Fixed-Charge Transportation on a Path: Linear Programming Formulations

The fixed-charge transportation problem is a fixed-charge network flow problem on a bipartite graph. This problem appears as a subproblem in many hard transportation problems, and is also both a special case and a strong relaxation of the challenging bigbucket multi-item lot-sizing problem. In this paper, we provide a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path. 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 suband super-modularity of an associated set function, thereby providing an interesting link with flow-cover type inequalities. The second proof is by showing that the projection of an O(n)-size extended linear programming formulation 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 feasible solutions to this problem. We finally show how to solve the separation problem associated to the path-modular inequalities in O(n) time.