A Path-size Weibit Stochastic User Equilibrium Model

Songyot KitthamkesornDepartment of Civil & Environmental Engineering

Utah State UniversityLogan, UT 84322-4110, USA

Email: songyot.k@aggiemail.usu.edu

Adviser: Anthony ChenEmail: anthony.chen@usu.edu

2

Outline

Review of closed-form route choice/network equilibrium models

Weibit route choice modelWeibit stochastic user equilibrium modelNumerical resultsConcluding Remarks

3

Outline

Review of closed-form route choice/network equilibrium models

Weibit route choice modelWeibit stochastic user equilibrium modelNumerical resultsConcluding Remarks

Deterministic User Equilibrium (DUE) Principle

Wardrop’s First Principle

“The journey costs on all used routes are equal, and less than those which would be experienced by a single vehicle on any unused route.”

Assumptions: All travelers have the same behavior and perfect knowledge of network travel costs.

4

Stochastic User Equilibrium (SUE) Principle and Conditions

“At stochastic user equilibrium, no travelers can improve his or her perceived travel cost by unilaterally changing routes.”

Daganzo and Sheffi (1977)

Pr , and , r ,ij ij ij ij ijr r l ij ijP G G l R r l R ij IJ g g

Daganzo, C.F., Sheffi, Y., 1977. On stochastic models of traffic assignment. Transportation Science, 11(3), 253-274.

, ,ij ijr r ij ijf P q r R ij IJ g

, ij

ijr ij

r R

f q ij IJ

1.0, ij

ijr

r R

P ij IJ

g

5

Probabilistic Route Choice Models

Multinomial logit (MNL) route choice model Dial (1971)

Multinomial probit (MNP) route choice model Daganzo and Sheffi

(1977)Closed form

1 1.. ,.., ..ij ij

ij ij ij ij ijr R R

P f t t dt dtNon-closed form

exp

expij

ijrij

r ijk

k R

gP

g

Perceived travel cost

6

Gumbel Normal

Dial, R., 1971. A probabilistic multipath traffic assignment model which obviates path enumeration. Transportation Research, 5(2), 83-111.

Daganzo, C.F. and Sheffi, Y., 1977. On stochastic models of traffic assignment. Transportation Science, 11(3), 253-274.

Gumbel Distribution

7

CDF ijrG

F t 1 exp , ,ij ijr rt

e t

Mean route travel cost ijrg ij

r ijr

Route perception variance 2ijr 2

2

6 ijr

PDF

Perceived travel cost

Gumbel

Location parameter

Scale parameter

Euler constant

Variance is a function of scale parameter only!!!

MNL Model and Closed-form Probability Expression

8

Under the independently distributed assumption, we have the joint survival function:

expij ij ijr r r

ij

t

r R

H e

expij ijijij ij ij

rr r r k k

ij

ttij ij ijr r r

k R

P e e dt

Then, the choice probability can be determined by

To obtain a closed-form, is fixed for all routes

expijijij ij

rr r k

ij

ttij ijr r

k R

P e e dt

Finally, we have

exp

expij

ijrij

r ijk

k R

gP

g

Identically distributed assumption

Independently and Identically distributed

(IID) assumption

9

Independently Distributed Assumption: Route Overlapping

DestinationOrigin

(100-x)(x)

(Travel cost)

(100-x)

(100)

i j

0.3

0.34

0.38

0.42

0.46

0.5

0 20 40 60 80 100

Prob

. of c

hoos

ing l

ower

rout

e

x

MNL solution

MNLIndependently distributed

Route overlapping

1

100

2 3 100 100 100

13

ij ij ij eP P Pe e e

MNP

10

Identically Distributed Assumption: Homogeneous Perception Variance

Origin Destination

(10)

(5)

(Travel cost)

0.1 5

0.1 5 0.1 10 0.1 5

1 0.621

ijl

ePe e e

DestinationOrigin

(125)

(120)

(Travel cost)

0.1 120

0.1 120 0.1 125 0.1 5

1 0.621

ijl

ePe e e

=

i j i j

MNL (=0.1)

PDF

Perceived travel cost 5 10 120 125

Same perception variance of2

26

MNP0.85ij

lP 0.54ijlP

Absolute cost difference

>

Existing Models

11

MNL1. Gumbel

Closed form

MNP2. Normal

Overla

pping

EXTENDED LOGIT

Closed form

Overla

pping

Diff. trip length

Extended Logit Models

12

MNLGumbel

Closed form

EXTENDED LOGIT

Closed form

Overla

pping

ij ij ijr r rU g

Modification of the deterministic term• C-logit (Cascetta et al., 1996)• Path-size logit (PSL) (Ben-Akiva and Bierlaire, 1999)

