route choice (transportation engineering)

CE

E 3

20W

inte

r 20

06

Route Choice

CEE 320Steve Muench

CE

E 3

20W

inte

r 20

06

Outline

1. General

2. HPF Functional Forms

3. Basic Assumptions

4. Route Choice Theoriesa. User Equilibrium

b. System Optimization

c. Comparison

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

HPF Functional FormsT

rave

l Tim

e

Traffic Flow (veh/hr)Capacity

Linear

Non-LinearFreeFlow

+=

β

αc

vTT 10

Common Non-linear HPF

from the Bureau of Public Roads (BPR)

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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. 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”

CE

E 3

20W

inte

r 20

06

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 = Flow

xn ≥ 0 for all routes

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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.

CE

E 3

20W

inte

r 20

06

Example: Mathematical Solution

( ) ( )dwwtxSnx

nn

∫∑=0

minmize ( ) ∫ ∫ +++=1 2

0 0

2 )4()46(x x

dwwdwwxS

( )2

1

0

3

0

2

3426

xx w

wwwxS +++= ( )

+++=3

42632

2211

xxxxxS

5.421 =+ xx ( )3

5.45.43

5.441826

312

112

3

1211

xxxxxxxS −+−+−++=

0925.20446 :minimum afor 2111 =−+−−+= xxx

dx

dS

⇒

⇒

⇒

025.1813 :gsimplifyin 121 =+− xx ⇒ Same equation as before

CE

E 3

20W

inte

r 20

06

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.

CE

E 3

20W

inte

r 20

06

Formulating the SO Problem

( ) ( )nn

nn xtxxS ∑=

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

xn = Flow on a specific route

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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 ++−+−=

CE

E 3

20W

inte

r 20

06

Example: Solution1. Minimize the SO function

2. Solve the minimized function

3. Find the total vehicular delay

( ) ( ) ( )[ ] 03415.4246 222 =++−−+−= xx

dx

dS

03883 222 =−+ xx

⇓

467.22 =x 033.21 =x

( ) ( ) ( ) ( ) ( ) minutes- vehicle592,53246708.10203313.141

≅+== ∑=

n

iii xtxS

CE

E 3

20W

inte

r 20

06

Compare UE and SO Solutions

• User equilibrium– t1 = 12.4 minutes

– t2 = 12.4 minutes

– x1 = 1,600 vehicles


– tixi = 55,800 veh-min

• System optimization– t1 = 14.3 minutes

– t2 = 10.08 minutes



– tixi = 53,592 veh-min

Route 1

Route 2City Suburb

CE

E 3

20W

inte

r 20

06

Primary Reference

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