Appendix
Proof of Proposition 3.1
The comparison of a stochastic customer required lead time value L to a deterministic WAW value X leads to a random assignment of orders to type 1 and type 2 orders. Type 1 orders have \( L\leq X \) with probability p and type 2 orders have \( L{>}X \) with probability \( \left( {1-p} \right) \). Based on the splitting property of the Poisson input stream with rate \( \lambda \), the two resulting streams of events are again Poisson streams with rates \( \varphi =p\lambda \) and \( \psi =\left( {1-p} \right)\lambda \) respectively (see Tijms (2003)). The Poisson stream 1 (\( L\leq X \)) with rate \( \varphi =p\lambda \) directly feeds the M/M/1 production system. The Poisson stream 2 with rate \( \psi =\left( {1-p} \right)\lambda \) feeds the buffer of units waiting to be released to the system. This WAW buffer can be transformed to an M/G/∞ queuing system since the service time has an arbitrary distribution, which is based on the arbitrary customer required lead time distribution, and each order has an own server since the waiting time is \( L-X \). For such an M/G/∞ queuing system, the output process is a Poisson process with the same rate as the input process, which is \( \psi =\left( {1-p} \right)\lambda \) (see Mirasol (1963) and Newell (1966)). This output process of the WAW buffer feeds the M/M/1 production system. Based on the merging property of two Poisson processes, the input process for the M/M/1 production system is Poisson with rate \( \lambda =\varphi +\psi \) (see Tijms (2003)).
Derivation of Equation (3.6)
Figure 3.4 visualizes The probability of an order being delivered on time in the production lead time \( W \) and customer required lead time \( L \) space. The shaded area indicates when an order is delivered on time. Assuming that the distribution of \( W \) and \( L \) are independent of each other (which holds true for FIFS dispatching discipline) the following equation for the service level holds:
$$ \begin{array}{llll} \eta =\int\limits_0^{\infty } {\int\limits_{\tau}^{\infty } {{f_W}\left( \tau \right)} {f_L}\left( \theta \right)d\theta d\tau } =\int\limits_0^{\infty } {{f_W}\left( \tau \right)\int\limits_{\tau}^{\infty } {{f_L}\left( \theta \right)d\theta d\tau } } =\int\limits_0^{\infty } {{f_W}\left( \tau \right)\left( {1-{F_L}\left( \tau \right)} \right)d\tau } \hfill \\ =1-\int\limits_0^{\infty } {{f_W}\left( \tau \right){F_L}\left( \tau \right)d\tau } =\left. {{F_W}\left( \tau \right){F_L}\left( \tau \right)} \right|_0^{\infty }-\int\limits_0^{\infty } {{f_W}\left( \tau \right){F_L}\left( \tau \right)d\tau } \hfill \\ =\int\limits_0^{\infty } {{F_W}\left( \tau \right){f_L}\left( \tau \right)d\tau } \hfill \cr \mathrm{ with}\ {F_L}(0)=0;\ {F_L}\left( \infty \right)=1;\ {F_W}(0)=0;\ {F_W}\left( \infty \right)=1; \end{array} $$
(3.17)
Derivation of Equation (3.7)
