NETWORK EQUILIBRIUM: OPTIMIZATION FORMULATIONS WITH BOTH QUANTITIES AND PRICES AS VARIABLES

The optimization formulation of the traffic assignment problem is usually stated in terms of flow (quantity) variables but can also be stated entirely in terms of price variables (the dual formulation), and recently it has also been stated in terms of a combination of both quantity and price variables. Here we consider properties and problems associated with the latter optimization formulations and relate these formulations to the traffic equilibrium conditions and to the purely quantity formulation. In the usual optimization formulation of the traffic assignment problem, developed and presented by Beckmann et al. (1956), Dafermos (1971), Florian (1976), and others, all the variables represent flows on arcs or paths: see the recent useful surveys by Femandez and Freisz (1983) and Friesz (1985). We will refer to this formulation as the quantity-quantity formulation, since both the travel demand side and the travel time or cost side of the objective function are expressed in terms of flow (quantity) variables. There is also an equivalent dual formulation of the traffic assignment problem (Carey (1985), Fukushima (1984)) in which all variables represent travel times or costs for arcs or paths. We will refer to this as the price-price formulation. If the travel demand equations, and the arc travel/cost equations, are invertable then the solution of the quantity-quantity formulation can be obtained directly from the solution of the price-price formulation, and vice-versa. As well as the above two optimization formulations we can construct two less well-known optimization formulations, which we will refer to as a quantity-price formulation and a price- quantity formulation. A quantity-price formulation is presented in Aashtiani (1979) Ch. 5 and in the survey paper Femandez and Friesz (1983), pp. 164-165. In the quantity-price formulation the cost part of the objective function is stated in terms of quantity variables and the demand part of the objective function is stated in terms of price variables. The quantity variables are then related to the price variables by explicitly including travel demands as functions of prices in the constraints. A price-quantity formulation can be stated analogously, the cost part of the objective function being stated in terms of price variables, the demand part in terms of quantity variables, and the price and quantity variables related to each other by including the arc travel time/cost equations explicitly in the constraints. The purpose of the present paper is to set out some properties and difficulties associated with the quantity-price and price-quantity formulations, and in particular to relate these models to the traffic equilibrium conditions, and to the quantity-quantity formulations. In the recent research literature, variational inequality formulations of the traffic equilibrium problem have generated much more interest than optimization formulations, since the former are more recent and more general formulations of the equilibrium problem than are the latter. However, it is still important to consider problems peculiar to optimization formulations since: a. When an optimization formulation is possible (when the Jacobian matrices of the travel demand functions and travel cost functions are symmetric) then it has great computational advantages over the variational inequality formulation. (See for example Fisk and Boyce (1983)). b. Even when the variational inequality problem cannot be reduced to a single equivalent optimization formulation (the asymmetric case), it may be solved by solving a sequence of variational inequality problems each of which reduces to an optimization problem. Thus even in this case the various optimization formulations are important.