Template:Crow extended model rga: Difference between revisions

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==Crow Extended Model==
#REDIRECT [[Reliability_Growth_Planning#Crow_Extended_Model]]
===Introduction===
The Crow Extended model for reliability growth planning is a revised and improved version of the MIL-HDBK-189 growth curve [13]. MIL-HDBK-189 can be considered as the growth curve based on the Crow-AMSAA (NHPP) model. Using MIL-HDBK-189 for reliability growth planning assumes that the corrective actions for the observed failure modes are incorporated during the test and at the specific time of failure. However, in actual practice, fixes may be delayed until after the completion of the test or some fixes may be implemented during the test while others are delayed and some are not fixed at all. The Crow Extended model for reliability growth planning provides additional input to be able to account for a specific management strategy and delayed fixes with specified effectiveness factors.
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===Definitions===
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====Management Strategy Ratio & Initial Failure Intensity====
When a system is tested and failure modes are observed, management can make one of two possible decisions, either to fix or not fix the failure mode. Therefore, the management strategy places failure modes into two categories: A modes and B modes. A modes are all failure modes such that, when seen during the test, no corrective action will be taken. This accounts for all modes for which management determines that it is not economically or otherwise justified to take a corrective action. B modes are either corrected during the test or the corrective action is delayed to a later time. The management strategy is defined by what portion of the failures will be fixed.
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Let  <math>{{\lambda }_{I}}</math>  be the initial failure intensity of the system in test.  <math>{{\lambda }_{A}}</math>  is defined as the A mode initial failure intensity and  <math>{{\lambda }_{B}}</math>  is defined as the B mode initial failure intensity.  <math>{{\lambda }_{A}}</math>  is the failure intensity of the system that will not be addressed by corrective actions even if a failure mode is seen during test.  <math>{{\lambda }_{B}}</math>  is the failure intensity of the system that will be addressed by corrective actions if a failure mode is seen during testing.
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Then, the initial failure intensity of the system is:
 
 
::<math>{{\lambda }_{I}}={{\lambda }_{A}}+{{\lambda }_{B}}</math>
 
 
The initial system MTBF is:
 
 
::<math>{{M}_{I}}=\frac{1}{{{\lambda }_{I}}}</math>
 
 
Based on the initial failure intensity definitions, the management strategy ratio is defined as:
 
 
::<math>msr=\frac{{{\lambda }_{B}}}{{{\lambda }_{A}}+{{\lambda }_{B}}}</math>
 
 
The  <math>msr</math>  is the portion of the initial system failure intensity that will be addressed by corrective actions, if seen during the test.
The Type A and B failure mode initial failure mode intensity is:
 
 
::<math>\begin{align}
  & {{\lambda }_{A}}= & \left( 1-msr \right)\cdot {{\lambda }_{I}} \\
& {{\lambda }_{B}}= & msr\cdot {{\lambda }_{I}} 
\end{align}</math>
 
====Effectiveness Factor====
 
When a delayed corrective action is implemented for a Type B failure mode, in other words a BD mode, the failure intensity for that mode is reduced if the corrective action is effective. Once a BD mode failure mode is discovered, it is rarely totally eliminated by a corrective action. After a BD mode has been found and fixed, a certain percentage of the failure intensity will be removed, but a certain percentage of the failure intensity will generally remain. The fraction decrease in the BD mode failure intensity due to corrective actions,  <math>d</math> ,  <math>\left( 0<d<1 \right),</math>  is called the effectiveness factor. A study on EFs showed that an average EF,  <math>d,</math>  was about 70%. Therefore, typically about 30%, i.e.  <math>100(1-d)%</math> , of the BD mode failure intensity will remain in the system after all of the corrective actions have been implemented. However, individual EFs for the failure modes may be larger or smaller than the average. This average value of 70% can be used for planning purposes, or if such information is recorded, an average effectiveness factor from a previous reliability growth program can be used.
 
====MTBF Goal====
 
When putting together a reliability growth plan, a goal MTBF  <math>{{M}_{G}}</math>  (or goal failure intensity  <math>{{\lambda }_{G}}</math> ) is defined as the requirement or target for the product at the end of the growth program.
 
 
====Growth Potential====
The failure intensity remaining in the system at the end of the test will depend on the management strategy given by the classification of the Type A and Type B failure modes. The engineering effort applied to the corrective actions determines the effectiveness factors. In addition, the failure intensity depends on  <math>h(t)</math> , which is the rate at which problem failure modes are being discovered during testing. The rate of discovery drives the opportunity to take corrective actions based on the seen failure modes and it is an important factor in the overall reliability growth rate. The reliability growth potential is the limiting value of the failure intensity as time  <math>T</math>  increases. This limit is the maximum MTBF that can be attained with the current management strategy. The maximum MTBF will be attained when all Type B modes have been observed and fixed.
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If all seen Type B modes are corrected by time  <math>T</math> , that is, no deferred corrective actions at time  <math>T</math> , then the growth potential is the maximum attainable with the Type B designation of the failure modes and the corresponding assigned effectiveness factors. This is called the nominal growth potential. In other words, the nominal growth potential is the maximum attainable growth potential assuming corrective actions are implemented for every mode that is planned to be fixed. In reality, some fixes to modes might be implemented at a later time due to schedule, budget, engineering, etc.
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If some seen Type B modes are not corrected at the end of the current test phase then the prevailing growth potential is below the maximum attainable with the Type B designation of the failure modes and the corresponding assigned effectiveness factors.
If all Type B failure modes are seen and corrected with an average effectiveness factor,  <math>d</math> , then the maximum reduction in the initial system failure intensity is the growth potential failure intensity:
 
 
::<math>{{\lambda }_{GP}}={{\lambda }_{A}}+\left( 1-d \right){{\lambda }_{B}}</math>
 
 
The growth potential MTBF is:
 
 
::<math>{{M}_{GP}}=\frac{1}{{{\lambda }_{GP}}}</math>
 
 
Note that based Eqns. (lambda GP), (lambda initial) and (msr), the initial failure intensity is equal to:
 
 
::<math>{{\lambda }_{I}}=\frac{{{\lambda }_{GP}}}{1-d\cdot msr}</math>
 
 
====Growth Potential Design Margin====
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The Growth Potential Design Margin ( <math>GPDM</math> ) can be considered as a safety margin when setting target MTBF values for the reliability growth plan. It is common for systems to degrade in terms of reliability when a prototype product is going into full manufacturing due to variation in material, processes, etc. Furthermore, the in-house reliability growth testing usually overestimates the actual product reliability, since the field usage conditions may not be perfectly simulated during growth testing. Typical values for the  <math>GPDM</math>  are around 1.2. Higher values yield less risk for the program, but require a more rigorous reliability growth test plan. Lower values imply higher program risk, with less safety margin.
 
During the planning stage, the growth potential MTBF,  <math>{{M}_{GP}},</math>  can be calculated based on the goal MTBF,  <math>{{M}_{G}},</math>  and the growth potential design margin,  <math>GPDM</math> :
 
 
::<math>{{M}_{GP}}=GPDM\cdot {{M}_{G}}</math>
 
 
or in terms of failure intensity:
 
 
::<math>{{\lambda }_{GP}}=\frac{{{\lambda }_{G}}}{GPDM}</math>
 
 
{{nominal idealized growth curve}}
 
{{actual idealized growth curve}}

Latest revision as of 01:40, 27 August 2012

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