AbstractIn this paper, an identical parallel machine scheduling problem with step-deteriorating jobs is considered to minimize the weighted sum of tardiness cost and extra energy consumption cost. In particular, the actual processing time of a job is assumed to be a step function of its starting time and its deteriorating threshold. When the starting time of a job is later than its deteriorating threshold, the job faces two choices: (1) maintaining its status in holding equipment and being processed with a base processing time and (2) consuming an extra penalty time to finish its processing. The two work patterns need different amounts of energy consumption. To implement energy-efficient scheduling, the selection of the pre-processing patterns must be carefully considered. In this paper, a mixed integer linear programming (MILP) model is proposed to minimize the total tardiness cost and the extra energy cost. Decomposition approaches based on logic-based Benders decomposition (LBBD) are developed by reformulating the studied problem into a master problem and some independent sub-problems. The master problem is relaxed by only making assignment decisions. The sub-problems are to find optimal schedules in the job-to-machine assignments given by the master problem. Moreover, MILP and heuristic based on Tabu search are used to solve the sub-problems. To evaluate the performance of our methods, three groups of test instances were generated inspired by both real-world applications and benchmarks from the literature. The computational results demonstrate that the proposed decomposition approaches can compute competitive schedules for medium- and large-size problems in terms of solution quality. In particular, the LBBD with Tabu search performs the best among the suggested four methods.
Single machine quadratic penalty function scheduling with deteriorating jobs and group technology
