A DYNAMIC DISEQUILIBRIUM INTERREGIONAL COMMODITY FLOW MODEL TERRY L. FRIESZ* Department of Systems Engineering and Operations Research, George Mason University, Fairfax, VA, 22030-4444, U.S.A. ZHONG-GUI SUO PB Farradyne Inc., Rockville, MD, U.S.A. and DAVID H. BERNSTEIN Department of Operations Research and Civil Engineering, Princeton University, Princeton, NJ, U.S.A. (Received 2 May 1997; in revised form 3 February 1998) AbstractÐWe are concerned in this paper with creating a dynamic description of interregional commodity movements which has steady states consistent with the traditional Samuelson±Takayama±Judge (STJ) static spatial price equilibrium model. This is accomplished by introducing a disequilibrium adjustment mechanism which diers from that of other network spatial price tatonnement models in that prices and interregional ¯ows follow distinct signals, and constraints ensuring balanced trade ¯ows are not enforced prior to attaining an equilibrium. The disequilibria arising from our relaxation of the ¯ow balance constraints are seen to pro- vide a foundation for describing failures of spatial markets to clear. Also notable among the features of our model is its treatment of a general network topology and non-separable cost, supply and demand functions. We show through numerical experiments that the proposed adjustment process is seen to hold promise as means of calculating static spatial price equilibria. Furthermore, we discuss the circumstances under which disequilibrium states visited by the adjustment process describe actual time-varying commodity ¯ows and prices, allowing the model to be used for predictive dynamic interregional freight modeling. # 1998 Elsevier Science Ltd. All rights reserved Keywords: dynamic neturic ¯ows, spatial price, (dis)equilibrium 1. INTRODUCTION A foundation for the prediction of interregional freight ¯ows is provided by the spatial price equilibrium paradigm put forward by Samuelson (1952) and Takayama and Judge (1971). y This paradigm has been extended and analyzed in great detail in numerous papers appearing in the economics, regional science and transportation literatures. See, for example, the various formula- tions in Friesz et al. (1983a), Friesz et al. (1984) and Friesz and Bernstein (1993), as well as the review by Friesz (1985), to cite but four references. Moreover, the spatial price equilibrium para- digm has been employed in the modeling of shippers within more complicated shipper±carrier predictive freight network models (see Friesz and Harker, 1985; Harker and Friesz, 1986a,b). Indeed, many of the shortcomings of the present state-of-the-art in predictive freight network models (see Friesz et al., 1983b) may be traced to the Samuelson±Takayama±Judge (STJ) para- digm. In this paper we are concerned with overcoming one of these: namely the failure to describe the disequilibria encountered in an actual spatially separated economy connected by a freight Transpn Res.-B, Vol. 32, No. 7, pp. 467±483, 1998 # 1998 Elsevier Science Ltd. All rights reserved Pergamon Printed in Great Britain PII: S0191-2615(98)00012-5 0191-2615/98 $19.00+0.00 467 *Author for correspondence. Fax: 001 703 993 1521; e-mail: tfriesz@os¯.gmu.edu y The literature on freight network models is a rich one. Work which either extends or departs from the Samuelson± Takayama±Judge paradigm includes Harker (1987), Crainic et al. (1990), Kornhauser et al. (1979), and Kresge and Roberts (1971).