travel time for a vehicle on link a There are other congestion functions. The CATS has long used a function different from that used by the BPR, but there seems to be little difference between results when the CATS and BPR functions are compared. Equilibrium assignment To assign traffic to paths and links we have to have rules, and there are the well-known Wardrop equilibrium conditions. The essence of these is that travelers will strive to find the shortest (least resistance) path from origin to destination, and network equilibrium occurs when no traveler can decrease travel effort by shifting to a new path. These are termed user optimal conditions, for no user will gain from changing travel paths once the system is in equilibrium. The user optimum equilibrium can be found by solving the following nonlinear programming problem subject to: where is the number of vehicles on path r from origin i to destination j. So constraint (2) says that all travel must take place –i = 1 ... n; j = 1 ... n = 1 if link a is on path r from i to j ; zero otherwise. So constraint (1) sums traffic on each link. There is a constraint for each link on the network. Constraint (3) assures no negative traffic. Example An example from Eash, Janson, and Boyce (1979) will illustrate the solution to the nonlinear program problem. There are two links from node 1 to node 2, and there is a resistance function for each link (see Figure 1). Areas under the curves in Figure 2 correspond to the integration from 0 to a in equation 1, they sum to 220,674. Note that the function for link b is plotted in the reverse direction. Figure 1: Two Route Network Figure 2: Graphical Solution to the Equilibrium Assignment Problem Figure 3: Allocation of Vehicles not Satisfying the Equilibrium Condition At equilibrium there are 2,152 vehicles on link a and 5847 on link b. Travel time is the same on each route: about 63. Figure 3 illustrates an allocation of vehicles that is not consistent with the equilibrium solution. The curves are unchanged. But with the new allocation of vehicles to routes the shaded area has to be included in the solution, so the Figure 3 solution is larger than the solution in Figure 2 by the area of the shaded area. Integrating travel choices The