Simultaneous Capacity and Planned Lead Time Optimization

Abstract

The literature on simultaneous capacity and inventory investment problems (see Chap. 2) shows that, especially for the MTS case, there are several contributions available, but for the MTO case, there is still a research gap about the influence of capacity investment on optimal planned lead time. A simultaneous approach to capacity and planned lead time setting is presented in this chapter based on costs for capacity, WIP at each stage, FGI, and backorders. This extends existing planned lead time approaches by introducing a distributed customer required lead time rather than applying a single value and a variable production lead time that can be optimized by investments. Furthermore, it extends existing capacity investment approaches by including a distributed customer required lead time and by rationing the work release, and therefore the inventory, with the application of planned lead times. First the simultaneous optimization problem is stated for a general setting before detailed results for a single- and a two-stage production system, assuming M/M/1 queues for the processing steps, are developed. For the two-stage production system consisting of M/M/1 queues and an exponentially distributed customer required lead time it is shown that only very little cost improvement can be gained by including a planned lead time for the upstream processing stage. This extends the findings of Yano (1987), Buzacott and Shanthikumar (1994), and Karaesmen et al. (2004) who analyze an MTO system with a constant customer required lead time. Furthermore, a large cost improvement potential is found when capacity investment and planned lead times are simultaneously optimized.

Appendix

Proof of Proposition 4.3

Based on Eq. (4.18) the following first-order optimality conditions can be stated:

$$ \begin{array}{llll} \frac{{dC\left( {X,\mu } \right)}}{dX }=0\Leftrightarrow {F_W}(X)=\frac{{{c_c}}}{{{c_f}+{c_c}}}\Leftrightarrow {X^{*}}=\frac{1}{{\mu -\lambda }}\ln \left( {\frac{{{c_f}+{c_c}}}{{{c_f}}}} \right) \hfill \\ \frac{{dC\left( {X,\mu } \right)}}{{d\mu }}=0\Leftrightarrow \frac{{{c_{\mu }}}}{\lambda }+\left( {{c_f}+{c_c}} \right)\int\limits_0^X {\tau {e^{{-\left( {\mu -\lambda } \right)\tau }}}\left( {1-{F_L}\left( \tau \right)} \right)d\tau } =\frac{{\left( {{c_y}+{c_c}} \right)}}{{{{{\left( {\mu -\lambda } \right)}}^2}}} \hfill \\ \Leftrightarrow \frac{{{c_{\mu }}}}{\lambda }+\left( {{c_f}+{c_c}} \right)\int\limits_0^X {\tau {e^{{-\left( {\mu -\lambda } \right)\tau }}}{e^{{-\beta \tau }}}} d\tau =\frac{{\left( {{c_y}+{c_c}} \right)}}{{{{{\left( {\mu -\lambda } \right)}}^2}}} \hfill \\ \Leftrightarrow \frac{{{c_{\mu }}}}{\lambda }+\frac{{\left( {{c_f}+{c_c}} \right)}}{{\left( {\mu +\beta -\lambda } \right)}}\left( {\frac{{1-{e^{{-\left( {\mu +\beta -\lambda } \right)X}}}}}{{\left( {\mu +\beta -\lambda } \right)}}-{e^{{-\left( {\mu +\beta -\lambda } \right)X}}}X} \right)=\frac{{\left( {{c_y}+{c_c}} \right)}}{{{{{\left( {\mu -\lambda } \right)}}^2}}} \end{array} $$

(4.32)

The second derivative of the cost function with respect to the processing rate $\mu$ is positive. Therefore, the cost function is convex and the first-order condition defines a unique minimum:

$$ \begin{array}{llll} \frac{{dC\left( {X,\mu } \right)}}{{{d^2}\mu }}=\frac{{2\lambda \left( {{c_y}+{c_c}} \right)}}{{{{{\left( {\mu -\lambda } \right)}}^3}}} \hfill \\ -\left( {{c_c}+{c_f}} \right)\lambda \frac{{\left( {-2+{e^{{-\left( {\mu -\lambda +\beta } \right)X}}}\left( {2+2X\left( {\mu -\lambda +\beta } \right)+{X^2}{{{\left( {\mu -\lambda +\beta } \right)}}^2}} \right)} \right)}}{{{{{\left( {\mu -\lambda +\beta } \right)}}^3}}}\mathop{>}\limits^{!}0 \hfill \\ \Leftrightarrow \frac{{2\left( {{c_y}+{c_c}} \right)}}{{\left( {{c_c}+{c_f}} \right)}} \hfill \\ -\frac{{{{{\left( {\mu -\lambda } \right)}}^3}}}{{{{{\left( {\mu -\lambda +\beta } \right)}}^3}}}\left( {-2+{e^{{-\left( {\mu -\lambda +\beta } \right)X}}}\left( {2+2X\left( {\mu -\lambda +\beta } \right)+{X^2}{{{\left( {\mu -\lambda +\beta } \right)}}^2}} \right)} \right)\mathop{>}\limits^{!}0 \hfill \\ \mathrm{ with}\ \frac{{2\left( {{c_y}+{c_c}} \right)}}{{\left( {{c_c}+{c_f}} \right)}}>1\ \mathrm{ from}\ {c_c}>{c_f} \hfill \\ \Leftrightarrow \frac{{{{{\left( {\mu -\lambda } \right)}}^3}}}{{{{{\left( {\mu -\lambda +\beta } \right)}}^3}}}\left( {-2+{e^{{-\left( {\mu -\lambda +\beta } \right)X}}}\left( {2+2X\left( {\mu -\lambda +\beta } \right)+{X^2}{{{\left( {\mu -\lambda +\beta } \right)}}^2}} \right)} \right)\mathop{<}\limits^{!}1 \hfill \\ \mathrm{ with}\ \frac{{{{{\left( {\mu -\lambda +\beta } \right)}}^3}}}{{{{{\left( {\mu -\lambda } \right)}}^3}}}>1 \hfill \\ \Leftrightarrow {e^{{-\left( {\mu -\lambda +\beta } \right)X}}}\left( {1+X\left( {\mu -\lambda +\beta } \right)+\frac{1}{2}{X^2}{{{\left( {\mu -\lambda +\beta } \right)}}^2}} \right)\mathop{<}\limits^{!}\frac{3}{2} \hfill \\ \Leftrightarrow {e^{{\left( {\mu -\lambda +\beta } \right)X}}}+\frac{1}{2}{e^{{\left( {\mu -\lambda +\beta } \right)X}}}\mathop{>}\limits^{!}1+X\left( {\mu -\lambda +\beta } \right)+\frac{1}{2}{X^2}{{\left( {\mu -\lambda +\beta } \right)}^2} \end{array} $$

