Skip to main content
SpringerLink
Log in

Traffic assignment problem under tradable credit scheme in a bi-modal stochastic transportation network: A cumulative prospect theory approach

可交易路票策略下的双模式随机网络交通分配问题: 累积前景理论方法

  • Published:
Journal of Central South University Aims and scope Submit manuscript

Abstract

The traffic equilibrium assignment problem under tradable credit scheme (TCS) in a bi-modal stochastic transportation network is investigated in this paper. To describe traveler’s risk-taking behaviors under uncertainty, the cumulative prospect theory (CPT) is adopted. Travelers are assumed to choose the paths with the minimum perceived generalized path costs, consisting of time prospect value (PV) and monetary cost. At equilibrium with a given TCS, the endogenous reference points and credit price remain constant, and are consistent with the equilibrium flow pattern and the corresponding travel time distributions of road sub-network. To describe such an equilibrium state, the CPT-based stochastic user equilibrium (SUE) conditions can be formulated under TCS. An equivalent variational inequality (VI) model embedding a parameterized fixed point (FP) model is then established, with its properties analyzed theoretically. A heuristic solution algorithm is developed to solve the model, which contains two-layer iterations. The outer iteration is a bisection-based contraction method to find the equilibrium credit price, and the inner iteration is essentially the method of successive averages (MSA) to determine the corresponding CPT-based SUE network flow pattern. Numerical experiments are provided to validate the model and algorithm.

摘要

本文研究了可交易路票策略(Tradable Credit Scheme, TCS)下双模式随机网络中的交通均衡配流问题. 采用累积前景理论(Cumulative Prospect Theory, CPT)来描述出行者在不确定环境下的风险决策行为. 假设出行者选择理解的广义路径费用(包括时间前景值和货币费用)最小的路径进行出行. 在给定路票策略下的交通均衡状态, 内生的参考点和路票价格保持不变, 而且与道路子网络中的均衡流量形态和对应的出行时间概率分布一致. 为了描述这种交通均衡状态, 本文构建了路票策略下基于 CPT 的随机用户均衡(Stochastic User Equilibrium, SUE)条件. 然后, 建立了一个嵌套参数型不动点(Fixed Point, FP)模型的等价变分不等式(Variational Inequality, VI)模型, 并在理论上分析了该模型的相关特性. 本文设计了一种启发式算法来求解该模型, 该算法包含两层迭代过程. 其中, 外层迭代是一个基于二分的收缩算法, 用于寻找均衡路票价格; 内层迭代本质上是相继平均算法(Method of Successive Averages, MSA), 用于确定对应的基于 CPT 的 SUE 网络流量形态. 通过数值实验, 本文验证了模型和算法的正确性和有效性.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

G :

A general strongly connected road network

G′ :

Rail network

N :

Set of nodes

N′ :

Set of nodes on the rail network

A :

Set of directed links, a ∊ A

A′ :

Directed links on the rail network

W :

Set of O-D pairs, w ∊ W

\(q_{{\rm{auto}}}^w\) :

Traffic demand on road network for O-D pair, w ∊ W

R w :

Set of all paths between O-D pair, w ∊ W

\(f_r^w\) :

Flow on path, r ∊ Rw

v a :

Flow on link, a ∊ A

\(\delta _{a,r}^w\) :

Element of the link/path incidence matrix

\({\widehat{t}^w}\) :

Travel time on metro line between O-D pair, w ∊ W

\({\widehat\tau ^w}\) :

Fare on metro line between O-D pair, w ∊ W

\(\widehat{q}_{{\rm{metro}}}^w\) :

Traffic demand on rail network for O-D pair, w ∊ W

K :

Total amount of credits issued

ϕ w :

Initial credit amount distributed to each traveler between O-D pair, w ∊ W

k a :

Credit charges on link, a ∊ A

k :

Credit charge scheme on road network, k=[ka, a ∊ A]

t a :

Mean travel time on link, a ∊ A

\(t_r^w\) :

Path travel time on route, r ∊ Rw

\(T_r^w\) :

Random path travel time on path, r ∊ Rw

\({(\sigma _r^w)^2}\) :

Variance of path travel time on path, r ∊ Rw

ζ:

Lower limit of travelers’ desired on-time arrival probability

\(b_r^w\) :

The minimal budgeted time for taking path, r ∊ Rw

ϖ w :

Path-travel-time reference point between O-D pair, w ∊ W

g w(·):

Value function

ψ(φ):

Perceived probability of an event

φ :

Actual probability of an event

\(u_r^w\) :

Time prospect value for choosing path, r ∊ Rw

û w :

Time prospect values on the metro line between O-D pair, w ∊ W

\(\underline t _r^w\) :

Lower bounds of the travel time on path, r ∊ Rw

ṯ:

Upper bounds of the travel time on path, \(\overline t _r^w\)

