1. CEE 320 Fall 2008 Route Choice CEE 320 Steve Muench

2. CEE 320 Fall 2008 Route Choice Final step in sequential approach –Trip generation (number of trips) –Trip distribution (origins and destinations) –Mode choice (bus, train, etc.) –Route choice (specific roadways used) Desired output from the traffic forecasting process: how many vehicles at any time on a roadway

3. CEE 320 Fall 2008 Complexity Route choice decisions are primarily a function of travel times, which are determined by traffic flow Travel time Traffic flow Relationship captured by highway performance function

4. CEE 320 Fall 2008 Outline 1.General 2.HPF Functional Forms 3.Basic Assumptions 4.Route Choice Theories a.User Equilibrium b.System Optimization c.Comparison

5. CEE 320 Fall 2008 Basic Assumptions 1.Travelers select routes on the basis of route travel times only –People select the path with the shortest TT –Premise: TT is the major criterion, quality factors such as “scenery” do not count –Generally, this is reasonable 2.Travelers know travel times on all available routes between their origin and destination –Strong assumption: Travelers may not use all available routes, and may base TTs on perception 3. Travelers all make this choice at the same time

6. CEE 320 Fall 2008 HPF Functional Forms Travel Time Traffic Flow (veh/hr) Capacity Linear Non-Linear Free Flow Common Non-linear HPF from the Bureau of Public Roads (BPR)

7. CEE 320 Fall 2008 Speed vs. Flow Flow (veh/hr) Speed (mph) u f Free Flow Speed Highest flow, capacity, q m Uncongested Flow Congested Flow umum q m is bottleneck discharge rate

8. CEE 320 Fall 2008 Theory of User Equilibrium Travelers will select a route so as to minimize their personal travel time between their origin and destination. User equilibrium (UE) is said to exist when travelers at the individual level cannot unilaterally improve their travel times by changing routes. Frank Knight, 1924

9. CEE 320 Fall 2008 Wardrop Wardrop’s 1st principle “ The journey times in all routes actually used are equal and less than those which would be experienced by a single vehicle on any unused route ” Wardrop’s 2nd principle “ At equilibrium the average journey time is minimum ”

10. CEE 320 Fall 2008 Theory of System-Optimal Route Choice Preferred routes are those, which minimize total system travel time. With System-Optimal (SO) route choices, no traveler can switch to a different route without increasing total system travel time. Travelers can switch to routes decreasing their TTs but only if System-Optimal flows are maintained. Realistically, travelers will likely switch to non-System-Optimal routes to improve their own TTs.

11. CEE 320 Fall 2008 Formulating the UE Problem Finding the set of flows that equates TTs on all used routes can be cumbersome. Alternatively, one can minimize the following function: n=Route between given O-D pair t n (w)dw=HPF for a specific route as a function of flow w=Flow xnxn ≥0 for all routes

12. CEE 320 Fall 2008 Formulating the UE Problem n=Route between given O-D pair t n (w)dw=HPF for a specific route as a function of flow w=Flow xnxn ≥0 for all routes

13. CEE 320 Fall 2008 Example (UE) Two routes connect a city and a suburb. During the peak-hour morning commute, a total of 4,500 vehicles travel from the suburb to the city. Route 1 has a 60-mph speed limit and is 6 miles long. Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are as follows: Route 1 HPF increases at the rate of 4 minutes for every additional 1,000 vehicles per hour. Route 2 HPF increases as the square of volume of vehicles in thousands per hour.. Route 1 Route 2City Suburb

14. CEE 320 Fall 2008 Example: Compute UE travel times on the two routes Determine HPFs –Route 1 free-flow TT is 6 minutes, since at 60 mph, 1 mile takes 1 minute. –Route 2 free-flow TT is 4 minutes, since at 45 mph, 1 mile takes 4/3 minutes. –TT 1 = 6 + 4x 1 –TT 2 = 4 + x 2 2 –Flow constraint: x 1 + x 2 = 4.5 Route 1 HPF increases at the rate of 4 minutes for every additional 1,000 vehicles per hour. Route 2 HPF increases as the square of volume of vehicles in thousands per hour..

