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| ==Arrhenius-Weibull==
| | #REDIRECT [[Arrhenius_Relationship]] |
| <br>
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| The <math>pdf</math> for 2-parameter Weibull distribution is given by:
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| <br>
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| ::{{weibull2pdf}}
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| <br>
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| The scale parameter (or characteristic life) of the Weibull distribution is <math>\eta </math> .
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| <br>
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| The Arrhenius-Weibull model pdf can then be obtained by setting <math>\eta =L(V)</math> in Eqn. (arrhenius):
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| <br>
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| ::<math>\eta =L(V)=C\cdot {{e}^{\tfrac{B}{V}}}</math>
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| <br>
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| and substituting for <math>\eta </math> in Eqn. (Weibullpdf):
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| <br>
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| ::<math>f(t,V)=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}</math>
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| <br>
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| An illustration of the <math>pdf</math> for different stresses is shown in Fig. 6. As expected, the <math>pdf</math> at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3 in Fig. 6). This behavior is observed when the parameter <math>B</math> of the Arrhenius model is positive.
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| [[Image:ALTA6.6.gif|thumb|center|300px|Behavior of the probability density function at different stresses and with the parameters held constant.]] | |
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| <br>
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| The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution is presented in greater detail in Chapter 5.
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| ===Arrhenius-Weibull Statistical Properties Summary===
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| ====Mean or MTTF====
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| The mean, <math>\overline{T}</math> (also called <math>MTTF</math> by some authors), of the Arrhenius-Weibull relationship is given by:
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| <br>
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| ::<math>\overline{T}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>
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| <br>
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| where <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)</math> is the gamma function evaluated at the value of <math>\left( \tfrac{1}{\beta }+1 \right)</math> .
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| <br>
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| <br>
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| ====Median====
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| <br>
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| The median, <math>\breve{T},</math>
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| for the Arrhenius-Weibull model is given by:
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| <br>
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| ::<math>\breve{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
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| ====Mode====
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| <br>
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| The mode, <math>\tilde{T},</math>
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| for the Arrhenius-Weibull model is given by:
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| <br>
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| ::<math>\tilde{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
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| ====Standard Deviation====
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| <br>
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| The standard deviation, <math>{{\sigma }_{T}},</math> for the Arrhenius-Weibull model is given by:
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| <br>
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| ::<math>{{\sigma }_{T}}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}</math>
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| ====Arrhenius-Weibull Reliability Function====
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| <br>
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| The Arrhenius-Weibull reliability function is given by:
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| ::<math>R(T,V)={{e}^{-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}</math>
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| If the parameter <math>B</math> is positive, then the reliability increases as stress decreases.
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| <br>
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| [[Image:ALTA6.7.gif|thumb|center|300px|Behavior of the reliability function at different stress and constant parameter values.]]
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| The behavior of the reliability function of the Weibull distribution for different values of <math>\beta </math> was illustrated in Chapter 5. In the case of the Arrhenius-Weibull model, however, the reliability is a function of stress also. A 3D plot such as the ones shown in Fig. 8 is now needed to illustrate the effects of both the stress and <math>\beta .</math>
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| <br>
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| ::<math></math>
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| <br>
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| [[Image:ALTA6.8.gif|thumb|center|300px|Reliability function for <math>\Beta<1 </math>, <math>\Beta=1 </math>, and <math>\Beta>1 </math>.]]
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| <br>
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| ====Conditional Reliability Function====
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| <br>
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| The Arrhenius-Weibull conditional reliability function at a specified stress level is given by:
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| ::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}</math>
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| <br>
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| or:
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| ::<math>R(T,t,V)={{e}^{-\left[ {{\left( \tfrac{T+t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }} \right]}}</math>
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| <br>
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| ====Reliable Life====
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| For the Arrhenius-Weibull relationship, the reliable life, <math>{{t}_{R}}</math> , of a unit for a specified reliability and starting the mission at age zero is given by:
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| <br>
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| ::<math>{{t}_{R}}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left\{ -\ln \left[ R\left( {{t}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>
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| <br>
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| This is the life for which the unit will function successfully with a reliability of <math>R({{t}_{R}})</math> . If <math>R({{t}_{R}})=0.50</math> then <math>{\breve{T}</math>,
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| the median life, or the life by which half of the units will survive.
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| <br>
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| ====Arrhenius-Weibull Failure Rate Function====
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| <br>
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| The Arrhenius-Weibull failure rate function, <math>\lambda (T)</math> , is given by:
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| <br>
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| ::<math>\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}</math>
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| <br>
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| [[Image:ALTA6.9.gif|thumb|center|300px|Failure rate function for <math>\Beta<1 </math>, <math>\Beta=1 </math>, and <math>\Beta>1 </math>.]]
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| <br>
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| ===Parameter Estimation===
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| ====Maximum Likelihood Estimation Method====
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| <br>
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| The Arrhenius-Weibull log-likelihood function is as follows:
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| <br>
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| ::<math>\begin{align}
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| & \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}} \right] \\
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| & & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
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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>R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>
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| <br>
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| ::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>
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| <br>
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| and:
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| • <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
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| • .. is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.
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| • <math>\beta </math> is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
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| • <math>B</math> is the Arrhenius parameter (unknown, the second of three parameters to be estimated).
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| • <math>C</math> is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).
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| • <math>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.
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| • <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
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| • <math>S</math> is the number of groups of suspension data points.
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| • <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
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| • <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
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| • <math>FI</math> is the number of interval data groups.
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| • <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.
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| • <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.
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| • <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.
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| The solution (parameter estimates) will be found by solving for <math>\widehat{\beta },</math> <math>\widehat{B},</math> <math>\widehat{C}</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math> <math>\tfrac{\partial \Lambda }{\partial B}=0</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:
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| <br>
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| ::<math>\begin{align}
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| & \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
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| & & -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
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| & & -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
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| & & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Li}^{\prime \prime }-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
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| \end{align}</math>
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| <br>
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| ::<math>\begin{align}
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| & \frac{\partial \Lambda }{\partial B}= & -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}+\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}{{\left( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }} \\
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| & & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
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| \end{align}</math>
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| <br>
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| ::<math>\begin{align}
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| & \frac{\partial \Lambda }{\partial C}= & -\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} \\
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| & & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
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| \end{align}</math>
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