ρ s :

TC rate of selling credits, ρs ∊ [0, 1]

ρ b :

TC rate of buying credits, ρb ∊ [0, 1]

ρ :

Credit price measured in money unit

\(\widetilde{C}_r^w\) :

Perceived generalized travel cost on path, r ∊ Rw

\(\widetilde{c}_r^w\) :

Expected generalized travel cost on path, r ∊ Rw

κ :

Conversion coefficient between time PV and monetary cost

\(\xi _r^w\) :

Traveler’s perception error of road sub-network

ĉ w :

Expected generalized travel cost on metro line between O-D pair, w ∊ W

\({\widehat\xi ^w}\) :

Traveler’s perception error of rail sub-network

\(\overline \theta \) :

Dispersion parameter of mode choice

θ :

Dispersion parameter of route choice

\(\overline c _{{\rm{auto}}}^w\) :

Weighted average of the expected generalized path costs between O-D pair, w ∊ W

\(P_r^w({\widetilde{\boldsymbol{c}}^w})\) :

Choice probability of the path, r ∊ Rw evaluated at \({\widetilde{\boldsymbol{c}}^w} = [\widetilde{c}_r^w,r \in {R_w}]\)

λ w :

Exogenous attractiveness of metro line between O-D pair, w ∊ W

\(S_{{\rm{auto}}}^w\) :

Satisfaction function between O-D pair, w ∊ W

References

  1. YANG H, WANG X L. Managing network mobility with tradable credits [J]. Transportation Research Part B, 2011, 45(3): 580–594. DOI: https://doi.org/10.1016/j.trb.2010.10.002.

    Article  Google Scholar 

  2. WU D, YIN Y, LAWPHONGPANICH S, YANG H. Design of more equitable congestion pricing and tradable credit schemes for multimodal transportation networks [J]. Transportation Research Part B, 2012, 46(9): 1273–1287. DOI: https://doi.org/10.1016/j.trb.2012.05.004.

    Article  Google Scholar 

  3. AKAMATSU T, WADA K. Tradable network permits: A new scheme for the most efficient use of network capacity [J]. Transportation Research Part C: Emerging Technologies, 2017, 79: 178–195. DOI: https://doi.org/10.1016/j.trc.2017.03.009.

    Article  Google Scholar 

  4. NIE Y, YIN Y. Managing rush hour travel choices with tradable credits scheme [J]. Transportation Research Part B, 2013, 50(4): 1–19. DOI: https://doi.org/10.1016/j.trb.2013.01.004.

    Article  Google Scholar 

  5. TIAN L J, YANG H, HUANG H J. Tradable credits schemes for managing bottleneck congestion and modal split with heterogeneous users [J]. Transportation Research Part E, 2013, 54(8): 1–13. DOI: https://doi.org/10.1016/j.tre.2013.04.002.

    Article  Google Scholar 

  6. XIAO F, QIAN Z, ZHANG H M. Managing bottleneck congestion with tradable credits [J]. Transportation Research Part B, 2013, 56: 1–14. DOI: https://doi.org/10.1016/j.trb.2013.06.016.

    Article  Google Scholar 

  7. XIAO L L, LIU T L, HUANG H J. Tradable permit schemes for managing morning commute with carpool under parking space constraint [J]. Transportation, 2019, 46(1): 1–24. DOI: https://doi.org/10.1007/s11116-019-09982-w.

    Article  Google Scholar 

  8. XU M, WANG G, GRANT-MULLER, GAO Z. Joint road toll pricing and capacity development in discrete transport network design problem [J]. Transportation, 2017, 44: 1–22. DOI: https://doi.org/10.1007/s11116-015-9674-2.

    Article  Google Scholar 

  9. WANG G, GAO Z, XU M. Integrating link-based discrete credit charging scheme into discrete network design problem [J]. European Journal of Operational Research, 2019, 272(1): 176–187. DOI: https://doi.org/10.1016/j.ejor.2018.05.069.

    Article  MathSciNet  Google Scholar 

  10. WANG G, XU M, GRANT-MULLER S, GAO Z. Combination of tradable credit scheme and link capacity improvement to balance economic growth and environmental management in sustainable-oriented transport development: A bi-objective bi-level programming approach [J]. Transportation Research Part A: Policy and Practice, 2018. DOI: https://doi.org/10.1016/j.tra.2018.10.031.

  11. LI Y, UKKUSURI S V, FAN J. Managing congestion and emissions in transportation networks with dynamic carbon credit charge scheme [J]. Computers & Operations Research, 2018, 99: 90–108. DOI: https://doi.org/10.1016/j.cor.2018.06.014.

    Article  MathSciNet  Google Scholar 

  12. HAN F, CHENG L. Stochastic user equilibrium model with a tradable credit scheme and application in maximizing network reserve capacity [J]. Engineering Optimization, 2017, 49(4): 549–564. DOI: https://doi.org/10.1080/0305215X.2016.1193357.