(4.33)

which is fulfilled with:

$$ \begin{array}{llll} {e^a}-1> a\Rightarrow {e^{{\left( {\mu -\lambda +\beta } \right)X}}}>1+X\left( {\mu -\lambda +\beta } \right) \hfill \\ \mathrm{ and}\ {e^a}>{a^2}\Rightarrow {e^{{\left( {\mu -\lambda +\beta } \right)X}}}>{{\left( {X\left( {\mu -\lambda +\beta } \right)} \right)}^2} \end{array} $$

(4.34)

The first derivative of ${X^{*}}$ with respect to $\beta$ is obviously negative, so ${X^{*}}$ increases in mean customer required lead time $1/\beta$.

Implicit differentiation of the optimality condition (4.20) leads to:

$$ \begin{array}{lllll} \Phi \left( {{\mu^{*}},{\mu^{*}}\left( \beta \right)} \right)={c_{\mu }}-\frac{{\left( {{c_y}+{c_c}} \right)\lambda }}{{{{{\left( {{\mu^{*}}-\lambda } \right)}}^2}}}+\left( {{c_f}+{c_c}} \right)\lambda \int\limits_0^{{{X^{*}}}} {\tau {e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau =0 \hfill \\ \frac{{d\Phi \left( {{\mu^{*}}\left( \beta \right)} \right)}}{{d{\mu^{*}}\left( \beta \right)}}=\frac{{2\left( {{c_y}+{c_c}} \right)\lambda }}{{{{{\left( {{\mu^{*}}-\lambda } \right)}}^3}}}+\left( {{c_f}+{c_c}} \right)\lambda \frac{{d{X^{*}}}}{{d{\mu^{*}}\left( \beta \right)}}{X^{*}}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right){X^{*}}}}} \hfill \\ -\left( {{c_f}+{c_c}} \right)\lambda \int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau \hfill \\ \frac{{d\Phi \left( {{\mu^{*}}\left( \beta \right)} \right)}}{{d\beta }}=-\left( {{c_f}+{c_c}} \right)\lambda \int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau \hfill \\ \mathrm{with}\ \frac{{d\Phi \left( {{\mu^{*}}\left( \beta \right)} \right)}}{{d{\mu^{*}}\left( \beta \right)}}\frac{{d{\mu^{*}}\left( \beta \right)}}{{d\beta }}+\frac{{d\Phi \left( {{\mu^{*}}\left( \beta \right)} \right)}}{{d\beta }}=0 \hfill \\ \Rightarrow \frac{{2\left( {{c_y}+{c_c}} \right)}}{{{{{\left( {{\mu^{*}}-\lambda } \right)}}^3}}}\frac{{d{\mu^{*}}\left( \beta \right)}}{{d\beta }}+\left( {{c_f}+{c_c}} \right)\frac{{d{X^{*}}}}{{d{\mu^{*}}\left( \beta \right)}}{X^{*}}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right){X^{*}}}}}\frac{{d{\mu^{*}}\left( \beta \right)}}{{d\beta }} \hfill \\ -\left( {{c_f}+{c_c}} \right)\int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau \left( {1+\frac{{d{\mu^{*}}\left( \beta \right)}}{{d\beta }}} \right)=0 \hfill \\ \end{array} $$

(4.35)

$$ \begin{array}{llll} \ \mathrm{ with}\ \frac{{d{X^{*}}}}{{d{\mu^{*}}\left( \beta \right)}}=\frac{-1 }{{{{{\left( {{\mu^{*}}-\lambda } \right)}}^2}}}\ln \left( {\frac{{{c_f}+{c_c}}}{{{c_c}}}} \right)\Leftrightarrow \frac{{d{X^{*}}}}{{d{\mu^{*}}\left( \beta \right)}}=\frac{{-{X^{*}}}}{{\left( {{\mu^{*}}-\lambda } \right)}} \hfill \\ \Rightarrow \frac{{d{\mu^{*}}\left( \beta \right)}}{{d\beta }}=\frac{{\left( {{c_f}+{c_c}} \right)\int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau }}{{\frac{{2\left( {{c_y}+{c_c}} \right)}}{{{{{\left( {{\mu^{*}}-\lambda } \right)}}^3}}}-\left( {{c_f}+{c_c}} \right)\left( {\int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau +\frac{{{{{\left( {{X^{*}}} \right)}}^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right){X^{*}}}}}}}{{\left( {{\mu^{*}}-\lambda } \right)}}} \right)}} \end{array} $$

