Novel integer L-shaped method for parallel machine scheduling problem under uncertain sequence-dependent setups

Abstract

We study scheduling problems on unrelated parallel machines with uncertainty in job processing and sequence-dependent setup times. We first formulate this problem as a two-stage stochastic program using a scenario-based approach, in which the first-stage decisions involve job-machine assignments, and the second-stage deals with job sequencing. We then propose an innovative modified integer L-shaped method, uniquely designed to decompose the primary problem into independent subproblems at both the scenario and machine levels. We have augmented this method with a specialized integer optimality cut, which is further strengthened using a sequential lifting procedure. To broaden the applicability of our model and solution, we have adapted it to a distributionally robust optimization (DRO) framework under Wasserstein ambiguity. This process involves reformulating the DRO model into a mixed-integer linear program (MILP) using conic duality theory. Additionally, we integrate a distribution separation program to define the worst-case probability distribution during each iteration of the proposed integer L-shaped method, enhancing the computational speed for solving the DRO model. We conducted a series of numerical experiments to illustrate the effectiveness and scalability of the proposed solution methodology, which demonstrated highly promising results. Our innovative decomposition algorithms showed superior performance, achieving solution times that are significantly faster (up to two orders of magnitude) compared to the traditional integer L-shaped method for the stochastic scheduling model and the direct MILP reformulation of the DRO model. These results were obtained for problem instances solved to the exactness within the time limit. Moreover, the solution derived from the DRO model exhibited greater robustness in out-of-sample analysis with simulated data compared to the solution obtained from the stochastic model.