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, jliu215@asu.edu Arizona State University Xuesong Zhou, xzhou74@asu.edu He Wei, Beijing Institute of City Planning, China

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

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)

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)

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

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.

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., 2016. Capacitated transit service network design with boundedly rational agents. Transportation research part B 93, 225-250.

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

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

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 .

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

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., 2009. Equivalent gap function-based reformulation and solution algorithm for the dynamic user equilibrium problem. Transportation research part B 43, 345-364.

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)

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.

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