A Lagrangian relaxation algorithm for order acceptance and scheduling problem: a globalised robust optimisation approach

In this paper, a simultaneous order acceptance and scheduling problem in a non-identical parallel machines environment is considered. The order is defined by their due date, revenue, tardiness penalty, different processing times on the machines, and sequence-dependent set-up times. A mixed-integer linear programming (MILP) formulation is presented to maximise profit. Furthermore, it is assumed that the revenue from an accepted order and the processing times are uncertain; the globalized robust counterpart (GRC) of the proposed MILP model is presented such that the normal range of the perturbation is the intersection of a box and a polyhedral. The problem is computationally intractable. Therefore, the Lagrangian relaxation algorithm is developed to solve it. A cutting plane method is used to update the Lagrangian multipliers and a heuristic method is presented to obtain feasible solutions. Through numerical experiments on randomly generated large instances with up to 40 orders and six machines, the authors demonstrate that the proposed Lagrangian algorithm outperforms the monolithic MILP model. Furthermore, a simulation study demonstrates that, on average, the GRC of the MILP model provides slightly better results in comparison with its conventional robust counterpart.