A distributionally robust approach for the two-machine permutation flow shop scheduling

We consider the two-machine permutation flow shop scheduling problem with uncertain job processing time, which is sampled from no specific distribution type. For the ease of discussion, an ambiguity set with a priori mean and support set information is constructed. We then introduce a distributionally robust optimization (DRO) perspective to handle the uncertainty. To the best of our knowledge, this is the first time that a DRO method is applied to this problem setting. Given that the original DRO model is nonlinear and intractable in nature, we first reformulate the inner maximization problem into a linear programming model with a fixed sequence, based on the duality theory and optimality conditions. By including the sequence decision, we further transform it into an equivalent mixed-integer linear programming (MILP) problem via incorporating the valid lower and upper bounds and McCormick inequalities. The obtained MILP could be solved to optimality with the off-the-shelf commercial solvers. In the numerical study, it is demonstrated that the DRO-based model could effectively solve the large scale instances with up to 100 jobs optimally within 30 s. Compared with the SLP, DRO model always triumphs on the worst-case indicator. And as the problem scale increases, the DRO model gradually outperforms the SLP in terms of the Up-90% and Up-75% indicators. Furthermore, the optimal sequence obtained by the deterministic model is less stable than the DRO model, which can enhance the robustness of the manufacturing system against processing uncertainty.