Warranty Data Analysis - Revision history

Miklos Szidarovszky: /* Example */

2017-08-16T15:37:46Z

Example

← Older revision		Revision as of 15:37, 16 August 2017
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	\end{matrix}\,\!</math></center>		\end{matrix}\,\!</math></center>

	Now, for this example, <math>n=6\,\!</math>, <math>{{\bar{e}}_{i,j}}=-0.~~5432~~\,\!</math> and <math>s=1.~~6890~~.\,\!</math>		Now, for this example, <math>n=6\,\!</math>, <math>{{\bar{e}}_{i,j}}=-0.0904\,\!</math> and <math>s=2.1366.\,\!</math>

	Thus the <math>z_{i,j}\,\!</math> values are:		Thus the <math>z_{i,j}\,\!</math> values are:

Sharon Honecker: /* Example */

2017-02-08T19:57:22Z

Example

← Older revision		Revision as of 19:57, 8 February 2017
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	\|}		\|}

	For example, for the month of September, the expected return number is given by:		For example, for the month of September, the expected return number from the June shipment is given by:

	::<math>{{\hat{F}}_{Jun,3}}=(100-6)\cdot \left( 1-\frac{R(3)}{R(2)} \right)=94\cdot 0.08239=7.7447\,\!</math>		::<math>{{\hat{F}}_{Jun,3}}=(100-6)\cdot \left( 1-\frac{R(3)}{R(2)} \right)=94\cdot 0.08239=7.7447\,\!</math>

Kate Racaza: /* Nevada Chart Format */

2015-12-01T18:12:00Z

Nevada Chart Format

← Older revision		Revision as of 18:12, 1 December 2015
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	These calculations can be performed automatically in Weibull++.		These calculations can be performed automatically in Weibull++.

			<div class="noprint">
	{{Examples Box\|Weibull++_Examples\|<p>More Nevada chart format warranty analysis examples are available! See also:</p>		{{Examples Box\|Weibull++_Examples\|<p>More Nevada chart format warranty analysis examples are available! See also:</p>
	{{Examples Both\|http://www.reliasoft.com/Weibull/examples/rc5/index.htm\|Warranty Analysis Example\|http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_5.html\|Watch the video...}}<nowiki/>		{{Examples Both\|http://www.reliasoft.com/Weibull/examples/rc5/index.htm\|Warranty Analysis Example\|http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_5.html\|Watch the video...}}<nowiki/>
	}}		}}
			</div>

	==Time-to-Failure Format==		==Time-to-Failure Format==

John Leavitt: /* Nevada Chart Format */

2015-01-27T18:22:06Z

Nevada Chart Format

← Older revision		Revision as of 18:22, 27 January 2015
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	==Nevada Chart Format==		==Nevada Chart Format==
	The Nevada format allows the user to convert shipping and warranty return data into the standard reliability data form of failures and suspensions so that it can easily be analyzed with traditional life data analysis methods. For each time period in which a number of products are shipped, there will be a certain number of returns or failures in subsequent time periods, while the rest of the population that was shipped will continue to operate in the following time periods. For example, if 500 units are shipped in May, and 10 of those units are warranty returns in June, that is equivalent to 10 failures at a time of one month. The other ~~450~~ units will go on to operate and possibly fail in the months that follow. This information can be arranged in a diagonal chart, as shown in the following figure.		The Nevada format allows the user to convert shipping and warranty return data into the standard reliability data form of failures and suspensions so that it can easily be analyzed with traditional life data analysis methods. For each time period in which a number of products are shipped, there will be a certain number of returns or failures in subsequent time periods, while the rest of the population that was shipped will continue to operate in the following time periods. For example, if 500 units are shipped in May, and 10 of those units are warranty returns in June, that is equivalent to 10 failures at a time of one month. The other 490 units will go on to operate and possibly fail in the months that follow. This information can be arranged in a diagonal chart, as shown in the following figure.

