Reliability Data - Logistic Model - Revision history

Lisa Hacker at 21:30, 18 September 2023

2023-09-18T21:30:39Z

← 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 [~~[Logistic\|~~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 18:24, 29 May 2014

2014-05-29T18:24:19Z

← Older revision		Revision as of 18:24, 29 May 2014
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Logistic\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Logistic\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

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	<li>The figure below shows the entered data and the estimated parameters using the standard Gompertz model.		<li>The figure below shows the entered data and the estimated parameters using the standard Gompertz model.

	[[Image:rga8.1.png\|center\|~~400px~~\|Estimated Standard Gompertz parameters for Example 1.]]		[[Image:rga8.1.png\|center\|650px\|Estimated Standard Gompertz parameters for Example 1.]]

	:Therefore:		Therefore:

	::<math>\begin{align}		:<math>\begin{align}
	& \widehat{a}= & 0.9999 \\		& \widehat{a}= & 0.9999 \\
	& \widehat{b}= & 0.2485 \\		& \widehat{b}= & 0.2485 \\
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	\end{align}\,\!</math>		\end{align}\,\!</math>

			:<math>R=(0.9999){{(0.2485)}^{{{0.6858}^{T}}}}\,\!</math>

	~~::<math>R=(0.9999){{(0.2485)}^{{{0.6858}^{T}}}}\,\!</math>~~		The values of the predicted reliabilities are plotted in the figure below.


	:The values of the predicted reliabilities are plotted in the figure below.

	[[Image:rga8.2.png\|center\|400px\|Gompertz Reliability vs. Time plot.]]		[[Image:rga8.2.png\|center\|400px\|Gompertz Reliability vs. Time plot.]]

			Notice how the standard Gompertz model is not really capable of handling the S-shaped characteristics of this data.
	:Notice how the standard Gompertz model is not really capable of handling the S-shaped characteristics of this data.

	</li><li>The least squares estimators of the Logistic growth curve parameters are given by Crow [[RGA_References\|[9]]]:		</li><li>The least squares estimators of the Logistic growth curve parameters are given by Crow [[RGA_References\|[9]]]:

	:where:		where:

	::<math>\begin{align}		:<math>\begin{align}
	{{{\hat{b}}}_{1}}= & \frac{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,{{T}_{i}}{{Y}_{i}}-N\cdot \bar{T}\cdot \bar{Y}}{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,T_{i}^{2}-N\cdot {{{\bar{T}}}^{2}}} \\ \\		{{{\hat{b}}}_{1}}= & \frac{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,{{T}_{i}}{{Y}_{i}}-N\cdot \bar{T}\cdot \bar{Y}}{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,T_{i}^{2}-N\cdot {{{\bar{T}}}^{2}}} \\ \\
	{{{\hat{b}}}_{0}}= & \bar{Y}-{{{\hat{b}}}_{1}}\bar{T} \\ \\		{{{\hat{b}}}_{0}}= & \bar{Y}-{{{\hat{b}}}_{1}}\bar{T} \\ \\
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	\bar{Y}= & \frac{1}{N}\underset{i=0}{\overset{N-1}{\mathop \sum }}\,{{Y}_{i}}		\bar{Y}= & \frac{1}{N}\underset{i=0}{\overset{N-1}{\mathop \sum }}\,{{Y}_{i}}
	\end{align}\,\!</math>		\end{align}\,\!</math>


	In this example <math>N=9\,\!</math>, which gives:		In this example <math>N=9\,\!</math>, which gives:

	::<math>\begin{align}		:<math>\begin{align}
	\overline{Y}&=\frac{1}{9}\sum_{i=0}^{8}ln\left (\frac{1}{R_{i}}-1 \right ) \\		\overline{Y}&=\frac{1}{9}\sum_{i=0}^{8}ln\left (\frac{1}{R_{i}}-1 \right ) \\
	&= -1.7355 \\ \\		&= -1.7355 \\ \\
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	\sum_{i=0}^{8}T_{i}Y_{i} &= -106.8630		\sum_{i=0}^{8}T_{i}Y_{i} &= -106.8630
	\end{align}\,\!</math>		\end{align}\,\!</math>


	From the equations for <math>b_{i}\,\!</math> and <math>\hat{b_{0}}\,\!</math>:		From the equations for <math>b_{i}\,\!</math> and <math>\hat{b_{0}}\,\!</math>:

	::<math>\begin{align}		:<math>\begin{align}
	\hat{b_{1}} &= \frac{-106.8630 - 9(4)(-1.7355)}{204-9(4)_{2}} \\		\hat{b_{1}} &= \frac{-106.8630 - 9(4)(-1.7355)}{204-9(4)_{2}} \\
	& = 0.7398 \\		& = 0.7398 \\
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	&= 1.2235		&= 1.2235
	\end{align}\,\!</math>		\end{align}\,\!</math>


