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April  2013, 9(2): 341-363. doi: 10.3934/jimo.2013.9.341

Using emission functions in modeling environmentally sustainable traffic assignment policies

O. İlker Kolak 1, , Orhan Feyzioğlu 1, , Ş. İlker Birbil 2, , Nilay Noyan 2, and  Semih Yalçindağ 3,

1. 

Faculty of Engineering and Technology, Galatasaray University, Çirağan Caddesi No:36 Ortaköy, 34357 Istanbul, Turkey, Turkey
2. 

Faculty of Engineering and Natural Sciences, Sabanci University, Üniversite Caddesi No:27 Tuzla, 34956 Istanbul, Turkey, Turkey
3. 

Faculty of Engineering and Natural Sciences, Sabanci University, niversite Caddesi No:27 Tuzla, 34956 Istanbul, Turkey

Received  November 2011 Revised  July 2012 Published  February 2013

Transport systems play a crucial role for sustainable development, and hence, sustainable urban transportation has recently become a major research area. Most of the existing studies propose evaluation methods that use simulation tools to assess the sustainability of different transportation policies. Although there are some recent studies, considering the sustainability dimension and the resulting policies through mathematical programming models is still an open research area. In this study, we focus on controlling the gas emissions for the environmental sustainability and propose several mathematical programming models that incorporate the measurements of gas emissions over a traffic network. We define emission functions in terms of the traffic flow so that the accumulated emission amounts can be modeled accurately, particularly in case of congestion. Using these emission functions, we introduce alternate objective functions and develop optimization models under various policies which are based on the well-known toll pricing and capacity enhancement. The proposed models both reflect the route choice decisions of the network users and the decisions of the transportation managers that aim at making the transport systems more sustainable through the policies of interest. We conduct a computational study on a well-known testing network and present numerical results to evaluate the proposed alternate models. We conclude that simultaneously applying the toll pricing and capacity enhancement policies is in general more effective in serving the travel demand and reducing the emission amounts compared to implementing these policies individually.
Keywords: toll pricing, traffic assignment, Transport, emission functions, capacity enhancement., mathematical programs with complementarity constraints.
Mathematics Subject Classification: Primary: 90B20, 90C33; Secondary: 90B1.
Citation: O. İlker Kolak, Orhan Feyzioğlu, Ş. İlker Birbil, Nilay Noyan, Semih Yalçindağ. Using emission functions in modeling environmentally sustainable traffic assignment policies. Journal of Industrial & Management Optimization, 2013, 9 (2) : 341-363. doi: 10.3934/jimo.2013.9.341
References:
[1]

M. Abdulaal and L. J. LeBlanc, Continuous equilibrium network design models, Transportation Research Part B: Methodological, 13 (1979), 19-32. Google Scholar

[2]

R. Akçelik, "Speed-Flow Models for Uninterrupted Traffic Facilities," Tech. Report, Akcelik Associates Pty Ltd, 2003. Google Scholar

[3]

R. Akçelik, "Speed-Flow and Bunching Relationships for Uninterrupted Flows," Proceeding of the 5th International Symposium on Highway Capacity and Quality of Service, Yokohama, Japan, 2006. Google Scholar

[4]

R. Arnott and K. Small, "The Economics of Traffic Congestion," Boston College working papers in economics, Boston College Department of Economics, 1993. Google Scholar

[5]

F. Babonneau and J.-P. Vial, An efficient method to compute traffic assignment problems with elastic demands, Transportation Science, 42 (2008), 249-260. Google Scholar

[6]

C. Benedek and L. Rilett, Equitable traffic assignment with environmental cost functions, Journal of Transportation Engineering, 124 (1998), 16-22. Google Scholar

[7]

Ş. I. Birbil, G. Gürkan and O. Listeş, Solving stochastic mathematical programs with complementarity constraints using simulation, Mathematics of Operations Research, 31 (2006), 739-760. doi: 10.1287/moor.1060.0215.  Google Scholar

[8]

, "Traffic Assignment Manual for Application with a Large, High Speed Computer,", U.S. Dept. of Commerce, (1964).   Google Scholar

[9]

L. Brotcorne, M. Labbe, P. Marcotte and G. Savard, A bilevel model for toll optimization on a multicommodity transportation network, Transportation Science, 35 (2001), 345-358. Google Scholar

[10]

, "Methods to Find the Cost-Effectiveness of Funding Air Quality Projects,", Tech. Report, (2010).   Google Scholar

[11]

