A globally convergent algorithm for transportation continuous network design problem Article First Online: 10 July 2007 DOI :
10.1007/s11081-007-9015-1

Cite this article as: Gao, Z., Sun, H. & Zhang, H. Optim Eng (2007) 8: 241. doi:10.1007/s11081-007-9015-1
16
Citations
187
Downloads
Abstract
The continuous network design problem (CNDP) is characterized by a bilevel programming model, in which the upper level problem is generally to minimize the total system cost under limited expenditure, while at the lower level the network users make choices with regard to route conditions following the user equilibrium principle. In this paper, the bilevel programming model for CNDP is transformed into a single level convex programming problem by virtue of an optimal-value function tool and the relationship between System Optimum (SO) and User Equilibrium (UE). By exploring the inherent nature of the CNDP, the optimal-value function for the lower level user equilibrium problem is proved to be continuously differentiable and its derivative in link capacity enhancement can be obtained efficiently by implementing user equilibrium assignment subroutine. However, the reaction (or response) function between the upper and lower level problem is implicit and its gradient is difficult to obtain. Although, here we approximately express the gradient with the difference concept at each iteration, based on the method of successive averages (MSA), we propose a globally convergent algorithm to solve the single level convex programming problem. Comparing with widely used heuristic algorithms, such as sensitivity analysis based (SAB) method, the proposed algorithm needs not strong hypothesis conditions and complex computation for the inverse matrix. Finally, a numerical example is presented to compare the proposed method with some existing algorithms.

Keywords Continuous network design problem B-level programming Method of successive averages Global convergence

References
Abdulaal M, LeBlanc LJ (1979) Continuous equilibrium network design models. Transp Res B 13:19–32

CrossRef Google Scholar
Boyce DE (1984) Urban transportation network equilibrium and design models: recent achievements and future prospectives. Environ Plan A 16:1445–1474

CrossRef Google Scholar
Chiou SW (1999) Optimization of area traffic control for equilibrium network flows. Transp Sci 33:279–289

MATH Google Scholar
Cho HJ (1988) Sensitivity analysis of equilibrium network flows and its application to the development of solution methods for equilibrium network design problems. PhD dissertation. University of Pennsylvania, Philadelphia

Cree ND, Maher MJ (1998) The continuous equilibrium optimal network design problem: a genetic approach. In: Transportation networks: recent methodological advances. Elsevier, Netherlands, pp 163–174

Google Scholar
Dafermos S (1980) Traffic equilibria and variational inequalities. Transp Sci 14:42–54

MathSciNet Google Scholar
Dafermos S, Nagurney A (1984) Sensitivity analysis for the asymmetric network equilibrium problem. Math Program 28:174–184

MATH CrossRef MathSciNet Google Scholar
Friesz TL (1981) An equivalent optimization problem with combined multiclass distribution assignment and modal split which obviates symmetry restriction. Transp Res B 15:361–369

CrossRef MathSciNet Google Scholar
Friesz TL (1985) Transportation network equilibrium, design and aggregation: key developments and research opportunities. Transp Res A 19:413–427

CrossRef Google Scholar
Friesz TL, Harker PT (1985) Properties of the iterative optimization equilibrium algorithm. Civ Eng Syst 2:142–154

Google Scholar
Friesz TL et al (1990) Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints. Math Program 48:265–284

Friesz TL et al (1993) The multiobjective equilibrium network design problem revisited: a simulated annealing approach. Eur J Oper Res 65:44–57

MATH CrossRef Google Scholar
Gao ZY, Song YF (2002) A reserve capacity model of optimal signal control with user-equilibrium route choice. Transp Res B 36:313–323

CrossRef Google Scholar
Gao ZY, Song YF Si BF (2000) Urban transportation continuous equilibrium network design problem: theory and method. China Railway Press, Beijing

Google Scholar
Gao ZY, Wu JJ, Sun HJ (2005) Solution algorithm for the bi-level discrete network design problem. Transport Res B 39:479–495

