Fixed-charge transportation on a path: optimization, LP formulations and separation

In the fixed-charge transportation problem, the goal is to optimally transport goods from depots to clients when there is a fixed cost associated to transportation or, equivalently, to opening an arc in the underlying bipartite graph. We further motivate its study by showing that it is both a special case and a strong relaxation of the big-bucket multi-item lot-sizing problem, and a generalization of a simple variant of the single-node flow set. This paper is essentially a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path. We give a $O(n^3)$-time optimization algorithm and a $O(n^2)$-size linear programming extended formulation. We describe a new class of inequalities that we call "path-modular" inequalities. We give two distinct proofs of their validity. The first one is direct and crucially relies on sub- and super-modularity of an associated set function. The second proof is by showing that the projection of the extended linear programming formulations onto the original variable space yields exactly the polyhedron described by the path-modular inequalities. Thus we also show that these inequalities suffice to describe the convex hull of the set of feasible solutions