The stochastic guaranteed service model with recourse for multi-echelon warehouse management

Abstract

The guaranteed service model (GSM) computes optimal order-points in multi-echelon inventory control under the assumptions that delivery times can be guaranteed and the demand is bounded. Our new stochastic guaranteed service model (SGSM) with Recourse covers also scenarios that violate these assumptions. Simulation experiments on real-world data of a large German car manufacturer show that policies based on the SGSM dominate GSM-policies.

Appendices

Appendix 1: Proof of Theorem 1

We repeat the theorem here for convenience.

Theorem 1

Let $(\varXi , p)$ be a finite lead time/demand distribution with the total-order property and positive demands. Let $n^{\mathrm{target}}$ be a target service level. Moreover, let $\bigl ((s^{in})^{\mathrm{GSM}}, (s^{in})^{\mathrm{GSM}}, x^{\mathrm{GSM}}, y^{\mathrm{GSM}}\bigr )$ be optimal for the GSM with induced lead times $L^*_i \ge 0$ and demand rates $\alpha ^*_i > 0$. Then there are marginal expediting costs $t_i$ and marginal outsourcing costs $c_i$ such that for the corresponding SGSM the following solution induced by the GSM is optimal:

$$\begin{aligned} (s_i^{in})^{\mathrm{SGSM}}&= (s_i^{in})^{\mathrm{GSM}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall i \in N(G),\end{aligned}$$

(67)

$$\begin{aligned} (s_i^{out})^{\mathrm{SGSM}}&= (s_i^{out})^{\mathrm{GSM}}\,\,\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall i \in N(G),\end{aligned}$$

(68)

$$\begin{aligned} x_i^{\mathrm{SGSM}}&= x_i^{\mathrm{GSM}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall i \in N(G),\end{aligned}$$

(69)

$$\begin{aligned} y_i^{\mathrm{SGSM}}&= y_i^{\mathrm{GSM}}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall i \in N(G),\end{aligned}$$

(70)

$$\begin{aligned} (r_i^{\omega })^{\mathrm{SGSM}}&= \max \bigl (0, L_i^{\omega } - x_i^{\mathrm{SGSM}} + (s_i^{in})^{\mathrm{SGSM}} - (s_i^{out})^{\mathrm{SGSM}}\bigr ) \qquad \forall i \in N(G),\qquad \end{aligned}$$

(71)

$$\begin{aligned} (q_i^{\omega })^{\mathrm{SGSM}}&= \max \bigl (0, \alpha _i^{\omega } x_i^{\mathrm{SGSM}} - y_i^{\mathrm{SGSM}}\bigr ) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall i \in N(G). \end{aligned}$$

(72)

Proof

We prove the assertion by constructing marginal expediting and outsourcing costs from the complementary slackness condition of a primal-dual optimal solution to the GSM. From this, we construct a primal-dual solution to the corresponding SGSM that satisfies complementary slackness condition of the SGSM.

We first list the GSM and its dual DGSM with the assumptions of this section (no integrality constraints, constant demand rates), where all variables appear on the left-hand side and all constants on the right-hand side. With this, the GSM reads as follows:

$$\begin{aligned} \min&\!\!\! \sum _{{i \in N(G)}} h_i y_i \end{aligned}$$

(73)

$$\begin{aligned} -s_i^{\mathrm{out}}&\ge -\bar{s}_i^{\mathrm{out}} \qquad \forall i \in D(G),\end{aligned}$$

(74)

$$\begin{aligned} s_j^{\mathrm{in}} {} - s_i^{\mathrm{out}}&\ge 0 \ \qquad \qquad \forall {ij}\in A(G),\end{aligned}$$

(75)

$$\begin{aligned} -s_i^{\mathrm{in}} {} + s_i^{\mathrm{out}} {} + x_i&\ge L^*_i \ \qquad \quad \forall i \in N(G),\end{aligned}$$

