A fix-and-optimize heuristic for the Unrelated Parallel Machine Scheduling Problem

Abstract

This paper proposes and evaluates a matheuristic approach for the Unrelated Parallel Machine Scheduling Problem (UPMSP). The UPMSP consists of assigning jobs to unrelated parallel machines considering different processing times for the same job in different machines. Additionally, a setup time is considered between the execution of jobs in the same machine. The problem is addressed by a fix-and-optimize matheuristic that iteratively selects a subset of variables to be fixed to their current values so that the remaining variables will compose a subproblem to be optimized by a mathematical programming solver. In the proposed approach, each subproblem consists of a subset of jobs that are assigned to a subset of machines in the incumbent solution. The subproblems are solved by the state-of-the-art exact algorithm for the UPMSP. In the experiments conducted on benchmark instances, the proposed fix-and-optimize algorithm achieved remarkable results. It outperformed the standalone exact algorithm by a large margin and resulted in competitive solutions when compared to the literature’s best-performing heuristic method for the problem. The proposed algorithm obtained the best solution for 669 out of the 1000 instances addressed in this work. Among them, 338 are new best-known solutions. In general, the proposed approach excels at solving instances with a high number of jobs per machine — it resulted in the best solution for 89% of the instances with a ratio of 10 or more jobs per machine in total.