Solution of a bilevel optimistic scheduling problem on parallel machines

TL;DR

This paper addresses a complex bilevel scheduling problem on parallel machines with two speed options, demonstrating NP-hardness, and proposing a MIP and branch-and-bound solution with computational tests.

Contribution

It introduces the first formulation and solution approach for a bilevel optimistic scheduling problem with two machine speeds, including complexity analysis and algorithms.

Findings

Problem is NP-hard in the strong sense.

Proposed algorithms solve instances with up to 80 jobs.

Computational experiments validate the effectiveness of the methods.

Abstract

We consider the uniform parallel machines scheduling problem in the context of optimistic bilevel optimization, where two speed options are considered. In this scenario, the leader aims to minimize the weighted number of tardy jobs, while the follower seeks to minimize the total completion time on a set of uniform machines. This problem has practical applications in Industry 4.0. We show that this problem is NP-hard in the strong sense by providing a reduction from the Numerical 3-Dimensional Matching problem and we provide a moderately exponential-time dynamic programming algorithm. The problem is solved by means of a concise MIP formulation and a branch-and-bound algorithm that embeds a column generation approach for the lower bound computation. Computational experiments are presented for instances with up to 80 jobs and 4 machines while larger problems are out of reach for the proposed approaches.