15. CEE 320 Fall 2008 Example: Compute UE travel times on the two routes Route use check (will both routes be used?) –All or nothing assignment on Route 1 –All or nothing assignment on Route 2 –Therefore, both routes will be used If all the traffic is on Route 1 then Route 2 is the desirable choice If all the traffic is on Route 2 then Route 1 is the desirable choice

16. CEE 320 Fall 2008 Example: Solution Apply Wardrop’s 1st principle requirements. All routes used will have equal times. 6 + 4x 1 = 4 + x 2 2 x 1 + x 2 = 4.5 Substituting and solving: 6 + 4x 1 = 4 + (4.5 – x 1 ) 2 6 + 4x 1 = 4 + 20.25 – 9x 1 + x 1 2 x 1 2 – 13x 1 + 18.25 = 0 x 1 = 1.6 or 11.4 (total is 4.5 so x 1 = 1.6 or 1,600 vehicles) x 2 = 4.5 – 1.6 = 2.9 or 2,900 vehicles Check answer: TT 1 = 6 + 4(1.6) = 12.4 minutes TT 2 = 4 + (2.9) 2 = 12.4 minutes

17. CEE 320 Fall 2008 Example: Mathematical Solution     Same equation as before

18. CEE 320 Fall 2008 Theory of System-Optimal Route Choice Preferred routes are those, which minimize total system travel time. With System-Optimal (SO) route choices, no traveler can switch to a different route without increasing total system travel time. Travelers can switch to routes decreasing their TTs but only if System-Optimal flows are maintained. Realistically, travelers will likely switch to non-System-Optimal routes to improve their own TTs. Not stable because individuals will be tempted to choose different route.

19. CEE 320 Fall 2008 Formulating the SO Problem Finding the set of flows that minimizes the following function: n=Route between given O-D pair t n (x n )=travel time for a specific route xnxn =Flow on a specific route

20. CEE 320 Fall 2008 Example (SO) Two routes connect a city and a suburb. During the peak-hour morning commute, a total of 4,500 vehicles travel from the suburb to the city. Route 1 has a 60-mph speed limit and is 6 miles long. Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are as follows: Route 1 HPF increases at the rate of 4 minutes for every additional 1,000 vehicles per hour. Route 2 HPF increases as the square of volume of vehicles in thousands per hour. Compute UE travel times on the two routes. Route 1 Route 2City Suburb

21. CEE 320 Fall 2008 Example: Solution 1.Determine HPFs as before –HPF 1 = 6 + 4x 1 –HPF 2 = 4 + x 2 2 –Flow constraint: x 1 + x 2 = 4.5 2.Formulate the SO equation –Use the flow constraint(s) to get the equation into one variable

22. CEE 320 Fall 2008 Example: Solution 1.Minimize the SO function 2.Solve the minimized function

23. CEE 320 Fall 2008 Example: Solution 1.Solve the minimized function 2.Find travel times 3.Find the total vehicular delay

24. CEE 320 Fall 2008 Compare UE and SO Solutions User equilibrium –t 1 = 12.4 minutes –t 2 = 12.4 minutes –x 1 = 1,600 vehicles –x 2 = 2,900 vehicles –t i x i = 55,800 veh-min System optimization –t 1 = 14.3 minutes –t 2 = 10.08 minutes –x 1 = 2,033 vehicles –x 2 = 2,467 vehicles –t i x i = 53,592 veh-min Route 1 Route 2City Suburb

25. CEE 320 Fall 2008 Why are the solutions different? Why is total travel time shorter? Notice in SO we expect some drivers to take a longer route. How can we force the SO? Why would we want to force the SO?

26. CEE 320 Fall 2008 UE SO

27. CEE 320 Fall 2008 Total Travel time is Minimum at SO

28. CEE 320 Fall 2008 By asking one driver to take 3 minutes longer, I save more than 3 minutes in the reduced travel time of all drivers (non- linear) Total travel time if X1=1600 is 55829 Total travel time if X1=1601 is 55819