A Bilevel Optimization Approach for a Class of Combinatorial Problems with Disruptions and Probing

We consider linear combinatorial optimization problems under uncertain disruptions that increase the cost coefficients of the objective function. A decision maker, or planner, can invest resources to probe the components (i.e., the coefficients) in order to learn their disruption status. In the proposed probing optimization problem, the planner, knowing just the disruptions’ probabilities, selects which components to probe subject to a probing budget in a first decision stage. Then, the uncertainty realizes, and the planner observes the disruption status of the probed components, after which the planner solves the combinatorial problem in the second stage. In contrast to standard two-stage stochastic optimization, the planner does not have access to the full uncertainty realization in the second stage. Consequently, the planner cannot directly optimize the second-stage objective function, which is given by the actual cost after disruptions, and the decisions have to be made based on an estimate of the cost. By assuming that the estimate is given by the conditional expected cost given the information revealed by probing, we reformulate the probing optimization problem as a bilevel problem with multiple followers and propose an exact algorithm based on a value function reformulation and three heuristic algorithms. We derive theoretical results that bound the value of information and the price of not having full information and a bound on the required probing budget that attains the same performance as full information. Our extensive computational experiments suggest that probing a fraction of the components is sufficient to yield large improvements in the optimal value, that our exact algorithm is competitive for small- to medium-scale instances, and that the proposed heuristics find high-quality solutions in large-scale instances.