An exact decomposition method for unrelated parallel machine scheduling with order acceptance and setup times

In this paper, an unrelated parallel machine scheduling problem is studied, where order acceptance, sequence and machine-dependent setup times and the maximum available times of machines are additionally considered. The objective is to maximize the profit, which is the difference between the total revenues of accepted jobs and the cost associated with makespan. A mixed integer programming (MIP) model is formulated. To tackle this problem efficiently, an exact decomposition method, which is a two-layer logic-based Benders decomposition (LBBD) based (denoted by TL-LBBD) method, is developed, where an inner LBBD is embedded into an outer LBBD. Specifically, in the outer LBBD method, the master problem is used to determine the acceptance of jobs, whereas the subproblem examines the schedule given the accepted jobs from the master problem. The subproblem could be further decomposed into an assignment master problem and a sequencing subproblem by the inner LBBD method as well. Extensive computational experiments are conducted and the results show that the developed TL-LBBD method produce better quality solutions in significantly less computation time than solving the MIP model directly and the classic LBBD method. Moreover, the maximum scales of the problem instances that could be solved to optimality by the developed TL-LBBD method within 30 min are also evaluated.