# Traffic equilibrium with a continuously distributed bound on travel weights: the rise of range anxiety and mental account

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## Abstract

A new traffic network equilibrium problem with continuously distributed bounds on path weights is introduced in this paper, as an emerging modeling tool for evaluating traffic networks in which the route choice behavior of individual motorists is subject to some physical or psychological upper limit of a travel weight. Such a problem may arise from at least two traffic network instances. First, in a traffic network serving electric vehicles, the driving range of these vehicles is subject to a distance constraint formed by onboard battery capacities and electricity consumption rates as well as network-wide battery-recharging opportunities, which cause the *range anxiety* issue in the driving population. Second, in a tolled traffic network, while drivers take into account both travel time and road toll in their route choice decisions, many of them implicitly or explicitly set a budget constraint in their *mental account* for toll expense, subject to their own income levels and other personal and household socio-economic factors. In both cases, we model the upper limit of the path travel weight (i.e., distance or toll) as a continuously distributed stochastic parameter across the driving population, to reflect the diverse heterogeneity of vehicle- and/or motorist-related travel characteristics. For characterizing this weight-constrained network equilibrium problem, we proposed a convex programming model with a finite number of constraints, on the basis of a newly introduced path flow variable named *interval path flow rate*. We also analyzed the problem’s optimality conditions for the case of path distance limits, and studied the existence of optimal tolls for the case of path toll limits. A linear approximation algorithm was further developed for this complex network equilibrium problem, which encapsulates an efficient weight-constrained *k*-minimum time path search procedure to perform the network loading. Numerical results obtained from conducting quantitative analyses on example networks clearly illustrate the applicability of the modeling and solution methods for the proposed problem and reveal the mechanism of stochastic weight limits reshaping the network equilibrium.

## Keywords

Traffic assignment with side constraints Network equilibrium Stochastic weight limit Electric vehicles Range anxiety Road tolls Mental account## Notes

### Acknowledgements

The authors greatly benefited in the review process from the comments offered by the editors and four anonymous referees. This study is jointly supported by research grants through the Young Talent Award from the China Recruitment Program of Global Experts, the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 2013-007312-0069), the National Natural Science Foundation of China (Grant No. 71471111, 71771150), the Science and Technology Commission of the Shanghai Municipality (Grant No. 17692108500). This research was also partially supported by the U.S. National Science Foundation (Grant No. CMMI-1254921, CMMI-1562291) and the Data-Supported Transportation Operations and Planning Center.

