Exact Algorithms for Scheduling Problems on Parallel Identical Machines with Conflict Jobs

TL;DR

This paper develops exact algorithms for scheduling conflict jobs on parallel machines, addressing NP-hard problems with mixed integer programming, constraint programming, and binary search, and verifies their effectiveness through numerical experiments.

Contribution

It introduces novel exact algorithms and models for conflict job scheduling on parallel machines, solving multiple objectives to optimality.

Findings

Algorithms efficiently solve small instances to optimality.

Mixed integer and constraint programming models are effective.

Numerical results demonstrate the practicality of the proposed methods.

Abstract

Machine scheduling problems involving conflict jobs can be seen as a constrained version of the classical scheduling problem, in which some jobs are conflict in the sense that they cannot be proceeded simultaneously on different machines. This conflict constraint naturally arises in several practical applications and has recently received considerable attentions in the research community. In fact, the problem is typically NP-hard (even for approximation) and most of algorithmic results achieved so far have heavily relied on special structures of the underlying graph used to model the conflict-job relation. Our focus is on three objective functions: minimizing the makespan, minimizing the weighted summation of the jobs' completion time, and maximizing the total weights of completed jobs; the first two of which have been intensively studied in the literature. For each objective function considered, we present several mixed integer linear programming models and a constraint programming model, from which we can solve the problems to optimality using dedicated solvers. Binary search-based algorithms are also proposed to solve the makespan problem. The results of numerical experiments performed on randomly generated data sets with up to 32 jobs and 6 machines are reported and analysed to verify the performance of the proposed methods.