	[[Image:Nevada-Chart-Illustration.png\|center\|450px\| ]]		[[Image:Nevada-Chart-Illustration.png\|center\|450px\| ]]

Richard House at 21:56, 26 September 2012

2012-09-26T21:56:55Z

Show changes

Richard House: /* Usage Format */

2012-09-05T17:20:31Z

Usage Format

← Older revision		Revision as of 17:20, 5 September 2012
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	where:		where:

	::<math>Q()\,\!</math> is the usage distribution Cumulative Density Function, ~~<math>~~cdf~~</math>~~ .		::<math>Q()\,\!</math> is the usage distribution Cumulative Density Function, ''cdf''.

	::<math>x\,\!</math> represents the intervals used in apportioning the suspended population.		::<math>x\,\!</math> represents the intervals used in apportioning the suspended population.

Kate Racaza: /* Example */

2012-08-26T21:30:57Z

Example

← Older revision		Revision as of 21:30, 26 August 2012
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	A company keeps track of its shipments and warranty returns on a month-by-month basis. The following table records the shipments in June, July and August, and the warranty returns through September:		A company keeps track of its shipments and warranty returns on a month-by-month basis. The following table records the shipments in June, July and August, and the warranty returns through September:


	{\|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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	\|Aug. 2010\|\|150\|\|-\|\|-\|\|4		\|Aug. 2010\|\|150\|\|-\|\|-\|\|4
	\|}		\|}


	We will examine the data month by month. In June 100 units were sold, and in July 3 of these units were returned. This gives 3 failures at one month for the June shipment, which we will denote as <math>{{F}_{JUN,1}}=3\,\!</math> . Likewise, 3 failures occurred in August and 5 occurred in September for this shipment, or <math>{{F}_{JUN,2}}=3\,\!</math> and <math>{{F}_{JUN,3}}=5\,\!</math>. Consequently, at the end of our three-month analysis period, there were a total of 11 failures for the 100 units shipped in June. This means that 89 units are presumably still operating, and can be considered suspensions at three months, or <math>{{S}_{JUN,3}}=89\,\!</math>. For the shipment of 140 in July, 2 were returned the following month, or <math>{{F}_{JUL,1}}=2\,\!</math> , and 4 more were returned the month after that, or <math>{{F}_{JUL,2}}=4\,\!</math>. After two months, there are 134 ( <math>140-2-4=134\,\!</math> ) units from the July shipment still operating, or <math>{{S}_{JUL,2}}=134\,\!</math>. For the final shipment of 150 in August, 4 fail in September, or <math>{{F}_{AUG,1}}=4\,\!</math>, with the remaining 146 units being suspensions at one month, or <math>{{S}_{AUG,1}}=146\,\!</math>.		We will examine the data month by month. In June 100 units were sold, and in July 3 of these units were returned. This gives 3 failures at one month for the June shipment, which we will denote as <math>{{F}_{JUN,1}}=3\,\!</math> . Likewise, 3 failures occurred in August and 5 occurred in September for this shipment, or <math>{{F}_{JUN,2}}=3\,\!</math> and <math>{{F}_{JUN,3}}=5\,\!</math>. Consequently, at the end of our three-month analysis period, there were a total of 11 failures for the 100 units shipped in June. This means that 89 units are presumably still operating, and can be considered suspensions at three months, or <math>{{S}_{JUN,3}}=89\,\!</math>. For the shipment of 140 in July, 2 were returned the following month, or <math>{{F}_{JUL,1}}=2\,\!</math> , and 4 more were returned the month after that, or <math>{{F}_{JUL,2}}=4\,\!</math>. After two months, there are 134 ( <math>140-2-4=134\,\!</math> ) units from the July shipment still operating, or <math>{{S}_{JUL,2}}=134\,\!</math>. For the final shipment of 150 in August, 4 fail in September, or <math>{{F}_{AUG,1}}=4\,\!</math>, with the remaining 146 units being suspensions at one month, or <math>{{S}_{AUG,1}}=146\,\!</math>.

