Minimum Convex Cost Dynamic Network Flows

Abstract

This paper presents and solves in polynomial time the minimum convex cost dynamic network flow problem, an infinite horizon integer programming problem which involves network flows evolving over time. The model is a finite network in which each arc has an associated transit time for flow to pass through it. An integral amount of flow is to be sent through arcs of the network in each period over an infinite horizon so as to satisfy conservation of flow from some fixed period on. Furthermore, the net amount of flow “in transit” is assumed fixed over the infinite horizon. The objective is to minimize the average convex cost per period of sending flow. This problem is a generalization of the minimum convex cost network flow problem, the maximum throughput dynamic network flow problem, and the minimum cost-to-time ratio circuit problem. Furthermore, the model has applications in periodic production and transshipment, airplane scheduling, cyclic capacity scheduling, and cyclic staffing. To solve the dynamic network flow problem, one first obtains an optimal continuous-valued flow which repeats every period. The fractional variables of this solution may be rounded in such a way as to obtain an optimal integral flow which repeats every q periods, where q is the least common denominator of the fractional parts of the continuous flow. Furthermore, this integral flow is also an optimal continuous flow.