	And from the least squares estimators for <math>\hat{b}\,\!</math> and <math>\hat{k}\,\!</math>:		And from the least squares estimators for <math>\hat{b}\,\!</math> and <math>\hat{k}\,\!</math>:

	::<math>\begin{align}		:<math>\begin{align}
	\widehat{b}= & {{e}^{1.2235}} \\		\widehat{b}= & {{e}^{1.2235}} \\
	= & 3.3991 \\		= & 3.3991 \\
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	= & 0.7398		= & 0.7398
	\end{align}\,\!</math>		\end{align}\,\!</math>


	Therefore, the Logistic reliability growth curve that represents this data set is given by:		Therefore, the Logistic reliability growth curve that represents this data set is given by:

	::<math>R=\frac{1}{1+3.3991\,{{e}^{-0.7398\,T}}}\,\!</math>		:<math>R=\frac{1}{1+3.3991\,{{e}^{-0.7398\,T}}}\,\!</math>


	The following figure shows the Reliability vs. Time plot. The plot shows that the observed data set is estimated well by the Logistic reliability growth curve, except in the region closely surrounding the inflection point of the observed reliability. This problem can be overcome by using the [[Gompertz_Models#The_Modified_Gompertz_Model\|modified Gompertz model]].		The following figure shows the Reliability vs. Time plot. The plot shows that the observed data set is estimated well by the Logistic reliability growth curve, except in the region closely surrounding the inflection point of the observed reliability. This problem can be overcome by using the [[Gompertz_Models#The_Modified_Gompertz_Model\|modified Gompertz model]].

	[[Image:rga8.3.png\|center\|400px\|Logistic Reliability vs. Time plot.]]</li></ol>		[[Image:rga8.3.png\|center\|400px\|Logistic Reliability vs. Time plot.]]</li></ol>

Chris Kahn at 18:35, 14 May 2014

2014-05-14T18:35:12Z

← Older revision		Revision as of 18:35, 14 May 2014
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	#Find a Gompertz curve that represents the data and plot it with the raw data.		#Find a Gompertz curve that represents the data and plot it with the raw data.

	#Find a Logistic reliability growth curve that represents the data and plot it with the raw data.		#Find a Logistic reliability growth curve that represents the data and plot it with the raw data.

Chris Kahn at 18:34, 14 May 2014

2014-05-14T18:34:43Z

← Older revision		Revision as of 18:34, 14 May 2014
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	Using the reliability growth data given in the table below, do the following:		Using the reliability growth data given in the table below, do the following:

	~~:1)~~ Find a Gompertz curve that represents the data and plot it with the raw data.		#Find a Gompertz curve that represents the data and plot it with the raw data.

	~~:2)~~ Find a Logistic reliability growth curve that represents the data and plot it with the raw data.		#Find a Logistic reliability growth curve that represents the data and plot it with the raw data.

	{\|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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	'''Solution'''		'''Solution'''
			<ol>
	~~:1)~~ The figure below shows the entered data and the estimated parameters using the standard Gompertz model.		<li>The figure below shows the entered data and the estimated parameters using the standard Gompertz model.

	[[Image:rga8.1.png\|center\|400px\|Estimated Standard Gompertz parameters for Example 1.]]		[[Image:rga8.1.png\|center\|400px\|Estimated Standard Gompertz parameters for Example 1.]]
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	:Notice how the standard Gompertz model is not really capable of handling the S-shaped characteristics of this data.		:Notice how the standard Gompertz model is not really capable of handling the S-shaped characteristics of this data.

	~~:2.~~ The least squares estimators of the Logistic growth curve parameters are given by Crow [[RGA_References\|[9]]]:		</li><li>The least squares estimators of the Logistic growth curve parameters are given by Crow [[RGA_References\|[9]]]:

	:where:		:where:
Line 110:		Line 110:
	The following figure shows the Reliability vs. Time plot. The plot shows that the observed data set is estimated well by the Logistic reliability growth curve, except in the region closely surrounding the inflection point of the observed reliability. This problem can be overcome by using the [[Gompertz_Models#The_Modified_Gompertz_Model\|modified Gompertz model]].		The following figure shows the Reliability vs. Time plot. The plot shows that the observed data set is estimated well by the Logistic reliability growth curve, except in the region closely surrounding the inflection point of the observed reliability. This problem can be overcome by using the [[Gompertz_Models#The_Modified_Gompertz_Model\|modified Gompertz model]].

