Capacitated dynamic production and remanufacturing planning under demand and return uncertainty

Abstract

This paper considers a stochastic dynamic multi-product capacitated lot sizing problem with remanufacturing. Finished goods come from two sources: a standard production resource using virgin material and a remanufacturing resource that processes recoverable returns. Both the period demands and the inflow of returns are random. For this integrated stochastic production and remanufacturing problem, we propose a nonlinear model formulation that is approximated by sample averages and a piecewise linear approximation model. In the first approach, the expected values of random variables are replaced by sample averages. The idea of the piecewise linear approximation model is to replace the nonlinear functions with piecewise linear functions. The resulting mixed-integer linear programs are solved to create robust (re)manufacturing plans.

Appendix: Description of test instances

At first, for each product k, the average demand $\overline{d}_k$ is generated randomly by following a uniform distribution in the interval [50, 150]. Based on the given average demand $\overline{d}_k$, a single series of the expected demand is provided based on a normal distribution. The demand series is given in Table 9. Analogously, three return series for each product k are generated. Three return series are generated randomly analogously to the demand series in which the average return $\overline{r}_k$ is related to the average demand $\overline{d}_k$ of product k, according to the assumed mean return rate rr. The return series are given in Tables 10, 11 and 12.

The holding costs hc$_k$ of the serviceables are assumed to be 0.05 for each product k. The holding costs hc$^r_k$ of the recoverables are assumed to be lower than the holding costs hc$_k$ of the serviceables. The production costs pc$_k$ are 10. For the holding costs hc$^r_k$ and remanufacturing costs pc$^r_k$, three scenarios are defined (see Table 4). The processing times pt$_k$ and the remanufacturing times pt$^r_k$ are both equal to 1.

For the determination of setup costs sc$_k$, we use Harris’ formula for the optimal lot size in the case of static demand. Based on the average demand $\overline{d}_k$ of product k and the respective time between orders (TBO), setup costs $sc_k$ can be derived as follows:

$$\begin{aligned}&\mathrm{sc}_k = \frac{\overline{d}_k \cdot \mathrm{TBO}^2 \cdot \mathrm{hc}_k}{2} \qquad \forall \, k. \end{aligned}$$

(41)

In the case of the recoverables, the setup costs sc$^r_k$ for remanufacturing equal the portion of remanufacturing costs in the setup costs sc$_k$.

The setup times st$_k$ (and st$^r_k$) are assumed to be equal to 20 for all products. To derive the capacity $c_t$, we assume a setup activity in each period t for all products and allow the production of the average demand $\overline{d}_k$ to be reduced by the average return $\overline{r}_k$; this leads to

$$\begin{aligned}&c_t = \sum _k \left( \mathrm{pt}_k \cdot (\overline{d}_k - \overline{r}_k) + \mathrm{st}_k \right) \qquad \forall \, t. \end{aligned}$$

(42)

Additionally, the capacity $c^r_t$ is defined according to the average return $\overline{r}_k$ and by allowing a setup activity in each period:

$$\begin{aligned}&c^r_t = \sum _k \left( \mathrm{pt}^r_k \cdot \overline{r}_k + \mathrm{st}^r_k \right) \qquad \forall \, t. \end{aligned}$$

(43)

To guarantee a large amount of feasible solutions, the available capacity is then extended with respect to the given utilization Util

$$\begin{aligned} c_t = \frac{c_t}{\mathrm{Util}} \qquad \forall \, t. \end{aligned}$$

(44)

The capacity $c^r_t$ of the remanufacturing system is extended analogously. The overtime costs oc are 10,000.