A Lower Bound for Uncapacitated, Multicommodity Fixed Charge Network Design Problems
Network design problems concern flows over networks in which a fixed charge must be incurred before an arc becomes available for use. The uncapacitated, multicommodity network design problem is modeled with (i) aggregate, and (ii) disaggregate "forcing" constraints. [Forcing constraints ensure logical relationships between the fixed charge-related and the flow-related decision variables.] A new lower bound for this problem referred to as the capacity improvement (CI) bound-is presented; and an efficient implementation scheme (using shortest path and linearized knapsack programs) is described. A key feature of the CI lower bound is that based on the LP relaxation of the aggregate version of the problem. A numerical example illustrates that the CI lower bound (i) can be as tight as the disaggregate LP relaxation, and (ii) can converge to the optimal objective function value of the IP formulation.