This work focuses on a broad class of facility location problems in the context of adaptive robust stochastic optimization under the state-dependent demand uncertainty. The demand is assumed to be significantly affected by related state information, such as the seasonal or socio-economic information. In particular, a state-wise ambiguity set is adopted for modeling the distributional uncertainty associated with the demand in different states. The conditional distributional characteristics in each state are described by a support, as well as by mean and dispersion measures, which are assumed to be conic representable. A robust sensitivity analysis is performed, in which, on the one hand, we analyze the impact of the change in ambiguity-set parameters (e.g., state probabilities, mean value abounds, and dispersion bounds in different states) onto the optimal worst-case expected total cost using the ambiguity dual variables. On the other hand, we analyze the impact of the change in location design onto the worst-case expected second-stage cost and show that the sensitivity bounds are fully described as the worst-case expected shadow-capacity cost. As for the solution approach, we propose a nested Benders decomposition algorithm for solving the model exactly, which leverages the subgradients of the worst-case expected second-stage cost at the location decisions formed insightfully by the associated worst-case distributions. The nested Benders decomposition approach ensures a finite-step convergence, which can also be regarded as an extension of the classic L-shaped algorithm for two-stage stochastic programming to our state-wise, robust stochastic facility location problem with conic representable ambiguity. Finally, the results of a series of numerical experiments are presented that justify the value of the state-wise distributional information incorporated in our robust stochastic facility location model, the robustness of the model, and the performance of the exact solution approach.

In this paper, a heuristic procedure is proposed for the facility location problem with general Bernoulli demands. This is a discrete facility location problem with stochastic demands that can be formulated as a two-stage stochastic program with recourse. In particular, facility locations and customer assignments must be decided here and now, i.e., before knowing the customers who will actually require to be served. In a second stage, service decisions are made according to the actual requests. The heuristic proposed consists of a greedy randomized adaptive search procedure followed by a path relinking. The heterogeneous Bernoulli demands make prohibitive the computational effort for evaluating feasible solutions. Thus the expected cost of a feasible solution is simulated when necessary. The results of extensive computational tests performed for evaluating the quality of the heuristic are reported, showing that high-quality feasible solutions can be obtained for the problem in fairly small computational times. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0755 .