Dynamic pricing in an urban freight environment Terry L. Friesz a, * , Reetabrata Mookherjee b , Jose ´ Holguı ´n-Veras c , Matthew A. Rigdon a a Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, United States b Zilliant Inc., Austin, TX, United States c Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, United States Received 19 February 2007; received in revised form 7 August 2007; accepted 7 August 2007 Abstract In this paper, we propose a dynamic, game theoretic model of dynamic pricing in an urban freight environment with three main entities: sellers, transporters and receivers. The sellers and transporters are modelled as non-cooperative Cour- not–Nash agents. The sellers compete to capture receiver input factor demands, while the transporters compete to capture the transportation demand generated by the seller/receiver transactions. Each competing agent’s extremal problem is for- mulated as an optimal control problem and the set of these coupled optimal control problems is transformed into a dif- ferential variational inequality representing the general Nash equilibrium problem. A nonlinear complementarity problem is also formulated and used to solve a numerical example. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Dynamic pricing; Urban freight; Dynamic game; Differential variational inequality 1. Introduction This paper discusses a model of dynamic pricing of freight services in an urban environment that follows the paradigm set in the field of revenue management for nonlinear pricing in a dynamic, game theoretic setting. There are many applications of dynamic pricing in a game theoretic setting, however, as far as we know, this is the first direct application of results from revenue management to urban freight transport and city logistics. Altman and Wynter (2004) give an overview of pricing in transportation and telecommunication networks while Lederer (2003) examines static price and production competition between profit maximizing firms that are spatially distributed. Zhang et al. (2005) discuss dynamic game theoretic models of infrastruc- ture networks including freight transportation networks. Some examples of dynamic pricing in a game 0191-2615/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2007.08.001 * Corresponding author. Tel.: +1 814 863 2445. E-mail addresses: tlf13@psu.edu (T.L. Friesz), reeto.mookherjee@gmail.com (R. Mookherjee), jhv@rpi.edu (J. Holguı ´n-Veras), mar409@psu.edu (M.A. Rigdon). Available online at www.sciencedirect.com Transportation Research Part B 42 (2008) 305–324 www.elsevier.com/locate/trb