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Route Choice

CEE 320Steve Muench

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Outline

1. General2. HPF Functional Forms3. Basic Assumptions4. Route Choice Theories

a. User Equilibriumb. System Optimizationc. Comparison

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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

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HPF Functional FormsTr

avel

Tim

e

Traffic Flow (veh/hr) Capacity

Linear

Non-LinearFreeFlow

cvTT 10

Common Non-linear HPF

from the Bureau of Public Roads (BPR)

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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

– Some studies say perception bias is small

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Theory of User EquilibriumTravelers 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 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”

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Formulating the UE Problem

dwwtxSnx

nn

0

min

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

w = Flowxn ≥ 0 for all routes

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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

Route 2City Suburb

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Example: Solution1. 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

2. 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

minutes245.4461 TT minutes404 2

2 TT

minutes60461 TT minutes25.245.44 2

2 TT

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

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Example: Solution3. Equate TTs

– Apply Wardrop’s 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.

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Example: Mathematical Solution dwwtxS

nx

nn

0

minmize 1 2

0 0

2 )4()46(x x

dwwdwwxS

2

1

0

3

0

2

3426

xx wwwwxS

3

42632

2211

xxxxxS

5.421 xx 3

5.45.435.441826

312

112

3

1211

xxxxxxxS

0925.20446 :minimum afor 2111 xxx

dxdS

025.1813 :gsimplifyin 121 xx Same equation as before

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Theory of System-Optimal Route Choice

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.

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Formulating the SO Problem

nn

nn xtxxS

Finding the set of flows that minimizes the following function:

n = Route between given O-D pairtn(xn) = travel time for a specific route

xn = Flow on a specific route

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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

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Example: Solution1. Determine HPFs as before

– HPF1 = 6 + 4x1

– HPF2 = 4 + x22

– Flow constraint: x1 + x2 = 4.5

2. Formulate the SO equation

– Use the flow constraint(s) to get the equation into one variable

22211

1

446 xxxxxtxSn

iii

322

222 4)5.4(4)5.4(6)( xxxxxS

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Example: Solution1. Minimize the SO function

2. Solve the minimized function

3. Find the total vehicular delay

03415.4246 222 xx

dxdS

03883 222 xx

467.22 x 033.21 x

minutes- vehicle592,53246708.10203313.141

n

iii xtxS

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Compare UE and SO Solutions

• 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

Route 1

Route 2City Suburb

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Primary Reference

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