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Machine scheduling problems involving conflict jobs can be seen as aconstrained version of the classical scheduling problem, in which some jobs areconflict in the sense that they cannot be proceeded simultaneously on differentmachines. This conflict constraint naturally arises in several practicalapplications and has recently received considerable attentions in the researchcommunity. In fact, the problem is typically NP-hard (even for approximation)and most of algorithmic results achieved so far have heavily relied on specialstructures of the underlying graph used to model the conflict-job relation. Ourfocus is on three objective functions: minimizing the makespan, minimizing theweighted summation of the jobs' completion time, and maximizing the totalweights of completed jobs; the first two of which have been intensively studiedin the literature. For each objective function considered, we present severalmixed 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 setswith up to 32 jobs and 6 machines are reported and analysed to verify theperformance of the proposed methods.

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