1. Prepared for 16th TRB National Transportation Planning Applications Conference
Outline
Gap Value in Simulation-Based Dynamic Traffic Assignment (DTA) Models: Why Does It Exist and How to Reduce It?
Jiangtao Liu,
Arizona State University
Xuesong Zhou,
He Wei, Beijing Institute of City Planning, China

2. Outline Outline Gap value in DTA models
Why does the gap exist in DTA models
Simulation-based DTA models
How to reduce the gap in simulation-based DTA models
Conclusions

3. 1. Gap value in DTA models Simplified Sioux Falls traffic network
Number of nodes 24
Number of links 76
Number of zones
Number of vehicles
360,600 (17:00-19:00)

4. 1. Gap value in DTA models A subarea of Beijing traffic network
Number of nodes
2,502
Number of links
5,397
Number of zones
236
Number of vehicles
460,777 (6:00am-9:30am)

5. 1. Gap value in DTA models
Wardrop’s first principle (1952): each traveler seeks to minimize his/her cost of transportation
User equilibrium condition:
All used routes have equal and least travel cost
No user may lower his travel cost through unilateral action.
Beckmann, McGuire, and Winsten (1956) first proposed a mathematical program for static user equilibrium
Smith (1993) gave an equivalent minimization program of the DUE problem by minimizing the path-based user objective function

6. 1. Gap value in DTA models DUE is reached when Gap(r*,p*) = 0.
Requirements for DUE condition in mathematical models
the path cost function: continuous and strictly monotone
path flows: finite and convex compact set
Those properties of path cost functions might not be satisfied in general road networks with complex traffic dynamics
For example, tight road capacity constraint can make path/link cost function not continuous. Tight road capacity constraint is considered to capture the physical queue and queue spillback in simulation-based DTA.

7. 2. Why does the gap exist in DTA models
Under tight link capacity constraint:
Unique solution:
Path cost of agent 1 is 5;
Path cost of agent 2 is 3.
Wardrop’s UE isn’t satisfied.
The Gap value is 2
The result is a Nash equilibrium
In order to finish their trips, travelers will form an indifference band.
The tight capacity constraint invokes the bounded rationality of travelers.
Reference: Liu, J., Zhou, X., Capacitated transit service network design with boundedly rational agents. Transportation research part B 93,

8. 2. Why does the gap exist in DTA models
Comparison on Gap Values under tight link capacity constraint
2 vehicles depart from node 1 to node 4

9. 2. Why does the gap exist in DTA models
Gap Value exists under tight capacity constraints
Solution 2
Solution 1

10. 2. Why does the gap exist in DTA models
Gap Value exists under tight capacity constraints
Wardrop’s first principle (1952): each traveler seeks to minimize his/her cost of transportation.
A smaller gap value may not represent a better assignment result .

11. 3. Simulation-based DTA models
Simulation iteration by iteration
Input
Simulation
Output
Process of simulation-based DTA models

12. 4. How to reduce the gap in simulation-based DTA models
Vehicles’ path switching rate at each iteration
Solution quality
Convergence and Oscillations
Computational time
Method of Successive Average (MSA)
Pre-determined switching rate
Specific path-swapping methods (Lu et al., 2009)
Mixed Step-size scheme
The difference in travel cost between the current path and the shortest path
There is still no systematic ways of determining the swapping rate (Lu et al., 2009)
Reference: Lu, C., Mahmassani, H., Zhou, X., Equivalent gap function-based reformulation and solution algorithm for the dynamic user equilibrium problem. Transportation research part B 43,

13. 4. How to reduce the gap in simulation-based DTA models
A subarea of Beijing traffic network
Number of nodes
2502
Number of links
5397
Number of zones
236
Number of vehicles
460,777 (6:00am-9:30am)

14. 5. Conclusions
Gap may always exist under tight capacity constraints in DTA models under Wardrop’s first principle (travel behavior)
Tight capacity constraints (traffic flow models/car-following model) can invoke bounded rationality of travelers
A smaller gap value may not represent a better assignment result under tight capacity constraints
Simulation-based DTA models obey the Wardrop’s first principle travel behavior and try to reduce the gap value at each iteration
The vehicles’ path switching rate at each iteration is very important for simulation-based DTA models to reduce the gap value.

15. Thanks!