A Polyhedral Study of the Static Probabilistic Lot-Sizing Problem

TL;DR

This paper analyzes the polyhedral structure of the static probabilistic lot-sizing problem, introduces valid inequalities, and proposes a new formulation that improves computational efficiency and solution quality.

Contribution

It provides a comprehensive polyhedral analysis, introduces inequalities that unify existing ones, and develops a reformulation exploiting simple recourse for better computational performance.

Findings

Proposed inequalities are facet-defining under certain conditions.

The new formulation reduces variables and constraints significantly.

Computational results demonstrate effectiveness of the inequalities and reformulation.

Abstract

We study the polyhedral structure of the static probabilistic lot-sizing problem and propose valid inequalities that integrate information from the chance constraint and the binary setup variables. We prove that the proposed inequalities subsume existing inequalities for this problem, and they are facet-defining under certain conditions. In addition, we show that they give the convex hull description of a related stochastic lot-sizing problem. We propose a new formulation that exploits the simple recourse structure, which significantly reduces the number of variables and constraints of the deterministic equivalent program. This reformulation can be applied to general chance-constrained programs with simple recourse. The computational results show that the proposed inequalities and the new formulation are effective for the the static probabilistic lot-sizing problems.