Mathematical Programming Approach to User Equilibrium Equating travel time on all used routes is a straightforward approach to user equilibrium, but can become cumbersome when many alternative routes are involved. The approach used to resolve this computational obstacle is to formulate the user-equilibrium problem as a mathematical program. Specifically, user-equilibrium route flows can be obtained by minimizing the following function [Sheffi, 1985]:
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min nx n h
S x t w dw= (8.8)
n = a specific route, and tn(w) = performance function corresponding to route n (w denotes flow, xn’s).
This function is subject to the constraints that the flow on all routes is greater than or equal to zero (xn ≥ 0) and that flow conservation holds (the flow on all routes between an origin and destination sums to the total number of vehicles, q, traveling between the origin and destination, nn
q x= ).
Formulating the user-equilibrium problem as a mathematical program allows an equilibrium solution to very complex highway networks (many origins and destinations) to be readily undertaken by computer.