(4.36)

which is positive if the denominator is positive:

$$ \frac{{2\left( {{c_y}+{c_c}} \right)}}{{{{{\left( {{\mu^{*}}-\lambda } \right)}}^3}}}\mathop{>}\limits^{!}\left( {{c_f}+{c_c}} \right)\left( {\int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau +\frac{{{{{\left( {{X^{*}}} \right)}}^2}}}{{\left( {{\mu^{*}}-\lambda } \right)}}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right){X^{*}}}}}} \right) $$

(4.37)

From the optimality condition (4.20):

$$ \begin{array}{lllll} \frac{{2{c_{\mu }}}}{{\lambda \left( {{\mu^{*}}-\lambda } \right)}}+\frac{2}{{\left( {{\mu^{*}}-\lambda } \right)}}\frac{{{c_f}+{c_c}}}{{{\mu^{*}}+\beta -\lambda }}\left( {\frac{{1-{e^{{-\left( {{\mu^{*}}+\beta -\lambda } \right){X^{*}}}}}}}{{\left( {{\mu^{*}}+\beta -\lambda } \right)}}-{e^{{-\left( {{\mu^{*}}+\beta -\lambda } \right){X^{*}}}}}{X^{*}}} \right)\mathop{>}\limits^{!} \hfill \\ \left( {{c_f}+{c_c}} \right)\left( {\int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau +\frac{{{{{\left( {{X^{*}}} \right)}}^2}}}{{\left( {{\mu^{*}}-\lambda } \right)}}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right){X^{*}}}}}} \right) \hfill \\ \Leftrightarrow -\frac{{{{{\left( {{\mu^{*}}+\beta -\lambda } \right)}}^2}}}{{\left( {{\mu^{*}}-\lambda } \right)}}\left( {\frac{{2{e^{{-\left( {{\mu^{*}}+\beta -\lambda } \right){X^{*}}}}}{X^{*}}+\left( {{\mu^{*}}+\beta -\lambda } \right){{{\left( {{X^{*}}} \right)}}^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right){X^{*}}}}}}}{{{{{\left( {{\mu^{*}}+\beta -\lambda } \right)}}^3}}}} \right) \hfill \\ +\frac{{\left( {{\mu^{*}}+\beta -\lambda } \right)}}{{\left( {{\mu^{*}}-\lambda } \right)}}\left( {\frac{{2-2{e^{{-\left( {{\mu^{*}}+\beta -\lambda } \right){X^{*}}}}}}}{{{{{\left( {{\mu^{*}}+\beta -\lambda } \right)}}^3}}}} \right)+\frac{{2{c_{\mu }}}}{{\lambda \left( {{\mu^{*}}-\lambda } \right)\left( {{c_f}+{c_c}} \right)}}\mathop{>}\limits^{!}\int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau \end{array} $$

(4.38)

with

$$ \begin{array}{llll} \int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau = \hfill \\ =\frac{{2-{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right){X^{*}}}}}\left( {2+2\left( {{\mu^{*}}-\lambda +\beta } \right){X^{*}}+{{{\left( {{\mu^{*}}-\lambda +\beta } \right)}}^2}{{{\left( {{X^{*}}} \right)}}^2}} \right)}}{{{{{\left( {{\mu^{*}}-\lambda +\beta } \right)}}^3}}} \end{array} $$

(4.39)

it follows that:

$$ \frac{{2{c_{\mu }}}}{{\lambda \left( {{\mu^{*}}-\lambda } \right)\left( {{c_f}+{c_c}} \right)}}+\left( {\frac{{\left( {{\mu^{*}}-\lambda +\beta } \right)}}{{\left( {{\mu^{*}}-\lambda } \right)}}-1} \right)\int\limits_0^{{{X^{*}}}} {{\tau^2}{e^{{-\left( {{\mu^{*}}-\lambda +\beta } \right)\tau }}}} d\tau \mathop{>}\limits^{!}0 $$

(4.40)

which obviously holds for any cost structure.

Derivation of Equations ( 4.22 ) and ( 4.23 )

Restating the cost function from Eq. (4.21) leads to:

$$ \begin{array}{llll} C\left( {{X_1},{X_2}} \right)=\ \lambda \left( {{c_{y,1 }}+{c_c}} \right)\left( {E\left[ {{W_1}} \right]+\int\limits_0^{{{X_2}}} {{F_{{{W_2}}}}\left( \tau \right)\left( {1-{F_L}\left( {\tau +{X_1}} \right)} \right)d\tau } } \right)+ \hfill \cr +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_X^{\infty } {{F_{{{W_2}}}}\left( {{X_2}} \right)\int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} } {f_L}\left( \theta \right)d\theta \hfill \cr +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_X^{\infty } {\int\limits_{{{X_2}}}^X {\int\limits_0^{{X-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {X-{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } } {f_L}\left( \theta \right)d\theta \hfill \cr +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_{{{X_1}}}^X {{F_{{{W_2}}}}\left( {\theta -{X_1}} \right)\int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} } {f_L}\left( \theta \right)d\theta \hfill \cr \ +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_{{{X_1}}}^X {\int\limits_{{\theta -{X_1}}}^{\theta } {\int\limits_0^{{\theta -{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {\theta -{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } } {f_L}\left( \theta \right)d\theta \hfill \cr +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_0^{{{X_1}}} {\int\limits_0^{\theta } {\int\limits_0^{{\theta -{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {\theta -{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } } {f_L}\left( \theta \right)d\theta \hfill \cr -\lambda {c_c}\int\limits_0^X {\left( {1-{F_L}\left( \tau \right)} \right)d\tau } -{\chi_1}{X_1}-{\chi_2}{X_2} \end{array} $$