$$ \begin{array}{lllll} E[I] =\int\limits_0^{\infty } {\int\limits_0^{\theta } {{f_W}\left( \tau \right)\left( {\theta -\tau } \right)d\tau } } {f_L}\left( \theta \right)d\theta =\int\limits_0^{\infty } {\frac{{\theta k+{e^{{-k\theta }}}-1}}{k}} \beta {e^{{-\beta \theta }}}d\theta \hfill \cr \qquad\,=\frac{\beta }{k}\left( {\int\limits_0^{\infty } {\theta k{e^{{-\beta \theta }}}} d\theta +\int\limits_0^{\infty } {{e^{{-\left( {k+\beta } \right)\theta }}}} d\theta -\int\limits_0^{\infty } {{e^{{-\beta \theta }}}} d\theta } \right)\cr \qquad\,=\frac{\beta }{k}\left( {\int\limits_0^{\infty } {\theta k{e^{{-\beta \theta }}}} d\theta +\frac{1}{{k+\beta }}-\frac{1}{\beta }} \right) \hfill \cr \qquad\,=\frac{\beta }{k}\left( {\frac{k}{{{\beta^2}}}+\frac{1}{{k+\beta }}-\frac{1}{\beta }} \right)=\frac{1}{\beta }-\frac{1}{{k+\beta }} \end{array} $$
(3.18)
Derivation of Equation (3.9)
$$ \begin{array}{lllll} \eta =\int\limits_0^X {{F_W}\left( \tau \right){f_L}\left( \tau \right)d\tau } +\int\limits_X^{\infty } {{F_W}(X){f_L}\left( \tau \right)d\tau } \hfill \cr \;\;\;=\int\limits_0^X {\left( {1-{e^{{-k\tau }}}} \right)\beta {e^{{-\beta \tau }}}d\tau } +\int\limits_X^{\infty } {\left( {1-{e^{-kX }}} \right)\beta {e^{{-\beta \tau }}}d\tau } \hfill \cr \;\;\;=\beta \left( {\left. {\frac{{{e^{{-\beta \tau }}}}}{{-\beta }}} \right|_0^X-\left. {\frac{{{e^{{-\left( {k+\beta } \right)\tau }}}}}{{-\left( {k+\beta } \right)}}} \right|_0^X} \right)+\left( {1-{e^{-kX }}} \right)\beta \left. {\frac{{{e^{{-\beta \tau }}}}}{{-\beta }}} \right|_X^{\infty } \hfill \cr \;\;\;=1+\frac{{\beta {e^{{-\left( {k+\beta } \right)X}}}-\beta }}{{k+\beta }}-\frac{{\left( {k+\beta } \right){e^{{-\left( {k+\beta } \right)X}}}}}{{k+\beta }}=1-\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{k+\beta }} \end{array} $$
(3.19)
Derivation of Equation (3.10)
$$ \begin{array}{lllll} E[I] =\int\limits_0^X {\int\limits_0^{\theta } {{f_W}\left( \tau \right)\left( {\theta -\tau } \right)d\tau } } {f_L}\left( \theta \right)d\theta +\int\limits_X^{\infty } {\int\limits_0^X {{f_W}\left( \tau \right)\left( {X-\tau } \right)d\tau } } {f_L}\left( \theta \right)d\theta \hfill \cr \qquad\,=\int\limits_0^X {\frac{{\theta k+{e^{{-k\theta }}}-1}}{k}} \beta {e^{{-\left( {\beta \theta } \right)}}}d\theta +\int\limits_X^{\infty } {\frac{{Xk+{e^{-kX }}-1}}{k}} \beta {e^{{-\beta \theta }}}d\theta \hfill \cr \qquad\,=\frac{\beta }{k}\left( {\int\limits_0^X {\theta k{e^{{-\beta \theta }}}} d\theta +\int\limits_0^X {{e^{{-\left( {k+\beta } \right)\theta }}}} d\theta -\int\limits_0^X {{e^{{-\beta \theta }}}} d\theta } \right)+\beta \frac{{Xk+{e^{-kX }}-1}}{k}\int\limits_X^{\infty } {{e^{{-\beta \theta }}}d\theta } \hfill \cr \qquad\,=\frac{\beta }{k}\left( {Xk\frac{{{e^{{-\beta X}}}}}{{-\beta }}-\frac{{k\left( {{e^{{-\beta X}}}-1} \right)}}{{{\beta^2}}}-\frac{{{e^{{-\left( {k+\beta } \right)X}}}-1}}{{\left( {k+\beta } \right)}}+\frac{{{e^{{-\beta X}}}-1}}{\beta }} \right) \hfill \cr \qquad\,\,+\frac{{Xk{e^{{-\beta X}}}+{e^{{-\left( {k+\beta } \right)X}}}-{e^{{-\beta X}}}}}{k} \hfill \cr \qquad\,=\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{k\left( {k+\beta } \right)}}-\frac{{k\left( {{e^{{-\beta X}}}-1} \right)+\beta }}{{k\beta }}=\frac{{\beta {e^{{-\left( {k+\beta } \right)X}}}-\left( {k+\beta } \right){e^{{-\beta X}}}+k}}{{\beta \left( {k+\beta } \right)}} \end{array} $$
(3.20)
Limes calculation \( \rho \to 1 \):
