The HCM delay relationships are discontinuous, nonmonotonic, and nonintegratable. The only method of equilibrium traffic assignment known to be able to handle similarly difficult delay relationships is most often referred to as "one-over-kay" assignment or "equilibrium/incremental" assignment or "method of successive averages". The method finds an unweighted average of many all-or-nothing assignments, where the delay found prior to any iteration (k+l) is calculated from the average of volumes from the preceding (k) assignments. Equilibrium/incremental assignment produces identical results to Frank-Wolfe decomposition (LeBlanc, et. al, 1975) on networks with simple (such as the BPR) delay relationships (Powell and Sheffi, 1982; Horowitz, 1990); however, convergence is slightly slower.
This algorithm has not yet been extensively tested on networks where delay can be a function of several volumes.
The UTOWN network, originally created for testing UTPS, was modified by incorporating signalized intersection and two-way stops, primarily at freeway off-ramps. The modified UTOWN network is shown in Figure 1.
Convergence to an equilibrium solution needs to be checked, but the standard methods derived from Frank-Wolfe decomposition will not work in this case. We are looking for a user-optimal assignment. In such an assignment each trip is assigned to a shortest path between its origin and destination. Therefore, it is possible to determine when equilibrium has been achieved by checking whether the used paths are indeed the shortest paths. A simple test can be devised that compares total travel time between two assignments.
Step 1. Run the assignment algorithm through the desired number of iterations. Obtain estimates of volumes. Recalculate the link travel times. Compute total travel time with the estimates of link volumes and the new travel times.
Step 2. Using the new travel times and averaged trip table from Step 1, perform an all-or-nothing assignment. Do not recalculate link travel times. Compute total travel time.
Step 3. Compare the total travel times from Steps 1 and 2. The total travel time from Step 2 will always be the smallest. If they are nearly the same, convergence to an equilibrium solution has been achieved. If they differ significantly, there could be two causes: (1) more iterations are required; or (2) the algorithm failed.
This test is similar to one ("S1 - S2") found in UTPS.
The test was performed on the UTOWN network (containing HCM delay relationships) for varying numbers of iterations of equilibrium/incremental assignment. As seen in Table 1, the equilibrium/incremental assignment algorithm will produce an equilibrium solution on a network with traffic controls. After 200 iterations the difference between Steps 1 and 2 was inconsequential. Equilibrium was effectively achieved after about 20 iterations. This rate of convergence is similar to Frank-Wolfe decomposition.
A significant body of research is being assembled on "asymmetric" traffic assignment problems, which include assignments where delay is a function of several link volumes. It is likely that even faster (and perhaps surer) algorithms will be developed within the next few years.
An inspection of the assigned volumes revealed that similar results would have been difficult to obtain with conventional delay/volume relationships. The assigned volumes on approximately half of the links in the original UTOWN network (without traffic controls) were considerably different from those of the modified UTOWN network (Figure 1). For example, the volumes for one particular freeway link differed by a factor of more than two. The other half of the links had surprisingly similar volumes across the two networks. One striking difference between the two assignments was the higher arterial volumes on congested links in the modified network. The algorithm gave these links more green time, thus more capacity. The original network, of course, had to provide equal signalization priority to each approach, regardless of need.
The UTOWN network is artificial and exaggerates problems with assignment algorithms. Still, it adequately demonstrates the importance of having precise estimates of intersection capacity.
A traffic assignment involving complex intersection delay relationships, such as those in the HCM, is adaptive in the same sense as an actuated signal, which can adjust itself to the existing traffic volumes. The algorithm allocates capacity to an approach according to its volume and competing volumes. Approaches with relatively large volumes receive more green time, and thus capacity, than approaches with small volumes. Theoretically, the maximum capacity of an approach is its saturation flow rate, less any possible flow lost during phase changes. In practice, however, a small amount of green time must be given to conflicting approaches, even when there is very little traffic.
Such an assignment is quite realistic, but there is one unfortunate side effect - the solution may not be unique. It is entirely possible for an adaptive traffic assignment to have two or more equally valid equilibrium solutions. Under such circumstances, one cannot judge which solution is the correct one. Indeed, all solutions may be correct. Differences would be due to small variations in signalization - something that is impossible to predict.