	It is now a simple matter to add up the number of failures for 1, 2, and 3 months, then add the suspensions to get our reliability data set:		It is now a simple matter to add up the number of failures for 1, 2, and 3 months, then add the suspensions to get our reliability data set:


	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	\end{matrix}</math></center>		\end{matrix}</math></center>

	These calculations ~~are~~ automatically ~~performed~~ in Weibull++. ~~View the~~ Weibull++ ~~solution in '''[~~http://www.reliasoft.com/Weibull/examples/rc5/index.htm ~~HTML]''' or '''[~~http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_5.html ~~Video]'''~~
			These calculations can be performed automatically in Weibull++.


			{{Examples Box\|Weibull++_Examples\|<p>More Nevada chart format warranty analysis examples are available! See also:</p>
			{{Examples Both\|http://www.reliasoft.com/Weibull/examples/rc5/index.htm\|Warranty Analysis Example\|http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_5.html\|Watch the video...}}<nowiki/>
			}}

	==Time-to-Failure Format==		==Time-to-Failure Format==

Richard House: /* Example */

2012-08-21T02:37:27Z

Example

← Older revision		Revision as of 02:37, 21 August 2012
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	::<math>{{e}_{Jun,3}}={{\hat{F}}_{Jun,3}}-{{F}_{Jun,3}}=7.7447-5=2.7447.</math>		::<math>{{e}_{Jun,3}}={{\hat{F}}_{Jun,3}}-{{F}_{Jun,3}}=7.7447-5=2.7447.</math>

	This can then be repeated for each cell, yielding the following table for <math>{{e}_{i,j}}</math> :		This can then be repeated for each cell, yielding the following table for <math>{{e}_{i,j}}\,\!</math> :

	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	\end{matrix}</math></center>		\end{matrix}</math></center>

	Now, for this example, <math>n=6</math> , <math>{{\bar{e}}_{i,j}}=-0.5432</math> and <math>s=1.6890.</math>		Now, for this example, <math>n=6\,\!</math> , <math>{{\bar{e}}_{i,j}}=-0.5432\,\!</math> and <math>s=1.6890.\,\!</math>

	Thus the <math>z_{i,j}</math> values are:		Thus the <math>z_{i,j}\,\!</math> values are:

	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	\end{matrix}</math></center>		\end{matrix}</math></center>

	The <math>z_{i,j}^{2}</math> values, for each cell, are given in the following table.		The <math>z_{i,j}^{2}\,\!</math> values, for each cell, are given in the following table.

	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	\end{matrix}</math></center>		\end{matrix}</math></center>

	If the critical value is set at <math>\alpha =</math> ~~0.01~~ and the caution value is set at <math>\alpha =</math> ~~0.1~~, then the critical and caution <math>{{\chi }^{2}}</math> values will be:		If the critical value is set at <math>\alpha = 0.01\,\!</math> and the caution value is set at <math>\alpha = 0.1\,\!</math>, then the critical and caution <math>{{\chi }^{2}}\,\!</math> values will be:

	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	\end{matrix}</math></center>		\end{matrix}</math></center>

	If we consider the sales periods as the basis for outlier detection, then after comparing the above table to the sum of <math>z_{i,j}^{2}</math> <math>(\chi _{1}^{2})</math> values for each sales period, we find that all the sales values do not exceed the critical and caution limits. For example, the total <math>{{\chi }^{2}}</math> value of the sale month of July is 0.6085. Its degrees of freedom is 2, so the corresponding caution and critical values are 4.6052 and 9.2103 respectively. Both values are larger than 0.6085, so the return numbers of the July sales period do not deviate (based on the chosen significance) from the model's predictions.		If we consider the sales periods as the basis for outlier detection, then after comparing the above table to the sum of <math>z_{i,j}^{2}\,\!</math> <math>(\chi _{1}^{2})\,\!</math> values for each sales period, we find that all the sales values do not exceed the critical and caution limits. For example, the total <math>{{\chi }^{2}}\,\!</math> value of the sale month of July is 0.6085. Its degrees of freedom is 2, so the corresponding caution and critical values are 4.6052 and 9.2103 respectively. Both values are larger than 0.6085, so the return numbers of the July sales period do not deviate (based on the chosen significance) from the model's predictions.

