Unrelated parallel machine scheduling with machine processing cost

In practical scheduling problems, some factors such as depreciation cost, green costs like the amount of energy consumption or carbon emission, other resources consumption, raw material cost, etc., are not explicitly related to the machine processing times. Most of these factors can be generally considered as machine costs. Considering the machine cost as another objective alongside the other classical time-driven decision objectives can be an attractive work in scheduling problems. However, this subject has not been discussed thoroughly in the literature for the case the machines have fixed processing costs. This paper investigates a general unrelated parallel machine scheduling problem with the machine processing cost. In this problem, it is assumed that processing a job on a machine incurs a particular cost in addition to processing time. The considered objectives are the makespan and the total cost, which are minimized simultaneously to obtain Pareto optimal solutions. The efficacy of the mathematical programming approach to solve the considered problem is evaluated rigorously in this paper. In this respect, a multiobjective solution procedure is proposed to generate a set of appropriate Pareto solutions for the decision-maker based on the mathematical programming approach. In this procedure, the ϵ-constraint method is first used to convert the bi-objective optimization problem into single-objective problems by transferring the makespan to the set of constraints. Then, the single-objective problems are solved using the CPLEX software. Moreover, some strategies are also used to reduce the solution time of the problem. At the end of the paper, comprehensive numerical experiments are conducted to evaluate the performance of the proposed multiobjective solution procedure. A vast range of problem sizes is selected for the test problems, up to 50 machines and 500 jobs. Furthermore, some rigorous analyses are performed to significantly restrict the patterns of generating processing time and cost parameters for the problem instances. The experimental results demonstrate the mathematical programming solution approach's efficacy in solving the problem. It is observed that even for large-scale problems, a diverse set of uniformly distributed Pareto solutions can be generated in a reasonable time with the gaps from the optimality less than 0.03 most of the time.