(76)

$$\begin{aligned} -\alpha ^*_i x_i {} + y_i&\ge 0 \ \qquad \qquad \forall i\in N(G),\end{aligned}$$

(77)

$$\begin{aligned} s_i^{\mathrm{in}}, s_i^{\mathrm{out}}, x_i, y_i&\ge 0 \ \qquad \qquad \forall i\in N(G). \end{aligned}$$

(78)

With dual variables $\pi _i$, $\rho _{{ij}}$, $\sigma _i$, and $\tau _i$ corresponding in that order to the four sets of restrictions, the dual DGSM of the GSM, with restrictions ordered according to $s_i^{\mathrm{in}}$, $s_i^{\mathrm{out}}$, $x_i$, $y_i$ reads as follows:

$$\begin{aligned}&\max \sum _{{i \in D(G)}} (-\bar{s}_i^{\mathrm{out}}) \pi _i + \sum _{{i \in N(G)}} L_i^* \sigma _i \end{aligned}$$

(79)

$$\begin{aligned} \sum _{{j: {ji} \in A(G)}} \rho _{{ji}} {} - \sigma _i&\le 0 \qquad \forall i \in N(G),\end{aligned}$$

(80)

$$\begin{aligned} -\pi _i {} - \sum _{{j: {ij} \in A(G)}} \rho _{{ij}} {} + \sigma _i&\le 0 \ \qquad \forall i \in N(G),\end{aligned}$$

(81)

$$\begin{aligned} \sigma _i {} - \alpha _i^* \tau _i&\le 0 \ \qquad \forall i \in N(G),\end{aligned}$$

(82)

$$\begin{aligned} \tau _i&\le h_i \ \qquad \forall i \in N(G),\end{aligned}$$

(83)

$$\begin{aligned} \pi _i, \rho _{{ij}}, \sigma _i, \tau _i&\ge 0 \ \qquad \forall i\in N(G), \nonumber \\ \ \forall {ij} \in A(G). \end{aligned}$$

(84)

From this we derive the optimality conditions via complementary slackness in primal-dual pairs of feasible solutions:

$$\begin{aligned} \pi _i \bigl ( - s_i^{\mathrm{out}} + \bar{s}_i^{\mathrm{out}} \bigr )&= 0 \qquad \forall i \in D(G),\end{aligned}$$

(85)

$$\begin{aligned} \rho _{{ij}} \bigl ( s_j^{\mathrm{in}} - s_i^{\mathrm{out}} \bigr )&= 0 \qquad \forall {ij} \in A(G),\end{aligned}$$

(86)

$$\begin{aligned} \sigma _i \bigl ( x_i - s_i^{\mathrm{in}} + s_i^{\mathrm{out}} - L^*_i \bigr )&= 0 \qquad \forall i \in N(G),\end{aligned}$$

(87)

$$\begin{aligned} \tau _i\bigl ( -\alpha ^*_i x_i + y_i \bigr )&= 0\qquad \forall i \in N(G),\end{aligned}$$

(88)

$$\begin{aligned} s_i^{\mathrm{in}} \left( \; \sum _{{j: {ji} \in A(G)}} \rho _{{ji}} - \sigma _i \right)&= 0\qquad \forall i \in N(G),\end{aligned}$$

(89)

$$\begin{aligned} s_i^{\mathrm{out}} \left( -\pi _i - \sum _{{j: {ij} \in A(G)}} \rho _{{ij}} + \sigma _i \right)&= 0\qquad \forall i \in N(G),\end{aligned}$$

(90)

$$\begin{aligned} x_i \bigl ( \sigma _i - \alpha _i^* \tau _i \bigr )&= 0\qquad \forall i \in N(G),\end{aligned}$$

(91)

$$\begin{aligned} y_i \bigl ( \tau _i - h_i \bigr )&= 0 \qquad \forall i \in N(G). \end{aligned}$$