CrossRef Google Scholar
Kim TJ (1990) Advanced transport and spatial systems models: applications to Korea. Springer, New York

Google Scholar
Kim TJ, Suh S (1988) Toward developing a national transportation planning model: a bilevel programming approach for Korea. Ann Reg Sci XXSPED:65–80

CrossRef Google Scholar
Luo ZQ, Pang JS, Ralph D (1996) Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge

Google Scholar
Magnanti TL, Wong RT (1984) Network design and transportation planning: models and algorithms. Transp Sci 18:1–55

Google Scholar
Mangasarian OL, Rosen JB (1964) Inequalities for stochastic nonlinear programming problems. Oper Res 12:143–154

MATH MathSciNet Google Scholar
Marcotte P (1983) Network optimization with continuous control parameters. Transp Sci 17:181–197

Google Scholar
Marcotte P (1986) Network design problem with congestion effects: a case of bi-level programming. Math Program 34:142–162

MATH CrossRef MathSciNet Google Scholar
Marcotte P, Marquis G (1992) Efficient implementation of heuristics for the continuous network design problem. Ann Oper Res 34:163–176

MATH CrossRef Google Scholar
Marcotte P, Zhu DL (1996) Exact and inexact penalty methods for the generalized bilevel programming problems. Math Program 74:141–157

MathSciNet Google Scholar
Meng Q, Yang H, Bell MGH (2001) An equivalent continuously differentiable model and a locally convergent algorithm for the continuous networks design problem. Transp Res B 35:83–105

CrossRef Google Scholar
Patriksson M (1994) The traffic assignment problem models and methods. VSB BV, Netherlands

Google Scholar
Powell WB, Sheffi Y (1982) The convergence of equilibrium algorithms with predetermined step size. Transp Sci 6:5–55

MathSciNet Google Scholar
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton

MATH Google Scholar
Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice–Hall, Englewood Cliffs

Google Scholar
Shimizu K, Ishizuka Y, Bard JF (1997) Nondifferentiable and two-level mathematical programming. Kluwer Academic, Massachusetts

MATH Google Scholar
Suwansirikul C, Friesz TL, Tobin RL (1987) Equilibrium decomposed optimization: a heuristic for the continuous equilibrium network design problem. Transp Sci 21:254–263

MATH Google Scholar
Tan HN, Gershwin SB, Athans M (1979) Hybrid optimization in urban traffic networks. Report No. DOT-TSC-RSPA-79-7. Laboratory for Information and Decision System, MIT, Cambridge, MA

Tobin RL, Friesz TL (1988) Sensitivity analysis for equilibrium network flows. Transp Sci 22:242–250

MATH CrossRef MathSciNet Google Scholar
Wong SC, Yang H (1997) Reserve capacity of a signal-controlled road network. Transp Res Part B 31:397–402

CrossRef Google Scholar
Yang H (1995) Sensitivity analysis for queuing equilibrium network flow and its application to traffic control. Math Comput Model 22:247–258

MATH Google Scholar
Yang H (1997) Sensitivity analysis for the elastic demand network equilibrium problem with applications. Transp Res B 31:55–70

CrossRef Google Scholar
Yang H, Bell MGH (1998) Models and algorithm for road network design: a review and some new developments. Transp Rev 18(3):257–278

CrossRef Google Scholar
Yang H, Yagar S (1994) Traffic assignment and traffic control in general freeway-arterial corridor systems. Transp Res B 28:463–486

CrossRef Google Scholar
Yang H, Meng Q, Liu GS (2004) The generalized transportation network optimization problem: models and algorithms. Working Paper, The Hong Kong University of Science and Technology

© Springer Science+Business Media, LLC 2007

Authors and Affiliations 1. State Key Laboratory of Rail Traffic Control and Safety, School of Traffic and Transportation Beijing Jiaotong University Beijing Peoples Republic of China