## References

- Arezki, Y., & Van Vliet, D. (1990). A full analytical implementation of the PARTAN Frank–Wolfe algorithm for equilibrium assignment.
*Transportation Science,**24*(1), 58–62.Google Scholar - Bahrami, S., Asshtiani, H. Z., Nourinejad, M., & Roorda, M. J. (2017). A complementarity equilibrium model for electric vehicles with charging.
*International Journal of Transportation Science and Technology,**6*(4), 255–271.Google Scholar - Ban, X. J., Ferris, M. C., Tang, L., & Lu, S. (2013). Risk-neutral second best toll pricing.
*Transportation Research Part B,**48,*67–87.Google Scholar - Batarce, M., & Ivaldi, M. (2014). Urban travel demand model with endogenous congestion.
*Transportation Research Part A,**59,*331–345.Google Scholar - Beckmann, M. J., McGuire, C. B., & Winsten, C. B. (1956).
*Studies in the economics of transportation*. New Haven, CT: Yale University Press.Google Scholar - Brotcorne, L., Labbe, M., Marcotte, P., & Savard, G. (2001). A bilevel model for toll optimization on a multicommodity transportation network.
*Transportation Science,**35*(4), 345–358.Google Scholar - Cheng, L., Yasunori, I., Nobuhiro, U., & Wang, W. (2003). Alternative quasi-Newton methods for capacitated user equilibrium assignment.
*Transportation Research Record,**1857,*109–116.Google Scholar - Daganzo, C. F. (1977a). On the traffic assignment problem with flow dependent costs—I.
*Transportation Research,**11*(6), 433–437.Google Scholar - Daganzo, C. F. (1977b). On the traffic assignment problem with flow dependent costs—II.
*Transportation Research,**11*(6), 439–441.Google Scholar - Dial, R. B. (1979). A probabilistic multipath traffic assignment problem which obviates path numeration.
*Transportation Research,**5*(2), 83–111.Google Scholar - Dial, R. B. (1996). Bicriterion traffic assignment: Basic theory and elementary algorithms.
*Transportation Science,**30*(2), 93–111.Google Scholar - Dial, R. B. (1997). Bicriterion traffic assignment: Efficient algorithms plus examples.
*Transportation Research Part B,**31*(5), 357–379.Google Scholar - Dial, R. B. (1999a). Network-optimized road pricing: Part I: A parable and a model.
*Operations Research,**47*(1), 54–64.Google Scholar - Dial, R. B. (1999b). Network-optimized road pricing: Part II: Algorithms and examples.
*Operations Research,**47*(2), 327–336.Google Scholar - Florian, M., Constantin, I., & Florian, D. (2009). A new look at projected gradient method for equilibrium assignment.
*Transportation Research Record,**2090,*10–16.Google Scholar - Florian, M., Guelat, J., & Spiess, H. (1987). An efficient implementation of the “PARTAN” variant of the linear approximation method for the network equilibrium problem.
*Networks,**17*(3), 319–339.Google Scholar - Frank, M., & Wolfe, P. (1956). An algorithm for quadratic programming.
*Naval Research Logistics Quarterly,**3*(1–2), 95–110.Google Scholar - Franke, T., & Krems, J. F. (2013). What drives range preferences in electric vehicle users?
*Transport Policy,**30,*56–62.Google Scholar - Gabriel, S. A., & Bernstein, D. (1997). The traffic equilibrium problem with nonadditive path costs.
*Transportation Science,**31*(4), 337–348.Google Scholar - Goodwin, P. B. (1981). The usefulness of travel budgets.
*Transportation Research Part A,**15*(1), 97–106.Google Scholar - Gulipalli, S., & Kockelman, K. M. (2008). Credit-based congestion pricing: A Dallas–Fort Worth application.
*Transport Policy,**15*(1), 23–32.Google Scholar - He, F., Yin, Y., & Lawphongpanich, S. (2014). Network equilibrium models with battery electric vehicles.
*Transportation Research Part B,**67,*306–319.Google Scholar - Hearn, D. W. (1980).
*Bounding flows in traffic assignment models, research report 80-4*. Gainesville, FL: Department of Industrial and Systems Engineering, University of Florida.Google Scholar - Himanen, V., Lee-Gosselin, M., & Perrels, A. (2005). Sustainability and the interactions between external effects of transport.
*Journal of Transport Geography,**13*(1), 23–28.Google Scholar - Holmberg, K., & Yuan, D. (2003). A multicommodity network flow problem with side constraints on paths solved by column generation.
*INFORMS Journal on Computing,**15*(1), 42–57.Google Scholar - Jahn, O., Möhring, R. H., Schulz, A. S., & Stier-Moses, N. E. (2005). System-optimal routing of traffic flows with user constraints in networks with congestion.
*Operations Research,**53*(4), 600–616.Google Scholar - Jiang, N., & Xie, C. (2014). Computing and analyzing mixed equilibrium network flows with gasoline and electric vehicles.
*Computer-Aided Civil and Infrastructure Engineering,**29*(8), 626–641.Google Scholar - Jiang, N., Xie, C., Duthie, J. C., & Waller, S. T. (2013). A network equilibrium analysis on destination, route and parking choices with mixed gasoline and electric vehicular flows.
*EURO Journal on Transportation and Logistics,**3*(1), 55–92.Google Scholar - Jiang, N., Xie, C., & Waller, S. T. (2012). Path-constrained traffic assignment: Model and algorithm.
*Transportation Research Record,**2283,*25–33.Google Scholar - Kitamura, R., & Lam, T. N. (1984). A model of constrained binary choice. In
*Proceedings of the 9th international symposium on transportation and traffic theory*, Delft, July 11–13, 1984.Google Scholar - Larsson, T., & Patriksson, M. (1995). An augmented Lagrangean dual algorithm for link capacity side constrained traffic assignment problems.
*Transportation Research Part B,**29*(6), 433–455.Google Scholar - Larsson, T., & Patriksson, M. (1999). Side constrained traffic equilibrium models—analysis, computation and applications.
