Differential Multiplant Monopoly on a Freight Network

In this paper we give a mathematical statement of a differential spatial monopoly wherein transactions are facilitated by a freight transportation network that is based on an underlying graph. The monopolist has a presence at every node, and freight services are available for paths connecting stipulated origin-destination (OD) pairs. The monopolist produces a single homogeneous product that may be manufactured and inventoried, as well as sold, at every node. The dynamics take the form of ordinary differential equations that describe flow conservation. We present the necessary conditions of the monopolist’s optimal control problem and observe their interpretation is that marginal revenue equals marginal cost at each node for each instant of continuous time. Existence of an optimal solution to the monopolist’s problem is proven and a numerical example solved using two algorithms: one implemented in discrete time and the other in continuous time.