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Route Choice

Route Choice. CEE 320 Steve Muench. 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)

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Route Choice

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  1. Route Choice CEE 320Steve Muench

  2. 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. Complexity • Route choice decisions are primarily a function of travel times, which are determined by traffic flow Traffic flow Travel time Relationship captured by highway performance function

  4. Outline • General • HPF Functional Forms • Basic Assumptions • Route Choice Theories • User Equilibrium • System Optimization • Comparison

  5. Basic Assumptions • 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 • 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. HPF Functional Forms Common Non-linear HPF Linear Travel Time FreeFlow Non-Linear from the Bureau of Public Roads (BPR) Capacity Traffic Flow (veh/hr)

  7. Speed vs. Flow ufFree Flow Speed Uncongested Flow um Speed (mph) Highest flow, capacity, qm Congested Flow Flow (veh/hr) qm is bottleneck discharge rate

  8. 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. 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. 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. 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:

  12. Formulating the UE Problem

  13. 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 Suburb City Route 2

  14. Example: Compute UE travel times on the two routes • 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.. 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. • TT1 = 6 + 4x1 • TT2 = 4 + x22 • Flow constraint: x1 + x2 = 4.5

  15. 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. Example: Solution Apply Wardrop’s 1st principle requirements. All routes used will have equal times. 6 + 4x1 = 4 + x22 x1 + x2 = 4.5 Substituting and solving: 6 + 4x1 = 4 + (4.5 – x1)2 6 + 4x1 = 4 + 20.25 – 9x1 + x12 x12 – 13x1 + 18.25 = 0 x1 = 1.6 or 11.4 (total is 4.5 so x1 = 1.6 or 1,600 vehicles) x2 = 4.5 – 1.6 = 2.9 or 2,900 vehicles Check answer: TT1 = 6 + 4(1.6) = 12.4 minutes TT2 = 4 + (2.9)2 = 12.4 minutes

  17. Example: Mathematical Solution     Same equation as before

  18. 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. Formulating the SO Problem Finding the set of flows that minimizes the following function:

  20. 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 Suburb City Route 2

  21. Example: Solution • Determine HPFs as before • HPF1 = 6 + 4x1 • HPF2 = 4 + x22 • Flow constraint: x1 + x2 = 4.5 • Formulate the SO equation • Use the flow constraint(s) to get the equation into one variable

  22. Example: Solution • Minimize the SO function • Solve the minimized function

  23. Example: Solution • Solve the minimized function • Find travel times • Find the total vehicular delay

  24. User equilibrium t1 = 12.4 minutes t2 = 12.4 minutes x1 = 1,600 vehicles x2 = 2,900 vehicles tixi = 55,800 veh-min System optimization t1 = 14.3 minutes t2 = 10.08 minutes x1 = 2,033 vehicles x2 = 2,467 vehicles tixi = 53,592 veh-min Compare UE and SO Solutions Route 1 Suburb City Route 2

  25. 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. UE SO

  27. Total Travel time is Minimum at SO

  28. 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

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