(92)

Next we do the same for the SGSM. The primal reads as follows:

$$\begin{aligned}&\min \sum _{{i \in N(G)}} h_i y_i {} + {\mathop {\mathop {\sum }\limits _{\omega \in \varOmega }}\limits _{i \in N(G)}} \bigl (p^{\omega } t_i r_i^{\omega } {} + p^{\omega } c_i q_i^{\omega }\bigr )\end{aligned}$$

(93)

$$\begin{aligned} - s_i^{\mathrm{out}}&\ge -\bar{s}_i^{\mathrm{out}} \ \quad \forall i \in D(G),\end{aligned}$$

(94)

$$\begin{aligned} s_j^{\mathrm{in}} {} - s_i^{\mathrm{out}}&\ge 0 \ \qquad \quad \forall {ij}\in A(G),\end{aligned}$$

(95)

$$\begin{aligned} - s_i^{\mathrm{in}} {} + s_i^{\mathrm{out}} {} + x_i {} + r_i^{\omega }&\ge L^*_i\ \qquad \forall i \in N(G),\nonumber \\ \ \forall \omega \in \varOmega ,\end{aligned}$$

(96)

$$\begin{aligned} -\alpha ^{\omega }_i x_i {} + y_i {} + q_i^{\omega }&\ge 0 \ \quad \qquad \forall i\in N(G),\nonumber \\ \ \qquad \quad \forall \omega \in \varOmega ,\end{aligned}$$

(97)

$$\begin{aligned} s_i^{\mathrm{in}}, s_i^{\mathrm{out}}, x_i, y_i, r_i^{\omega } , q_i^{\omega }&\ge 0 \ \qquad \quad \forall i\in N(G),\end{aligned}$$

(98)

$$\begin{aligned} \ \forall \omega \in \varOmega . \end{aligned}$$

(99)

With dual variables $\pi _i$, $\rho _{{ij}}$, $\sigma _i^{\omega }$, and $\tau _i^{\omega }$ corresponding in that order to the four sets of restrictions, the dual DSGSM of the SGSM, with restrictions ordered according to $s_i^{\mathrm{in}}$, $s_i^{\mathrm{out}}$, $x_i$, $y_i$, $r_i^{\omega }$, $q_i^{\omega }$, reads as follows:

$$\begin{aligned} \max&\sum _{{i \in D(G)}} (-\bar{s}_i^{\mathrm{out}}) \pi _i {} + \sum _{{i \in N(G)}} L_i^* \sigma _i^{\omega } \end{aligned}$$

(100)

$$\begin{aligned} \sum _{{j: {ji} \in A(G)}} \rho _{{ji}} {} - \sum _{{\omega \in \varOmega }} \sigma _i^{\omega }&\le 0 \ \qquad \forall i \in N(G),\end{aligned}$$

(101)

$$\begin{aligned} -\pi _i {} - \sum _{{j: {ij} \in A(G)}} \rho _{{ij}} {} + \sum _{{\omega \in \varOmega }} \sigma _i^{\omega }&\le 0 \ \qquad \forall i \in N(G),\end{aligned}$$

(102)

$$\begin{aligned} \sum _{{\omega \in \varOmega }} \sigma _i^{\omega } {} - \sum _{{\omega \in \varOmega }} \alpha _i^{\omega } \tau _i^{\omega }&\le 0 \quad \ \forall i \in N(G),\end{aligned}$$

(103)

$$\begin{aligned} \sum _{{\omega \in \varOmega }} \tau _i^{\omega }&\le h_i \ \qquad \forall i \in N(G),\end{aligned}$$

(104)

$$\begin{aligned} \sigma _i^{\omega }&\le p^{\omega } t_i \ \quad \forall i \in N(G),\end{aligned}$$

