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This paper is meant as a guide for researchers interested in dynamic modeling of commodity flows from the perspective of spatial price equilibrium. In particular, we present a type of dynamic spatial price equilibrium (DSPE) in continuous time as a basis for modeling freight flows in a network economy. We consider the circumstance of a known matrix of travel times between all pairs of markets (origindestination pairs) within a network for which paths (routes) are articulated. We also consider the unit cost of transport to be the sum of the price for freight services and a surcharge for backorders. Prices for freight services follow a nonlinear operator explained herein. That operator allows consideration of break-point pricing, as well as other forms of nonlinear pricing. The DSPE model considered is expressed four different ways. The first formulation is a nonlinear complementarity problem with explicit embedded dynamics describing the rate of change of inventories at each node as the net of production, consumption, import, and export, with explicit time shifts that account for shipping latencies. We also provide three alternative formulations: a differential complementarity system, a differential variation inequality, and a variational inequality based on a state operator. We discuss algorithms appropriate to each formulation and close with a discussion of future research needed to make DSPE models applicable to freight systems planning and the pricing of freight services. 

In this paper we provide a statement of dynamic spatial price equilibrium (DSPE) in continuous time as a basis for modeling freight flows in a network economy. The model presented describes a spatial price equilibrium due to its reliance on the notion that freight movements occur in response to differences between the local and distant prices of goods for which there is excess demand; moreover, local and distant delivered prices are equated at equilibrium. We propose and analyze a differential variational inequality (DVI) associated with dynamic spatial price equilibrium to study the Nash-like aggregate game at the heart of DSPE using the calculus of variations and optimal control theory. Our formulation explicitly considers inventory and the time lag between shipping and demand fulfillment. We stress that such a time lag cannot be readily accommodated in a discrete-time formulation. We provide an in-depth analysis of the DVI’s necessary conditions that reveals the dynamic user equilibrium nature of freight flows obtained from the DVI, alongside the role played by freight transport in maintaining equilibrium commodity prices and the delivered-price-equals-local-price property of spatial price equilibrium. By intent, our contribution is wholly theoretical in nature, focusing on a mathematical statement of the defining equations and inequalities for dynamic spatial price equilibrium (DSPE), while also showing there is an associated differential variational inequality (DVI), any solution of which is a DSPE. The model of spatial price equilibrium we present integrates the theory of spatial price equilibrium in a dynamic setting with the path delay operator notion used in the theory of dynamic user equilibrium. 

This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dy- namic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE; then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples. 

Dynamic user equilibrium (DUE) is the most widely studied form of dynamic traffic assignment (DTA), in which road travelers engage in a non-cooperative Nash-like game with departure time and route choices. DUE models describe and predict the time-varying traffic flows on a network consistent with traffic flow theory and travel behavior. This paper documents theoretical and numerical advances in synthesizing traffic flow theory and DUE modeling, by presenting a holistic computational theory of DUE, which is numerically implemented in a MATLAB package. In particular, the dynamic network loading (DNL) sub-problem is formulated as a system of differential algebraic equations based on the Lighthill-Whitham-Richards fluid dynamic model, which captures the formation, propagation and dissipation of physical queues as well as vehicle spillback on networks. Then, the fixed-point algorithm is employed to solve the DUE problems with simultaneous route and departure time choices on several large-scale networks. We make openly available the MATLAB package, which can be used to solve DUE problems on user-defined networks, aiming to not only facilitate benchmarking a wide range of DUE algorithms and solutions, but also offer researchers a platform to further develop their own models and applications. The MATLAB package and computational examples are available online.