Reliability Allocation - Revision history

Lisa Hacker at 22:02, 18 September 2023

2023-09-18T22:02:24Z

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	<noinclude>{{Banner BlockSim Articles}}{{Navigation box}}		<noinclude>{{Banner BlockSim Articles}}{{Navigation box}}
	''This article also appears in the [~~[Reliability_Importance_and_Optimized_Reliability_Allocation_(Analytical)#Component_Reliability_Importance\|~~System ~~Analysis Reference~~]~~] book~~.''		''This article also appears in the [https://help.reliasoft.com/reference/system_analysis System analysis reference].''
	</noinclude>		</noinclude>
	In the process of developing a new product, the engineer is often faced with the task of designing a system that conforms to a set of reliability specifications. The engineer is given the goal for the system and must then develop a design that will achieve the desired reliability of the system, while performing all of the system's intended functions at a minimum cost. This involves a balancing act of determining how to allocate reliability to the components in the system so the system will meet its reliability goal while at the same time ensuring that the system meets all of the other associated performance specifications.		In the process of developing a new product, the engineer is often faced with the task of designing a system that conforms to a set of reliability specifications. The engineer is given the goal for the system and must then develop a design that will achieve the desired reliability of the system, while performing all of the system's intended functions at a minimum cost. This involves a balancing act of determining how to allocate reliability to the components in the system so the system will meet its reliability goal while at the same time ensuring that the system meets all of the other associated performance specifications.

Richard House: /* Other Notes on User-Defined Cost Functions */

2016-01-06T16:28:45Z

Other Notes on User-Defined Cost Functions

← Older revision		Revision as of 16:28, 6 January 2016
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	Another method for developing a reliability cost function would be to obtain different samples of components from different suppliers and test the samples to determine the reliability of each sample type. From this data, a curve could be fitted through standard regression techniques and an equation defining the cost as a function of reliability could be developed. The following figure shows such a curve.		Another method for developing a reliability cost function would be to obtain different samples of components from different suppliers and test the samples to determine the reliability of each sample type. From this data, a curve could be fitted through standard regression techniques and an equation defining the cost as a function of reliability could be developed. The following figure shows such a curve.

	[[Image:6.21.png\|center\|~~500px~~\|Typical reliability growth curve generated using ReliaSoft's Reliability Growth software.\|link=]]		[[Image:6.21.png\|center\|650px\|Typical reliability growth curve generated using ReliaSoft's Reliability Growth software.\|link=]]

	Lastly, and in cases where a reliability growth program is in place, the simplest way of obtaining a relationship between cost and reliability is by associating a cost to each development stage of the growth process. Reliability growth models such as the Crow (AMSAA), Duane, Gompertz and Logistic models can be used to describe the cost as a function of reliability.		Lastly, and in cases where a reliability growth program is in place, the simplest way of obtaining a relationship between cost and reliability is by associating a cost to each development stage of the growth process. Reliability growth models such as the Crow (AMSAA), Duane, Gompertz and Logistic models can be used to describe the cost as a function of reliability.

	If a reliability growth model has been successfully implemented, the development costs over the respective development time stages can be applied to the growth model, resulting in equations that describe reliability/cost relationships. These equations can then be entered into BlockSim as user-defined cost functions (feasibility policies). The only potential drawback to using growth model data is the lack of flexibility in applying the optimum results. Making the cost projection for future stages of the project would require the assumption that development costs will be accrued at a similar rate in the future, which may not always be a valid assumption. Also, if the optimization result suggests using a high reliability value for a component, it may take more time than is allotted for that project to attain the required reliability given the current reliability growth of the project.		If a reliability growth model has been successfully implemented, the development costs over the respective development time stages can be applied to the growth model, resulting in equations that describe reliability/cost relationships. These equations can then be entered into BlockSim as user-defined cost functions (feasibility policies). The only potential drawback to using growth model data is the lack of flexibility in applying the optimum results. Making the cost projection for future stages of the project would require the assumption that development costs will be accrued at a similar rate in the future, which may not always be a valid assumption. Also, if the optimization result suggests using a high reliability value for a component, it may take more time than is allotted for that project to attain the required reliability given the current reliability growth of the project.

