Exploiting the Structure of Two-Stage Robust Optimization Models with Integer Adversarial Variables

This paper addresses a class of two-stage robust optimization models with integer adversarial variables. We discuss the importance of this class of problems in modeling two-stage robust resource planning problems where some tasks have uncertain arrival times and duration periods. We apply Dantzig-Wolfe decomposition to exploit the structure of these models and show that the original problem reduces to a single-stage robust problem. We propose a Benders algorithm for the reformulated single-stage problem. Since the master problem and subproblem in the Benders algorithm are mixed integer programs, it is computationally demanding to solve them optimally at each iteration of the algorithm. Therefore, we develop novel stopping conditions for these mixed integer programs and provide the relevant convergence proofs. We also develop a heuristic algorithm called dual algorithm. In this heuristic, we dualize the linear programming relaxation of the adversarial problem in the reformulated problem and iteratively generate cuts to shape the convex hull of the uncertainty set. We combine this heuristic with the Benders algorithm to create a more effective algorithm called Benders-dual algorithm. Extensive computational experiments on a two-stage nurse planning problem are performed to compare these algorithms.