$$ \mathop{\lim}\limits_{{\rho \to 1}}\left( {\frac{{\beta {e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}-\left( {\mu \left( {1-\rho } \right)+\beta } \right){e^{{-\beta X}}}+\mu \left( {1-\rho } \right)}}{{\beta \left( {\mu \left( {1-\rho } \right)+\beta } \right)}}} \right)=\frac{{\beta {e^{{-\beta X}}}-\beta {e^{{-\beta X}}}}}{{{\beta^2}}}=0 $$
(3.21)
Limes calculation \( \rho \to 0 \):
$$ \begin{array}{lllll} \mathop{\lim}\limits_{{\rho \to 0}}\left( {\frac{{\beta {e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}-\left( {\mu \left( {1-\rho } \right)+\beta } \right){e^{{-\beta X}}}+\mu \left( {1-\rho } \right)}}{{\beta \left( {\mu \left( {1-\rho } \right)+\beta } \right)}}} \right) \hfill \\=\frac{{\beta {e^{{-\left( {\mu +\beta } \right)X}}}-\left( {\mu +\beta } \right){e^{{-\beta X}}}+\mu }}{{\beta \left( {\mu +\beta } \right)}}=\frac{{\beta {e^{{-\left( {\mu +\beta } \right)X}}}-\left( {\mu +\beta } \right){e^{{-\beta X}}}}}{{\beta \left( {\mu +\beta } \right)}}+\frac{\mu }{{\beta \left( {\mu +\beta } \right)}} \hfill \\\mathrm{ with}\ {e^{{-\left( {\mu +\beta } \right)X}}}<{e^{{-\beta X}}}\Rightarrow \beta {e^{{-\left( {\mu +\beta } \right)X}}}-\beta {e^{{-\beta X}}}<0 \hfill \\\Rightarrow \frac{{\beta {e^{{-\left( {\mu +\beta } \right)X}}}-\beta {e^{{-\beta X}}}-\mu {e^{{-\beta X}}}}}{{\beta \left( {\mu +\beta } \right)}}+\frac{\mu }{{\beta \left( {\mu +\beta } \right)}}<\frac{\mu }{{\beta \left( {\mu +\beta } \right)}} \end{array} $$
(3.22)
Proof of Proposition 3.2
For the pdf of production lead time being independent of the WAW X, the service level with WAW policy is lower than without WAW policy:
$$ \begin{array}{lllll} \int\limits_0^X {{F_W}\left( \tau \right){f_L}\left( \tau \right)d\tau } +\int\limits_X^{\infty } {{F_W}(X){f_L}\left( \tau \right)d\tau } \mathop{<}\limits^{!}\int\limits_0^{\infty } {{F_W}\left( \tau \right){f_L}\left( \tau \right)d\tau } \hfill \\\Leftrightarrow \int\limits_X^{\infty } {{F_W}(X){f_L}\left( \tau \right)d\tau } \mathop{<}\limits^{!}\int\limits_X^{\infty } {{F_W}\left( \tau \right){f_L}\left( \tau \right)d\tau } \end{array} $$
(3.23)
which holds since any cdf is an increasing function.
For the pdf of production lead time being independent of the WAW X, the expected FGI lead time with WAW policy is lower than without WAW policy:
$$ \begin{array}{lllll} \int\limits_0^X {\int\limits_0^{\theta } {{f_W}\left( \tau \right)\left( {\theta -\tau } \right)d\tau } } {f_L}\left( \theta \right)d\theta +\int\limits_X^{\infty } {\int\limits_0^X {{f_W}\left( \tau \right)\left( {X-\tau } \right)d\tau } } {f_L}\left( \theta \right)d\theta \hfill \\ \qquad\qquad\qquad\qquad\qquad\quad\qquad\mathop{<}\limits^{!}\int\limits_0^{\infty } {\int\limits_0^{\theta } {{f_W}\left( \tau \right)\left( {\theta -\tau } \right)d\tau } } {f_L}\left( \theta \right)d\theta \hfill \\\Leftrightarrow \int\limits_X^{\infty } {\int\limits_0^X {{f_W}\left( \tau \right)\left( {X-\tau } \right)d\tau } } {f_L}\left( \theta \right)d\theta \mathop{<}\limits^{!}\int\limits_X^{\infty } {\int\limits_0^{\theta } {{f_W}\left( \tau \right)\left( {\theta -\tau } \right)d\tau } } {f_L}\left( \theta \right)d\theta \end{array} $$
(3.24)
which holds for any production lead time and customer required lead time distribution since \( \theta \geq X \) on the right hand side of the last line in inequality (3.24).
