Variable Neighborhood Search for Minimizing the Makespan in a Uniform Parallel Machine Scheduling

Abstract

This paper investigates a uniform parallel machine scheduling problem for makespan minimization. Due to the problem’s NP-hardness, much effort from researchers has been directed toward proposing heuristic and metaheuristic algorithms that can find an optimal or a near-optimal solution in a reasonable amount of time. This work proposes two versions of a variable neighborhood search (VNS) algorithm with five neighborhood structures, differing in their initial solution generation strategy. The first uses the longest processing time (LPT) rule, while the second introduces a novel element by utilizing a randomized longest processing time (RLPT) rule. The neighborhood structures for both versions were modified from the literature to account for the variable processing times in uniform parallel machines. We evaluated the performance of both VNS versions using a numerical example, comparing them against a genetic algorithm and a tabu search from existing literature. Results showed that the proposed VNS algorithms were competitive and obtained the optimal solution with much less effort. Additionally, we assessed the performance of the VNS algorithms on randomly generated instances. For small-sized instances, we compared their performance against the optimal solution obtained from a mathematical formulation, and against lower bounds derived from the literature for larger instances. Computational results showed that the VNS version with the randomized LPT rule (RLPT) as the initial solution (RVNS) outperformed that with the LPT rule as the initial solution (LVNS). Moreover, RVNS found the optimal solution in 90.19% of the small instances and yielded an average relative gap of about 0.15% for all cases.

Keywords

• longest processing time
• scheduling
• uniform parallel machines
• variable neighborhood search