CEE 320 Spring 2008 Route Choice CEE 320

CEE 320 Spring 2008 Route Choice CEE 320 Anne

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

Route Choice • Equilibrium problem for alternate routes • Requires relationship between: – Travel time (TT) – Traffic flow (TF) • Highway Performance Function (HPF) CEE 320 Spring 2008 – Common term for this relationship

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

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 – CEE 320 Spring 2008 – Strong assumption: Travelers may not use all available routes, and may base TTs on perception Some studies say perception bias is small

Theory of User Equilibrium CEE 320 Spring 2008 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. Wadrop definition: A. K. A. Wardrop’s 1 st principle “The travel time between a specified origin & destination on all used routes is equal, and less than or equal to the travel time that would be experienced by a traveler on any unused route”

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 tn(w)dw = HPF for a specific route as a function of flow CEE 320 Spring 2008 w = Flow xn ≥ 0 for all routes

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. Compute UE travel times on the two routes. CEE 320 Spring 2008 Route 1 City Route 2 Suburb

Example: Solution 1. 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. HPF 1 = 6 + 4 x 1 HPF 2 = 4 + x 22 Flow constraint: x 1 + x 2 = 4. 5 2. Route use check (will both routes be used? ) – All or nothing assignment on Route 1 If all the traffic is on Route 1 then Route 2 is the desirable choice – All or nothing assignment on Route 2 CEE 320 Spring 2008 If all the traffic is on Route 2 then Route 1 is the desirable choice – Therefore, both routes will be used

Example: Solution 3. Equate TTs CEE 320 Spring 2008 – Apply Wardrop’s 1 st principle requirements. All routes used will have equal times, and ≤ those on unused routes. Hence, if flows are distributed between Route 1 and Route 2, then both must be used on travel time equivalency bases.

Example: Mathematical Solution CEE 320 Spring 2008 Same equation as before

Theory of System-Optimal Route Choice CEE 320 Spring 2008 Wardrop’s Second Principle: 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.

Formulating the SO Problem Finding the set of flows that minimizes the following function: n = Route between given O-D pair tn(xn) = travel time for a specific route CEE 320 Spring 2008 xn = Flow on a specific route

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. CEE 320 Spring 2008 Route 1 City Route 2 Suburb

Example: Solution 1. Determine HPFs as before – – – HPF 1 = 6 + 4 x 1 HPF 2 = 4 + x 22 Flow constraint: x 1 + x 2 = 4. 5 2. Formulate the SO equation CEE 320 Spring 2008 – Use the flow constraint(s) to get the equation into one variable

Example: Solution 1. Minimize the SO function 2. Solve the minimized function CEE 320 Spring 2008 3. Find the total vehicular delay

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 tixi = 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 tixi = 53, 592 veh-min CEE 320 Spring 2008 Route 1 City Route 2 Suburb

Primary Reference CEE 320 Spring 2008 • Mannering, F. L. ; Kilareski, W. P. and Washburn, S. S. (2005). Principles of Highway Engineering and Traffic Analysis, Third Edition. Chapter 8