    Article  MathSciNet  Google Scholar 

  13. WANG X L, YANG H, ZHU D L, LI C. Tradable travel credits for congestion management with heterogeneous users [J]. Transportation Research Part E, 2012, 48(2): 426–437. DOI: https://doi.org/10.1016/j.tre.2011.10.007.

    Article  Google Scholar 

  14. ZHU D L, YANG H, LI C M, WANG X L. Properties of the multiclass traffic network equilibria under a tradable credit scheme [J]. Transportation Science, 2015, 49(3): 519–534. DOI: https://doi.org/10.1287/trsc.2013.0508.

    Article  Google Scholar 

  15. LIU Y, NIE Y M. A credit-based congestion management scheme in general two-mode networks with multiclass users [J]. Networks and Spatial Economics, 2017, 17(3): 681–711. DOI: https://doi.org/10.1007/s11067-017-9340-7.

    Article  MathSciNet  Google Scholar 

  16. WANG J P, LIU T L, HUANG H J. Tradable OD-based travel permits for bi-modal traffic management with heterogeneous users [J]. Transportation Research Part E: Logistics and Transportation Review, 2018, 118: 589–605. DOI: https://doi.org/10.1016/j.tre.2018.08.015.

    Article  Google Scholar 

  17. NIE Y. Transaction costs and tradable mobility credits [J]. Transportation Research Part B, 2012, 46(1): 189–203. DOI: https://doi.org/10.1016/j.trb.2011.10.002.

    Article  Google Scholar 

  18. BAO Y, GAO Z, XU M, YANG H. Tradable credit scheme for mobility management considering travelers’ loss aversion [J]. Transportation Research Part E, 2014, 68: 138–154. DOI: https://doi.org/10.1016/j.tre.2014.05.007.

    Article  Google Scholar 

  19. HE F, YIN Y F, SHIRMOHAMMADI N, NIE Y M. Tradable credit schemes on networks with mixed equilibrium behaviors [J]. Transportation Research Part B, 2013, 57: 47–65. DOI: https://doi.org/10.1016/j.trb.2013.08.016.

    Article  Google Scholar 

  20. GUO R Y, HUANG H J, YANG H. Tradable credit scheme for control of evolutionary traffic flows to system optimum: Model and its convergence [J]. Networks and Spatial Economics, 2019, 19(3): 833–868. DOI: https://doi.org/10.1007/s11067-018-9432-z.

    Article  Google Scholar 

  21. XIAO F, LONG J, LI L, KOU G, NIE Y. Promoting social equity with cyclic tradable credits [J]. Transportation Research Part B: Methodological, 2019, 121: 56–73. DOI: https://doi.org/10.1016/j.trb.2019.01.002.

    Article  Google Scholar 

  22. DOGTEROM N, ETTEMA D, DIJST M. Tradable credits for managing car travel: A review of empirical research and relevant behavioural approaches [J]. Transport Reviews, 2017. DOI: https://doi.org/10.1080/01441647.2016.1245219.

    Article  Google Scholar 

  23. YIN Y, IEDA H. Assessing performance reliability of road networks under non-recurrent congestion [J]. Transportation Research Record, 2001, 1771: 148–155. DOI: https://doi.org/10.3141/1771-19.

    Article  Google Scholar 

  24. LO H K, LUO X W, SIU B W Y. Degradable transport network: Travel time budget of travelers with heterogeneous risk aversion [J]. Transportation Research Part B, 2006, 40(9): 792–806. DOI: https://doi.org/10.1016/j.trb.2005.10.003.

    Article  Google Scholar 

  25. SHAO H, LAM W H K, TAM M L. A reliability-based stochastic assignment model for network with multiple user classes under uncertainty in demand [J]. Network and Spatial Economics, 2006, 6(3, 4): 173–204. DOI: https://doi.org/10.1007/s11067-006-9279-6.

    Article  MathSciNet  Google Scholar 

  26. SUN C, CHENG L, ZHU S, HAN F, CHU Z. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance [J]. Transportation Research Part D, 2019. DOI: https://doi.org/10.1016/j.trd.2017.03.002.

    Article  Google Scholar 

  27. WATLING D. User equilibrium traffic network assignment with stochastic travel times and late arrival penalty [J]. European Journal of Operational Research, 2006, 175(3): 1539–1556. DOI: https://doi.org/10.1016/j.ejor.2005.02.039.

    Article  MathSciNet  Google Scholar 

  28. ZHOU Z, CHEN A. Comparative analysis of three user equilibrium models under stochastic demand [J]. Journal of Advanced Transportation, 2008, 42(3): 239–263. DOI: https://doi.org/10.1002/atr.5670420304.