(4.41)

Taking the first derivative of the restated cost function with respect to ${X_1}$ shows:

$$ \begin{array}{llllllllll} \frac{{dC\left( {{X_1},{X_2}} \right)}}{{d{X_1}}}=-\lambda \left( {{c_{y,1 }}+{c_c}} \right)\int\limits_0^{{{X_2}}} {{F_{{{W_2}}}}\left( \tau \right){f_L}\left( {\tau +{X_1}} \right)d\tau } -\lambda {c_c}\left( {1-{F_L}(X)} \right) \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right){F_{{{W_2}}}}\left( {{X_2}} \right)\left( {\int\limits_X^{\infty } {{f_L}\left( \theta \right)\int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)d{\tau_1}} } d\theta -{f_L}(X)\int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} } \right) \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_X^{\infty } {{f_L}\left( \theta \right)d\theta } \left( {\int\limits_{{{X_2}}}^X {\int\limits_0^{{X-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)d{\tau_1}} d{\tau_2}} } \right) \hfill \\ -\lambda \left( {{c_f}+{c_c}} \right){f_L}(X)\int\limits_{{{X_2}}}^X {\int\limits_0^{{X-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {X-{\tau_2}-{\tau_1}} \right)d{\tau_1}} d{\tau_2}} \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_{{{X_1}}}^X {{F_{{{W_2}}}}\left( {\theta -{X_1}} \right){f_L}\left( \theta \right)d\theta } \int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)d{\tau_1}} \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right){F_{{{W_2}}}}\left( {X-{X_1}} \right){f_L}(X)\int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} \hfill \\ -\lambda \left( {{c_f}+{c_c}} \right)\int\limits_{{{X_1}}}^X {{f_{{{W_2}}}}\left( {\theta -{X_1}} \right){f_L}\left( \theta \right)d\theta } \int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right){f_L}(X)\int\limits_{{X-{X_1}}}^X {\int\limits_0^{{X-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {X-{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } \hfill \\ -\lambda \left( {{c_f}+{c_c}} \right){f_L}\left( {{X_1}} \right)\int\limits_0^{{{X_1}}} {\int\limits_0^{{{X_1}-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_{{{X_1}}}^X {{f_L}\left( \theta \right)\int\limits_0^{{{X_1}}} {{f_{{{W_2}}}}\left( {\theta -{X_1}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} d\theta } \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right){f_L}\left( {{X_1}} \right)\int\limits_0^{{{X_1}}} {\int\limits_0^{{{X_1}-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } -{\chi_1}=0 \hfill \\ \Leftrightarrow \left( {{c_f}+{c_c}} \right){F_{{{W_1}}}}\left( {{X_1}} \right)\int\limits_{{{X_1}}}^{{{X_1}+{X_2}}} {{F_{{{W_2}}}}\left( {\tau -{X_1}} \right){f_L}\left( \tau \right)d\tau } \hfill \\ +\left( {{c_f}+{c_c}} \right)\left( {1-{F_L}(X)} \right)\int\limits_{{{X_2}}}^{{{X_1}+{X_2}}} {{F_{{{W_2}}}}\left( \tau \right)} {f_{{{W_1}}}}\left( {{X_1}+{X_2}-\tau } \right)d\tau \hfill \\ -\left( {{c_{y,1 }}+{c_c}} \right)\int\limits_0^{{{X_2}}} {{F_{{{W_2}}}}\left( \tau \right){f_L}\left( {\tau +{X_1}} \right)d\tau } -{c_c}\left( {1-{F_L}(X)} \right)-\frac{{{\chi_1}}}{\lambda }=0 \hfill \\ \Leftrightarrow \left( {{c_f}+{c_c}} \right){F_{{{W_1}}}}\left( {{X_1}} \right)\left( {{F_{{{W_2}}}}\left( {{X_2}} \right){F_L}(X)-\int\limits_{{{X_1}}}^X {{f_{{{W_2}}}}\left( {\tau -{X_1}} \right){F_L}\left( \tau \right)d\tau } } \right) \hfill \\ +\left( {{c_f}+{c_c}} \right)\left( {1-{F_L}(X)} \right)\int\limits_{{{X_2}}}^{{{X_1}+{X_2}}} {{F_{{{W_2}}}}\left( \tau \right)} {f_{{{W_1}}}}\left( {{X_1}+{X_2}-\tau } \right)d\tau -{c_c}\left( {1-{F_L}(X)} \right)-\frac{{{\chi_1}}}{\lambda } \hfill \\ =\left( {{c_{y,1 }}+{c_c}} \right)\left( {{F_{{{W_2}}}}\left( {{X_2}} \right){F_L}(X)-\left( {{F_{{{W_2}}}}\left( {{X_2}} \right){F_L}(X)-\int\limits_0^{{{X_2}}} {{F_{{{W_2}}}}\left( \tau \right){f_L}\left( {\tau +{X_1}} \right)d\tau } } \right)} \right) \hfill \\ \end{array} $$

