Crow-AMSAA Model - Helicopter System Example - Revision history

Lisa Hacker at 21:28, 18 September 2023

2023-09-18T21:28:42Z

← Older revision		Revision as of 21:28, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Crow-AMSAA (NHPP)\|~~Reliability ~~Growth and Repairable System Analysis Reference]~~]''.		''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

David J. Groebel at 18:34, 30 May 2014

2014-05-30T18:34:05Z

← Older revision		Revision as of 18:34, 30 May 2014
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	The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.10.png\|center\|~~400px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.~~]]		[[Image:rga5.10.png\|center\|450px]]

	[[Image:rga5.11.png\|center\|~~400px\|Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds.~~]]		[[Image:rga5.11.png\|center\|450px]]

	The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:		The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:
Line 117:		Line 117:
	The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.12.png\|center\|~~400px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.~~]]		[[Image:rga5.12.png\|center\|450px]]

	[[Image:rga5.13.png\|center\|~~00px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.~~]]		[[Image:rga5.13.png\|center\|450px]]
	</li>		</li>
	</ol>		</ol>

David J. Groebel at 00:12, 23 April 2014

2014-04-23T00:12:31Z

← Older revision		Revision as of 00:12, 23 April 2014
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

Line 93:		Line 93:
	The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.10.png\|center\|~~500px~~\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.10.png\|center\|400px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]

			[[Image:rga5.11.png\|center\|400px\|Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]
	[[Image:rga5.11.png\|center\|~~500px~~\|Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]

	The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:		The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:
Line 118:		Line 117:
	The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.12.png\|center\|~~500px~~\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.12.png\|center\|400px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]


	[[Image:rga5.13.png\|center\|~~500px~~\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.13.png\|center\|00px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]
	</li>		</li>
	</ol>		</ol>

Chris Kahn at 23:19, 31 January 2014

2014-01-31T23:19:57Z

← Older revision		Revision as of 23:19, 31 January 2014
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	Do the following:		Do the following:

	1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.		#Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
			#Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.
	2) Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.

	'''Solution'''		'''Solution'''
			<ol>
	1) Using RGA, the value of <math>\hat{\beta }\,\!</math>, must be solved numerically. Once <math>\hat{\beta }\,\!</math> has been estimated then the value of <math>\hat{\lambda }\,\!</math> can be determined. The parameter values are displayed below:		<li>Using RGA, the value of <math>\hat{\beta }\,\!</math>, must be solved numerically. Once <math>\hat{\beta }\,\!</math> has been estimated then the value of <math>\hat{\lambda }\,\!</math> can be determined. The parameter values are displayed below:

	:<math>\hat{\beta }= 0.81361\,\!</math>		:<math>\hat{\beta }= 0.81361\,\!</math>
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	= & 0.53697		= & 0.53697
	\end{align}\,\!</math>		\end{align}\,\!</math>
			</li>
	2) The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for <math>T=500\,\!</math> hour are:		<li>The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for <math>T=500\,\!</math> hour are:

	:<math>\begin{align}		:<math>\begin{align}
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	[[Image:rga5.13.png\|center\|500px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.13.png\|center\|500px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]
			</li>
			</ol>

Kate Racaza: Created page with '{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. A new hel…'

2014-01-31T17:35:56Z

Created page with '<noinclude>{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. </noinclude> A new hel…'

New page

<noinclude>{{Banner RGA Examples}}
''This example appears in the [[Crow-AMSAA (NHPP)|Reliability Growth and Repairable System Analysis Reference book]]''.
</noinclude>

A new helicopter system is under development. System failure data has been collected on five helicopters during the final test phase. The actual failure times cannot be determined since the failures are not discovered until after the helicopters are brought into the maintenance area. However, total flying hours are known when the helicopters are brought in for service, and every 2 weeks each helicopter undergoes a thorough inspection to uncover any failures that may have occurred since the last inspection. Therefore, the cumulative total number of flight hours and the cumulative total number of failures for the 5 helicopters are known for each 2-week period. The total number of flight hours from the test phase is 500, which was accrued over a period of 12 weeks (six 2-week intervals). For each 2-week interval, the total number of flight hours and total number of failures for the 5 helicopters were recorded. The grouped data set is displayed in the following table.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+'''Grouped Data for a New Helicopter System'''
!Interval
!Interval Length
!Failures in Interval
|-
|1|| 0 - 62|| 12
|-
|2|| 62 -100|| 6
|-
|3|| 100 - 187|| 15
|-
|4|| 187 - 210|| 3
|-
|5|| 210 - 350|| 18
|-
|6|| 350 - 500|| 16
|}

Do the following:

1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.

2) Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.

'''Solution'''

1) Using RGA, the value of <math>\hat{\beta }\,\!</math>, must be solved numerically. Once <math>\hat{\beta }\,\!</math> has been estimated then the value of <math>\hat{\lambda }\,\!</math> can be determined. The parameter values are displayed below:

:<math>\hat{\beta }= 0.81361\,\!</math>

:<math>\hat{\lambda }= 0.44585\,\!</math>

The [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|grouped Fisher Matrix confidence bounds]] can be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level by:

:<math>\begin{align}
{{\beta }_{L}} = & \hat{\beta }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}} \\
= & 0.6546 \\
{{\beta }_{U}} = & \hat{\beta }{{e}^{-{{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}} \\
= & 1.0112
\end{align}\,\!</math>

and:

:<math>\begin{align}
{{\lambda }_{L}} = & \hat{\lambda }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} \\
= & 0.14594 \\
{{\lambda }_{U}} = & \hat{\lambda }{{e}^{-{{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} \\
= & 1.36207
\end{align}\,\!</math>

Crow confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level, as:

:<math>\begin{align}
{{\beta }_{L}} = & \hat{\beta }(1-S) \\
= & 0.63552 \\
{{\beta }_{U}} = & \hat{\beta }(1+S) \\
= & 0.99170
\end{align}\,\!</math>

and:

:<math>\begin{align}
{{\lambda }_{L}} = & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\
= & 0.36197 \\
{{\lambda }_{U}} = & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot T_{k}^{\beta }} \\
= & 0.53697
\end{align}\,\!</math>

2) The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for <math>T=500\,\!</math> hour are:

:<math>\begin{align}
{{[{{m}_{c}}(T)]}_{L}} = & {{{\hat{m}}}_{c}}(t){{e}^{{{z}_{\alpha /2}}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}} \\
= & 5.8680 \\
{{[{{m}_{c}}(T)]}_{U}} = & {{{\hat{m}}}_{c}}(t){{e}^{-{{z}_{\alpha /2}}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}} \\
= & 8.6947
\end{align}\,\!</math>

and:

:<math>\begin{align}
{{[MTB{{F}_{i}}]}_{L}} = & {{{\hat{m}}}_{i}}(t){{e}^{{{z}_{\alpha /2}}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}} \\
= & 6.6483 \\
{{[MTB{{F}_{i}}]}_{U}} = & {{{\hat{m}}}_{i}}(t){{e}^{-{{z}_{\alpha /2}}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}} \\
= & 11.5932
\end{align}\,\!</math>

The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.

[[Image:rga5.10.png|center|500px|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]

[[Image:rga5.11.png|center|500px|Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]

The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:

:<math>\begin{align}
{{[{{m}_{c}}(T)]}_{L}} = & \frac{1}{C{{(t)}_{U}}} \\
= & 5.85449 \\
{{[{{m}_{c}}(T)]}_{U}} = & \frac{1}{C{{(t)}_{L}}} \\
= & 8.79822
\end{align}\,\!</math>

and:

:<math>\begin{align}
{{[MTB{{F}_{i}}]}_{L}} = & {{\hat{m}}_{i}}(1-W) \\
= & 6.19623 \\
{{[MTB{{F}_{i}}]}_{U}} = & {{\hat{m}}_{i}}(1+W) \\
= & 11.36223
\end{align}\,\!</math>

The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

[[Image:rga5.12.png|center|500px|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]

[[Image:rga5.13.png|center|500px|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]