Minimum-aggregate-concave-cost multicommodity flows in strong-series-parallel networks

This work develops a polynomial-time dynamic-programming algorithm for minimum-aggregate-concave-cost multicommodity flow problems in strong-series-parallel networks. Many important problems from production and inventory management, capacity planning, network design and transportation can be formulated in these terms. Our results include a characterization of extreme flows in strong-series-parallel networks and an algorithm based on this characterization that searches extreme hows efficiently to find an optimal one. The algorithm runs in time proportional to A(N + K), where A, N and K are respectively the: numbers of arcs, nodes and commodities in the network and appears to be the first to solve the problem in polynomial time. When applied to the dynamic economic-order-quantity problem, the algorithm matches the performance of that of Wagner and Whitin (1958). Moreover, our algorithm has broader applications, including the multi-division capacity expansion problem and the generalization of the dynamic economic-order-quantity problem to series-parallel production processes in which subassemblies are assembled into a finished product.