Risk-pooling strategy, lead time, delivery reliability and inventory control decisions in a stochastic multi-objective supply chain network design

Abstract

A three-echelon supply chain network design is considered and a multi-objective mixed integer non-linear problem is modeled. Risk-pooling, lead time between facilities and the concept of the reliability are used to modeling. The objectives of the model are minimization of the supply chain’s total cost and maximization the delivery reliability, which can be equal to minimize the risk of not delivering the products with regard to stochastic demand. To solve the model a heuristic hierarchical algorithm is presented. Numerical examples are considered to test and verify the Pareto front. The results show the effectiveness and efficiency of the heuristic algorithm.

Appendix

Here, Eq. (3.10) is proved which is used in the first objective function of the model and shows the annual inventory cost at each manufacturer, $H_i$ based on the economic production quantity (EPQ).

Equation (8.1) implies the optimal order quantity for each manufacturer based on the EPQ procedure where, $D_i$ is the daily demand and $Q_i^*$ is the optimal order quantity at manufacturer i.

$$\begin{aligned}&Q_i^*=\sqrt{\frac{2{ ND }_i A_i }{h_i }}\times \sqrt{\frac{1}{1-\frac{{ ND }_i }{{ NM }_i }}},\quad \forall i \end{aligned}$$

(8.1)

$$\begin{aligned}&D_i =\sum _{j=1}^n {\sum _{k=1}^l {X_{ij} Y_{jk} \mu _k } } ,\quad \forall i \end{aligned}$$

(8.2)

$$\begin{aligned}&\sum _{i=1}^n {X_{ij} Y_{jk} } =Z_{ik} ,\quad \forall i,k \end{aligned}$$

(8.3)

According to Eqs. (8.1), (8.2) and (8.3) we can compute the value of the $H_i$ as a closed-form as follows.

$$\begin{aligned} H_i= & {} \frac{Q_i^*}{2}h_i \left( {1-\frac{ ND _i }{{ NM }_i }} \right) =\sqrt{\frac{2 ND _i A_i }{h_i }}\times \sqrt{\frac{1}{1-\frac{{ ND }_i }{{ NM }_i }}}\times \frac{h_i }{2}\left( {1-\frac{{ ND }_i }{{ NM }_i }} \right) \nonumber \\= & {} \sqrt{2{ ND }_i A_i }\times \frac{\sqrt{h_i \left( {1-\frac{D_i }{M_i }} \right) }}{2}\nonumber \\= & {} \frac{\sqrt{2{ ND }_i A_i h_i \left( {1-\frac{D_i }{M_i }} \right) }}{2}=\frac{\sqrt{2{ ND }_i A_i h_i -\frac{2{ ND }_i^2 A_i h_i }{M_i }}}{2} \nonumber \\= & {} \frac{\sqrt{2N{\mathop {\sum }\nolimits _{j=1}^n} {{\mathop {\sum }\nolimits _{k=1}^l} {X_{ij} Y_{jk} \mu _k A_i h_i } } -2N\frac{\left( {{\mathop {\sum }\nolimits _{j=1}^n} {{\mathop {\sum }\nolimits _{k=1}^l} {X_{ij} Y_{jk} \mu _k } } } \right) ^{2}A_i h_i }{M_i }}}{2} \nonumber \\= & {} \frac{\sqrt{2N\sum _{k=1}^l {Z_{ik} \mu _k A_i h_i } -2N\frac{\left( {{\mathop {\sum }\nolimits _{k=1}^l} {Z_{ik} \mu _k } } \right) ^{2}A_i h_i }{M_i }}}{2} \nonumber \\&\frac{\sqrt{2N{\mathop {\sum }\nolimits _{k=1}^l} {Z_{ik} \mu _k A_i h_i } -2N\frac{\left( {{\mathop {\sum }\nolimits _{k=1}^l} {Z_{ik} \mu _k^2 } +2{\mathop {\sum }\nolimits _{k=1}^{l-1}} {{\mathop {\sum }\nolimits _{t=k+1}^l} {Z_{ik} Z_{it} \mu _k \mu _t } } } \right) A_i h_i }{M_i }}}{2} \nonumber \\= & {} \frac{\sqrt{2NA_i h_i \left( {\sum _{k=1}^l {Z_{ik} \mu _k -\frac{\left( {{\mathop {\sum }\nolimits _{k=1}^l} {Z_{ik} \mu _k^2 } +2{\mathop {\sum }\nolimits _{k=1}^{l-1}} {{\mathop {\sum }\nolimits _{t=k+1}^l} {Z_{ik} Z_{it} \mu _k \mu _t } } } \right) }{M_i }} } \right) }}{2},\quad \forall i \end{aligned}$$

(8.4)

Therefore, based on Eq. (8.4), the annual inventory cost at each manufacturer is derived from Eq. (8.5).

$$\begin{aligned} H_i =\frac{\sqrt{2NA_i h_i \left( {{\mathop {\sum }\nolimits _{k=1}^l} {Z_{ik} \mu _k } -\frac{\left( {{\mathop {\sum }\nolimits _{k=1}^l} {Z_{ik} \mu _k^2 } +2{\mathop {\sum }\nolimits _{k=1}^{l-1}} {{\mathop {\sum }\nolimits _{t=k+1}^l} {Z_{ik} Z_{it} \mu _k \mu _t } } } \right) }{M_i }} \right) }}{2},\quad \forall i \end{aligned}$$

(8.5)