(105)

$$\begin{aligned} \ \forall \omega \in \varOmega ,\end{aligned}$$

(106)

$$\begin{aligned} \tau _i^{\omega }&\le p^{\omega } c_i \ \qquad \forall i \in N(G),\end{aligned}$$

(107)

$$\begin{aligned} \ \forall \omega \in \varOmega ,\end{aligned}$$

(108)

$$\begin{aligned} \pi _i, \rho _{{ij}}, \sigma _i^{\omega }, \tau _i^{\omega }&\ge 0 \ \qquad \quad \forall i\in N(G),\nonumber \\ \ \qquad \qquad \qquad \forall {ij} \in A(G),\nonumber \\ \ \qquad \qquad \qquad \forall \omega \in \varOmega . \end{aligned}$$

(109)

The resulting optimality conditions for the SGSM are as follows:

$$\begin{aligned} \pi _i \bigl ( - s_i^{\mathrm{out}} + \bar{s}_i^{\mathrm{out}} \bigr )&= 0 \qquad \forall i \in D(G),\end{aligned}$$

(110)

$$\begin{aligned} \rho _{{ij}} \bigl ( s_j^{\mathrm{in}} - s_i^{\mathrm{out}} \bigr )&= 0 \qquad \forall {ij} \in A(G),\end{aligned}$$

(111)

$$\begin{aligned} \sigma _i^{\omega } \bigl (x_i - s_i^{\mathrm{in}} + s_i^{\mathrm{out}} + r_i^{\omega } - L^{\omega }_i \bigr )&= 0 \qquad \forall i \in N(G), \omega \in \varOmega ,\end{aligned}$$

(112)

$$\begin{aligned} \tau _i^{\omega } \bigl (-\alpha ^{\omega }_i x_i + y_i + q_i^{\omega } \bigr )&= 0 \qquad \forall i \in N(G), \omega \in \varOmega ,\end{aligned}$$

(113)

$$\begin{aligned} s_i^{\mathrm{in}} \big ( \; \sum _{{j: {ji} \in A(G)}} \rho _{{ji}} - \sum _{{\omega \in \varOmega }} \sigma _i^{\omega } \big )&= 0 \qquad \forall i \in N(G),\end{aligned}$$

(114)

$$\begin{aligned} s_i^{\mathrm{out}} \left( -\pi _i - \sum _{{j: {ij} \in A(G)}} \rho _{{ij}} + \sum _{{\omega \in \varOmega }} \sigma _i^{\omega } \right)&= 0 \qquad \forall i \in N(G),\end{aligned}$$

(115)

$$\begin{aligned} x_i \left( \; \sum _{{\omega \in \varOmega }} \sigma _i^{\omega } - \sum _{{\omega \in \varOmega }}\alpha _i^{\omega } \tau _i^{\omega } \right)&= 0 \qquad \forall i \in N(G),\end{aligned}$$

(116)

$$\begin{aligned} y_i \left( \; \sum _{{\omega \in \varOmega }} \tau _i^{\omega } - h_i \right)&= 0\qquad \forall i \in N(G),\end{aligned}$$

(117)

$$\begin{aligned} r_i^{\omega } \bigl ( \sigma _i^{\omega } - p^{\omega } t_i \bigr )&= 0\qquad \forall i \in N(G), \omega \in \varOmega ,\end{aligned}$$

(118)

$$\begin{aligned} q_i^{\omega } \bigl ( \tau _i^{\omega } - p^{\omega } c_i \bigr )&= 0 \qquad \forall i \in N(G), \omega \in \varOmega . \end{aligned}$$

(119)

Consider now a primal-dual pair of optimal solutions to the GSM and DGSM, respectively, with target service level $n^{\mathrm{target}} > 0$ and actual service level $n^* > 0$, denoted by

$$\begin{aligned} \bigl ( x^{\mathrm{GSM}}, y^{\mathrm{GSM}}, (s^{in})^{\mathrm{GSM}}, (s^{in})^{\mathrm{GSM}} \bigr ), \quad \bigl ( \pi ^{\mathrm{GSM}}, \rho ^{\mathrm{GSM}}, \sigma ^{\mathrm{GSM}}, \tau ^{\mathrm{GSM}} \bigr ). \end{aligned}$$