	If we consider returns periods as the basis for outliers detection, then after comparing the above table to the sum of <math>z_{i,j}^{2}</math> <math>(\chi _{1}^{2})</math> values for each return period, we find that all the return values do not exceed the critical and caution limits. For example, the total <math>{{\chi }^{2}}</math> value of the sale month of August is 3.7157. Its degree of freedom is 3, so the corresponding caution and critical values are 6.2514 and 11.3449 respectively. Both values are larger than 3.7157, so the return numbers for the June return period do not deviate from the model's predictions.		If we consider returns periods as the basis for outliers detection, then after comparing the above table to the sum of <math>z_{i,j}^{2}\,\!</math> <math>(\chi _{1}^{2})\,\!</math> values for each return period, we find that all the return values do not exceed the critical and caution limits. For example, the total <math>{{\chi }^{2}}\,\!</math> value of the sale month of August is 3.7157. Its degree of freedom is 3, so the corresponding caution and critical values are 6.2514 and 11.3449 respectively. Both values are larger than 3.7157, so the return numbers for the June return period do not deviate from the model's predictions.

	This analysis can be automatically performed in Weibull++ by entering the alpha values in the Statistical Process Control page of the control panel and selecting which period to color code, as shown next.		This analysis can be automatically performed in Weibull++ by entering the alpha values in the Statistical Process Control page of the control panel and selecting which period to color code, as shown next.
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	[[Image:Warranty Example 5 SPC settings.png\|center\|250px\| ]]		[[Image:Warranty Example 5 SPC settings.png\|center\|250px\| ]]

	To view the table of chi-squared values ( <math>z_{i,j}^{2}</math> or <math>\chi _{1}^{2}</math> values), click the '''Show Results (...)''' button.		To view the table of chi-squared values ( <math>z_{i,j}^{2}\,\!</math> or <math>\chi _{1}^{2}\,\!</math> values), click the '''Show Results (...)''' button.

	[[Image:Warranty Example 5 Chi-square.png\|center\|450px\| ]]		[[Image:Warranty Example 5 Chi-square.png\|center\|450px\| ]]

Richard House: /* Basic Analysis Method */

2012-08-21T02:34:08Z

Basic Analysis Method

Richard House: /* Example */

2012-08-21T02:29:41Z

Example

← Older revision		Revision as of 02:34, 21 August 2012
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	::<math>{{e}_{i,j}}={{\hat{F}}_{i,j}}-{{F}_{i,j}}</math>		::<math>{{e}_{i,j}}={{\hat{F}}_{i,j}}-{{F}_{i,j}}</math>

	where <math>{{\hat{F}}_{i,j}}</math> is the estimated number of failures based on the estimated distribution parameters for the sales period <math>i</math> and the return period <math>j</math> , which is calculated using the equation for the conditional probability, and <math>{{F}_{i,j}}</math> is the actual number of failure for the sales period <math>i</math> and the return period <math>j</math> .		where <math>{{\hat{F}}_{i,j}}\,\!</math> is the estimated number of failures based on the estimated distribution parameters for the sales period <math>i</math> and the return period <math>j\,\!</math> , which is calculated using the equation for the conditional probability, and <math>{{F}_{i,j}}\,\!</math> is the actual number of failure for the sales period <math>i\,\!</math> and the return period <math>j\,\!</math> .

	Since we are assuming that the model is accurate, <math>{{e}_{i,j}}</math> should follow a normal distribution with mean value of zero and a standard deviation <math>s</math> , where:		Since we are assuming that the model is accurate, <math>{{e}_{i,j}}\,\!</math> should follow a normal distribution with mean value of zero and a standard deviation <math>s\,\!</math> , where:

	::<math>{{\bar{e}}_{i,j}}=\frac{\underset{i}{\mathop{\sum }}\,\underset{j}{\mathop{\sum }}\,{{e}_{i,j}}}{n}=0</math>		::<math>{{\bar{e}}_{i,j}}=\frac{\underset{i}{\mathop{\sum }}\,\underset{j}{\mathop{\sum }}\,{{e}_{i,j}}}{n}=0</math>

	and <math>n</math> is the total number of return data (total number of residuals).		and <math>n\,\!</math> is the total number of return data (total number of residuals).