Modification of the random error term

• Cross Nested logit (CNL) (Bekhor and Prashker, 1999)• Paired Combinatorial logit (PCL) (Bekhor and Prashker,

1999)• Generalized Nested logit (GNL) (Bekhor and Prashker, 2001)

Ben-Akiva, M. and Bierlaire, M., 1999. Discrete choice methods and their applications to short term travel decisions. Handbook of Transportation Science, R.W. Halled, Kluwer Publishers.

Cascetta, E., Nuzzolo, A., Russo, F., Vitetta, A., 1996. A modified logit route choice model overcoming path overlapping problems: specification and some calibration results for interurban networks. In Proceedings of the 13th International Symposium on Transportation and Traffic Theory, Leon, France, 697-711.

Bekhor, S., Prashker, J.N., 1999. Formulations of extended logit stochastic user equilibrium assignments. Proceedings of the 14 th International Symposium on Transportation and Traffic Theory, Jerusalem, Israel, 351-372.Bekhor S., Prashker, J.N., 2001. A stochastic user equilibrium formulation for the generalized nested logit model. Transportation Research Record 1752, 84-90.

13

Independently Distributed Assumption: Route Overlapping

DestinationOrigin

(100-x)(x)

(Travel cost)

(100-x)

(100)

i j

0.3

0.34

0.38

0.42

0.46

0.5

0 20 40 60 80 100

Prob

. of c

hoos

ing l

ower

rout

e

x

MNLC-logitPSLPCLCNLGNL

MNP

MNL

14

Scaling Technique

Origin Destination

(10)

(5)

(Travel cost)

0.51 5

0.51 5 0.51 10 0.93ijl

ePe e

DestinationOrigin

(125)

(120)

(Travel cost)

0.04 120

0.04 120 0.04 125 0.52ijl

ePe e

i j i j

(=0.51)

PDF

Perceived travel cost 5 10 120 125

Same perception variance

CV = 0.5

Chen, A., Pravinvongvuth, S., Xu, X., Ryu, S. and Chootinan, P., 2012. Examining the scaling effect and overlapping problem in logit-based stochastic user equilibrium models. Transportation Research Part A, 46(8), 1343-1358.

(=0.02)

Same perception variance

>

3rd Alternative

15

MNL1. Gumbel

Closed form

MNP2. Normal

Overla

pping

Diff. trip length

EXTENDED LOGIT

Closed form

Overla

pping

MNW3. Weibull

Closed form

Multinomial weibit model(Castillo et al., 2008)

PSW

Closed form

Castillo et al. (2008) Closed form expressions for choice probabilities in the Weibull case. Transportation Research Part B 42(4), 373-380

Overla

pping

Path-size weibit model

Modification of the deterministic term

Diff. trip length

Diff. trip length

16

Outline

Review of closed-form route choice/network equilibrium models

Weibit route choice modelWeibit stochastic user equilibrium modelNumerical resultsConcluding Remarks

17

Weibull Distribution

CDF ijrG

F t 1 exp , 0,

ijrij

rijr

tt

Mean travel cost ijrg

11ij ijr r ij

r

Route perception variance 2ijr 2 221ij ij ij

r r rijr

g

Variance is a function of route cost!!!

PDF

Perceived travel cost

Weibull

Gamma function

Location parameter

Shapeparameter

Scaleparameter

Multinomial Weibit (MNW) Model and Closed-form Prob. Expression

18

Under the independently distributed assumption, we have the joint survival function:

Then, the choice probability can be determined by

To obtain a closed-form, and are fixed for all routes

Finally, we have

Since the Weibull variance is a function of route cost, the identically distributed assumption does NOT apply

exp

ijr

ij

ij ijr r

ijr R r

tH

1

exp exp

ijij ijr r l

ijr ij

ij ij ij ij ij ijijr r r r r lij ijr

r rij ij ij ijl rr r r ll R

t t tP dt

1

exp

ij ij

ij ij

ij ij ij ijijr rij ij

r rij ij ijk Rr r k

t tP dt

ij

ij

ij

ij ijrij

rij ijk

k R

gP

g

Castillo et al. (2008) Closed form expressions for choice probabilities in the Weibull case. Transportation Research Part B 42(4), 373-380