(120)

Case 1: $n^* = 1$. In this case, the lead times and demand rates in all scenarios are bounded by $L_i^*$ and $\alpha _i^*$, respectively. We claim that for all $c_i >\frac{h_i}{p^{\omega ^*}}$ and $t_i > \alpha _i^* c_i$ the given SGSM solution is optimal. We show first, that the SGSM optimality conditions imply that $r_i^{\omega } = 0$ and $q_i^{\omega } = 0$ for all $\omega \in \varOmega$ and all $i \in N(G)$.

Indeed: Assume, for the sake of contradiction, that there is an optimal solution to the SGSM/DSGSM with $q_i^{\omega } > 0$. Then, by the total-order property, $q_i^{\omega ^*} > 0$. Equation (119) implies $\tau _i^{\omega } = p^{\omega ^*} c_i$. Therefore, we have

$$\begin{aligned} \sum _{{\omega \in \varOmega }} \tau _i^{\omega } \ge \tau _i^{\omega ^*} = p^{\omega ^*} c_i > h_i, \end{aligned}$$

(121)

which contradicts the feasibility of $\tau _i^{\omega ^*}$ in DSGSM. Thus, $q_i^{\omega } = 0$ for all $\omega \in \varOmega$.

Assume next, for the sake of contradiction, that there is an optimal solution to the SGSM/DSGSM with $r_i^{\omega } > 0$. Again, this implies that $r_i^{\omega ^*} > 0$. Then Eq. (118), $\alpha _i^* \ge \alpha _i^{\omega }$, and the feasibility of $\tau _i^{\omega }$ in DSGSM imply the following:

$$\begin{aligned} \sum _{{\omega \in \varOmega }} \sigma _i^{\omega } - \sum _{{\omega \in \varOmega }} \alpha _i^{\omega } \tau _i^{\omega }&\ge \sigma _i^{\omega ^*} - \sum _{{\omega \in \varOmega }} \alpha _i^{\omega } \tau _i^{\omega }\end{aligned}$$

(122)

$$\begin{aligned}&\ge p^{\omega ^*} t_i - \alpha _i^* \sum _{{\omega \in \varOmega }} \tau _i^{\omega }\end{aligned}$$

(123)

$$\begin{aligned}&\ge p^{\omega ^*} t_i -\alpha _i^* h_i\end{aligned}$$

(124)

$$\begin{aligned}&> p^{\omega ^*} \alpha _i^* \frac{h_i}{p^{\omega ^*}} - \alpha _i^* h_i\end{aligned}$$

(125)

$$\begin{aligned}&= 0, \end{aligned}$$

(126)

which this time contradicts the feasibility of $\sigma _i^{\omega ^*}$ in DSGSM.

Thus, in the SGSM with the given marginal costs for expediting and outsourcing, expediting and outsourcing quantities $r_i^{\omega }$ and $q_i^{\omega }$, respectively, can be fixed to zero. The resulting SGSM is identical to the GSM. Hence, the SGSM solution from the assertion, whose first-stage part equals an optimal GSM solution, is optimal.

Case 2: $0 < n^* < 1$. As an abbreviation for the following, we define

$$\begin{aligned} \bar{n} := \sum _{{\omega > \omega ^*}} p^{\omega } = 1 - n^* > 0 \quad \text {and} \quad \bar{\alpha }_i := \sum _{{\omega > \omega ^*}} \alpha _i^{\omega } p^{\omega } > 0. \end{aligned}$$

(127)

Define marginal costs for expediting and outsourcing as follows:

$$\begin{aligned} c_i := \frac{\tau ^{\mathrm{GSM}}}{\bar{n}} \quad \text {and}\quad t_i := \frac{\bar{\alpha }_i}{\bar{n}} c_i. \end{aligned}$$

(128)

Note that the definition of $c_i$ corresponds to the standard penalty to model chance constraints with stochastic right-hand side, whereas the definition of $t_i$ is different because we have to keep under control the stochastic coefficient $\alpha _i^{\omega }$ in front of $x_i$.

