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 loss 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. 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 $NP$-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 considered approach; we also demonstrate that the MIP reformulation of DRSPP can be solved effectively using off-the-shelf solvers.

翻译：在这项研究中,我们考虑了最短路径问题,即弧成本受分布不确定性的影响。基本上,决策者试图在符合决策者最初信息的候选人分布的模棱两可(或一个家庭)中,将最坏的预期损失降到最低,因为候选人分布的模棱两可(或一个家庭)与决策者最初的信息一致。这个模棱两可是由所有符合关于弧子子和对特定弧的个别概率限制的线性第一阶限制的分布式分布式第一阶限制所形成的。根据一些额外的假设,由此产生的分布稳健的最短路径问题(DRSPP)承认了相当强和混合整数的重新编程(MIP)。有力的重订显示是硬值的,而没有第一阶限制的问题则证明是多式的。我们进行了数字实验,以说明考虑的方法的优点;我们还表明,对DRSPPP的重新拟订的MIP可以使用现成的溶液有效解决。