For the pdf of production lead time being independent of the WAW X, the expected tardiness with WAW policy is higher than without WAW policy:
$$ \begin{array}{lllll} \int\limits_0^X {\int\limits_{\theta}^{\infty } {{f_W}\left( \tau \right)\left( {\tau -\theta } \right)d\tau } } {f_L}\left( \theta \right)d\theta +\int\limits_X^{\infty } {\int\limits_X^{\infty } {{f_W}\left( \tau \right)\left( {\tau -X} \right)d\tau } } {f_L}\left( \theta \right)d\theta \hfill \\ \qquad\qquad\qquad\qquad\qquad\quad\qquad\mathop{>}\limits^{!}\int\limits_0^{\infty } {\int\limits_{\theta}^{\infty } {{f_W}\left( \tau \right)\left( {\tau -\theta } \right)d\tau } } {f_L}\left( \theta \right)d\theta \hfill \\\Leftrightarrow \int\limits_X^{\infty } {\int\limits_X^{\infty } {{f_W}\left( \tau \right)\left( {\tau -X} \right)d\tau } } {f_L}\left( \theta \right)d\theta \mathop{>}\limits^{!}\int\limits_X^{\infty } {\int\limits_{\theta}^{\infty } {{f_W}\left( \tau \right)\left( {\tau -\theta } \right)d\tau } } {f_L}\left( \theta \right)d\theta \end{array} $$
(3.25)
which holds for any production lead time and customer required lead time distribution since \( \theta \geq X \) on the right hand side of the last line in inequality (3.25).
Derivation of Equation (3.13)
$$ \begin{array}{lllll} E[G]=\frac{{\rho \mu \left( {{e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}-1} \right)}}{{\mu \left( {1-\rho } \right)+\beta }}-\frac{{\rho \mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta } \hfill \cr \frac{dE[G] }{{d\rho }}=\frac{{\mu \left( {{e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}-1} \right)}}{{\mu \left( {1-\rho } \right)+\beta }}+\frac{{X\rho {\mu^2}{e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}}}{{\mu \left( {1-\rho } \right)+\beta }}+\frac{{\rho {\mu^2}\left( {{e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}-1} \right)}}{{{{{\left( {\mu \left( {1-\rho } \right)+\beta } \right)}}^2}}} \hfill \cr -\frac{{\mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta }=0 \hfill \cr \Leftrightarrow \frac{{\mu {e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}\left( {1+X\rho \mu } \right)-\mu }}{{\mu \left( {1-\rho } \right)+\beta }}+\frac{{\rho {\mu^2}\left( {{e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}-1} \right)}}{{{{{\left( {\mu \left( {1-\rho } \right)+\beta } \right)}}^2}}}=\frac{{\mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta } \hfill \cr \Leftrightarrow \frac{{\mu {e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}\left( {1+X\rho \mu } \right)}}{{\mu \left( {1-\rho } \right)+\beta }}+\frac{{\rho {\mu^2}{e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}}}{{{{{\left( {\mu \left( {1-\rho } \right)+\beta } \right)}}^2}}}=\frac{{\mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta } \hfill \cr +\frac{\mu }{{\mu \left( {1-\rho } \right)+\beta }}+\frac{{\rho {\mu^2}}}{{{{{\left( {\mu \left( {1-\rho } \right)+\beta } \right)}}^2}}} \end{array} $$
