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.
Highlights
IntroductionOther relevant applications arise in the context of health-care, where, for example, patients have to be assigned to surgery rooms that must be equipped by considering the type (i.e., the family) of surgery to be performed
Let J = {1, . . . , n} be a set of jobs to be scheduled on a set M = {1, . . . , m} of identical parallel machines, with m ≤ n
Preprint submitted to Elsevier i ∈ F = {1, . . . , f }, with f ≤ n, so the set of jobs can be partitioned as J = ∪i∈F Ji, in such a way that each set Ji contains the jobs of family i ∈ F
Summary
Other relevant applications arise in the context of health-care, where, for example, patients have to be assigned to surgery rooms that must be equipped by considering the type (i.e., the family) of surgery to be performed In such cases, the weight usually models a level of urgency for the patient. The most recent works on parallel machine scheduling problem with family setup and total weighted completion time minimization are the ones of Liao et al (2012) and Tseng and Lee (2017). Our results prove that one of the AF formulations and the SC one are able to solve to proven optimality several open instances from the literature Despite these good results, the problem remains very challenging, especially because of the family setup times. The extensive computational experiments that we performed are presented and discussed in detail in Section 4, Section 5 reports the concluding remarks and future research directions
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