1. Multi-modal Bi-criterion Highway Assignment for Toll Roads Jian Zhang Andres Rabinowicz Jonathan Brandon Caliper Corporation
12/9/2018

2. Traditional Traffic Assignment
Assuming uniformity among users’：
Access to network
Link cost function (VDF)
Value of time (VOT)
Out-of-pocket cost 2

3. Advantages of MMA (multi-modal assignment)
Simulating user variety in：
Vehicle type (car，bus，light / heavy trucks, etc.)
Occupancy（HOV，LOV)
Driver characteristics (cost or time sensitive)
Class-specific VOT’s 3

4. Advantages of MMA
Simulating vehicle-type-specific network accessibility：
Lane usage (HOV, HOT)
Height/Weight limits
Toll charge
Access to network status info. 4

5. MMA Model Analysis Standardized volume standardized volume :
vak — total PCE on link a for type k
αk — PCE conversion factor for type k
Cost function based on
standardized volume :
cak — cost for type k (e.g. toll payment)
βk — VOT for type k 5

6. T2 Traffic Assignment Model
Developed by Robert Dial, features include:
VOT as a random variable in calculating link cost:
P — path,α— random VOT variable with known probability density
Bi-criterion multiple path building
Fast path calculation using “Efficient Frontier”
One user class (|K|=1) 6

7. New MMA-T2 Model Combining MMA and T2: Multiple user classes
VOT of each user class as a random variable with its unique mean and distribution
Bi-criterion multiple path building for each user type 7

8. MMA-T2 Model Advantages
Representing various user income levels
Considering VOT variation within each user group
Allowing more realistic simulation of user response to different highway toll policies. 8

9. MMA-T2 Model Structure (variational inequality )9

10. MMA-T2 Model Solution Model Solution Equilibrium Conditions:
gpk — travel cost on path p for type k
uwk — min. travel cost between w-th OD pair for type k
fpk — type k volume on path p
Note: Only min.-cost paths have flow, which satisfies multi-modal traffic assignment equilibrium condition 10

11. MMA-T2 Model Solution Algorithm
Step 0: Initialization: do non-equilibrium T2 assignment for each user type
Step 1: Renew link times
Step 2: Conduct non-equilibrium T2 assignment for each user type based on new link times so as to get auxiliary link flows.
Step 3: Compute step size and combine current and auxiliary link flows (MSA)
Step 4: Check convergence. Stop if converged, otherwise go to Step 1 11

12. Example 1: Network Capacity and Free-flow Speed
Free-flow Time and Cost 12

13. Example 1: Users
VOT Probability Density
O-D Demand 13

14. Example 1: Results Class 1 Flows Class 2 Flows Class 3 Flows
Total Flows 14

15. Example 2: Network
Mass. Highway Network (5627 links, 2025 nodes, 106 toll links) 15

16. Example 2: User Demands Classes : Car, Light Truck, Heavy Truck
# of O-D pairs = 19,460
Total demand = 416,426 (70%, 23%, 7%) 16

17. Example 2: Toll Scenarios
Original Toll Matrices
Toll Scenarios : 1 – original 2 – original * 2
3 – original * 3 4 – original * 4 17

18. Example 2: User VOT Distribution Scenarios
SD = 5
SD = 10
SD = 15
Pearson Type III Distribution
Means: Car = 20, Light Truck = 40, Heavy Truck = 60 ($/hr) 18

19. Example 2 Result: Toll Road Usages19

20. Example 2 Results: Average Trip Time20

21. Observations
When toll charge is high, MMA model tends to underestimate toll road usages than predicted by MMA-T2 model
When toll charge is high, MMA model tends to overestimate average trip times than forecast by MMA-T2 model
As user VOT distributions become wider, toll road usages increase and average trip times drop 21

22. Conclusion
MMA-T2 model has the advantages of both MMA and T2 (bi-criterion multi-path building, multi-user-class specifications, etc.)
The new model is able to capture heterogeneous users’ response to toll charges for a more accurate evaluation of different toll scenarios 22

23. Thanks! 23