# An Illustration of Variable Value-of-Time Pricing Modeling

DynusT incorporates a breakthrough methodology to enable Dynamic User Equilibrium (DUE) assignments with variable Value-of-Times (VoT), for each vehicle in the system. This feature allows users to incorporate a continuous VOT distribution in evaluating static or dynamic pricing scenarios, without any increase in run time compared to traditional approaches where only an average or a discrete VoT distribution of values are permitted. This breakthrough also allows different VoTs to be associated with different trip purposes (e.g., work trips have different VoTs from social, recreational trips) thus making DynusT the best DTA model to model tolling schemas and, congestion pricing, or to be integrated with fine-grained activity-based-models (AMBs).

The purpose of this example is to demonstrate the DUE benefits utilizing continuously distributed VoTs.

The example uses a test network with one OD pair and two connecting routes, as illustrated in Figure 1. The following three VOT distributions cases are tested:

Fixed VOT of $20/hr, for all travelers (Uniform VOT scenario)

Variable VOT for three groups of travelers - 50% were assigned a VOT of $20/hr, 25% were assigned a VOT of $30/hr, and the remaining 25% were assigned a VOT of $10/hr (Discrete VOT scenario).

Continuously uniformly distributed VOT for all travelers: (Continuous VOT scenario). A lognormal distribution would be more realistic to represent the real VOT distribution, but the uniform distribution is utilized herein for the purpose of demonstration.

A demand of 1,000 vehicles is assumed to maintain free flow without congestion in the network. All vehicles depart from node A and head toward node C. Only two route options are available, i.e., A->C or A->B->C. It is assumed that link A->C, which the shortest travel time route, is a toll road with a fixed toll.

Figure 1: Illustrative Network

Figure 2 illustrates how the flow on the toll road (route A->C) varies as the toll rate increases. In the uniform VOT case, when the toll rate is below $2.76, all 1,000 vehicles choose the toll road because the toll road’s generalized cost ( is lower than that of A->B->C and by the definition of user equilibrium, all users will use the lowest cost route. When the toll rate is slightly higher than $2.76, all travelers switch from the toll road to route A->B->C.

Figure 2: Toll road usage vs. toll rate

In the Discrete VOT case, similar behavior is observed with increasing toll rates. Specifically, when the toll rate is around $1.38, 25 % of the travelers with VOT equal to $10/hr will switch from the toll road to route A->B->C, since paying the toll will be equivalent to 8.3 min (8.3=60*$1.38/$10/hr) of additional travel time. Similarly, when the toll rate increases to $2.76 and then to $4.14, travelers with VOT’s $20/hr and $30/hr respectively will abruptly switch to route A->B->C. In contrast, in the Continuous VOT case, the toll road usage is gradually reduced with increasing toll rates as expected, starting when the toll rate is around $1.38 (when those travelers with $10/hr VOT are affected) and ending when the toll rate is around $4.14, when all travelers, including those with VOT, equal to $30/hr, decide to switch.

The results demonstrate that the new variable VOT capability models route choices between tolled and non-tolled facilities more realistically, as compared to the traditional approaches.