(3.26)
With defining
$$ \begin{array}{lllll} lhs\left( \rho \right)=\frac{{\mu {e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}\left( {1+X\rho \mu } \right)}}{{\mu \left( {1-\rho } \right)+\beta }}+\frac{{\rho {\mu^2}{e^{{-\left( {\mu \left( {1-\rho } \right)+\beta } \right)X}}}}}{{{{{\left( {\mu \left( {1-\rho } \right)+\beta } \right)}}^2}}} \hfill \cr rhs\left( \rho \right)=\frac{{\mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta }+\frac{\mu }{{\mu \left( {1-\rho } \right)+\beta }}+\frac{{\rho {\mu^2}}}{{{{{\left( {\mu \left( {1-\rho } \right)+\beta } \right)}}^2}}} \end{array} $$
(3.27)
it follows that Eq. (3.26) has one unique solution which is a maximum since \( \frac{d}{{d\rho }}lhs\left( \rho \right)>0 \), \( \frac{d}{{d\rho }}rhs\left( \rho \right)>0 \), \( \frac{d}{{{d^2}\rho }}lhs\left( \rho \right)>0 \), \( \frac{d}{{{d^2}\rho }}rhs\left( \rho \right)>0 \) and \( lhs\left( \rho \right){>}rhs\left( \rho \right) \) for \( \rho =0 \) as well as \( lhs\left( \rho \right){<}rhs\left( \rho \right) \) for \( \rho =1 \). For \( \rho =0 \) it follows
$$ lhs(0){>}rhs(0)\Leftrightarrow \frac{{\left( {{e^{{-\left( {\mu +\beta } \right)X}}}-1} \right)}}{{\left( {\mu +\beta } \right)X}}>\frac{{\left( {{e^{{-\beta X}}}-1} \right)}}{{\beta X}} $$
(3.28)
which holds since
$$ \frac{d}{da}\left( {\frac{{{e^{-a }}-1}}{a}} \right)>0\Leftrightarrow -\frac{{{e^{-a }}}}{a}-\frac{{{e^{-a }}-1}}{{{a^2}}}>0\Leftrightarrow {e^a}-1{>}a $$
(3.29)
and for \( \rho =1 \) it holds:
$$ \begin{array}{lllll} lhs(1){<}rhs(1)\Leftrightarrow \frac{{\mu {e^{{-\beta X}}}\left( {1+X\mu } \right)}}{\beta }+\frac{{{\mu^2}\left( {{e^{{-\beta X}}}-1} \right)}}{{{\beta^2}}}<\frac{{\mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta }+\frac{\mu }{\beta } \hfill \cr \Leftrightarrow {e^{{\beta X}}}-1>\beta X \end{array} $$
(3.30)
which is fulfilled with \( {e^x}-1{>}x \).
Derivation of Equations (3.14) and (3.15)
Based on Eqs. (3.6) and (3.9) for the service level in a system without and with WAW policy the following can be stated provided the service level in both systems is equal:
$$ \begin{array}{lllll} \eta =1-\frac{\beta }{{\left( {k+\beta } \right)}}\Leftrightarrow k\left( {1-\eta } \right)-\beta \eta =0 \hfill \cr \mathrm{ and} \hfill \cr {\eta_X}=1-\frac{{{k_X}{e^{{-\left( {{k_X}+\beta } \right)X}}}+\beta }}{{{k_X}+\beta }}\Leftrightarrow {k_X}\left( {1-{\eta_X}} \right)-{k_X}{e^{{-\left( {{k_X}+\beta } \right)X}}}-\beta {\eta_X}=0 \hfill \cr \mathrm{ with}\ \eta ={\eta_X} \hfill \cr \Rightarrow k\left( {1-\eta } \right)-\beta \eta ={k_X}\left( {1-\eta } \right)-{k_X}{e^{{-\left( {{k_X}+\beta } \right)X}}}-\beta \eta \hfill \cr \Leftrightarrow \left( {k-{k_X}} \right)\left( {1-\eta } \right)=-{k_X}{e^{{-\left( {{k_X}+\beta } \right)X}}} \hfill \cr \mathop{\Leftrightarrow}\limits_{\begin{array}{lllll}\scriptstyle k=\mu \left( {1-\rho } \right) \\{k_X}=\mu \left( {1-{\rho_X}} \right) \end{array}}\rho ={\rho_X}+\frac{{1-{\rho_X}}}{{1-\eta }}{e^{{-\left( {\mu \left( {1-{\rho_X}} \right)+\beta } \right)X}}} \end{array}$$
