Enhanced arc-flow formulations to minimize weighted completion time on identical parallel machines

TL;DR

This paper introduces new arc-flow formulations for scheduling jobs on identical parallel machines to minimize total weighted completion time, significantly improving the size of solvable instances compared to existing methods.

Contribution

The paper presents novel arc-flow formulations that efficiently solve larger instances of the scheduling problem using a capacitated network representation and mixed integer linear programming.

Findings

Solves instances with up to 400 jobs efficiently.

Achieves very low optimality gaps for instances up to 1000 jobs.

Outperforms existing formulations in computational tests.

Abstract

We consider the problem of scheduling a set of jobs on a set of identical parallel machines, with the aim of minimizing the total weighted completion time. The problem has been solved in the literature with a number of mathematical formulations, some of which require the implementation of tailored branch-and-price methods. In our work, we solve the problem instead by means of new arc-flow formulations, by first representing it on a capacitated network and then invoking a mixed integer linear model with a pseudo-polynomial number of variables and constraints. According to our computational tests, existing formulations from the literature can solve to proven optimality benchmark instances with up to 100 jobs, whereas our most performing arc-flow formulation solves all instances with up to 400 jobs and provides very low gap for larger instances with up to 1000 jobs.