A mixed integer programming formulation for the stochastic lot sizing problem with controllable processing times

Abstract

In this study, we address the capacitated stochastic lot-sizing problem under α service level constraints. We assume that processing times can be decreased in return for compression cost that follows a convex function. We consider this problem under the static uncertainty strategy suggesting to determine replenishment plans at the beginning of the planning horizon. We develop an extended mixed integer programming (MIP) formulation built on a predefined piecewise linear approximation. Then, we adopt the so-called dynamic cut generation approach to be able to use the proposed MIP formulation with no prior approximation of the cost function. Also, we demonstrate how to extend the dynamic cut generation approach to consider the exact inventory cost in the objective function. We show the computational performance of the proposed MIP model with the dynamic cut generation approach in an extensive numerical study where second order cone programming formulations developed in the literature are used as benchmark. The results reveal that the proposed MIP model deployed with the dynamic cut generation yields a superior computational performance as compared to the benchmark formulations especially when the order of compression cost function is higher.