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A Path-size Weibit Stochastic User Equilibrium Model

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.edu. Outline.

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A Path-size Weibit Stochastic User Equilibrium Model

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  1. A Path-size Weibit Stochastic User Equilibrium Model SongyotKitthamkesorn 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.edu

  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., 1977. On stochastic models of traffic assignment. Transportation Science, 11(3), 253-274.

  6. Probabilistic Route Choice Models Perceived travel cost Gumbel Normal Multinomial probit (MNP)route choice model Daganzoand Sheffi (1977) Multinomial logit (MNL)route choice model Dial (1971) Non-closed form Closed form 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.

  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 distributedassumption, 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 Overlapping j i MNP Independently distributed MNL Route overlapping

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

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

  12. Extended Logit Models MNL EXTENDED LOGIT Overlapping Gumbel Closed form Closed form Modification of the deterministic term • C-logit(Cascettaet 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., 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. 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. Bekhor, S., Prashker, J.N., 1999. Formulations of extended logit stochastic user equilibrium assignments. Proceedings of the 14th 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 j i MNP MNL

  14. Scaling Technique CV = 0.5 j i j i (=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., 2012. Examining the scaling effect and overlapping problem in logit-based stochastic user equilibrium models. Transportation Research Part A, 46(8), 1343-1358.

  15. 3rd Alternative MNL EXTENDED LOGIT Overlapping 1. Gumbel Closed form Closed form MNP Overlapping Diff. trip length 2. Normal PSW MNW Modification of the deterministic term Diff. trip length Diff. trip length Overlapping 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), 373-380

  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 parameter 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 distributedassumption, 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), 373-380

  19. Identically Distributed Assumption: Homogeneous Perception Variance CV = 0.5 j i j i 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., 1999. 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. andBekhor,S., 2012. C-logit stochastic user equilibrium model: formulations and solution algorithm. Transportmetrica, 8(1), 17-41.

  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

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