Dominance inequalities for scheduling around an unrestrictive common due date

TL;DR

This paper introduces dominance inequalities to improve the exact solution process for scheduling problems with a common due date, significantly reducing computation time by strengthening the mixed integer programming formulation.

Contribution

It develops new dominance inequalities based on neighborhood properties to eliminate non-locally optimal solutions, enhancing the efficiency of solving scheduling problems with a mixed integer program.

Findings

Adding dominance inequalities reduces solving time significantly.

The inequalities effectively eliminate non-optimal solutions.

The approach improves the computational efficiency of exact methods.

Abstract

The problem considered in this work consists in scheduling a set of tasks on a single machine, around an unrestrictive common due date to minimize the weighted sum of earliness and tardiness. This problem can be formulated as a compact mixed integer program (MIP). In this article, we focus on neighborhood-based dominance properties, where the neighborhood is associated to insert and swap operations. We derive from these properties a local search procedure providing a very good heuristic solution. The main contribution of this work stands in an exact solving context: we derive constraints eliminating the non locally optimal solutions with respect to the insert and swap operations. We propose linear inequalities translating these constraints to strengthen the MIP compact formulation. These inequalities, called dominance inequalities, are different from standard reinforcement inequalities. We provide a numerical analysis which shows that adding these inequalities significantly reduces the computation time required for solving the scheduling problem using a standard solver.