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28 Pages

### Route_Choice

Course: CEE 320, Fall 2008
School: Washington
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Word Count: 1331

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Choice CEE Route 320 Fall 2008 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) Desired output from the traffic forecasting process: how many vehicles at any time on a roadway CEE 320 Fall 2008 Complexity Route choice decisions are...

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Choice CEE Route 320 Fall 2008 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) Desired output from the traffic forecasting process: how many vehicles at any time on a roadway CEE 320 Fall 2008 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 CEE 320 Fall 2008 Outline 1. 2. 3. 4. General HPF Functional Forms Basic Assumptions Route Choice Theories a. User Equilibrium b. System Optimization c. Comparison 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 1. 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 CEE 320 Fall 2008 HPF Functional Forms Travel Time Linear Common Non-linear HPF v T = T0 1 + c Free Flow Non-Linear Capacity from the Bureau of Public Roads (BPR) Traffic Flow (veh/hr) CEE 320 Fall 2008 Speed vs. Flow 2 u q = k j u - uf uf Free Flow Speed Speed (mph) Uncongested Flow um Congested Flow CEE 320 Fall 2008 Highest flow, Flow (veh/hr) capacity, qm qm is bottleneck discharge rate 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 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" 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. 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: min n xn t ( w) dw n 0 n = Route between given O-D pair tn(w)dw = HPF for a specific route as a function of flow w = Flow CEE 320 Fall 2008 xn 0 for all routes Formulating the UE Problem min n xn t ( w) dw = n 0 n n min t n ( w) dw 0 xn = n xn min t ( w) dw 0 n = Route between given O-D pair tn(w)dw = HPF for a specific route as a function of flow w = Flow CEE 320 Fall 2008 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.. Route 1 CEE 320 Fall 2008 City Route 2 Suburb 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 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 TT1 = 6 + 4( 4.5) = 24 minutes TT2 = 4 + ( 0 ) = 4 minutes 2 If all the traffic is on Route 1 then Route 2 is the desirable or choice All nothing assignment on Route 2 TT1 = 6 + 4( 0 ) = 6 minutes TT2 = 4 + ( 4.5) = 24.25 minutes 2 If all the traffic is on Route 2 then Route 1 is the desirable choice Therefore, both routes will be used CEE 320 Fall 2008 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 CEE 320 Fall 2008 Example: Mathematical Solution minmize S ( x ) = n xn t n ( w) dw 0 S ( x ) = (6 + 4w)dw + (4 + w 2 )dw 0 0 3 x2 2 S ( x ) = 6 x1 + 2 x1 + 4 x 2 + 3 x1 x2 S ( x ) = 6w + 2w 2 x1 0 w + 4w + 3 3 x2 0 x1 + x2 = 4.5 x13 4.5 3 2 2 S ( x ) = 6 x1 + 2 x + 18 - 4 x1 + - 4.5 x1 + 4.5 x1 - 3 3 2 1 for a minimum : CEE 320 Fall 2008 dS = 6 + 4 x1 - 4 - 20.25 + 9 x1 - x12 = 0 dx simplifying : x12 - 13 x1 + 18.25 = 0 Same equation as before 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. CEE 320 Fall 2008 Not stable because individuals will be tempted to choose different route. Formulating the SO Problem Finding the set of flows that minimizes the following function: min xnt n ( xn ) = min xnt n ( xn ) n n n = Route between given O-D pair tn(xn) = travel time for a specific route xn = Flow on a specific route 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 CEE 320 Fall 2008 City Route 2 Suburb Example: Solution 1. Determine HPFs as before HPF1 = 6 + 4x1 HPF2 = 4 + x22 Flow constraint: x1 + x2 = 4.5 1. Formulate the SO equation 2 S ( x ) = t i xi = ( 6 + 4 x1 ) x1 + 4 + x 2 x 2 i =1 n ( ) Use the flow constraint(s) to get the equation into o...

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