A Path-size - PowerPoint Presentation

Slide1

A Path-size

Weibit Stochastic User Equilibrium Model

Songyot

Kitthamkesorn

Department of Civil & Environmental Engineering

Utah State University

Logan, UT 84322-4110, USA

Email:

songyot.k@aggiemail.usu.edu

Adviser: Anthony Chen

Email:

anthony.chen@usu.eduSlide2

2

OutlineReview of closed-form route choice/network equilibrium models

Weibit

route choice model

Weibit

stochastic user equilibrium model

Numerical results

Concluding RemarksSlide3

3

Outline

Review of closed-form route choice/network equilibrium models

Weibit

route choice model

Weibit

stochastic user equilibrium model

Numerical results

Concluding RemarksSlide4

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.

4Slide5

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)

Daganzo, C.F., Sheffi, Y., 1977. On stochastic models of traffic assignment.

Transportation Science

, 11(3), 253-274.

5Slide6

Probabilistic Route Choice Models

Multinomial

logit

(MNL)

route choice

model Dial

(1971)

Multinomial

probit

(MNP) route choice model Daganzo

and

Sheffi

(1977)

Closed form

Non-closed form

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.Slide7

Gumbel

Distribution7

PDF

Perceived travel cost

Gumbel

Location parameter

Scale parameter

Euler constant

Variance is a function of scale parameter only!!!Slide8

MNL Model and Closed-form Probability Expression

8

Under the

independently distributed

assumption, we have the joint survival function:

Then, the choice probability can be determined by

To obtain a closed-form,

 is fixed for all routes

Finally, we have

Identically distributed

assumption

Independently and Identically distributed (IID) assumptionSlide9

9

Independently Distributed Assumption: Route Overlapping

i

j

MNL

Independently distributed

Route overlapping

MNPSlide10

10

Identically Distributed Assumption: Homogeneous Perception Variance

=

i

j

i

j

MNL (

=0.1)

PDF

Perceived travel cost

5

10

120

125

Same perception variance of

MNP

Absolute cost difference

>Slide11

Existing Models

11

MNL

1.

Gumbel

Closed form

MNP

2. Normal

Overlapping

EXTENDED LOGIT

Closed form

Overlapping

Diff. trip lengthSlide12

Extended Logit Models

12

MNL

Gumbel

Closed form

EXTENDED LOGIT

Closed form

Overlapping

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 13

th

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.Slide13

13

Independently Distributed Assumption: Route Overlapping

i

j

MNP

MNLSlide14

14

Scaling Technique

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

>Slide15

3rd Alternative

15

MNL

1.

Gumbel

Closed form

MNP

2. Normal

Overlapping

Diff. trip length

EXTENDED LOGIT

Closed form

Overlapping

MNW

3.

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

Overlapping

Path-size

weibit

model

Modification of the deterministic term

Diff. trip length

Diff. trip lengthSlide16

16

OutlineReview of closed-form route choice/network equilibrium models

Weibit

route choice model

Weibit

stochastic user equilibrium model

Numerical results

Concluding RemarksSlide17

17

Weibull Distribution

Variance is a function of route cost!!!

PDF

Perceived travel cost

Weibull

Gamma function

Location parameter

Shape

parameter

Scale

parameterSlide18

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

Castillo

et al. (2008) Closed form expressions for choice probabilities in the

Weibull

case.

Transportation Research Part B

42(4), 373-380Slide19

19

Identically Distributed Assumption: Homogeneous Perception Variance

i

j

i

j

MNW

model

PDF

Perceived travel cost

5

10

120

125

Route-specific perception variance

CV = 0.5

Relative cost difference

>Slide20

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

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.

which gives the PSW model: Slide21

21

Independently Distributed Assumption: Route Overlapping

MNL, MNW

MNP

PSWSlide22

22

OutlineReview of closed-form route choice/network equilibrium models

Weibit

route choice model

Weibit

stochastic user equilibrium model

Numerical results

Concluding RemarksSlide23

Comparison between MNL Model and MNW Model

23

Extreme value distribution

Gumbel

(type I)

Weibull

(type III)

Log

Weibull

Log

Transformation

IID

Independence

AssumeSlide24

24

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

Multiplicative Beckmann’s transformation

(

MBec

)

Relative cost difference

under

congestionSlide25

25

A MP Formulation for the PSW-SUE Model Slide26

26

Equivalency Condition

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

By constructing the Lagrangian function, we have

Then, we have the PSW route flow solution, i.e.,Slide27

27

Uniqueness Condition

The second derivative

By assuming , the route flow solution

of PSW-SUE is unique.Slide28

28

Path-Based Partial Linearization AlgorithmSlide29

29

OutlineReview of closed-form route choice/network equilibrium models

Weibit

route choice model

Weibit

stochastic user equilibrium model

Numerical results

Concluding RemarksSlide30

30

Real Network

Winnipeg network, Canada

154

zones, 2,535 links, and

4,345 O-D pairs.

Slide31

31

Convergence ResultsSlide32

32

Winnipeg Network ResultsSlide33

33

Link Flow Difference between MNW-SUE and PSW-SUE ModelsSlide34

34

Link Flow Difference between PSLs-SUE and PSW-SUE ModelsSlide35

Drawback: Insensitive to an Arbitrary Multiplier R

oute Cost35Slide36

Incorporating ij

36

Zh

ou,

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)

MNW model

PSW model

General route cost

Flow dependent Slide37

37

Concluding Remarks

Reviewed the probabilistic route choice/network equilibrium models

Presented a new closed-form route choice model

Provided a PSW-SUE mathematical programming formulation under congested networks

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

Demonstrated with a real networkSlide38

38

Thank You