The SGSM solution from the assertion of the theorem is obviously feasible for the SGSM. We claim that the following is a solution to the DSGM, which together with the given SGSM solution satisfies the optimality conditions for the SGSM:

$$\begin{aligned} \pi _i^{\mathrm{SGSM}}&:= \frac{\bar{\alpha }_i}{\alpha _i^* \bar{n}} \pi _i^{\mathrm{GSM}} \qquad \qquad \quad \,\,\,\,\forall i \in N(G),\end{aligned}$$

(129)

$$\begin{aligned} \rho _{{ij}}^{\mathrm{SGSM}}&:= \frac{\bar{\alpha }_i}{\alpha _i^* \bar{n}} \rho _{{ij}}^{\mathrm{GSM}} \qquad \qquad \quad \,\,\,\, \forall i \in N(G),\end{aligned}$$

(130)

$$\begin{aligned} (\sigma _i^{\omega })^{\mathrm{SGSM}}&:= \left\{ \begin{array}{ll} 0 &{} \text {if } \omega \le \omega ^*,\\ p^{\omega } t_i &{} \text {if } \omega > \omega ^*; \end{array}\right. \qquad \forall i \in N(G),\end{aligned}$$

(131)

$$\begin{aligned} (\tau _i^{\omega })^{\mathrm{SGSM}}&:= \left\{ \begin{array}{ll} 0 &{} \text {if } \omega \le \omega ^*,\\ p^{\omega } c_i &{} \text {if } \omega > \omega ^*; \end{array}\right. \qquad \forall i \in N(G). \end{aligned}$$

(132)

Since $(\sigma _i^{\omega })^{\mathrm{SGSM}}$ and $(\tau _i^{\omega })^{\mathrm{SGSM}}$ are only positive for $\omega > \omega ^*$, the validity of the SGSM optimality Eqs. (112) and (113) follows from the definitions of $(r_i^{\omega })^{\mathrm{SGSM}}$ and $(q_i^{\omega })^{\mathrm{SGSM}}$ and the validity of the GSM optimality Eqs. (87) and (88).

The validity of the SGSM optimality Eqs. (118) and (119) follows directly from the definitions of $(\sigma _i^{\omega })^{\mathrm{SGSM}}$ and $(\tau _i^{\omega })^{\mathrm{SGSM}}$.

Furthermore, we have

$$\begin{aligned} \sum _{{\omega \in \varOmega }} (\tau _i^{\omega })^{\mathrm{SGSM}} = \sum _{{\omega > \omega ^*}} (\tau _i^{\omega })^{\mathrm{SGSM}} = \sum _{{\omega > \omega ^*}} p^{\omega } c_i = \sum _{{\omega > \omega ^*}} p^{\omega } \frac{\tau ^{\mathrm{GSM}}}{\bar{n}} = \tau ^{\mathrm{GSM}}. \end{aligned}$$

(133)

Thus, since $y_i^{\mathrm{SGSM}} = y_i^{\mathrm{GSM}}$, the validity of all the remaining SGSM optimality Eqs. (117) containing $\sum _{\omega \in \varOmega } (\tau _i^{\omega })^{\mathrm{SGSM}}$ follows from the validity of the corresponding GSM optimality Eqs. (92) with $\tau _i^{\mathrm{GSM}}$. Moreover:

$$\begin{aligned} \sum _{{\omega \in \varOmega }} (\sigma _i^{\omega })^{\mathrm{SGSM}} = \sum _{{\omega > \omega ^*}} (\sigma _i^{\omega })^{\mathrm{SGSM}} = \sum _{{\omega > \omega ^*}} p^{\omega } t_i = \bar{n} t_i = \bar{\alpha }_i c_i = \sum _{{\omega > \omega ^*}} \alpha _i^{\omega } (\tau _i^{\omega })^{\mathrm{SGSM}}. \end{aligned}$$

(134)

This proves the validity of the SGSM optimality Eqs. (116).

