In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the incoming arc flows cannot be split or merged when routed through outgoing arcs. This so-called "no-split no-merge" requirement arises in unit train scheduling where train consists should remain intact at stations that lack necessary equipment and manpower to attach/detach them. Solving the unsplittable network flow problems with standard mixed-integer programming formulations is computationally difficult due to the large number of binary variables needed to determine matching pairs between incoming and outgoing arcs of nodes with no-split no-merge constraint. In this paper, we study a stochastic variant of the unit train scheduling problem where the demand is uncertain.

We develop a novel decision diagram (DD)-based framework that decomposes the underlying two-stage formulation into a master problem that contains the combinatorial requirements, and a subproblem that models a continuous network flow problem.

The master problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a substantial improvement in solution time. Similarly to the Benders decomposition technique, the subproblems output cutting planes that are used to refine the master DD. Computational experiments show a significant improvement in solution time of the DD framework compared with that of standard methods.

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