*Transportation Research Part B,**33*(4), 233–264.Google Scholar - Lawler, E. L. (1976).
*Combinatorial optimization: Networks and matroids, holt*. New York, NY: Rinehart & Winston.Google Scholar - LeBlanc, L. J., Helgason, R. V., & Boyce, D. E. (1985). Improved efficiency of the Frank–Wolfe algorithm for convex network programs.
*Transportation Science,**19*(4), 445–462.Google Scholar - Leurent, F. (1993). Cost versus time equilibrium over a network.
*European Journal of Operational Research,**71*(2), 205–221.Google Scholar - Lin, Z. (2014). Optimizing and diversifying electric vehicle driving range for U.S. drivers.
*Transportation Science,**48*(4), 635–650.Google Scholar - Lo, H. K., & Chen, A. (2000). Traffic equilibrium problem with route-specific costs: Formulation and algorithms.
*Transportation Research Part B,**34*(6), 493–513.Google Scholar - Marcotte, P., & Zhu, D. L. (2009). Existence and computation of optimal tolls in multiclass network equilibrium problems.
*Operations Research Letters,**27*(3), 211–214.Google Scholar - Marrow, K., Karner, D., & Francfort, J. (2008). Plug-in hybrid electric vehicle charging infrastructure review. In
*Report INL/EXT*-*08*-*15058*, U.S. Department of Energy, Washington, DC.Google Scholar - Mock, P., Schmid, S., & Friendrich, H. (2010). Market prospects of electric passenger vehicles. In G. Pistoria (Ed.),
*Electric and hybrid vehicles: Power sources, models, sustainability, infrastructure and the market*. Amsterdam: Elsevier.Google Scholar - Mokhtarian, P. L., & Chen, C. (2004). TTB or not TTB, that is the question: A review and analysis of the empirical literature on travel time and money budgets.
*Transportation Research Part A,**38*(9–10), 643–675.Google Scholar - Nie, Y., Zhang, H. M., & Lee, D. H. (2004). Models and algorithms for the traffic assignment problem with link capacity constraints.
*Transportation Research Part B,**38*(4), 285–312.Google Scholar - Raith, A., Wang, J. Y. T., Ehrgott, M., & Mitchell, S. (2014). Solving multi-objective traffic assignment.
*Annals of Operations Research,**222*(1), 483–516.Google Scholar - Schulz, A. S., & Stier-Moses, N. E. (2006). Efficiency and fairness of system-optimal routing with user constraints.
*Networks,**48*(4), 223–234.Google Scholar - Thaler, R. H. (1985). Mental accounting and consumer choice.
*Marketing Science,**4*(3), 199–214.Google Scholar - Thaler, R. H. (1990). Anomalies: Saving, fungibility, and mental accounts.
*Journal of Economic Perspectives,**4*(1), 193–205.Google Scholar - Thaler, R. H. (1999). Mental accounting matters.
*Journal of Behavioral Decision Making,**12*(3), 183–206.Google Scholar - Verhoef, E. (2002a). Second-best congestion pricing in general networks: Heuristic algorithms for finding second-best optimal toll levels and toll points.
*Transportation Research Part B,**36*(8), 707–729.Google Scholar - Verhoef, E. (2002b). Second-best congestion pricing in general static transportation networks with elastic demands.
*Regional Science and Urban Economics,**32*(3), 281–310.Google Scholar - Wang, J. Y. T., & Ehrgott, M. (2013). Modeling route choice behavior in a tolled road network with a time surplus maximization bi-objective user equilibrium model.
*Transportation Research Part B,**57,*342–360.Google Scholar - Wang, L., Lin, A., & Chen, Y. (2010a). Potential impact of recharging plug-in hybrid electric vehicles on locational marginal prices.
*Naval Research Logistics,**57*(8), 686–700.Google Scholar - Wang, J. Y. T., Raith, A., & Ehrgott, M. (2010b). Tolling analysis with bi-objective traffic assignment. In M. Ehrgott, B. Naujoks, T. Stewart, & J. Wallenius (Eds.),
*Multiple criteria decision making for sustainable energy and transportation systems*. Berlin: Springer.Google Scholar - Wang, T. G., Xie, C., Xie, J., & Waller, S. T. (2016). Path-constrained traffic assignment: A trip chain analysis under range anxiety.
*Transportation Research Part C,**68,*447–461.Google Scholar - Xie, C., & Jiang, N. (2016). Relay requirement and traffic assignment of electric vehicles.
*Computer-Aided Civil and Infrastructure Engineering,**31*(8), 580–598.Google Scholar - Xie, C., Wang, T. G., Pu, X., & Karoonsoontawong, A. (2017). Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety.
*Transportation Research Part B*,*103*, 136–157.Google Scholar - Yang, H., & Huang, H. J. (2004). The multi-class, multi-criteria traffic network equilibrium and systems optimum problem.
*Transportation Research Part B,**28*(1), 1–15.Google Scholar - Yang, H., Wang, X., & Yin, Y. (2012). The impact of speed limits on traffic equilibrium and system performance in networks.
*Transportation Research Part B,**46*(10), 1295–1307.Google Scholar - Yang, H., & Wong, S. C. (1999). Estimation of the most likely equilibrium traffic queueing pattern in a capacity-constrained network.
*Annals of Operations Research,**87,*73–85.Google Scholar - Yen, J. Y. (1971). Finding the
*k*shortest loopless paths in a network.*Management Science,**17*(11), 712–716.Google Scholar - Zahavi, Y. (1979). UMOT project. In
*Report DOT*-*RSPA*-*DPB*-*20*-*79*-*3*, U.S. Department of Transportation, Washington, DC.Google Scholar - Zahavi, Y., & Ryan, J. M. (1980). Stability of travel components over time.
*Transportation Research Record,**750,*13–19.Google Scholar - Zhang, H., & Ge, Y. (2004). Modeling variable demand equilibrium under second-best road pricing.
*Transportation Research Part B,**38*(8), 733–749.Google Scholar