Lloyd-Lipow Least Squares Example - Revision history

Lisa Hacker at 21:30, 18 September 2023

2023-09-18T21:30:30Z

← Older revision		Revision as of 21:30, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Lloyd-Lipow\|~~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>

Chris Kahn at 17:41, 12 June 2014

2014-06-12T17:41:35Z

← Older revision		Revision as of 17:41, 12 June 2014
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	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|+'''The Test Results and Reliabilities of Each Stage Calculated ~~From~~ Raw Data and the Predicted Reliability'''		\|+'''The Test Results and Reliabilities of Each Stage Calculated from Raw Data and the Predicted Reliability'''
	!Test Stage Number(<math>k\,\!</math>)		!Test Stage Number(<math>k\,\!</math>)
	!Number of Tests in Stage(<math>n_k\,\!</math>)		!Number of Tests in Stage(<math>n_k\,\!</math>)

David J. Groebel at 19:46, 29 May 2014

2014-05-29T19:46:42Z

← Older revision		Revision as of 19:46, 29 May 2014
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	:<math>{{R}_{k}}=0.8104-\frac{0.2201}{k}\,\!</math>		:<math>{{R}_{k}}=0.8104-\frac{0.2201}{k}\,\!</math>
			</li>
			<li>The reliabilities from the raw data and the reliabilities predicted from the Lloyd-Lipow reliability growth model are given in the last two columns of the table. The figure below shows the plot. Based on the given data, the model cannot do much more than to basically fit a line through the middle of the points.

	~~The reliabilities from the raw data and the reliabilities predicted from the Lloyd-Lipow reliability growth model are given in the last two columns of the table~~. ~~The figure below shows the plot. Based on the given data, the model cannot do much more than to basically fit a line through the middle of the points~~.		[[Image:rga6.1.png\|center\|450px]]

	~~[[Image:rga6.1.png\|center\|450px]]~~		</li>
			</ol>

David J. Groebel at 19:45, 29 May 2014

2014-05-29T19:45:39Z

← Older revision		Revision as of 19:45, 29 May 2014
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Lloyd-Lipow\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Lloyd-Lipow\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

	After a 20-stage reliability development test program, 20 groups of success/failure data were obtained and are given in the table below. Do the following:		After a 20-stage reliability development test program, 20 groups of success/failure data were obtained and are given in the table below. Do the following:

	~~:1)~~ Fit the Lloyd-Lipow model to the data using least squares.		#Fit the Lloyd-Lipow model to the data using least squares.
	~~:2)~~ Plot the reliabilities predicted by the Lloyd-Lipow model along with the observed reliabilities, and compare the results.		#Plot the reliabilities predicted by the Lloyd-Lipow model along with the observed reliabilities, and compare the results.


	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
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	\|20\|\| 8\|\| 7\|\| 0.875\|\| 0.7998		\|20\|\| 8\|\| 7\|\| 0.875\|\| 0.7998
	\|}		\|}


	'''Solution'''		'''Solution'''
			<ol>
			<li>The least squares estimates are:

	~~The least squares estimates are:~~		:<math>\begin{align}

	::<math>\begin{align}
	\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{1}{k}= & \underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{1}{k}=3.5977 \\		\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{1}{k}= & \underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{1}{k}=3.5977 \\
	\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{1}{{{k}^{2}}}= & \underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{1}{{{k}^{2}}}=1.5962 \\		\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{1}{{{k}^{2}}}= & \underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{1}{{{k}^{2}}}=1.5962 \\
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	:and:		and:

	~~::<math>\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{{{S}_{k}}}{k\cdot {{n}_{k}}}=\underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{{{S}_{k}}}{k\cdot {{n}_{k}}}=2.5632\,\!</math>~~


			:<math>\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{{{S}_{k}}}{k\cdot {{n}_{k}}}=\underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{{{S}_{k}}}{k\cdot {{n}_{k}}}=2.5632\,\!</math>

	Using these estimates to obtain <math>\hat{R_{\infty}}\,\!</math> and <math>\hat{\alpha}\,\!</math>yields:		Using these estimates to obtain <math>\hat{R_{\infty}}\,\!</math> and <math>\hat{\alpha}\,\!</math>yields:

			:<math>\begin{align}
	::<math>\begin{align}
	\text{ }{{\hat{R}}_{\infty }}= &\frac{\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,{{R}_{k}}-\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{{{R}_{k}}}{k}}{N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}-{{\left( \underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k} \right)}^{2}}} \\		\text{ }{{\hat{R}}_{\infty }}= &\frac{\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,{{R}_{k}}-\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{{{R}_{k}}}{k}}{N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}-{{\left( \underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k} \right)}^{2}}} \\

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	\end{align}\,\!</math>		\end{align}\,\!</math>

	:and:		and:

	::<math>\begin{align}		:<math>\begin{align}
	\hat{\alpha }= &\frac{\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,{{R}_{k}}-N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{{{R}_{k}}}{k}}{N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}-{{\left( \underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k} \right)}^{2}}}\\		\hat{\alpha }= &\frac{\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,{{R}_{k}}-N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{{{R}_{k}}}{k}}{N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}-{{\left( \underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k} \right)}^{2}}}\\
	= & \frac{(3.5977)(15.413)-(20)(2.5637)}{(20)(1.5962)-{{(3.5977)}^{2}}} \\		= & \frac{(3.5977)(15.413)-(20)(2.5637)}{(20)(1.5962)-{{(3.5977)}^{2}}} \\
	= & 0.2207		= & 0.2207
	\end{align}\,\!</math>		\end{align}\,\!</math>


	Therefore, the Lloyd-Lipow reliability growth model is as follows, where <math>k\,\!</math> is the test stage.		Therefore, the Lloyd-Lipow reliability growth model is as follows, where <math>k\,\!</math> is the test stage.