	[[Image:rga8.3.png\|center\|400px\|Logistic Reliability vs. Time plot.]]		[[Image:rga8.3.png\|center\|400px\|Logistic Reliability vs. Time plot.]]</li></ol>

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

2014-01-21T20:47:27Z

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

New page

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

Using the reliability growth data given in the table below, do the following:

:1) Find a Gompertz curve that represents the data and plot it with the raw data.

:2) Find a Logistic reliability growth curve that represents the data and plot it with the raw data.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+'''Development Time vs. Observed Reliability data and Predicted Reliabilities'''
!Time, months
!Raw Data Reliability (%)
!Gompertz Reliability (%)
!Logistic Reliablity (%)
|-
|0|| 31.00|| 24.85|| 22.73
|-
|1|| 35.50|| 38.48|| 38.14
|-
|2|| 49.30|| 51.95|| 56.37
|-
|3|| 70.10|| 63.82|| 73.02
|-
|4|| 83.00|| 73.49|| 85.01
|-
|5|| 92.20|| 80.95|| 92.24
|-
|6|| 96.40|| 86.51|| 96.14
|-
|7|| 98.60|| 90.54|| 98.12
|-
|8|| 99.00|| 93.41|| 99.09
|}

'''Solution'''

:1) The figure below shows the entered data and the estimated parameters using the standard Gompertz model.

[[Image:rga8.1.png|center|400px|Estimated Standard Gompertz parameters for Example 1.]]

:Therefore:

::<math>\begin{align}
& \widehat{a}= & 0.9999 \\
& \widehat{b}= & 0.2485 \\
& \widehat{c}= & 0.6858
\end{align}\,\!</math>

::<math>R=(0.9999){{(0.2485)}^{{{0.6858}^{T}}}}\,\!</math>

:The values of the predicted reliabilities are plotted in the figure below.

[[Image:rga8.2.png|center|400px|Gompertz Reliability vs. Time plot.]]

:Notice how the standard Gompertz model is not really capable of handling the S-shaped characteristics of this data.

:2. The least squares estimators of the Logistic growth curve parameters are given by Crow [[RGA_References|[9]]]:

:where:

::<math>\begin{align}
{{{\hat{b}}}_{1}}= & \frac{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,{{T}_{i}}{{Y}_{i}}-N\cdot \bar{T}\cdot \bar{Y}}{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,T_{i}^{2}-N\cdot {{{\bar{T}}}^{2}}} \\ \\
{{{\hat{b}}}_{0}}= & \bar{Y}-{{{\hat{b}}}_{1}}\bar{T} \\ \\
{{Y}_{i}}= & \ln \left( \frac{1}{{{R}_{i}}}-1 \right) \\ \\
\bar{Y}= & \frac{1}{N}\underset{i=0}{\overset{N-1}{\mathop \sum }}\,{{Y}_{i}}
\end{align}\,\!</math>

In this example <math>N=9\,\!</math>, which gives:

::<math>\begin{align}
\overline{Y}&=\frac{1}{9}\sum_{i=0}^{8}ln\left (\frac{1}{R_{i}}-1 \right ) \\
&= -1.7355 \\ \\
\overline{T}&=\frac{1}{9}\sum_{i=0}^{8}T_{i} = 4 \\
\sum_{i=0}^{8}T_{i}^{2} &= 204 \\
\sum_{i=0}^{8}T_{i}Y_{i} &= -106.8630
\end{align}\,\!</math>

From the equations for <math>b_{i}\,\!</math> and <math>\hat{b_{0}}\,\!</math>:

::<math>\begin{align}
\hat{b_{1}} &= \frac{-106.8630 - 9(4)(-1.7355)}{204-9(4)_{2}} \\
& = 0.7398 \\
\hat{b_{0}} &= -1.7355 - (-0.7398)(4)\\
&= 1.2235
\end{align}\,\!</math>

And from the least squares estimators for <math>\hat{b}\,\!</math> and <math>\hat{k}\,\!</math>:

::<math>\begin{align}
\widehat{b}= & {{e}^{1.2235}} \\
= & 3.3991 \\
\widehat{k}= & -(-0.7398) \\
= & 0.7398
\end{align}\,\!</math>

Therefore, the Logistic reliability growth curve that represents this data set is given by:

::<math>R=\frac{1}{1+3.3991\,{{e}^{-0.7398\,T}}}\,\!</math>

The following figure shows the Reliability vs. Time plot. The plot shows that the observed data set is estimated well by the Logistic reliability growth curve, except in the region closely surrounding the inflection point of the observed reliability. This problem can be overcome by using the [[Gompertz_Models#The_Modified_Gompertz_Model|modified Gompertz model]].

[[Image:rga8.3.png|center|400px|Logistic Reliability vs. Time plot.]]