(4.42)

Rearranging terms:

$$ \begin{array}{llll} \Leftrightarrow \left( {\left( {{c_f}+{c_c}} \right){F_{{{W_1}}}}\left( {{X_1}} \right)-\left( {{c_{y,1 }}+{c_c}} \right)} \right)\int\limits_0^{{{X_2}}} {{F_{{{W_2}}}}\left( \tau \right){f_L}\left( {\tau +{X_1}} \right)d\tau } \hfill \\ +\left( {{c_f}+{c_c}} \right)\left( {1-{F_L}\left( {{X_1}+{X_2}} \right)} \right)\int\limits_{{{X_2}}}^{{{X_1}+{X_2}}} {{F_{{{W_2}}}}\left( \tau \right)} {f_{{{W_1}}}}\left( {{X_1}+{X_2}-\tau } \right)d\tau \hfill \\ -{c_c}\left( {1-{F_L}\left( {{X_1}+{X_2}} \right)} \right)-\frac{{{\chi_1}}}{\lambda }=0 \hfill \\ \end{array} $$

(4.43)

Taking the first derivative of the restated cost function with respect to ${X_2}$ leads to:

$$ \begin{array}{lllllll} \frac{{dC\left( {{X_1},{X_2}} \right)}}{{d{X_2}}}=\lambda \left( {{c_{y,1 }}+{c_c}} \right){F_{{{W_2}}}}\left( {{X_2}} \right)\left( {1-{F_L}(X)} \right) \hfill \\ -\lambda \left( {{c_f}+{c_c}} \right){F_{{{W_2}}}}\left( {{X_2}} \right){f_L}(X)\int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right){f_{{{W_2}}}}\left( {{X_2}} \right)\int\limits_X^{\infty } {{f_L}\left( \theta \right)d\theta \int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} } \hfill \\ -\lambda \left( {{c_f}+{c_c}} \right)\int\limits_X^{\infty } {{f_L}\left( \theta \right)d\theta } \int\limits_0^{{X-{X_2}}} {{f_{{{W_2}}}}\left( {{X_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right)\int\limits_X^{\infty } {{f_L}\left( \theta \right)d\theta } \int\limits_{{{X_2}}}^X {\int\limits_0^{{X-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)d{\tau_1}} d{\tau_2}} \hfill \\ -\lambda \left( {{c_f}+{c_c}} \right){f_L}(X)\int\limits_{{{X_2}}}^X {\int\limits_0^{{X-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {X-{\tau_2}-{\tau_1}} \right)d{\tau_1}} d{\tau_2}} \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right){F_{{{W_2}}}}\left( {X-{X_1}} \right){f_L}(X)\int\limits_0^{{{X_1}}} {{f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {{X_1}-{\tau_1}} \right)d{\tau_1}} \hfill \\ +\lambda \left( {{c_f}+{c_c}} \right){f_L}(X)\int\limits_{{X-{X_1}}}^X {\int\limits_0^{{X-{\tau_2}}} {{f_{{{W_2}}}}\left( {{\tau_2}} \right){f_{{{W_1}}}}\left( {{\tau_1}} \right)\left( {X-{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } \hfill \\ -\lambda {c_c}\left( {1-{F_L}(X)} \right)-{\chi_2} \hfill \\ \Leftrightarrow \left( {{c_f}+{c_c}} \right)\left( {1-{F_L}(X)} \right)\int\limits_{{{X_2}}}^X {{f_{{{W_2}}}}\left( \tau \right)} {F_{{{W_1}}}}\left( {X-\tau } \right)d\tau \hfill \\ +\left( {{c_{y,1 }}+{c_c}} \right){F_{{{W_2}}}}\left( {{X_2}} \right)\left( {1-{F_L}(X)} \right)-{c_c}\left( {1-{F_L}(X)} \right)-\frac{{{\chi_2}}}{\lambda }=0 \hfill \\ \Leftrightarrow \left( {\left( {{c_{y,1 }}+{c_c}} \right)-\left( {{c_f}+{c_c}} \right){F_{{{W_1}}}}\left( {{X_1}} \right)} \right)\left( {1-{F_L}\left( {{X_1}+{X_2}} \right)} \right){F_{{{W_2}}}}\left( {{X_2}} \right) \hfill \\ +\left( {{c_f}+{c_c}} \right)\left( {1-{F_L}\left( {{X_1}+{X_2}} \right)} \right)\int\limits_{{{X_2}}}^{{{X_1}+{X_2}}} {{F_{{{W_2}}}}\left( \tau \right)} {f_{{{W_1}}}}\left( {{X_1}+{X_2}-\tau } \right)d\tau \hfill \\ -{c_c}\left( {1-{F_L}\left( {{X_1}+{X_2}} \right)} \right)-\frac{{{\chi_2}}}{\lambda }=0 \hfill \\ \end{array} $$

(4.44)

