Distributionally robust scheduling for the two-stage hybrid flowshop with uncertain processing time

In the present paper, we investigate the two-stage hybrid flowshop with uncertain processing time. The true probability distribution of the processing time is unknown, but the statistical features can be extracted from historical data, such as the mean, lower and upper bounds. To obtain the exact scheduling result, a distributionally robust optimization (DRO) model is built to minimize the worst-case expected makespan. Then the inner problem is further reformulated as a minimization problem with a fixed sequence based on duality theory and the totally unimodular property. In addition, valid lower and upper bounds are introduced to transform the DRO model into an equivalent mixed-integer linear programming (MILP) problem with McCormick inequalities, which can be handled directly with the off-the-shelf commercial solvers. The numerical analysis demonstrates the higher computational efficiency of the DRO-based model compared with its stochastic programming (SP) counterpart. In particular, the DRO model consistently outperforms the SP model in terms of worst-case indicators. And in most cases, the DRO model triumphs the SP model in terms of average, up-quartile and up-decile indicators. Moreover, the optimal schedule obtained by the DRO model demonstrates stronger stability compared with the deterministic model. These features shed light on the principles behind reliable schedules for the two-stage hybrid flowshop scheduling model, thereby enhancing the robustness of the manufacturing system in the face of process uncertainty.