    Article  Google Scholar 

  29. AVINERI E. The effect of reference point on stochastic network equilibrium [J]. Transportation Science, 2006, 40(4): 409–420. DOI: https://doi.org/10.2307/25769318.

    Article  Google Scholar 

  30. CONNORS R D, SUMALEE A. A network equilibrium model with travelers’ perception of stochastic travel times [J]. Transportation Research Part B, 2009, 43(6): 614–624. DOI: https://doi.org/10.1016/j.trb.2008.12.002.

    Article  Google Scholar 

  31. XU H, LOU Y, YIN Y, ZHOU J. A prospect-based user equilibrium model with endogenous reference points and its application in congestion pricing [J]. Transportation Research Part B, 2011, 45(2): 311–328. DOI: https://doi.org/10.1016/j.trb.2010.09.003.

    Article  Google Scholar 

  32. SUMALEE A, CONNORS R D, LUATHEP P. Network equilibrium under cumulative prospect theory and endogenous stochastic demand and supply [C]// Transportation and Traffic Theory 2009: Golden Jubilee. Boston, MA: Springer, 2009: 19–38. DOI: https://doi.org/10.1007/978-1-4419-0820-92.

    Chapter  Google Scholar 

  33. KOSZEGI B, RABIN M. A model of reference-dependent preferences [J]. Quarterly Journal of Economics, 2006, 121(4): 1133–1165. DOI: https://doi.org/10.1093/qje/121.4.1133.

    MATH  Google Scholar 

  34. PRELEC D. The probability weighting function [J]. Econometrica, 1998, 66(3): 497–527. DOI: https://doi.org/10.2307/2998573.

    Article  MathSciNet  Google Scholar 

  35. TVERSKY A, KAHNEMAN D. Advances in prospect theory: Cumulative representation of uncertainty [J]. Journal of Risk and Uncertainty, 1992, 5(4): 297–323. DOI: https://doi.org/10.2307/41755005.

    Article  Google Scholar 

  36. XU X, CHEN A, JANSUWAN S, YANG C, RYU S. Transportation network redundancy: Complementary measures and computational methods [J]. Transportation Research Part B, 2018, 114: 68–85. DOI: https://doi.org/10.1016/j.trb.2018.05.014.

    Article  Google Scholar 

  37. MENG Q, LIU Z. Impact analysis of cordon-based congestion pricing on mode-split for a bimodal transportation network [J]. Transportation Research Part C, 2012, 21(1): 134–147. DOI: https://doi.org/10.1016/j.trc.2011.06.007.

    Article  Google Scholar 

  38. WU Z X, LAM W H K, HUANGH J. Equity and efficiency analysis of pricing strategies in a bimodal network with heterogeneous user groups [J]. Transportation Research Record, 2008, 2089: 43–50. DOI: https://doi.org/10.3141/2089-06.

    Article  Google Scholar 

  39. CANTARELLA G E. A general fixed-point approach to multimode multi-user equilibrium assignment with elastic demand [J]. Transportation Science, 1997, 31(2): 107–128. DOI: https://doi.org/10.1287/trsc.31.2.107.

    Article  Google Scholar 

  40. MENG Q, LIU Z, WANG S. Asymmetric stochastic user equilibrium problem with elastic demand and link capacity constraints [J]. Transportmetrica A: Transport Science, 2014, 10(4): 304–326. DOI: https://doi.org/10.1080/23249935.2013.765929.

    Article  Google Scholar 

  41. FACCHINEI F, PANG J. Finite-dimensional variational inequalities and complementarity problems [M]. New York: Springer-Verlag, 2003. DOI: https://doi.org/10.1007/b97544.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Automobile, Chang’an University, Xi’an, 710064, China

    Fei Han  (韩飞)

  2. School of Information Engineering, Chang’an University, Xi’an, 710064, China

    Fei Han  (韩飞) & Xiang-mo Zhao  (赵祥模)

  3. School of Transportation, Southeast University, Nanjing, 210096, China

    Lin Cheng  (程琳)

Authors
  1. Fei Han  (韩飞)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiang-mo Zhao  (赵祥模)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lin Cheng  (程琳)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Han  (韩飞).

Additional information

Foundation item: Project(BX20180268) supported by National Postdoctoral Program for Innovative Talent, China; Project(300102228101) supported by Fundamental Research Funds for the Central Universities of China; Project(51578150) supported by the National Natural Science Foundation of China; Project(18YJCZH130) supported by the Humanities and Social Science Project of Chinese Ministry of Education

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, F., Zhao, Xm. & Cheng, L. Traffic assignment problem under tradable credit scheme in a bi-modal stochastic transportation network: A cumulative prospect theory approach. J. Cent. South Univ. 27, 180–197 (2020). https://doi.org/10.1007/s11771-020-4287-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11771-020-4287-0

Key words

关键词

Search

Navigation