Derivation of Equation ( 4.28 )

$$ \begin{array}{lllllll} E\left[ I \right]=\left( {1-{F_L}(X)} \right)\int\limits_0^{{{X_1}}} {\left( {1-{e^{{-\left( {{\mu_2}-\lambda } \right)\left( {\tau +{X_2}} \right)}}}} \right)\left( {1-{e^{{-\left( {{\mu_1}-\lambda } \right)\left( {{X_1}-\tau } \right)}}}} \right)d\tau } \hfill \\ +\int\limits_0^{{{X_1}}} {\left( {1-{e^{{-\left( {{\mu_1}-\lambda } \right)\tau }}}} \right)d\tau } \int\limits_0^{{{X_2}}} {\left( {{\mu_2}-\lambda } \right){e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}} {f_L}\left( {\theta +{X_1}} \right)d\theta \hfill \\ +\int\limits_{{{X_1}}}^X {\int\limits_{{\theta -{X_1}}}^{\theta } {\int\limits_0^{{\theta -{\tau_2}}} {\left( {{\mu_2}-\lambda } \right){e^{{-\left( {{\mu_2}-\lambda } \right){\tau_2}}}}\left( {{\mu_1}-\lambda } \right){e^{{-\left( {{\mu_1}-\lambda } \right){\tau_1}}}}\left( {\theta -{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } } {f_L}\left( \theta \right)d\theta \hfill \\ +\int\limits_0^{{{X_1}}} {\int\limits_0^{\theta } {\int\limits_0^{{\theta -{\tau_2}}} {\left( {{\mu_2}-\lambda } \right){e^{{-\left( {{\mu_2}-\lambda } \right){\tau_2}}}}\left( {{\mu_1}-\lambda } \right){e^{{-\left( {{\mu_1}-\lambda } \right){\tau_1}}}}\left( {\theta -{\tau_2}-{\tau_1}} \right)d{\tau_2}d{\tau_1}} } } {f_L}\left( \theta \right)d\theta \hfill \\ =\left( {1-{F_L}(X)} \right)\left( {{X_1}-\frac{{1-{e^{{-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_1}-\lambda }}+\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)X}}}}}{{{\mu_1}-{\mu_2}}}} \right. \hfill \\ \left. {+\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)X}}}}}{{{\mu_2}-\lambda }}-\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right){X_2}}}}}}{{{\mu_2}-\lambda }}-\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right){X_2}-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_2}-\lambda }}} \right) \hfill \\ +\left( {{X_1}-\frac{{1-{e^{{-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_1}-\lambda }}} \right)\left( {{\mu_2}-\lambda } \right)\int\limits_0^{{{X_2}}} {{e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}} {f_L}\left( {\theta +{X_1}} \right)d\theta \hfill \\ +\int\limits_{{{X_1}}}^X {{f_L}\left( \theta \right)\int\limits_{{\theta -{X_1}}}^{\theta } {\left( {{\mu_2}-\lambda } \right){e^{{-\left( {{\mu_2}-\lambda } \right){\tau_2}}}}\frac{{-1+{e^{{-\left( {{\mu_1}-\lambda } \right)\left( {\theta -{\tau_2}} \right)}}}+\left( {{\mu_1}-\lambda } \right)\left( {\theta -{\tau_2}} \right)}}{{\left( {{\mu_1}-\lambda } \right)}}d{\tau_2}} } d\theta \hfill \\ +\int\limits_0^{{{X_1}}} {{f_L}\left( \theta \right)\int\limits_0^{\theta } {\left( {{\mu_2}-\lambda } \right){e^{{-\left( {{\mu_2}-\lambda } \right){\tau_2}}}}\frac{{-1+{e^{{-\left( {{\mu_1}-\lambda } \right)\left( {\theta -{\tau_2}} \right)}}}+\left( {{\mu_1}-\lambda } \right)\left( {\theta -{\tau_2}} \right)}}{{\left( {{\mu_1}-\lambda } \right)}}d{\tau_2}} } d\theta \hfill \\ =\left( {1-{F_L}(X)} \right)\left( {{X_1}-\frac{{1-{e^{{-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_1}-\lambda }}+\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)X}}}}}{{{\mu_1}-{\mu_2}}}} \right. \hfill \\ \left. {+\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)X}}}}}{{{\mu_2}-\lambda }}-\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right){X_2}}}}}}{{{\mu_2}-\lambda }}-\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right){X_2}-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_2}-\lambda }}} \right) \hfill \\ +\left( {{X_1}-\frac{{1-{e^{{-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_1}-\lambda }}} \right)\left( {{\mu_2}-\lambda } \right)\int\limits_0^{{{X_2}}} {{e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}} {f_L}\left( {\theta +{X_1}} \right)d\theta \hfill \\ +\int\limits_{{{X_1}}}^X {{f_L}\left( \theta \right)} \left( {\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}-{e^{{-\left( {{\mu_2}-\lambda } \right)\left( {\theta -{X_1}} \right)}}}}}{{\left( {{\mu_1}-\lambda } \right)}}-\frac{{\left( {{\mu_2}-\lambda } \right)}}{{\left( {{\mu_1}-\lambda } \right)}}\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}-{e^{{-\left( {{\mu_2}-{\mu_1}} \right)\left( {\theta -{X_1}} \right)-\left( {{\mu_1}-\lambda } \right)\theta }}}}}{{\left( {{\mu_2}-{\mu_1}} \right)}}} \right. \hfill \\ \left. {+\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}-{e^{{-\left( {{\mu_2}-\lambda } \right)\left( {\theta -{X_1}} \right)}}}}}{{\left( {{\mu_2}-\lambda } \right)}}+{e^{{-\left( {{\mu_2}-\lambda } \right)\left( {\theta -{X_1}} \right)}}}\theta -{e^{{-\left( {{\mu_2}-\lambda } \right)\left( {\theta -{X_1}} \right)}}}\left( {\theta -{X_1}} \right)} \right)d\theta \hfill \\ +\int\limits_0^{{{X_1}}} {{f_L}\left( \theta \right)\left( {\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}-1}}{{\left( {{\mu_1}-\lambda } \right)}}-\frac{{\left( {{\mu_2}-\lambda } \right)}}{{\left( {{\mu_1}-\lambda } \right)}}\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}-{e^{{-\left( {{\mu_1}-\lambda } \right)\theta }}}}}{{\left( {{\mu_2}-{\mu_1}} \right)}}+\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)\theta }}}-1}}{{\left( {{\mu_2}-\lambda } \right)}}+\theta } \right)d\theta } \hfill \\ \end{array} $$