	The estimated standard deviation of the prediction errors can then be calculated by:		The estimated standard deviation of the prediction errors can then be calculated by:
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	::<math>s=\sqrt{\frac{1}{n-1}\underset{i}{\mathop \sum }\,\underset{j}{\mathop \sum }\,e_{i,j}^{2}}</math>		::<math>s=\sqrt{\frac{1}{n-1}\underset{i}{\mathop \sum }\,\underset{j}{\mathop \sum }\,e_{i,j}^{2}}</math>

	and <math>{{e}_{i,j}}</math> can be normalized as follows:		and <math>{{e}_{i,j}}\,\!</math> can be normalized as follows:

	::<math>{{z}_{i,j}}=\frac{{{e}_{i,j}}}{s}</math>		::<math>{{z}_{i,j}}=\frac{{{e}_{i,j}}}{s}</math>

	where <math>{{z}_{i,j}}</math> is the standardized error. <math>{{z}_{i,j}}</math> follows a normal distribution with <math>\mu =0</math> and <math>\sigma =1</math> .		where <math>{{z}_{i,j}}\,\!</math> is the standardized error. <math>{{z}_{i,j}}\,\!</math> follows a normal distribution with <math>\mu =0\,\!</math> and <math>\sigma =1\,\!</math> .

	It is known that the square of a random variable with standard normal distribution follows the <math>{{\chi }^{2}}</math> (Chi Square) distribution with 1 degree of freedom and that the sum of the squares of <math>m</math> random variables with standard normal distribution follows the <math>{{\chi }^{2}}</math> distribution with <math>m</math> degrees of freedom ~~<math>~~.~~</math>~~ This then can be used to help detect the abnormal returns for a given sales period, return period or just a specific cell (combination of a return and a sales period).		It is known that the square of a random variable with standard normal distribution follows the <math>{{\chi }^{2}}\,\!</math> (Chi Square) distribution with 1 degree of freedom and that the sum of the squares of <math>m\,\!</math> random variables with standard normal distribution follows the <math>{{\chi }^{2}}\,\!</math> distribution with <math>m\,\!</math> degrees of freedom. This then can be used to help detect the abnormal returns for a given sales period, return period or just a specific cell (combination of a return and a sales period).

	:* For a cell, abnormality is detected if <math>z_{i,j}^{2}=\chi _{1}^{2}\ge \chi _{1,\alpha }^{2}.</math>		:* For a cell, abnormality is detected if <math>z_{i,j}^{2}=\chi _{1}^{2}\ge \chi _{1,\alpha }^{2}.\,\!</math>
	:* For an entire sales period <math>i</math> , abnormality is detected if <math>\underset{j}{\mathop{\sum }}\,z_{i,j}^{2}=\chi _{J}^{2}\ge \chi _{\alpha ,J}^{2},</math> where <math>J</math> is the total number of return period for a sales period <math>i</math> .		:* For an entire sales period <math>i\,\!</math> , abnormality is detected if <math>\underset{j}{\mathop{\sum }}\,z_{i,j}^{2}=\chi _{J}^{2}\ge \chi _{\alpha ,J}^{2},\,\!</math> where <math>J\,\!</math> is the total number of return period for a sales period <math>i\,\!</math> .
	:* For an entire return period <math>j</math> , abnormality is detected if <math>\underset{i}{\mathop{\sum }}\,z_{i,j}^{2}=\chi _{I}^{2}\ge \chi _{\alpha ,I}^{2},</math> where <math>I</math> is the total number of sales period for a return period <math>j</math> .		:* For an entire return period <math>j\,\!</math> , abnormality is detected if <math>\underset{i}{\mathop{\sum }}\,z_{i,j}^{2}=\chi _{I}^{2}\ge \chi _{\alpha ,I}^{2},\,\!</math> where <math>I\,\!</math> is the total number of sales period for a return period <math>j\,\!</math> .
	Here <math>\alpha </math> is the criticality value of the <math>{{\chi }^{2}}</math> distribution, which can be set at critical value or caution value. It describes the level of sensitivity to outliers (returns that deviate significantly from the predictions based on the fitted model). Increasing the value of <math>\alpha </math> increases the power of detection, but this could lead to more false alarms.		Here <math>\alpha \,\!</math> is the criticality value of the <math>{{\chi }^{2}}\,\!</math> distribution, which can be set at critical value or caution value. It describes the level of sensitivity to outliers (returns that deviate significantly from the predictions based on the fitted model). Increasing the value of <math>\alpha \,\!</math> increases the power of detection, but this could lead to more false alarms.