Richard House: /* Implementing the Optimization */

2016-01-06T16:27:15Z

Implementing the Optimization

← Older revision		Revision as of 16:27, 6 January 2016
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	So, the reliability has increased by a value of 21% and the cost has increased by one dollar. In a similar fashion, we can continue to add more units in parallel, thus increasing the reliability and the cost. We now have an array of reliability values and the associated costs that we can use to develop a cost function for this fault tolerance scheme. The next figure shows the relationship between cost and reliability for this example.		So, the reliability has increased by a value of 21% and the cost has increased by one dollar. In a similar fashion, we can continue to add more units in parallel, thus increasing the reliability and the cost. We now have an array of reliability values and the associated costs that we can use to develop a cost function for this fault tolerance scheme. The next figure shows the relationship between cost and reliability for this example.

	[[Image:6.15.png\|center\|~~700px~~\|Cost function for redundant parallel units.\|link=]]		[[Image:6.15.png\|center\|650px\|Cost function for redundant parallel units.\|link=]]

	As can be seen, this looks quite similar to the general behavior cost model presented earlier. In fact, a standard regression analysis available in Weibull++ indicates that an exponential model fits this cost model quite well. The function is given by the following equation, where <math>C\,\!</math> is the cost in dollars and <math>R\,\!</math> is the fractional reliability value.		As can be seen, this looks quite similar to the general behavior cost model presented earlier. In fact, a standard regression analysis available in Weibull++ indicates that an exponential model fits this cost model quite well. The function is given by the following equation, where <math>C\,\!</math> is the cost in dollars and <math>R\,\!</math> is the fractional reliability value.

Richard House: /* Maximum Achievable Reliability */

2016-01-06T16:26:27Z

Maximum Achievable Reliability

← Older revision		Revision as of 16:26, 6 January 2016
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	As the component reliability, <math>{{R}_{i}}\,\!</math>, approaches the maximum achievable reliability, <math>{{R}_{i,max}}\,\!</math>, the cost function approaches infinity. The maximum achievable reliability acts as a scale parameter for the cost function. By decreasing <math>{{R}_{i,max}}\,\!</math>, the cost function is compressed between <math>{{R}_{i,min}}\,\!</math> and <math>{{R}_{i,max}}\,\!</math>, as shown in the figure below.		As the component reliability, <math>{{R}_{i}}\,\!</math>, approaches the maximum achievable reliability, <math>{{R}_{i,max}}\,\!</math>, the cost function approaches infinity. The maximum achievable reliability acts as a scale parameter for the cost function. By decreasing <math>{{R}_{i,max}}\,\!</math>, the cost function is compressed between <math>{{R}_{i,min}}\,\!</math> and <math>{{R}_{i,max}}\,\!</math>, as shown in the figure below.

	[[Image:6_12.png\|center\|~~700px~~\|Effect of the maximum achievable reliability on the cost function.\|link=]]		[[Image:6_12.png\|center\|650px\|Effect of the maximum achievable reliability on the cost function.\|link=]]

	===Cost Function===		===Cost Function===

Richard House: /* The Feasibility Term, f\,\! */

2016-01-06T16:25:44Z

The Feasibility Term, f\,\!

← Older revision		Revision as of 16:25, 6 January 2016
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	The feasibility term in <math>{{C}_{i}}({{R}_{i}})={{e}^{(1-f)\cdot \tfrac{{{R}_{i}}-{{R}_{\min ,i}}}{{{R}_{\max ,i}}-{{R}_{i}}}}}\ \,\!</math> is a constant (or an equation parameter) that represents the difficulty in increasing a component's reliability relative to the rest of the components in the system. Depending on the design complexity, technological limitations, etc., certain components can be very hard to improve. Clearly, the more difficult it is to improve the reliability of the component, the greater the cost. The following figure illustrates the behavior of the function defined in the above equation for different values of <math>f\,\!</math>. It can be seen that the lower the feasibility value, the more rapidly the cost function approaches infinity.		The feasibility term in <math>{{C}_{i}}({{R}_{i}})={{e}^{(1-f)\cdot \tfrac{{{R}_{i}}-{{R}_{\min ,i}}}{{{R}_{\max ,i}}-{{R}_{i}}}}}\ \,\!</math> is a constant (or an equation parameter) that represents the difficulty in increasing a component's reliability relative to the rest of the components in the system. Depending on the design complexity, technological limitations, etc., certain components can be very hard to improve. Clearly, the more difficult it is to improve the reliability of the component, the greater the cost. The following figure illustrates the behavior of the function defined in the above equation for different values of <math>f\,\!</math>. It can be seen that the lower the feasibility value, the more rapidly the cost function approaches infinity.

