\|450px\|The '''Analysis''' tab of the SimuMatic window.]]		[[Image:rga14.9.png\|thumb\|center\|450px\|The '''Analysis''' tab of the SimuMatic window.]]

			[[Image:rga14.9.5.png\|thumb\|center\|450px\|The '''Test Design''' tab of the SimuMatic window.]]

	[[Image:rga14.10.png\|thumb\|center\|450px\|The '''Results''' tab of the SimuMatic window.]]		[[Image:rga14.10.png\|thumb\|center\|450px\|The '''Results''' tab of the SimuMatic window.]]

Athanasios Gerokostopoulos: /* SimuMatic Example */

2013-11-27T19:25:22Z

SimuMatic Example

← Older revision		Revision as of 19:33, 27 November 2013
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	[[Image:rga14.9.png\|thumb\|center\|450px\|The '''Analysis''' tab of the SimuMatic window.]]		[[Image:rga14.9.png\|thumb\|center\|450px\|The '''Analysis''' tab of the SimuMatic window.]]

			[[Image:rga14.9.5.png\|thumb\|center\|450px\|The '''Test Design''' tab of the SimuMatic window.]]

	[[Image:rga14.10.png\|thumb\|center\|450px\|The '''Results''' tab of the SimuMatic window.]]		[[Image:rga14.10.png\|thumb\|center\|450px\|The '''Results''' tab of the SimuMatic window.]]

← Older revision		Revision as of 19:25, 27 November 2013
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	1) The next figure shows the SimuMatic window with all the appropriate inputs for creating the data sets.		1) The next figure shows the SimuMatic window with all the appropriate inputs for creating the data sets.

	[[Image:rga14.8.png\|thumb\|center\|~~300px~~\|The Main tab of the SimuMatic window.]]		[[Image:rga14.8.png\|thumb\|center\|450px\|The Main tab of the SimuMatic window.]]


	:The next two figures show the settings in the Analysis and Results tab of the SimuMatic window in order to obtain the desired outputs.		:The next two figures show the settings in the Analysis and Results tab of the SimuMatic window in order to obtain the desired outputs.

	[[Image:rga14.9.png\|thumb\|center\|~~300px~~\|The '''Analysis''' tab of the SimuMatic window.]]		[[Image:rga14.9.png\|thumb\|center\|450px\|The '''Analysis''' tab of the SimuMatic window.]]


	[[Image:rga14.10.png\|thumb\|center\|~~300px~~\|The '''Results''' tab of the SimuMatic window.]]		[[Image:rga14.10.png\|thumb\|center\|450px\|The '''Results''' tab of the SimuMatic window.]]


	:The following figure displays the results of the simulation. The columns labeled "Beta" and "Lambda" contain the different parameters obtained by calculating each data set generated via simulation for the 1,000 data sets. The "IMTBF(15,000)" column contains the instantaneous MTBF at 15,000 hours (the end of test time), given the parameters obtained by calculating each data set generated via simulation. The "Target DMTBF" column contains the time required for the MTBF to reach 1,000 hours, given the parameters obtained from the simulation.		:The following figure displays the results of the simulation. The columns labeled "Beta" and "Lambda" contain the different parameters obtained by calculating each data set generated via simulation for the 1,000 data sets. The "IMTBF(15,000)" column contains the instantaneous MTBF at 15,000 hours (the end of test time), given the parameters obtained by calculating each data set generated via simulation. The "Target DMTBF" column contains the time required for the MTBF to reach 1,000 hours, given the parameters obtained from the simulation.

	[[Image:rga14.11.png\|thumb\|center\|~~300px~~\|Simulation results using SimuMatic.]]		[[Image:rga14.11.png\|thumb\|center\|450px\|Simulation results using SimuMatic.]]


	2) The next figure shows the plot of the instantaneous MTBF with the 90% confidence bounds.		2) The next figure shows the plot of the instantaneous MTBF with the 90% confidence bounds.

	[[Image:rga14.12.png\|thumb\|center\|~~300px~~\|Instantaneous MTBF with 90% confidence bounds.]]		[[Image:rga14.12.png\|thumb\|center\|450px\|Instantaneous MTBF with 90% confidence bounds.]]


	3) The 1-sided 90% lower confidence bound on time, assuming MTBF = 1,000 hours, can be obtained from the results of the simulation. In the "Sorted" data sheet, this is the target DMTBF value that corresponds to 10.00%, as shown in the next figure. Therefore the 1-sided 90% lower confidence bound on time is 12,642.21 hours.		3) The 1-sided 90% lower confidence bound on time, assuming MTBF = 1,000 hours, can be obtained from the results of the simulation. In the "Sorted" data sheet, this is the target DMTBF value that corresponds to 10.00%, as shown in the next figure. Therefore the 1-sided 90% lower confidence bound on time is 12,642.21 hours.

	[[Image:rga14.13.png\|thumb\|center\|~~300px~~\|The 1-sided 90% lower confidence bound on time assuming 1,000 hours MTBF.]]		[[Image:rga14.13.png\|thumb\|center\|450px\|The 1-sided 90% lower confidence bound on time assuming 1,000 hours MTBF.]]


	4) The next figure shows the 1-sided 90% lower confidence bound on time in the instantaneous MTBF plot. This is indicated by the target lines on the plot.		4) The next figure shows the 1-sided 90% lower confidence bound on time in the instantaneous MTBF plot. This is indicated by the target lines on the plot.

	[[Image:rga14.14.png\|thumb\|center\|~~300px~~\|Instantaneous MTBF with the target.]]		[[Image:rga14.14.png\|thumb\|center\|450px\|Instantaneous MTBF with the target.]]


	5) The 1-sided 90% lower confidence bound on the instantaneous MTBF at the end of the test is again obtained from the "Sorted" data sheet by looking at the value in the "IMTBF(15,000)" column that corresponds to 10.00%. As seen in the simulation results shown above, the 1-sided 90% lower confidence bound on the instantaneous MTBF at the end of the test is 605.93 hours.		5) The 1-sided 90% lower confidence bound on the instantaneous MTBF at the end of the test is again obtained from the "Sorted" data sheet by looking at the value in the "IMTBF(15,000)" column that corresponds to 10.00%. As seen in the simulation results shown above, the 1-sided 90% lower confidence bound on the instantaneous MTBF at the end of the test is 605.93 hours.