19

Identically Distributed Assumption: Homogeneous Perception Variance

Origin Destination

(10)

(5)

(Travel cost)

DestinationOrigin

(125)

(120)

(Travel cost)i j i j

2.1

2.1 2.1 2.1

5 1 0.815 10 101

5

ijlP

2.1

2.1 2.1 2.1

120 1 0.52120 125 1251

120

ijlP

MNW model

PDF

Perceived travel cost 5 10 120 125

Route-specific perception variance

22 22 11 1

1 1

ijij rr ij ijij

g

CV = 0.5

Relative cost difference

0ij 2.1ij

>

20

Path-Size Weibit (PSW) Model

To handle the route overlapping problem, a path-size factor (Ben-Akiva and Bierlaire, 1999) is introduced, i.e.,

Path-size factor

MNW random utility maximization model

ij

ij ij ij ijr r rU g

Weibull distributed random error term

Ben-Akiva, M. and Bierlaire, M., 1999. Discrete choice methods and their applications to short term travel decisions. Handbook of Transportation Science, R.W. Halled, Kluwer Publishers.

ij

ij ijrij ij

r rijr

gU

which gives the PSW model:

ij

ij

ij

ij ij ijr rij

rij ij ijk k

k R

gP

g

1

r

ij

ij ar ij ij

a r akk R

lL

21

Independently Distributed Assumption: Route Overlapping

DestinationOrigin

(100-x)(x)

(Travel cost)

(100-x)

(100)

0.3

0.34

0.38

0.42

0.46

0.5

0 20 40 60 80 100

Prob

. of c

hoos

ing l

ower

rout

e

x

MNL solution

MNL, MNW

MNP

PSW

2.1

2.1 2.1 2.1

1 1000.33

1 100 1 100 1 100ij

lP

2.1

2.1 2.1 2.1

1 1000.5

1 100 0.5 100 0.5 100ij

lP

2.1

2.1 2.1 2.1

1 1000.4

1 100 0.75 100 0.75 100ij

lP

22

Outline

Review of closed-form route choice/network equilibrium models

Weibit route choice modelWeibit stochastic user equilibrium modelNumerical resultsConcluding Remarks

Comparison between MNL Model and MNW Model

23

ij

ij

ij

ijrij

rijk

k R

gP

g

exp

expij

ijrij

r ijk

k R

gP

g

Extreme value distribution

Gumbel (type I) Weibull (type III)Log Weibull

Log Transformation

IID Independence0ij Assume

0

1min ln ln 1a

ij

vij ij

a r rija A ij IJ r R

Z d f f

ij

ijr ij

r R

f q

0ijrf

s.t.

24

A Mathematical Programming (MP) Formulation for the MNW-SUE model

Multiplicative Beckmann’s transformation

(MBec)

ij

ij

ij

ijrij

rijk

k R

gP

g

Relative cost difference under congestion

25

A MP Formulation for the PSW-SUE Model

0

1 1min ln ln 1 lna

ij ij

vij ij ij ij

a r r r rij ija A ij IJ r R ij IJ r R

Z d f f f

ij

ijr ij

r R

f q

0ijrf

s.t.

ij

ij

ij

ij ijr rij

rij ijk k

k R

gP

g

26

Equivalency Condition

By setting the partial derivative w.r.t. route flow variable equal to zero, we have

ij

ijij ij r

ij IJ r R

L Z q f

By constructing the Lagrangian function, we have

expij

ij ij ij ijr ij r rf g

expij

ij ij

ij ij ij ijij r ij r r

r R r R

q f g

Then, we have the PSW route flow solution, i.e.,

ij

ij

ij

ij ijijr rij r

rij ijijk k

k R

gfPq g

27

Uniqueness Condition

The second derivative

2

1 0 ,

,

ijbbr ij

b A b rij ij

ij ijbr lbr bl

b A b

d r ldv fZdf f r ldv

By assuming , the route flow solution of PSW-SUE is unique.

