  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.
