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18 Pages

### Route Choice

Course: CEE 320, Fall 2008
School: Washington
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Word Count: 957

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Choice CEE Route 320 Spring 2007 CEE 320 Steve Muench Outline 1. 2. 3. 4. General HPF Functional Forms Basic Assumptions Route Choice Theories a. User Equilibrium b. System Optimization c. Comparison CEE 320 Spring 2007 Route Choice Equilibrium problem for alternate routes Requires relationship between: Travel time (TT) Traffic flow (TF) Highway Performance Function (HPF) Common term for this...

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Choice CEE Route 320 Spring 2007 CEE 320 Steve Muench Outline 1. 2. 3. 4. General HPF Functional Forms Basic Assumptions Route Choice Theories a. User Equilibrium b. System Optimization c. Comparison CEE 320 Spring 2007 Route Choice Equilibrium problem for alternate routes Requires relationship between: Travel time (TT) Traffic flow (TF) Highway Performance Function (HPF) Common term for this relationship CEE 320 Spring 2007 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 Spring 2007 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 CEE 320 Spring 2007 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 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. Wardrops 1st 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 CEE 320 Spring 2007 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 S ( x ) = 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 Spring 2007 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. Route 1 CEE 320 Spring 2007 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. HPF1 = 6 + 4x1 HPF2 = 4 + x22 Flow constraint: x1 + x2 = 4.5 1. Route use check (will both routes be used?) All or nothing assignment on Route 1 TT1 = 6 + 4( 4.5) = 24 minutes 2 TT2 = 4 + ( 0) = 4 minutes TT1 = 6 + 4( 0 ) = 6 minutes 2 TT2 = 4 + ( 4.5) = 24.25 minutes If all the traffic is on Route 1 then Route 2 is the desirable choice All or nothing assignment on Route 2 If all the traffic is on Route 2 then Route 1 is the desirable choice CEE 320 Spring 2007 Therefore, routes both will be used Example: Solution 1. Equate TTs Apply Wardrops 1st principle requirements. All routes used will have equal times, and those on unused routes. Hence, if flows are distributed between Route1 and Route 2, then both must be used on travel time equivalency bases. CEE 320 Spring 2007 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 Spring 2007 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 Wardrops 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. CEE 320 Spring 2007 Formulating the SO Problem Finding the set of flows that minimizes the following function: S ( x ) = xn t n ( xn ) n n = Route between given O-D pair tn(xn) = travel time for a specific route xn = Flow on a specific route CEE 320 Spring 2007 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 vehic...

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