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. Our distributional constraints can be designed from incomplete and partially observable data. 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 effectively using off-the-shelf solvers.

中文翻译：

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分布鲁棒最短路径问题的一种方法

在这项研究中，我们考虑了弧成本受分布不确定性影响的最短路径问题。基本上，决策者试图在与决策者的初始信息一致的候选分布的模糊集（或族）上最小化她的最坏情况预期损失。模糊集由满足关于弧子集的规定线性一阶矩约束和关于特定弧的个别概率约束的所有分布形成。我们的分布约束可以从不完整和部分可观察的数据中设计。在一些额外的假设下，由此产生的分布稳健的最短路径问题 (DRSPP) 承认等效的稳健和混合整数规划 (MIP) 重构。稳健的重构被证明是强$NP$-hard，而没有一阶矩约束的问题被证明是多项式可解的。我们进行了数值实验来说明所提出方法的优点；我们还证明可以使用现成的求解器有效地解决 DRSPP 的 MIP 重构。