(4.45)

With exponential distribution for customer required lead time:

$$ \begin{array}{lllllll} E\left[ I \right]={e^{{-\beta X}}}\left( {{X_1}-\frac{{1-{e^{{-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_1}-\lambda }}+\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)X}}}}}{{{\mu_1}-{\mu_2}}}+\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right)X}}}}}{{{\mu_2}-\lambda }}-\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right){X_2}}}}}}{{{\mu_2}-\lambda }}} \right. \hfill \\ \left. {-\frac{{{e^{{-\left( {{\mu_2}-\lambda } \right){X_2}-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_2}-\lambda }}} \right)+\left( {{X_1}-\frac{{1-{e^{{-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{{\mu_1}-\lambda }}} \right)\frac{{\left( {{e^{{-\beta {X_1}}}}-{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_2}-\beta {X_1}}}}} \right)\beta \left( {{\mu_2}-\lambda } \right)}}{{\left( {{\mu_2}-\lambda +\beta } \right)}} \hfill \\ +\frac{{\left( {{\mu_2}-\lambda } \right)\beta \left( {{e^{{-\left( {{\mu_1}-\lambda +\beta } \right){X_1}}}}-{e^{{-\left( {{\mu_1}-\lambda +\beta } \right){X_1}-\left( {{\mu_2}-\lambda +\beta } \right){X_2}}}}} \right)}}{{\left( {{\mu_2}-{\mu_1}} \right)\left( {{\mu_2}-\lambda +\beta } \right)\left( {{\mu_1}-\lambda } \right)}} \hfill \\ +\frac{{{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_2}-\beta {X_1}}}}\beta \left( {{\mu_1}+{\mu_2}-2\lambda -{X_1}\left( {{\mu_1}-\lambda } \right)\left( {{\mu_2}-\lambda } \right)} \right)}}{{\left( {{\mu_2}-\lambda +\beta } \right)\left( {{\mu_1}-\lambda } \right)\left( {{\mu_2}-\lambda } \right)}} \hfill \\ +\frac{{{e^{{-\left( {{\mu_2}-\lambda +\beta } \right)X}}}\beta \left( {\left( {{\mu_1}-\lambda } \right){\mu_2}+\lambda \left( {{\mu_2}-\lambda } \right)-{\mu_1}\left( {{\mu_1}+{\mu_2}-2\lambda } \right)} \right)}}{{\left( {{\mu_2}-{\mu_1}} \right)\left( {{\mu_2}-\lambda +\beta } \right)\left( {{\mu_1}-\lambda } \right)\left( {{\mu_2}-\lambda } \right)}} \hfill \\ +\frac{{{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_1}}}}\beta \left( {{\mu_1}\left( {{\mu_1}+{\mu_2}-2\lambda } \right)-\left( {{\mu_1}-\lambda } \right){\mu_2}-\lambda \left( {{\mu_2}-\lambda } \right)} \right)}}{{\left( {{\mu_2}-{\mu_1}} \right)\left( {{\mu_2}-\lambda +\beta } \right)\left( {{\mu_1}-\lambda } \right)\left( {{\mu_2}-\lambda } \right)}} \hfill \\ +\frac{{{e^{{-\beta {X_1}}}}\beta \left( {-\left( {{\mu_2}-\lambda } \right)+\left( {{\mu_1}-\lambda } \right)\left( {-1+{X_1}\left( {{\mu_2}-\lambda } \right)} \right)} \right)}}{{\left( {{\mu_2}-\lambda +\beta } \right)\left( {{\mu_1}-\lambda } \right)\left( {{\mu_2}-\lambda } \right)}}-\frac{{1-{e^{{-\beta {X_1}}}}}}{{\left( {{\mu_1}-\lambda } \right)}} \hfill \\ +\frac{{\beta \left( {1-{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_1}}}}} \right)}}{{\left( {{\mu_1}-\lambda } \right)\left( {{\mu_2}-\lambda +\beta } \right)}}+\frac{{\beta \left( {{\mu_2}-\lambda } \right)}}{{\left( {{\mu_1}-\lambda } \right)\left( {{\mu_2}-{\mu_1}} \right)}}\left( {\frac{{1-{e^{{-\left( {{\mu_1}-\lambda +\beta } \right){X_1}}}}}}{{\left( {{\mu_1}-\lambda +\beta } \right)}}-\frac{{1-{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_1}}}}}}{{\left( {{\mu_2}-\lambda +\beta } \right)}}} \right) \hfill \\ -\frac{{1-{e^{{-\beta {X_1}}}}}}{{\left( {{\mu_2}-\lambda } \right)}}+\beta \frac{{1-{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_1}}}}}}{{\left( {{\mu_2}-\lambda } \right)\left( {{\mu_2}-\lambda +\beta } \right)}}+\frac{{1-{e^{{-\beta {X_1}}}}\left( {1+\beta {X_1}} \right)}}{\beta } \hfill \\ \end{array} $$

