An equilibrium solution for the freight network problem is obtained. The system consists of multiple shippers and carriers, each acting as profit maximizing agents. The flow pattern maximizing total system profits is called the vertically efficient flows. We show that if the carrier or coalition of carriers use vertically efficient nonlinear pricing schedules, the vertically efficient flows are the equilibrium flows. The division of surplus associated with each shipment is obtained by solving a linear programming problem. Vertically efficient nonlinear tariff schedules are of a form that is commonly used in the transportation industry.
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Hurley et al. (1994) studied this question.