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| | [[Category: For Deletion]] |
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| | valign="middle" align="left" bgcolor=EEEDF7|[[Image: ALTA-Examples-banner.png|400px|center]]
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| <br>
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| Using the simple step-stress data given [[Time-Varying Stress Models#Model Formulation|here]], one would define <math>x(t)</math> as:
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| <center><math>\begin{align}
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| x(t)=\ & 2,\text{ }0<t\le 250 \\
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| =\ & 3,\text{ }250<t\le 350 \\
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| =\ & 4,\text{ }350<t\le 370 \\
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| =\ & 5,\text{ }370<t\le 380 \\
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| =\ & 6,\text{ }380<t\le 390 \\
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| =\ & 7,\text{ }390<t\le +\infty
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| \end{align}</math></center>
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| Assuming a power relation as the underlying life-stress relationship and the Weibull distribution as the underlying life distribution, one can then formulate the log-likelihood function for the above data set as,
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| ::<math>\begin{align}
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| & \ln (L)= & \Lambda =\overset{F}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,\ln \left\{ \beta {{\left[ \frac{x(t)}{a} \right]}^{n}}{{\left[ \mathop{}_{0}^{{{t}_{i}}}{{\left[ \frac{\left[ x(u) \right]}{a} \right]}^{n}}du \right]}^{\beta -1}} \right\} -\overset{F}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,\left\{ {{\left[ \mathop{}_{0}^{{{t}_{i}}}{{\left[ \frac{\left[ x(u) \right]}{a} \right]}^{n}}du \right]}^{\beta }} \right\}
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| \end{align}</math>
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| <br>
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| where:
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| <br>
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| • <math>F</math> is the number of exact time-to-failure data points.
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| <br>
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| • <math>\beta </math> is the Weibull shape parameter.
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| <br>
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| • <math>a</math> and <math>n</math> are the IPL parameters.
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| <br>
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| • <math>x(t)</math> is the stress profile function.
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| <br>
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| • <math>{{t}_{i}}</math> is the <math>{{i}^{th}}</math> time to failure.
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| <br>
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| The parameter estimates for <math>\hat{\beta }</math> , <math>\hat{a}</math> and <math>\hat{n}</math> can be obtained by simultaneously solving, <math>\tfrac{\partial \Lambda }{\partial a}=0</math> and <math>\tfrac{\partial \Lambda }{\partial n}=0</math> . Using ALTA, the parameter estimates for this data set are:
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| ::<math>\begin{align}
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| \widehat{\beta }=\ & 2.67829 \\
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| \widehat{\alpha }=\ & 9.842122 \\
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| \widehat{n}=\ & -3.998466
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| \end{align}</math>
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| <br>
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| Once the parameters are obtained, one can now determine the reliability for these units at any time <math>t</math> and stress <math>x(t)</math> from:
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| ::<math>R\left( t,x\left( t \right) \right)={{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}}</math>
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| or at a fixed stress level <math>x(t)=2V</math> and <math>t=300</math> ,
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| ::<math>R\left( t=300,x(t)=2 \right)={{e}^{-{{\left[ \mathop{}_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}}=97.5%</math>
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| The mean time to failure (MTTF) at any stress <math>x(t)</math> can be determined by:
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| ::<math>MTTF\left( x\left( t \right) \right)=\int_{0}^{\infty }t\left[ \left\{ \beta {{\left[ \frac{x\left( t \right)}{a} \right]}^{n}}{{\left[ \int_{0}^{t}{{\left[ \frac{x\left( u \right)}{a} \right]}^{n}}du \right]}^{\beta -1}} \right\}{{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}} \right]dt</math>
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| or at a fixed stress level <math>x\left( t \right)=2V</math> ,
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| ::<math>MTTF\left( x\left( t \right) \right)=1046.3hrs</math>
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| <br>
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| Any other metric of interest (e.g. failure rate, conditional reliability etc.) can also be determined using the basic definitions given in [[Appendix A: Brief Statistical Background|Appendix A]] and calculated automatically with ALTA.
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| <br>
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| <br>
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