An exact approach for the multicommodity network optimization problem with a step cost function

We investigate the Multicommodity Network Optimization Problem with a Step Cost Function (MNOP-SCF) where the available facilities to be installed on the edges have discrete step-increasing cost and capacity functions. This strategic long-term planning problem requires installing at most one facility capacity on each edge so that all the demands are routed and the total installation cost is minimized. We describe a path-based formulation that we solve exactly using an enhanced constraint generation based procedure combined with columns and new cuts generation algorithms. The main contribution of this work is the development of a new exact separation model that identifies the most violated bipartition inequalities coupled with a knapsack-based problem that derives additional cuts. To assess the performance of the proposed approach, we conducted computational experiments on a large set of randomly generated instances. The results show that it delivers optimal solutions for large instances with up to 100 nodes, 600 edges, and 4950 commodities while in the literature, the best developed approaches are limited to instances with 50 nodes, 100 edges, and 1225 commodities.