# Computing non-stationary $(s, S)$ policies using mixed integer linear programming

## TL;DR

This paper develops MILP-based models to compute near-optimal non-stationary $(s, S)$ policies for stochastic lot sizing, achieving high accuracy and reasonable computational efficiency.

## Contribution

It introduces a novel MILP reformulation for non-stationary $(s, S)$ policies and combines it with a binary search approach for larger instances.

## Key findings

- Optimality gaps around 0.3% of the optimal cost
- Models are easily implementable with off-the-shelf solvers
- Computational times are reasonable for practical use

## Abstract

This paper addresses the single-item single-stocking location stochastic lot sizing problem under the $(s, S) $ policy. We first present a mixed integer non-linear programming (MINLP) formulation for determining near-optimal $(s, S)$ policy parameters. To tackle larger instances, we then combine the previously introduced MINLP model and a binary search approach. These models can be reformulated as mixed integer linear programming (MILP) models which can be easily implemented and solved by using off-the-shelf optimisation software. Computational experiments demonstrate that optimality gaps of these models are around $0.3\%$ of the optimal policy cost and computational times are reasonable.