0,b bd dv b A

28

Path-Based Partial Linearization Algorithm

· n=0· f(0) = 0 à Free flow travel cost

Initialization

Flow Network

Search direction

Update flow f(n)

ij ijr ij ry n q P n g

Solve

Line search

0 1

arg minn Z n n n

f y f

Update

1n n n n n f f y f

Stopping criteria

Link flow Link travel cost

Route travel cost

Result

NOn = n+1

YES

n = n+1

PSW probability

29

Outline

Review of closed-form route choice/network equilibrium models

Weibit route choice modelWeibit stochastic user equilibrium modelNumerical resultsConcluding Remarks

30

Real Network

0 2 4 6

Kilometers

0 2 4 6

Kilometers

Winnipeg network, Canada154 zones, 2,535 links, and4,345 O-D pairs.

31

Convergence Results

1.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00

0 10 20 30 40 50

Iteration

MNW-SUEPSW-SUE

32

Winnipeg Network Results

92 301

2

345

6

100

141

2

34

56

52 501

23

Route PSLs-SUE MNW-SUE PSW-SUE1 0.585 0.439 0.4722 0.178 0.242 0.2483 0.237 0.319 0.280

O-D (50, 52)

Route PSLs-SUE MNW-SUE PSW-SUE1 0.140 0.191 0.1492 0.233 0.120 0.2263 0.139 0.176 0.1484 0.154 0.129 0.1385 0.193 0.196 0.1966 0.141 0.188 0.142

O-D (14, 100)

Route PSLs-SUE MNW-SUE PSW-SUE1 0.097 0.117 0.1262 0.269 0.194 0.2293 0.282 0.197 0.2524 0.173 0.139 0.1745 0.130 0.244 0.1346 0.050 0.109 0.087

O-D (92, 30)

33

Link Flow Difference between MNW-SUE and PSW-SUE Models

PSLs - PSW-500 to -300-300 to -100-100 to 100100 to 300300 to 500500 to 700

12

261

584

235115

70

200

400

600

800

-500 to -300

-300 to -100

-100 to 100

100 to 300

300 to 500

500 to 700

Num

ber o

f lin

ks

Flow difference (MNW-SUE - PSW-SUE)

CBD

2 31

503

37 7 00

200

400

600

800

-500 to -300

-300 to -100

-100 to 100

100 to 300

300 to 500

500 to 700

Num

ber o

f lin

ks

Flow difference (MNW-SUE - PSW-SUE)

Non-CBD

CBD

MNW - PSW-500 to -300-300 to -100-100 to 100100 to 300300 to 500500 to 700

34

Link Flow Difference between PSLs-SUE and PSW-SUE Models

1129

794

186104

00

200

400

600

800

-500 to -300

-300 to -100

-100 to 100

100 to 300

300 to 500

500 to 700

Num

ber o

f lin

ks

Flow difference (PSLs-SUE - PSW-SUE)

CBD

1 11

555

9 4 00

200

400

600

800

-500 to -300

-300 to -100

-100 to 100

100 to 300

300 to 500

500 to 700

Num

ber o

f lin

ks

Flow difference (PSLs-SUE - PSW-SUE)

Non-CBD

CBD

PSLs - PSW-500 to -300-300 to -100-100 to 100100 to 300300 to 500500 to 700

Drawback: Insensitive to an Arbitrary Multiplier Route Cost

35

2.1, 0;ij ij

2.1

2.1 2.1 2.1

5 1 0.8115 10 1 2

ijlP

2.1, 0;ij ij

2.1

2.1 2.1 2.1

50 1 0.81150 100 1 2

ijlP

2.1, 4;ij ij

2.1

2.1 2.1

5 40.977

5 4 10 4ij

lP

2.1, 4;ij ij

2.1

2.1 2.1

50 40.824

50 4 100 4ij

lP

a) Short network b) Long network

i jOrigin Destination

(10)

(5)

(Travel cost)

i jDestinationOrigin

(100)

(50)

(Travel cost)

Incorporating ij

36Zhou, Z., Chen, A. and Bekhor, S., 2012. C-logit stochastic user equilibrium model: formulations and solution algorithm. Transportmetrica, 8(1), 17-41.

Variational Inequality (VI)

* * * 0,T

f f f P g f q f

ij

ij

ij

ij ijr

ij ijk

k R

g

g

ij

ij

ij

ij ij ijr r

ij ij ijk k

k R

g

g

MNW model

PSW model

General route cost

Flow dependent

37

Concluding Remarks

Reviewed the probabilistic route choice/network equilibrium models

Presented a new closed-form route choice modelProvided a PSW-SUE mathematical

programming formulation under congested networks

Developed a path-based algorithm for solving the PSW-SUE model

Demonstrated with a real network

38

Thank You