A bi-objective mathematical model for an unrelated parallel machine scheduling problem with job-splitting

Studies dealing with parallel machine scheduling problems generally assume that jobs are not split. However, splitting of jobs can offer opportunities such as delivering the jobs on time or evenly distributing the jobs to the machines. In this study, two different multi-objective mathematical models are proposed for the unrelated parallel machine scheduling problem without and with job-splitting. The first model is for the unrelated parallel machine scheduling problem without job-splitting and it was developed based on the model proposed by Sarac and Tutumlu [1]. In the second model, in which the jobs can split, besides the machines to which the jobs will be assigned and their order, it is also determined how the jobs will split and in which ratios the jobs will be processed on which machines. The objectives of both models are to minimize the completion time of the last job and the number of machines to be used. Multi-objective models were transformed into a single-objective structure with the epsilon constraint method. Randomly generated test problems were solved with the proposed models using the GAMS/Cplex solver, and the results were compared. Considering the problems for which the optimum solution is obtained, the first model found a solution in an average of 85% less time than the model in the literature [1], and the second model shortens the completion time of the last job in an average of 14%. In addition, a mathematical heuristic algorithm is proposed for solving large-size problems.