On the other hand, for all $i \in N(G)$:

$$\begin{aligned} \sum _{{\omega \in \varOmega }} (\sigma _i^{\omega })^{\mathrm{SGSM}} = \bar{\alpha }_i c_i = \bar{\alpha }_i \frac{\bar{n}}{\bar{n}} c_i = \frac{\bar{\alpha }_i}{\bar{n}} (\bar{n} c_i) = \frac{\bar{\alpha }_i}{\bar{n}} \tau _i^{\mathrm{GSM}} = \frac{\bar{\alpha }_i}{\alpha _i^* \bar{n}} \sigma _i^{\mathrm{GSM}}. \end{aligned}$$

(135)

Moreover, recall that for all $i \in N(G)$ we have defined:

$$\begin{aligned} \pi _i^{\mathrm{SGSM}}&= \frac{\bar{\alpha }_i}{\alpha _i^* \bar{n}} \pi _i^{\mathrm{GSM}}\end{aligned}$$

(136)

$$\begin{aligned} \rho _{{ij}}^{\mathrm{SGSM}}&= \frac{\bar{\alpha }_i}{\alpha _i^* \bar{n}} \rho _{{ij}}^{\mathrm{GSM}}. \end{aligned}$$

(137)

Now, scale by $\frac{\bar{\alpha }_i}{\alpha _i^* \bar{n}} > 0$ the optimal GSM solutions in the homogeneous GSM optimality Eqs. (89) and (90) as well as (85), (86) and (87). In the resulting valid equations, using (135), (136), and (137), substitute $\frac{\alpha }{\alpha _i^* \bar{n}} \sigma _i^{\mathrm{GSM}}$ by $\sum _{\omega \in \varOmega } (\sigma _i^{\omega })^{\mathrm{SGSM}}$, $\frac{\bar{\alpha }_i}{\alpha _i^* \bar{n}} \pi _i^{\mathrm{GSM}}$ by $\pi _i^{\mathrm{SGSM}}$, and $\frac{\bar{\alpha }_i}{\alpha _i^* \bar{n}} \rho _{{ij}}^{\mathrm{GSM}}$ by $\rho _{{ij}}^{\mathrm{SGSM}}$. The resulting valid equations are exactly the corresponding SGSM optimality equations with the SGSM/DSGSM values that we have defined. Therefore, the SGSM optimality Eqs. (114) and (115) as well as (110), (111) and (112) are satisfied for the given DSGSM solution. Analogously, we can show that the given solution to the DSGSM is feasible for the DSGSM. Consequently, the asserted SGSM solution is optimal.$\square$

Appendix 2: Results of the individual simulation runs

In this section we show some of the numerical results in detail. The average costs of the ten simulation runs listed here are given in Tables 1 and 3. The Tables 9, 10, 11 and 12 include the results that lead to the average costs of Table 2. Tables 13 and 14 include the results for the single runs of GSM 96 % (4) and SGSM $200\rightarrow 50$, weeks, asym (10) of Table 3.

Table 9 No reduction ($3\rightarrow 3$)

Table 10 Symmetric reduction ($50\rightarrow 3$)

Table 11 Asymmetric reduction ($50\rightarrow 3$)

Table 12 No reduction ($50\rightarrow 50$)

Table 13 GSM with a prescribed service level of 96 %

Table 14 SGSM($200\rightarrow 3$) asymmetric reduction with time discretization in weeks