(3.31)
The index X indicates the system applying the WAW policy. The utilization loss can be calculated as:
$$ \Delta \rho =\rho -{\rho_X}=\frac{{1-{\rho_X}}}{{1-\eta }}{e^{{-\left( {\mu \left( {1-{\rho_X}} \right)+\beta } \right)X}}} $$
(3.32)
Based on Eqs. (3.12) and (3.4), the following can be stated for the FGI reduction:
$$ \begin{array}{lllll} \Delta E[G]=E[G]-E[{G_X}]=\frac{{k\mu \rho }}{{k+\beta }}-\frac{{{\rho_X}\mu \left( {{e^{{-\left( {{k_X}+\beta } \right)X}}}-1} \right)}}{{{k_X}+\beta }}+\frac{{{\rho_X}\mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta } \hfill \\=\frac{{{\rho_X}\mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta }+\frac{{\left( {{\rho_X}+\varDelta \rho } \right)\mu k}}{{\beta \left( {k+\beta } \right)}}-\frac{{{\rho_X}\mu \left( {{e^{{-\left( {{k_X}+\beta } \right)X}}}-1} \right)}}{{{k_X}+\beta }} \end{array} $$
(3.33)
Proof of Proposition 3.3
Service level increases with increasing \( k \) (decreases with increasing mean production lead time \( 1/k \)):
$$ \begin{array}{lllll} \eta =1-\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{k+\beta }} \hfill \cr \frac{{d\eta }}{dk }=\frac{{kX{e^{{-\left( {k+\beta } \right)X}}}-{e^{{-\left( {k+\beta } \right)X}}}}}{{k+\beta }}+\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{{{{\left( {k+\beta } \right)}}^2}}}\mathop{>}\limits^{!}0 \hfill \cr \Rightarrow {k^2}X{e^{{-\left( {k+\beta } \right)X}}}+\beta kX{e^{{-\left( {k+\beta } \right)X}}}+\beta \mathop{>}\limits^{!}\beta {e^{{-\left( {k+\beta } \right)X}}} \end{array} $$
(3.34)
which is fulfilled with \( \beta >\beta {e^{{-\left( {k+\beta } \right)X}}} \) and \( {k^2}X{e^{{-\left( {k+\beta } \right)X}}}+\beta kX{e^{{-\left( {k+\beta } \right)X}}}>0 \).
The expected FGI lead time increases with increasing \( k \) (decreases with increasing mean production lead time \( 1/k \)):
$$ \begin{array}{lllll} E\left[ I \right]=\frac{{\beta {e^{{-\left( {k+\beta } \right)X}}}-\left( {k+\beta } \right){e^{{-\beta X}}}+k}}{{\beta \left( {k+\beta } \right)}} \hfill \cr \frac{{dE\left[ I \right]}}{dk }=\frac{{1-{e^{{-\beta X}}}-X\beta {e^{{-\left( {k+\beta } \right)X}}}}}{{\beta \left( {k+\beta } \right)}}-\frac{{k+\beta {e^{{-\left( {k+\beta } \right)X}}}-\left( {k+\beta } \right){e^{{-\beta X}}}}}{{\beta {{{\left( {k+\beta } \right)}}^2}}}\mathop{>}\limits^{!}0 \hfill \cr \Rightarrow 1-\left( {k+\beta } \right)X{e^{{-\left( {k+\beta } \right)X}}}-{e^{{-\left( {k+\beta } \right)X}}}\mathop{>}\limits^{!}0\Rightarrow {e^{{\left( {k+\beta } \right)X}}}\mathop{>}\limits^{!}1+\left( {k+\beta } \right)X \end{array} $$
(3.35)
which is fulfilled with \( {e^x}-1{>}x \).