	::<math>{{R}_{k}}=0.8104-\frac{0.2201}{k}\,\!</math>		:<math>{{R}_{k}}=0.8104-\frac{0.2201}{k}\,\!</math>


	The reliabilities from the raw data and the reliabilities predicted from the Lloyd-Lipow reliability growth model are given in the last two columns of the table. The figure below shows the plot. Based on the given data, the model cannot do much more than to basically fit a line through the middle of the points.		The reliabilities from the raw data and the reliabilities predicted from the Lloyd-Lipow reliability growth model are given in the last two columns of the table. The figure below shows the plot. Based on the given data, the model cannot do much more than to basically fit a line through the middle of the points.

	[[Image:rga6.1.png\|center\|~~400px\|Comparison of the predicted reliability and the raw data.~~]]		[[Image:rga6.1.png\|center\|450px]]

Kate Racaza: Created page with '{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. After a 20-stag…'

2014-01-21T18:53:08Z

Created page with '<noinclude>{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. </noinclude> After a 20-stag…'

New page

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

After a 20-stage reliability development test program, 20 groups of success/failure data were obtained and are given in the table below. Do the following:

:1) Fit the Lloyd-Lipow model to the data using least squares.
:2) Plot the reliabilities predicted by the Lloyd-Lipow model along with the observed reliabilities, and compare the results.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+'''The Test Results and Reliabilities of Each Stage Calculated From Raw Data and the Predicted Reliability'''
!Test Stage Number(<math>k\,\!</math>)
!Number of Tests in Stage(<math>n_k\,\!</math>)
!Number of Successful Tests(<math>S_k\,\!</math>)
!Raw Data Reliability
!Lloyd-Lipow Reliability
|-
|1|| 9|| 6|| 0.667|| 0.7002
|-
|2|| 9|| 5|| 0.556|| 0.7369
|-
|3|| 8|| 7|| 0.875|| 0.7552
|-
|4|| 10|| 6|| 0.600|| 0.7662
|-
|5|| 9|| 7|| 0.778|| 0.7736
|-
|
|-
|6|| 10|| 8|| 0.800|| 0.7788
|-
|7|| 10|| 7|| 0.700|| 0.7827
|-
|8|| 10|| 6|| 0.600|| 0.7858
|-
|9|| 11|| 7|| 0.636|| 0.7882
|-
|10|| 11|| 9|| 0.818|| 0.7902
|-
|
|-
|11|| 9|| 9|| 1.000|| 0.7919
|-
|12|| 12|| 10|| 0.833|| 0.7933
|-
|13|| 12|| 9|| 0.750|| 0.7945
|-
|14|| 11|| 8|| 0.727|| 0.7956
|-
|15|| 10|| 7|| 0.700|| 0.7965
|-
|
|-
|16|| 10|| 8|| 0.800|| 0.7973
|-
|17|| 11|| 10|| 0.909|| 0.7980
|-
|18|| 10|| 9|| 0.900|| 0.7987
|-
|19|| 9|| 8|| 0.889|| 0.7992
|-
|20|| 8|| 7|| 0.875|| 0.7998
|}

'''Solution'''

The least squares estimates are:

::<math>\begin{align}
\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{1}{k}= & \underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{1}{k}=3.5977 \\
\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{1}{{{k}^{2}}}= & \underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{1}{{{k}^{2}}}=1.5962 \\
\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{{{S}_{k}}}{{{n}_{k}}}= & \underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{{{S}_{k}}}{{{n}_{k}}}=15.4131
\end{align}\,\!</math>

:and:

::<math>\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{{{S}_{k}}}{k\cdot {{n}_{k}}}=\underset{k=1}{\overset{20}{\mathop \sum }}\,\frac{{{S}_{k}}}{k\cdot {{n}_{k}}}=2.5632\,\!</math>

Using these estimates to obtain <math>\hat{R_{\infty}}\,\!</math> and <math>\hat{\alpha}\,\!</math>yields:

::<math>\begin{align}
\text{ }{{\hat{R}}_{\infty }}= &\frac{\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,{{R}_{k}}-\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{{{R}_{k}}}{k}}{N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}-{{\left( \underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k} \right)}^{2}}} \\

= & \frac{(1.5962)(15.413)-(3.5977)(2.5637)}{(20)(1.5962)-{{(3.5977)}^{2}}} \\
= & 0.8104
\end{align}\,\!</math>

:and:

::<math>\begin{align}
\hat{\alpha }= &\frac{\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,{{R}_{k}}-N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{{{R}_{k}}}{k}}{N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}-{{\left( \underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k} \right)}^{2}}}\\
= & \frac{(3.5977)(15.413)-(20)(2.5637)}{(20)(1.5962)-{{(3.5977)}^{2}}} \\
= & 0.2207
\end{align}\,\!</math>

Therefore, the Lloyd-Lipow reliability growth model is as follows, where <math>k\,\!</math> is the test stage.

::<math>{{R}_{k}}=0.8104-\frac{0.2201}{k}\,\!</math>

The reliabilities from the raw data and the reliabilities predicted from the Lloyd-Lipow reliability growth model are given in the last two columns of the table. The figure below shows the plot. Based on the given data, the model cannot do much more than to basically fit a line through the middle of the points.

[[Image:rga6.1.png|center|400px|Comparison of the predicted reliability and the raw data.]]