(4.46)

Simplifying and rearranging terms leads to:

$$ \begin{array}{llllll} E\left[ I \right]=\frac{{{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_2}-\left( {{\mu_2}-\lambda +\beta } \right){X_1}}}}\left( {{\mu_1}-\lambda } \right)}}{{\left( {{\mu_1}-{\mu_2}} \right)\left( {{\mu_2}-\lambda +\beta } \right)}}-\frac{{{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_2}-\left( {{\mu_1}-\lambda +\beta } \right){X_1}}}}\left( {{\mu_2}-\lambda } \right)}}{{\left( {{\mu_1}-{\mu_2}} \right)\left( {{\mu_2}-\lambda +\beta } \right)}} \hfill \\ -\frac{{{e^{{-\left( {{\mu_2}-\lambda +\beta } \right){X_2}-\beta {X_1}}}}}}{{\left( {{\mu_2}-\lambda +\beta } \right)}}+\frac{{{e^{{\left( {{\mu_1}-\lambda +\beta } \right){X_1}}}}\left( {{\mu_2}-\lambda } \right)}}{{\left( {{\mu_1}-\lambda +\beta } \right)\left( {{\mu_2}-\lambda +\beta } \right)}} \hfill \\ -\frac{{{e^{{-\beta {X_1}}}}\left( {{\mu_2}-\lambda } \right)}}{{\beta \left( {{\mu_2}-\lambda +\beta } \right)}}+\frac{{\left( {{\mu_1}-\lambda } \right)\left( {{\mu_2}-\lambda } \right)}}{{\beta \left( {{\mu_1}-\lambda +\beta } \right)\left( {{\mu_2}-\lambda +\beta } \right)}} \hfill \\ \end{array} $$

(4.47)

4.1.1 Optimality Conditions in Two-Stage M/M/1 System

The following optimality conditions for ${X_1}$ and ${X_2}$ can be stated with Eqs. (4.25) and (4.26):

$$ \begin{array}{llllll} X_2^{*}=0: \hfill \\ X_1^{*}:\int\limits_0^{{{X_1}}} {\left( {1-{e^{{-\left( {{\mu_2}-\lambda } \right)\tau }}}} \right)\left( {{\mu_1}-\lambda } \right){e^{{-\left( {{\mu_1}-\lambda } \right)\left( {{X_1}-\tau } \right)}}}} d\tau =\frac{{{c_c}}}{{{c_f}+{c_c}}} \hfill \\ \Leftrightarrow \frac{{{c_f}}}{{{c_f}+{c_c}}}=\frac{{\left( {{\mu_1}-\lambda } \right){e^{{-\left( {{\mu_2}-\lambda } \right){X_1}}}}-\left( {{\mu_2}-\lambda } \right){e^{{-\left( {{\mu_1}-\lambda } \right){X_1}}}}}}{{\left( {{\mu_1}-{\mu_2}} \right)}} \hfill \\ \end{array} $$

(4.48)

$$ \begin{array}{lllllll} X_2^{*}>0: \hfill \\ X_1^{*}:\frac{{{c_{y,1 }}+{c_c}}}{{{c_f}+{c_c}}}={F_{{{W_1}}}}\left( {{X_1}} \right)\Leftrightarrow X_1^{*}=\frac{1}{{\left( {{\mu_1}-\lambda } \right)}}\ln \left( {\frac{{{c_f}+{c_c}}}{{{c_f}-{c_{y,1 }}}}} \right) \hfill \\ X_2^{*}:\int\limits_{{{X_2}}}^{{{X_1}+{X_2}}} {\left( {1-{e^{{-\left( {{\mu_2}-\lambda } \right)\tau }}}} \right)\left( {{\mu_1}-\lambda } \right){e^{{-\left( {{\mu_1}-\lambda } \right)\left( {{X_1}+{X_2}-\tau } \right)}}}} d\tau =\frac{{{c_c}}}{{{c_f}+{c_c}}} \hfill \\ \Leftrightarrow X_2^{*}=\frac{1}{{{\mu_2}-\lambda }}\left( {\ln \left( {\frac{{{c_f}-{c_{y,1 }}}}{{{c_f}+{c_c}}}-{{{\left( {\frac{{{c_f}-{c_{y,1 }}}}{{{c_f}+{c_c}}}} \right)}}^{{\frac{{{\mu_2}-\lambda }}{{{\mu_1}-\lambda }}}}}} \right)-\ln \left( {\frac{{{c_{y,1 }}}}{{{c_f}+{c_c}}}\frac{{{\mu_2}-{\mu_1}}}{{{\mu_1}-\lambda }}} \right)} \right) \hfill \\ \end{array} $$

(4.49)