	====Example====		====Example====

← Older revision		Revision as of 02:29, 21 August 2012
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	::<math>Q(t\|T)=1-R(t\|T)=1-\frac{R(T+t)}{R(T)}</math>		::<math>Q(t\|T)=1-R(t\|T)=1-\frac{R(T+t)}{R(T)}</math>

	For the June shipment, there are 89 units that have successfully operated until the end of September ( <math>T=3~~</math> <math>~~months)</math> . The probability of one of these units failing in the next month ( <math>t=1~~</math> <math>~~month)</math> is then given by:		For the June shipment, there are 89 units that have successfully operated until the end of September ( <math>T=3 months)\,\!</math> . The probability of one of these units failing in the next month ( <math>t=1 month)\,\!</math> is then given by:

	::<math>Q(1\|3)=1-\frac{R(4)}{R(3)}=1-\frac{{{e}^{-{{\left( \tfrac{4}{6.70} \right)}^{2.49}}}}}{{{e}^{-{{\left( \tfrac{3}{6.70} \right)}^{2.49}}}}}=1-\frac{0.7582}{0.8735}=0.132</math>		::<math>Q(1\|3)=1-\frac{R(4)}{R(3)}=1-\frac{{{e}^{-{{\left( \tfrac{4}{6.70} \right)}^{2.49}}}}}{{{e}^{-{{\left( \tfrac{3}{6.70} \right)}^{2.49}}}}}=1-\frac{0.7582}{0.8735}=0.132</math>

	Once the probability of failure for an additional month of operation is determined, the expected number of failed units during the next month, from the June shipment, is the product of this probability and the number of units at risk ( <math>{{S}_{JUN,3}}=89)</math> or:		Once the probability of failure for an additional month of operation is determined, the expected number of failed units during the next month, from the June shipment, is the product of this probability and the number of units at risk ( <math>{{S}_{JUN,3}}=89)\,\!</math> or:

	::<math>{{\widehat{F}}_{JUN,4}}=89\cdot 0.132=11.748\text{, or 12 units}</math>		::<math>{{\widehat{F}}_{JUN,4}}=89\cdot 0.132=11.748\text{, or 12 units}</math>
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	::<math>Q(1\|2)=1-\frac{R(3)}{R(2)}=1-\frac{0.8735}{0.9519}=0.0824</math>		::<math>Q(1\|2)=1-\frac{R(3)}{R(2)}=1-\frac{0.8735}{0.9519}=0.0824</math>

	This value is multiplied by <math>{{S}_{JUL,2}}=134</math> to determine the number of failures, or:		This value is multiplied by <math>{{S}_{JUL,2}}=134\,\!</math> to determine the number of failures, or:

	::<math>{{\widehat{F}}_{JUL,3}}=134\cdot 0.0824=11.035\text{, or 11 units}</math>		::<math>{{\widehat{F}}_{JUL,3}}=134\cdot 0.0824=11.035\text{, or 11 units}</math>
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	::<math>Q(1\|1)=1-\frac{R(2)}{R(1)}=1-\frac{0.9519}{0.9913}=0.0397</math>		::<math>Q(1\|1)=1-\frac{R(2)}{R(1)}=1-\frac{0.9519}{0.9913}=0.0397</math>

	This value is multiplied by <math>{{S}_{AUG,1}}=146</math> to determine the number of failures, or:		This value is multiplied by <math>{{S}_{AUG,1}}=146\,\!</math> to determine the number of failures, or:

	::<math>{{\widehat{F}}_{AUG,2}}=146\cdot 0.0397=5.796\text{, or 6 units}</math>		::<math>{{\widehat{F}}_{AUG,2}}=146\cdot 0.0397=5.796\text{, or 6 units}</math>