	[[Image:6_11.png\|center\|~~700px~~\|Behavior of the cost function for different feasibility values.\|link=]]		[[Image:6_11.png\|center\|650px\|Behavior of the cost function for different feasibility values.\|link=]]

	Several methods can be used to obtain a feasibility value. Weighting factors for allocating reliability have been proposed by many authors and can be used to quantify feasibility. These weights depend on certain factors of influence, such as the complexity of the component, the state of the art, the operational profile, the criticality, etc. Engineering judgment based on past experience, supplier quality, supplier availability and other factors can also be used in determining a feasibility value. Overall, the assignment of a feasibility value is going to be a subjective process. Of course, this problem is negated if the relationship between the cost and the reliability for each component is known because one can use regression methods to estimate the parameter value.		Several methods can be used to obtain a feasibility value. Weighting factors for allocating reliability have been proposed by many authors and can be used to quantify feasibility. These weights depend on certain factors of influence, such as the complexity of the component, the state of the art, the operational profile, the criticality, etc. Engineering judgment based on past experience, supplier quality, supplier availability and other factors can also be used in determining a feasibility value. Overall, the assignment of a feasibility value is going to be a subjective process. Of course, this problem is negated if the relationship between the cost and the reliability for each component is known because one can use regression methods to estimate the parameter value.

Richard House: /* Defining a Feasibility Policy in BlockSim */

2016-01-05T15:48:37Z

Defining a Feasibility Policy in BlockSim

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	*Lastly, and since the evaluation is relative, it is preferable to use the predefined functions unless you have a compelling reason (or data) to do otherwise. The last section in this chapter describes cases where user-defined functions are preferred.		*Lastly, and since the evaluation is relative, it is preferable to use the predefined functions unless you have a compelling reason (or data) to do otherwise. The last section in this chapter describes cases where user-defined functions are preferred.

	[[Image:6.13.png\|center\|700px\|thumb\|Setting the default feasibility function in BlockSim with the feasibility slider. Note that the feasibility slider displays values, ''SV'', from 0.1 to 9.9 when moved by the user, with SV=9.9 being the hardest. The relationship between ''f'' and ''SV'' is "f = (1 - SV/10)"). ]]		[[Image:6.13.png\|center\|700px\|thumb\|Setting the default feasibility function in BlockSim with the feasibility slider. Note that the feasibility slider displays values, ''SV'', from 0.1 to 9.9 when moved by the user, with SV=9.9 being the hardest. The relationship between ''f'' and ''SV'' is "f = (1 - SV/10)").\|link= ]]


	[[Image:6.14.png\|center\|700px\|thumb\|Setting a user-defined feasibility function in BlockSim. Any user-defined equation can be entered as a function of ''R.'']]		[[Image:6.14.png\|center\|700px\|thumb\|Setting a user-defined feasibility function in BlockSim. Any user-defined equation can be entered as a function of ''R.''\|link=]]

	==Implementing the Optimization==		==Implementing the Optimization==

Richard House: /* Other Notes on User-Defined Cost Functions */

2016-01-04T22:31:23Z

Other Notes on User-Defined Cost Functions

← Older revision		Revision as of 22:31, 4 January 2016
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	Another method for developing a reliability cost function would be to obtain different samples of components from different suppliers and test the samples to determine the reliability of each sample type. From this data, a curve could be fitted through standard regression techniques and an equation defining the cost as a function of reliability could be developed. The following figure shows such a curve.		Another method for developing a reliability cost function would be to obtain different samples of components from different suppliers and test the samples to determine the reliability of each sample type. From this data, a curve could be fitted through standard regression techniques and an equation defining the cost as a function of reliability could be developed. The following figure shows such a curve.

	[[Image:6.21.png\|center\|~~400px~~\|Typical reliability growth curve generated using ReliaSoft's Reliability Growth software.]]		[[Image:6.21.png\|center\|500px\|Typical reliability growth curve generated using ReliaSoft's Reliability Growth software.\|link=]]

	Lastly, and in cases where a reliability growth program is in place, the simplest way of obtaining a relationship between cost and reliability is by associating a cost to each development stage of the growth process. Reliability growth models such as the Crow (AMSAA), Duane, Gompertz and Logistic models can be used to describe the cost as a function of reliability.		Lastly, and in cases where a reliability growth program is in place, the simplest way of obtaining a relationship between cost and reliability is by associating a cost to each development stage of the growth process. Reliability growth models such as the Crow (AMSAA), Duane, Gompertz and Logistic models can be used to describe the cost as a function of reliability.

