Heuristics for the Single-Item Dynamic Lot-Sizing Problem with Rework of Internal Returns

Abstract

While external product returns from customers are well-studied in the dynamic lot-sizing literature, the same is not true for internal returns resulting from imperfect production. We approach this problem by considering a basic dynamic single-product lot-sizing model in which some of the items produced do not meet quality requirements and, therefore, must be reworked. The objective is to minimize the sum of setup and inventory costs for new production and rework while fully satisfying demand. To this end, three heuristics are developed, based essentially on two production policies that can efficiently coordinate new production and rework for different parameter constellations. This is confirmed by a computational study in which the developed heuristics yielded highly competitive results compared to those obtained with a commercial solver.

Appendix

The following MIP formulation has been used to calculate the optimal solutions using Gurobi as the commercial solver.

$$\begin{aligned}&\min \sum ^{T}_{t=1} h_s\cdot I_{s,t} + h_d \cdot I_{d,t} + y_t \cdot R_p + z_t \cdot R_r \ \end{aligned}$$

(11)

$$\begin{aligned} \mathrm {subject~to} \end{aligned}$$

$$\begin{aligned} I_{s,t}= & {} I_{s,t-1} + (1-\beta )\cdot p_t + r_t - d_t \quad \forall t=1,\ldots , T \end{aligned}$$

(12)

$$\begin{aligned} I_{d,t}= & {} I_{d,t-1} + \beta \cdot p_t - r_t \quad \forall t=1,\ldots , T \end{aligned}$$

(13)

$$\begin{aligned} p_t\le & {} d_{t,T} \cdot y_t \quad \forall t=1,\ldots , T \end{aligned}$$

(14)

$$\begin{aligned} r_t\le & {} d_{t,T} \cdot z_t \quad \forall t=1,\ldots , T \end{aligned}$$

(15)

$$\begin{aligned} I_{s,0}= & {} I_{d,0} = I_{d,T} = I_{s,T} = 0 \end{aligned}$$

(16)

$$\begin{aligned} p_t,r_t,I_{s,t},I_{d,t}\ge & {} 0, d_t > 0 \quad \forall t=1,\ldots , T \end{aligned}$$

(17)

$$\begin{aligned} y_t,z_t\in & {} \{0;1\} \quad \forall t=1,\ldots , T \end{aligned}$$

(18)