Polynomial time algorithms for two special classes of the proportionate multiprocessor open shop

This paper describes a complex scheduling problem taken from a hospital diagnostic testing center that schedules hundreds of patients in an open shop environment consisting of multiple facilities and multiple processors. This scheduling problem, known as the multiprocessor open shop (MPOS) problem, is strongly NP-hard with few published results. Realizing that in many MPOS environments processing times are stage-dependent, not both job and stage-dependent, this paper examines a new class of problems for the MPOS--proportionate ones. This paper exploits the structural nature of the proportionate MPOS and defines new terms. Despite the enormous complexity of the MPOS problem, this work demonstrates that polynomial time algorithms exist for two special cases. Since other applications of this problem exist in service and manufacturing environments, solving the proportionate MPOS problem is not only significant in the theory of optimization, but also in many real-world applications.