Mathematical formulations for scheduling jobs on identical parallel machines with family setup times and total weighted completion time minimization

Arthur Kramer*, Manuel Iori*, and Philippe Lacomme**

*DISMI, University of Modena and Reggio Emilia, 42100 Reggio Emilia, Italy.

**ISIMA, University of Clermont-Auvergne, 63178 Aubičre, France.

Abstract

This paper addresses the parallel machine scheduling problem with family dependent setup times and total weighted completion time minimization. In this problem, when two jobs j and k are scheduled consecutively on the same machine, a setup time is performed between the finishing time of j and the starting time of k if and only if j and k belong to different families. The problem is strongly NP-hard and is commonly addressed in the literature by heuristic approaches and by branch-and-bound algorithms. Achieving proven optimal solution is a challenging task even for small size instances. Our contribution is to introduce five novel mixed integer linear programs based on concepts derived from one-commodity, arc-flow and set covering formulations. Numerical experiments on more than 13000 benchmark instances show that one of the arc-flow models and the set covering model are quite efficient, as they provide on average better solutions than state-of-the-art approaches, with shorter computation times, and solve to proven optimality a large number of open instances from the literature.

Key words

Scheduling, mathematical formulations, parallel machines, family setup times, weighted completion times.

 

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