A Path-size Weibit Stochastic User Equilibrium Model Songyot Kitthamkesorn Department of Civil &...

A Path-size Weibit Stochastic User Equilibrium Model

Songyot KitthamkesornDepartment of Civil & Environmental Engineering

Utah State UniversityLogan, UT 84322-4110, USA

Email: [email protected]

Adviser: Anthony ChenEmail: [email protected]

mailto:[email protected]

mailto:[email protected]

2

Outline

Review of closed-form route choice/network equilibrium models

Weibit route choice modelWeibit stochastic user equilibrium modelNumerical resultsConcluding Remarks

3

Outline



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

ijr ij

r

Route perception variance 2ijr 2

2

6 ijr

PDF


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

low

er ro

ute

x

MNL solution

MNL

Independently distributed

Route overlapping

1

100

2 3 100 100 100

1

3ij ij ij e

P 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

10.62

1ij

l

eP

e e e

DestinationOrigin

(125)

(120)

(Travel cost)

0.1 120

0.1 120 0.1 125 0.1 5

10.62

1ij

l

eP

e 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

Ove

rlapp

ing

EXTENDED LOGIT

Closed form

Ove

rlapp

ing

Diff. trip length

Extended Logit Models

12

MNLGumbel

Closed form

EXTENDED LOGIT

Closed form

Ove

rlapp

ing

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


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

low

er ro

ute

x

MNLC-logitPSLPCLCNLGNL

MNP

MNL

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Scaling Technique

Origin Destination

(10)

(5)

(Travel cost)

0.51 5

0.51 5 0.51 100.93ij

l

eP

e e

DestinationOrigin

(125)

(120)

(Travel cost)

0.04 120

0.04 120 0.04 1250.52ij

l

eP

e e

i j i j

(=0.51)

PDF


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

Ove

rlapp

ing

Diff. trip length

EXTENDED LOGIT

Closed form

Ove

rlapp

ing

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

Ove

rlapp

ing

Path-size weibit model

Modification of the deterministic term

Diff. trip length

Diff. trip length

16

Outline



17

Weibull Distribution

CDF ijrG

F t 1 exp , 0,

ijrij

r

ijr

tt

Mean travel cost ijrg

11ij ij

r r ijr

Route perception variance 2ijr 2 22

1ij ij ijr r rij

r

g

Variance is a function of route cost!!!

PDF


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 r

r 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 10.81

5 10 101

5

ijlP

2.1

2.1 2.1 2.1

120 10.52

120 125 1251

120

ijlP

MNW model

PDF


Route-specific perception variance

2

2 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

l

L

21


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

low

er ro

ute

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



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 Independence

0ij Assume

0

1min ln ln 1

a

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 ln

a

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

gfP

q g

27

Uniqueness Condition

The second derivative

2

10 ,

,

ijbbr ij

b A b r

ij ijij ijbr lbr bl

b A b

dr l

dv fZ

df fr l

dv

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



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 50

1

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 0.252

4 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 10.811

5 10 1 2ij

lP

2.1, 0;ij ij

2.1

2.1 2.1 2.1

50 10.811

50 100 1 2ij

lP

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

A Path-size Weibit Stochastic User Equilibrium Model Songyot Kitthamkesorn Department of Civil &...

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