On the Complexity of the Production-Transportation Problem

The production-transportation problem (PTP) is a generalization of the transportation problem. In PTP, we decide not only the level of shipment from each source to each sink but also the level of supply at each source. A concave production cost function is associated with the assignment of supplies to sources. Thus the objective function of PTP is the sum of the linear transportation costs and the production costs. We show that this problem is generally NP-hard and present some polynomial classes. In particular, we propose a polynomial algorithm for the case in which the transportation cost matrix has the Monge property and the number of sources is fixed. for the problem with two sources. 0. Introduction. It is known that the minimization problem over a polyhedron is polynomial when the objective function is convex [GLS88]. In contrast, many concave minimization problems are NP-hard. We consider here a concave minimization problem over transportation constraints called the production-transportation problem (PTP). PTP is a generalization of the transportation problem. In PTP we need to decide not only the level of shipment from each source to each sink but also the level of supply at each source. A concave production cost function is associated with the assignment of supplies to sources. The objective function is the sum of the linear transportation costs and the concave production costs.