Transportation Problem Allowing Sending and Bringing Back

Abstract

This paper considers a transportation problem on a weighted graph. The weights specify the amounts of commodities at nodes, which are positive if the amounts are stored at nodes and negative if the amounts are needed at nodes. To meet all demands we use vehicles, one at each node, with some loading capacity to and from neighbors. In a trip using a vehicle we can send commodities from a node to a neighbor along an edge and also bring back some other commodities from the neighbor. In this paper we are interested in feasibility problem, which is to decide whether there is a single round of trips that meet all demands. We prove the feasibility problem is NP-complete even in the easiest case of a one-commodity transportation problem with unbounded capacity. We also present several different polynomial-time algorithms for other cases.