1. A Path-size Weibit Stochastic User Equilibrium Model
Songyot Kitthamkesorn
Department of Civil & Environmental Engineering
Utah State University
Logan, UT , USA
Adviser: Anthony Chen

2. Outline Review of closed-form route choice/network equilibrium models
Weibit route choice model
Weibit stochastic user equilibrium model
Numerical results
Concluding Remarks

3. Outline Review of closed-form route choice/network equilibrium models
Weibit route choice model
Weibit stochastic user equilibrium model
Numerical results
Concluding Remarks

4. 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.

5. Stochastic User Equilibrium (SUE) Principle and Conditions
Daganzo and Sheffi (1977)
“At stochastic user equilibrium, no travelers can improve his or
her perceived travel cost by unilaterally changing routes.”
Daganzo, C.F., Sheffi, Y., On stochastic models of traffic assignment. Transportation Science, 11(3),

6. Probabilistic Route Choice Models
Perceived travel cost
Gumbel
Normal
Multinomial logit (MNL) route choice model Dial (1971)
Multinomial probit (MNP) route choice model Daganzo and Sheffi (1977)
Closed form
Non-closed form
Dial, R., A probabilistic multipath traffic assignment model which obviates path enumeration. Transportation Research, 5(2),
Daganzo, C.F. and Sheffi, Y., On stochastic models of traffic assignment. Transportation Science, 11(3),

7. Gumbel Distribution PDF Gumbel Perceived travel cost
Location parameter
Euler constant
Scale parameter
Variance is a function of scale parameter only!!!

8. MNL Model and Closed-form Probability Expression
Under the independently distributed assumption, we have the joint survival function:
Then, the choice probability can be determined by
Identically distributed
assumption
To obtain a closed-form,  is fixed for all routes
Finally, we have
Independently and Identically distributed (IID) assumption

9. Independently Distributed Assumption: Route Overlappingj
MNP
Independently distributed
MNL
Route overlapping

10. Identically Distributed Assumption: Homogeneous Perception Variancej i j
MNL (=0.1) =
Absolute cost difference
MNP
>
Same perception variance of
PDF 5 10
120
125
Perceived travel cost

11. Existing Models 1. Gumbel 2. Normal MNL MNP Overlapping Closed form
EXTENDED LOGIT
Overlapping
1. Gumbel
Closed form
Closed form
MNP
Overlapping
Diff. trip length
2. Normal

12. Extended Logit Models Gumbel MNL Overlapping Closed form Closed form
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)
Cascetta, E., Nuzzolo, A., Russo, F., Vitetta, A., 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,
Ben-Akiva, M. and Bierlaire, M., Discrete choice methods and their applications to short term travel decisions. Handbook of Transportation Science, R.W. Halled, Kluwer Publishers.
Bekhor, S., Prashker, J.N., Formulations of extended logit stochastic user equilibrium assignments. Proceedings of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem, Israel,
Bekhor S., Prashker, J.N., A stochastic user equilibrium formulation for the generalized nested logit model. Transportation Research Record 1752,

13. Independently Distributed Assumption: Route Overlappingj
MNP
MNL

14. Same perception variance Same perception variance
Scaling Technique
CV = 0.5 i j i j
(=0.51)
(=0.02)
>
Same perception variance
PDF
Same perception variance 5 10
120
125
Perceived travel cost
Chen, A., Pravinvongvuth, S., Xu, X., Ryu, S. and Chootinan, P., Examining the scaling effect and overlapping problem in logit-based stochastic user equilibrium models. Transportation Research Part A, 46(8),

15. 3rd Alternative 1. Gumbel 2. Normal 3. Weibull MNL MNP MNW PSW
EXTENDED LOGIT
Overlapping
1. Gumbel
Closed form
Closed form
MNP
Overlapping
Diff. trip length
2. Normal
MNW
Modification of the deterministic term
PSW
Diff. trip length
Overlapping
Diff. trip length
3. Weibull
Closed form
Closed form
Multinomial weibit model
(Castillo et al., 2008)
Path-size weibit model
Castillo et al. (2008) Closed form expressions for choice probabilities in the Weibull case. Transportation Research Part B 42(4),

16. Outline Review of closed-form route choice/network equilibrium models
Weibit route choice model
Weibit stochastic user equilibrium model
Numerical results
Concluding Remarks

17. Weibull Distribution PDF Location parameter Weibull Shape
Perceived travel cost
Scale
parameter
Gamma function
Variance is a function of route cost!!!

18. Multinomial Weibit (MNW) Model and Closed-form Prob. Expression
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
Since the Weibull variance is a function of route cost, the identically distributed assumption does NOT apply
Finally, we have
Castillo et al. (2008) Closed form expressions for choice probabilities in the Weibull case. Transportation Research Part B 42(4),

19. Identically Distributed Assumption: Homogeneous Perception Variance
CV = 0.5 i j i j
MNW model
>
Relative cost difference
Route-specific perception variance
PDF 5 10
120
125
Perceived travel cost

20. Path-Size Weibit (PSW) Model
MNW random utility maximization model
Weibull distributed random error term
To handle the route overlapping problem, a path-size factor (Ben-Akiva and Bierlaire, 1999) is introduced, i.e.,
Path-size factor
which gives the PSW model:
Ben-Akiva, M. and Bierlaire, M., Discrete choice methods and their applications to short term travel decisions. Handbook of Transportation Science, R.W. Halled, Kluwer Publishers.

21. Independently Distributed Assumption: Route Overlapping
PSW
MNP
MNL, MNW

22. Outline Review of closed-form route choice/network equilibrium models
Weibit route choice model
Weibit stochastic user equilibrium model
Numerical results
Concluding Remarks

23. Comparison between MNL Model and MNW Model
Extreme value distribution
Log
Weibull
Gumbel (type I)
Weibull (type III)
Assume
IID
Independence
Log
Transformation

24. A Mathematical Programming (MP) Formulation for the MNW-SUE model
Multiplicative Beckmann’s transformation
(MBec)
Relative cost difference under congestion

25. A MP Formulation for the PSW-SUE Model

26. Equivalency Condition
By constructing the Lagrangian function, we have
By setting the partial derivative w.r.t. route flow variable equal to zero, we have
Then, we have the PSW route flow solution, i.e.,

27. Uniqueness Condition The second derivative
By assuming , the route flow solution of PSW-SUE is unique.

28. Path-Based Partial Linearization Algorithm

29. Outline Review of closed-form route choice/network equilibrium models
Weibit route choice model
Weibit stochastic user equilibrium model
Numerical results
Concluding Remarks

30. Real Network Winnipeg network, Canada 154 zones, 2,535 links, and
4,345 O-D pairs.

31. Convergence Results

32. Winnipeg Network Results

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

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

35. Drawback: Insensitive to an Arbitrary Multiplier Route Cost

36. Incorporating ij Variational Inequality (VI) General route cost
MNW model
Flow dependent
PSW model
Zhou, Z., Chen, A. and Bekhor, S., C-logit stochastic user equilibrium model: formulations and solution algorithm. Transportmetrica, 8(1),

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 network

38. Thank You