# A global constraint for the capacitated single-item lot-sizing problem

## TL;DR

This paper introduces a new global constraint for the capacitated single-item lot-sizing problem, enabling more efficient solution methods within a constraint programming framework by leveraging a novel lower bound and time decomposition.

## Contribution

It formulates the lot-sizing problem as a global constraint with bound consistency, providing a new lower bound and filtering rules based on dynamic programming, improving solution efficiency.

## Key findings

- Global constraint achieves bound consistency in pseudo-polynomial time.
- Constraint programming outperforms decomposed models on difficult instances.
- The proposed approach is competitive with mixed integer linear programming.

## Abstract

The goal of this paper is to set a constraint programming framework to solve lot-sizing problems. More specifically, we consider a single-item lot-sizing problem with time-varying lower and upper bounds for production and inventory. The cost structure includes time-varying holding costs, unitary production costs and setup costs. We establish a new lower bound for this problem by using a subtle time decomposition. We formulate this NP-hard problem as a global constraint and show that bound consistency can be achieved in pseudo-polynomial time and when not including the costs, in polynomial time. We develop filtering rules based on existing dynamic programming algorithms, exploiting the above mentioned time decomposition for difficult instances. In a numerical study, we compare several formulations of the problem: mixed integer linear programming, constraint programming and dynamic programming. We show that our global constraint is able to find solutions, unlike the decomposed constraint programming model and that constraint programming can be competitive, in particular when adding combinatorial side constraints.