	If a reliability growth model has been successfully implemented, the development costs over the respective development time stages can be applied to the growth model, resulting in equations that describe reliability/cost relationships. These equations can then be entered into BlockSim as user-defined cost functions (feasibility policies). The only potential drawback to using growth model data is the lack of flexibility in applying the optimum results. Making the cost projection for future stages of the project would require the assumption that development costs will be accrued at a similar rate in the future, which may not always be a valid assumption. Also, if the optimization result suggests using a high reliability value for a component, it may take more time than is allotted for that project to attain the required reliability given the current reliability growth of the project.		If a reliability growth model has been successfully implemented, the development costs over the respective development time stages can be applied to the growth model, resulting in equations that describe reliability/cost relationships. These equations can then be entered into BlockSim as user-defined cost functions (feasibility policies). The only potential drawback to using growth model data is the lack of flexibility in applying the optimum results. Making the cost projection for future stages of the project would require the assumption that development costs will be accrued at a similar rate in the future, which may not always be a valid assumption. Also, if the optimization result suggests using a high reliability value for a component, it may take more time than is allotted for that project to attain the required reliability given the current reliability growth of the project.

Richard House: /* Implementing the Optimization */

2016-01-04T22:26:35Z

Implementing the Optimization

← Older revision		Revision as of 22:26, 4 January 2016
Line 162:		Line 162:
	So, the reliability has increased by a value of 21% and the cost has increased by one dollar. In a similar fashion, we can continue to add more units in parallel, thus increasing the reliability and the cost. We now have an array of reliability values and the associated costs that we can use to develop a cost function for this fault tolerance scheme. The next figure shows the relationship between cost and reliability for this example.		So, the reliability has increased by a value of 21% and the cost has increased by one dollar. In a similar fashion, we can continue to add more units in parallel, thus increasing the reliability and the cost. We now have an array of reliability values and the associated costs that we can use to develop a cost function for this fault tolerance scheme. The next figure shows the relationship between cost and reliability for this example.

	[[Image:6.15.png\|center\|~~400px~~\|Cost function for redundant parallel units.]]		[[Image:6.15.png\|center\|700px\|Cost function for redundant parallel units.\|link=]]

	As can be seen, this looks quite similar to the general behavior cost model presented earlier. In fact, a standard regression analysis available in Weibull++ indicates that an exponential model fits this cost model quite well. The function is given by the following equation, where <math>C\,\!</math> is the cost in dollars and <math>R\,\!</math> is the fractional reliability value.		As can be seen, this looks quite similar to the general behavior cost model presented earlier. In fact, a standard regression analysis available in Weibull++ indicates that an exponential model fits this cost model quite well. The function is given by the following equation, where <math>C\,\!</math> is the cost in dollars and <math>R\,\!</math> is the fractional reliability value.

Richard House: /* Maximum Achievable Reliability */

2016-01-04T22:23:02Z

Maximum Achievable Reliability

← Older revision		Revision as of 22:23, 4 January 2016
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	As the component reliability, <math>{{R}_{i}}\,\!</math>, approaches the maximum achievable reliability, <math>{{R}_{i,max}}\,\!</math>, the cost function approaches infinity. The maximum achievable reliability acts as a scale parameter for the cost function. By decreasing <math>{{R}_{i,max}}\,\!</math>, the cost function is compressed between <math>{{R}_{i,min}}\,\!</math> and <math>{{R}_{i,max}}\,\!</math>, as shown in the figure below.		As the component reliability, <math>{{R}_{i}}\,\!</math>, approaches the maximum achievable reliability, <math>{{R}_{i,max}}\,\!</math>, the cost function approaches infinity. The maximum achievable reliability acts as a scale parameter for the cost function. By decreasing <math>{{R}_{i,max}}\,\!</math>, the cost function is compressed between <math>{{R}_{i,min}}\,\!</math> and <math>{{R}_{i,max}}\,\!</math>, as shown in the figure below.


	[[Image:6_12.png\|center\|700px\|Effect of the maximum achievable reliability on the cost function.\|link=]]		[[Image:6_12.png\|center\|700px\|Effect of the maximum achievable reliability on the cost function.\|link=]]

Richard House: /* Maximum Achievable Reliability */

2016-01-04T22:22:45Z

Maximum Achievable Reliability

← Older revision		Revision as of 22:22, 4 January 2016
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	[[Image:6_12.png\|center\|~~400px~~\|Effect of the maximum achievable reliability on the cost function.]]		[[Image:6_12.png\|center\|700px\|Effect of the maximum achievable reliability on the cost function.\|link=]]

	===Cost Function===		===Cost Function===