The expected tardiness decreases with increasing \( k \) (increases with increasing mean production lead time \( 1/k \)):
$$ \begin{array}{lllll} E\left[ C \right]=\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{k\left( {k+\beta } \right)}} \hfill \cr \frac{{dE\left[ C \right]}}{dk }=\frac{{{e^{{-\left( {k+\beta } \right)X}}}-kX{e^{{-\left( {k+\beta } \right)X}}}}}{{k\left( {k+\beta } \right)}}-\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{k{{{\left( {k+\beta } \right)}}^2}}}-\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{{k^2}\left( {k+\beta } \right)}}\mathop{<}\limits^{!}0 \hfill \cr \Rightarrow \frac{{\left( {k+\beta } \right)k{e^{{-\left( {k+\beta } \right)X}}}-\left( {k+\beta } \right){k^2}X{e^{{-\left( {k+\beta } \right)X}}}}}{{{k^2}{{{\left( {k+\beta } \right)}}^2}}}-\frac{{{k^2}{e^{{-\left( {k+\beta } \right)X}}}+k\beta }}{{{k^2}{{{\left( {k+\beta } \right)}}^2}}} \hfill \cr -\frac{{\left( {k+\beta } \right)k{e^{{-\left( {k+\beta } \right)X}}}+\left( {k+\beta } \right)\beta }}{{{k^2}{{{\left( {k+\beta } \right)}}^2}}}\mathop{<}\limits^{!}0 \hfill \cr \Rightarrow -{k^3}X{e^{{-\left( {k+\beta } \right)X}}}-\beta {k^2}X{e^{{-\left( {k+\beta } \right)X}}}-{k^2}{e^{{-\left( {k+\beta } \right)X}}}-2k\beta -{\beta^2}\mathop{<}\limits^{!}0 \end{array} $$
(3.36)
which is fulfilled with \( k,X,\beta >0 \).
The expected FGI increases with increasing \( k \) (decreases with increasing mean production lead time \( 1/k \)):
$$ \begin{array}{lllll} E\left[ G \right]=\frac{{\lambda \left( {{e^{{-\left( {k+\beta } \right)X}}}-1} \right)}}{{k+\beta }}-\frac{{\lambda \left( {{e^{{-\beta X}}}-1} \right)}}{\beta } \hfill \cr \frac{{dE\left[ G \right]}}{dk }=-\frac{{\lambda \left( {{e^{{-\left( {k+\beta } \right)X}}}-1} \right)}}{{{{{\left( {k+\beta } \right)}}^2}}}-\frac{{X\lambda {e^{{-\left( {k+\beta } \right)X}}}}}{{k+\beta }}\mathop{>}\limits^{!}0 \hfill \cr \Rightarrow -\lambda \left( {{e^{{-\left( {k+\beta } \right)X}}}-1} \right)-\left( {k+\beta } \right)X\lambda {e^{{-\left( {k+\beta } \right)X}}}\mathop{>}\limits^{!}0 \hfill \cr \Rightarrow {e^{{\left( {k+\beta } \right)X}}}\mathop{>}\limits^{!}1+\left( {k+\beta } \right)X \end{array} $$
(3.37)
which is fulfilled with \( {e^x}-1{>}x \).
3.1 Service Level, FGI, FGI Lead Time, and Tardiness with Respect to Expected WIP
With \( k=\frac{\mu }{{1+E\left[ Y \right]}} \) and \( \rho =\frac{{E\left[ Y \right]}}{{1+E\left[ Y \right]}} \) from Eq. (3.1), the Eqs. (3.9, 3.10, 3.11 and 3.12) can be restated with respect to expected WIP when \( \mu \) is predefined:
$$ \begin{array}{lllll} \eta =1-\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{k+\beta }}=1-\frac{{\frac{\mu }{{1+E\left[ Y \right]}}{e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}+\beta }}{{\frac{\mu }{{1+E\left[ Y \right]}}+\beta }} \hfill \cr =1-\frac{{\mu {e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}+\beta \left( {1+E\left[ Y \right]} \right)}}{{\mu +\beta \left( {1+E\left[ Y \right]} \right)}} \end{array} $$
(3.38)
$$ \begin{array}{lllll} E[I] =\frac{{\beta {e^{{-\left( {k+\beta } \right)X}}}-\left( {k+\beta } \right){e^{{-\beta X}}}+k}}{{\beta \left( {k+\beta } \right)}} \hfill \cr =\frac{{\beta \left( {1+E\left[ Y \right]} \right){e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}-\left( {\mu +\beta \left( {1+E\left[ Y \right]} \right)} \right){e^{{-\beta X}}}+\mu }}{{\beta \left( {\mu +\beta \left( {1+E\left[ Y \right]} \right)} \right)}} \end{array} $$
