Effects of feasibility cuts in Lagrangian relaxation for a two-stage stochastic facility location and network flow problem

Abstract

This paper studies a two-stage stochastic facility location and network flow problem with uncertainty in demand, supply, network availability, transportation costs, and cost of facility activation. The goal is to design a minimum cost long-term network of facilities, in anticipation of uncertain supply, demand, and network structure. Facility location decisions are made in the first stage and facility activation and network flow decisions are made in the second stage. We develop a branch and price algorithm built on a Lagrangian relaxation with two subproblems, one per stage of decision making. To improve convergence, we show that the Lagrangian subproblems could be strengthened by using Benders decomposition. Namely, we add to the first stage Lagrangian subproblem, Benders feasibility cuts generated from the second stage Lagrangian subproblem, and guarantee that the former only generates optimality cuts. Numerical results show that feasibility cuts tighten the Lagrangian duality gap in the root node of the branch and price tree, and solutions generated by our algorithm improve over the literature in terms of computational time and number of solved instances.