An approach to the distributionally robust shortest path problem

In this study we consider the shortest path problem, where the arc costs are subject to distributional uncertainty. Basically, the decision-maker attempts to minimize her worst-case expected regret over an ambiguity set (or a family) of candidate distributions that are consistent with the decision-maker's initial information. The ambiguity set is formed by all distributions that satisfy prescribed linear first-order moment constraints with respect to subsets of arcs and individual probability constraints with respect to particular arcs. Our distributional constraints can be constructed in a unified manner from real-life data observations. In particular, the decision-maker may collect some new distributional information and thereby improve her solutions in the subsequent decision epochs. Under some additional assumptions the resulting distributionally robust shortest path problem (DRSPP) admits equivalent robust and mixed-integer programming (MIP) reformulations. The robust reformulation is shown to be strongly $NP$-hard, whereas the problem without the first-order moment constraints is proved to be polynomially solvable. We perform numerical experiments to illustrate the advantages of the proposed approach; we also demonstrate that the MIP reformulation of DRSPP can be solved reasonably fast using off-the-shelf solvers.