Abstract
This paper focuses on the vector traffic network equilibrium problem with demands uncertainty and capacity constraints of arcs, in which, the demands are not exactly known and assumed as a discrete set that contains finite scenarios. For different scenario, the demand may be changed, which seems much more reasonable in practical programming. By using the linear scalarization method, we introduce several definitions of parametric equilibrium flows and reveal their mutual relations. Meanwhile, the relationships between the scalar variational inequality as well as the (weak) vector equilibrium flows are explored, meanwhile, some necessary and sufficient conditions that ensure the (weak) vector equilibrium flows are also considered. Additionally, by means of nonlinear scalarization functionals, two kinds of equilibrium principles are derived. All of the derived conclusions contain the demands uncertainty and capacity constraints of arcs, thus the results proposed in this paper improved some existing works. Finally, some numerical examples are employed to show the merits of the improved conclusions.
Similar content being viewed by others
References
Wardrop J G. Some theoretical aspects of road traffic research. P I Civil Eng, 1952, 1: 325–362
Nagurney A. Network Economics, a Variational Inequality Approach. Dordrecht: Kluwer Academic Publishers, 1999
Nagurney A. On the relationship between supply chain and transportation network equilibria: A supernetwork equivalence with computations. Transport Res E-Log, 2006, 42: 293–316
Meng Q, Huang Y K, Cheu R L. A note on supply chain network equilibrium models. Transport Res E-Log, 2007, 43: 60–71
Athanasenas A. Traffic simulation models for rural road network management. Transport Res E-Log, 1997, 33: 233–243
Gabriel S A, Bernstein D. The traffic equilibrium problem with nonadditive path costs. Transport Sci, 1997, 31: 337–348
Chen G Y, Yen N D. On the variational inequality model for network equilibrium. Internal Report 3. 196 (724). Department of Mathemat-ics, University of Pisa, 1993
Daniele P, Maugeri A, Oettli W. Time-dependent traffic equilibria. J Optimiz Theory App, 1999, 103: 543–555
Khanh P Q, Luu L M. On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications break to traffic network equilibria. J Optimiz Theory App, 2004, 123: 533–548
Khanh P Q, Luu L M. Some existence results for vector quasivariational inequalities involving multifunctions and applications to traffic equilibrium problems. J Glob Optim, 2005, 32: 551–568
Yang X Q, Goh C J. On vector variational inequalities: Application to vector equilibria. J Optimiz Theory App, 1997, 95: 431–443
Nagurney A, Dong J. A multiclass, multicriteria traffic network equilibrium model with elastic demand. Transport Res B-Meth, 2002, 36: 445–469
Goh C J, Yang X Q. Vector equilibrium problem and vector optimization. Eur J Oper Res, 1999, 116: 615–628
Yang H, Huang H J. The multiclass, multi-criteria traffic network equilibrium and systems optimum problem. Transport Res B-Meth, 2004, 38: 1–15
Lu S. Sensitivity of static traffic user equilibria with perturbations in arc cost function and travel demand. Transport Sci, 2008, 42: 105–123
Connors R D, Sumalee A. A network equilibrium model with travellers’ perception of stochastic travel times. Transport Res B-Meth, 2009, 43: 614–624
Li S J, Teo K L, Yang X Q. A remark on a standard and linear vector network equilibrium problem with capacity constraints. Eur J Oper Res, 2008, 184: 13–23
Konnov I V. Vector network equilibrium problems with elastic de-mands. J Glob Optim, 2013, 57: 521–531
Tan Z, Yang H, Guo R. Pareto efficiency of reliability-based traffic equilibria and risk-taking behavior of travelers. Transport Res B-Meth, 2014, 66: 16–31
Luc D T, Phuong T T T. Equilibrium in multi-criteria transportation networks. J Optimiz Theory App, 2016, 169: 116–147
Lin Z. The study of traffic equilibrium problems with capacity constraints of arcs. Nonlinear Anal-Real World Appl, 2010, 11: 2280–2284
Lin Z. On existence of vector equilibrium flows with capacity constraints of arcs. Nonlinear Anal-Theory Methods Appl, 2010, 72: 2076–2079
Xu Y D, Li S J, Teo K L. Vector network equilibrium problems with capacity constraints of arcs. Transport Res E-Log, 2012, 48: 567–577
Li S J, Teo K L, Yang X Q. Vector equilibrium problems with elastic demands and capacity constraints. J Glob Optim, 2007, 37: 647–660
Jahn J. Vector Optimization-Theory, Applications, and Extensions. Berlin: Springer, 2011
Gerth C, Weidner P. Nonconvex separation theorems and some applications in vector optimization. J Optimiz Theory App, 1990, 67: 297–320
Göpfert A, Riahi H, Tammer C, et al. Variational Methods in Partially Ordered Spaces. New York: Springer, 2003
Khan A A, Tammer C, Zălinescu C. Set-valued Optimization: An Introduction with Applications. Berlin: Springer, 2015
Hiriart-Urruty J B. Tangent cones, generalized gradients and mathe-matical programming in Banach spaces. Math Oper Res, 1979, 4: 79–97
Zaffaroni A. Degrees of efficiency and degrees of minimality. SIAM J Control Optim, 2003, 42: 1071–1086
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Cao, J., Li, R., Huang, W. et al. Traffic network equilibrium problems with demands uncertainty and capacity constraints of arcs by scalarization approaches. Sci. China Technol. Sci. 61, 1642–1653 (2018). https://doi.org/10.1007/s11431-017-9172-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11431-017-9172-4