Integrated forward/reverse logistics network design under uncertainty with pricing for collection of used products

Abstract

This paper addresses design and planning of an integrated forward/reverse logistics network over a planning horizon with multiple tactical periods. In the network, demand for new products and potential return of used products are stochastic. Furthermore, collection amounts of used products with different quality levels are assumed dependent on offered acquisition prices to customer zones. A uniform distribution function defines the expected price of each customer zone for one unit of each used product. Using two-stage stochastic programming, a mixed-integer linear programming model is proposed. To cope with demand and potential return uncertainty, Latin Hypercube Sampling method is applied to generate fan of scenarios and then, backward scenario reduction technique is used to reduce the number of scenarios. Due to the problem complexity, a novel simulation-based simulated annealing algorithm is developed to address large-sized test problems. Numerical results indicate the applicability of the model as well as the efficiency of the solution approach. In addition, the performance of the scenario generation method and the importance of stochasticity are examined for the optimization problem. Finally, several numerical experiments including sensitivity analysis on main parameters of the problem are performed.

Appendices

Appendix 1

See Tables 9, 10.

Table 9 Parameters’ generation for the test problems

Table 10 Available budgets for the test problems

Appendix 2

In order to calculate EVPI and VSS, the optimization problem (33) for each scenario is defined as follows:

$$\begin{aligned} F\left( {\mathbf{x}, \zeta ^{s}} \right) =\mathop {\hbox {Min}}\limits _{\mathbf{x} \in X} \, \mathbf{c}^{T}{} \mathbf{x} + Q\left( {\mathbf{x}, \zeta ^{s}} \right) , \qquad \forall s\in S. \end{aligned}$$

(33)

Here, WS, wait-and-see solution, can be calculated as follows:

$$\begin{aligned} \hbox {WS}=\sum _{s\in \mathrm{S}} {\pi ^{s}\times F\left( {\mathbf{x}, \zeta ^{s}} \right) } \end{aligned}$$

The EVPI is the difference between the optimal objective value of the recourse problem (RP), optimization problem (5), and WS as:

$$\begin{aligned} \hbox {EVPI}=\hbox {RP}-\hbox {WS}. \end{aligned}$$

By considering the expected values of stochastic parameters instead of them in problem (5), the Expected Value problem (EV) is defined as:

$$\begin{aligned} \hbox {EV} = \mathop {\hbox {Min}}\limits _{\mathbf{x} \in X} \, \mathbf{c}^{T}{} \mathbf{x} + Q\left( {\mathbf{x}, {\bar{\varvec{\upzeta }}}} \right) . \end{aligned}$$

The optimal solution of the EV is ${\bar{\mathbf{x}}}\left( {{\bar{\varvec{\upzeta }}}} \right)$ and hence, the expected objective value of using the EV solution (EEV) and VSS can be determined as:

$$\begin{aligned} \hbox {EEV}= & {} \mathbf{c}^{T}{\bar{\mathbf{x}}}\left( {{\bar{\varvec{\upzeta }}}} \right) +\sum _{s\in \mathrm{S}} {\pi ^{s}\times } Q\left( {{\bar{\mathbf{x}}}\left( {{\bar{\varvec{\upzeta }}}} \right) , \zeta ^{s}} \right) , \\ \hbox {VSS}= & {} \hbox {EEV}-\hbox {RP}. \end{aligned}$$