An integer linear programming model for the constrained shortest path tour problem

Let D = (V, A) be a directed graph with set of vertices V and set of arcs A, and let each arc (i, j) ∈ A, with i, j ∈ V, be associated with a non-negative cost. The constrained shortest path tour problem (CSPTP) is NP-Hard and consists in finding a shortest path between two distinct vertices s ∈ V and t ∈ V such that the path does not include repeated arcs and must visit a sequence of vertex disjoint subsets T₁, …, T_N in this order. In this work, we formulate the CSPTP as an integer linear programming (ILP) model and present valid inequalities for the problem. Computational experiments performed on benchmark data sets from the literature show that our ILP model consistently outperforms existing exact algorithms for the CSPTP and finds optimal solutions for most instances.