(3.39)
$$ E[C]=\frac{{k{e^{{-\left( {k+\beta } \right)X}}}+\beta }}{{k\left( {k+\beta } \right)}}=\frac{{\mu \left( {1+E\left[ Y \right]} \right){e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}+\beta {{{\left( {1+E\left[ Y \right]} \right)}}^2}}}{{\mu \left( {\mu +\beta \left( {1+E\left[ Y \right]} \right)} \right)}} $$
(3.40)
$$ \begin{array}{lllll} E[G] =\frac{{\rho \mu \left( {{e^{{-\left( {k+\beta } \right)X}}}-1} \right)}}{{k+\beta }}-\frac{{\rho \mu \left( {{e^{{-\beta X}}}-1} \right)}}{\beta } \hfill \cr =\frac{{E\left[ Y \right]\mu \left( {{e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}-1} \right)}}{{\mu +\beta \left( {1+E\left[ Y \right]} \right)}}-\frac{{E\left[ Y \right]\mu \left( {{e^{{-\beta X}}}-1} \right)}}{{\beta \left( {1+E\left[ Y \right]} \right)}} \end{array} $$
(3.41)
Taking the first derivative of service level with respect to expected WIP leads to:
$$ \begin{array}{lllll} \frac{{d\eta }}{{dE\left[ Y \right]}}= \hfill \cr -\frac{{\left( {{\mu^2}X+\left( {1+E\left[ Y \right]} \right)\beta \left( {\left( {1+E\left[ Y \right]} \right)\left( {{e^{{\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}-1} \right)+\mu X} \right)} \right)}}{{{{{\left( {1+E\left[ Y \right]} \right)}}^2}{{{\left( {\beta \left( {1+E\left[ Y \right]} \right)+\mu } \right)}}^2}}}\mu {e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}} \end{array} $$
(3.42)
which is negative because \( {e^{{\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}>1 \).
The first derivative of expected FGI lead time with respect to expected WIP is negative as well:
$$ \begin{array}{lllll} \frac{dE[I] }{{dE\left[ Y \right]}}= \hfill \cr \frac{{\left( {\left( {1+E\left[ Y \right]} \right)\left( {1-{e^{{\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}} \right)+X\beta +E\left[ Y \right]X\beta +X\mu } \right)}}{{\left( {1+E\left[ Y \right]} \right){{{\left( {\mu +\beta \left( {1+E\left[ Y \right]} \right)} \right)}}^2}}}\mu {e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}\mathop{<}\limits^{!}0 \hfill \cr \Leftrightarrow \left( {1+E\left[ Y \right]} \right)\left( {X\beta +\frac{{X\mu }}{{1+E\left[ Y \right]}}+1-{e^{{\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}} \right)\mathop{<}\limits^{!}0 \hfill \cr \Leftrightarrow {e^{{\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}-1\mathop{>}\limits^{!}\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X \end{array} $$
(3.43)
which holds since \( {e^x}-1> x \).
The first derivative of the expected tardiness with respect to expected WIP is positive:
$$ \begin{array}{lllll} \frac{dE[C] }{{dE\left[ Y \right]}} =\frac{{{{{\left( {1+E\left[ Y \right]} \right)}}^3}{\beta^2}+{e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}{\mu^2}\left( {1+E\left[ Y \right]+X\mu } \right)}}{{\left( {1+E\left[ Y \right]} \right)\mu {{{\left( {\mu +\beta \left( {1+E\left[ Y \right]} \right)} \right)}}^2}}} \hfill \cr +\frac{{2\beta \left( {1+E\left[ Y \right]} \right)+X\mu \beta {e^{{-\left( {\frac{\mu }{{1+E\left[ Y \right]}}+\beta } \right)X}}}}}{{{{{\left( {\mu +\beta \left( {1+E\left[ Y \right]} \right)} \right)}}^2}}}>0 \end{array} $$
(3.44)