A. Chen and K. Subprasom, Analysis of regulation and policy of private toll roads in a build-operate-transfer scheme under demand uncertainty, Transportation Research Part A: Policy and Practice, 41 (2007), 537-558. Google Scholar

[12]

S.-W. Chiou, Bilevel programming for the continuous transport network design problem, Transportation Research Part B: Methodological, 39 (2005), 361-383. Google Scholar

[13]

J. Dargay, D. Gately and M. Sommer, Vehicle ownership and income growth, worldwide: 1960-2030, The Energy Journal, 28 (2007), 163-190. Google Scholar

[14]

G. de Ceuster, B. van Herbruggen, O. Ivanova, K. Carlier, A. Martino and D. Fiorello, "TREMOVE - Final Report," Tech. Report, Transport Mobility Leuven, 2007. Google Scholar

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E. Deakin, "Sustainable Development and Sustainable Transportation: Strategies for Economic Prosperity Environmental Quality and Equity," Tech. Report 2001'03, Institute of Urban and Regional Development, University of California, Berkeley, 2001. Google Scholar

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S. P. Dirkse and M. C. Ferris, Traffic modeling and variational inequalities using GAMS, in "Operations Research and Decision Aid Methodologies in Traffic and Transportation Management," NATO ASI Series F, Springer-Verlag, (1997), 136-163. Google Scholar

[17]

A. Drud, CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems, Mathematical Programming, 31 (1985), 153-191. doi: 10.1007/BF02591747.  Google Scholar

[18]

, "Major Pollutants from Petrol and Diesel Engines,", Tech. Report, (2010).   Google Scholar

[19]

, "User's Guide to Mobile 6.1 and Mobile 6.2 - Mobile Source Emission Factor Model,", Environmental Protection Agency, (2003).   Google Scholar

[20]

M. C. Ferris, S. P. Dirkse and A. Meeraus, Mathematical programs with equilibrium constraints: Automatic reformulation and solution via constrained optimization, in "Frontiers in Applied General Equilibrium Modeling" (eds. T. J.Kehoe, T. N. Srinivasan and J. Whalley), Cambridge University Press, (2005), 67-93. Google Scholar

[21]

T. L. Friesz, R. L. Tobin, H.-J. Cho and N. J. Mehta, Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints, Mathematical Programming, 48 (1990), 265-284. doi: 10.1007/BF01582259.  Google Scholar

[22]

, "GAMS-A User's Guide,", General Algebraic Modeling System, (2010).   Google Scholar

[23]

Z. Gao and Y. Song, A reserve capacity model of optimal signal control with user-equilibrium route choice, Transportation Research: Part B, 36 (2002), 313-323. Google Scholar

[24]

D. Gkatzoflias, C. Kouridis, L. Ntziachristos and Z. Samaras, "Copert 5 - Computer Programme to Calculate Emissions from Road Transport, User Manual, Version 5.0," European Environment Agency, 2007. Google Scholar

[25]

S. Gokhale and M. Khare, A review of deterministic, stochastic and hybrid vehicular exhaust emission models, International Journal of Transport Management, 2 (2004), 59-74. Google Scholar

[26]

R. L. Guensler and D. Sperling, Congestion pricing and motor vehicle emissions: An initial review, Transportation Research Board Special Report, 242 (1994), 356-379. Google Scholar

[27]

E. Gutt, V. Patton and N. Spencer, "Building on 30 Years of Clean Air Act Success: The Case for Reducing NOx Air Pollution," Tech. Report, Environmental Defense Fund, 2000. Google Scholar

[28]

D. W. Hearn and M. V. Ramana, Solving congestion toll pricing models, in "Equilibrium and Advanced Transportation Modeling" (eds. P. Marcotte and S. Nguyen), Kluwer Academic Publishers, (1998), 109-124. Google Scholar

[29]

C. M. Jeon and A. Amekudzi, Addressing sustainability in transportation systems: Definitions, indicators, and metrics, Journal of Infrastructure Systems, 11 (2005), 31-50. Google Scholar

[30]

O. Johansson-Stenman and T. Sterner, What is the scope for environmental road pricing?, in "Road Pricing, Traffic Congestion and the Environment: Issues of Efficiency and Social Feasibility" (eds. K. J. Button and E. T. Verhoef), Edward Elgar Pub, London, UK, (1998), 150-167. Google Scholar

[31]

M. Ketzel, P. Louka, P. Sahm, E. Guilloteau, J.-F. Sini and N. Moussiopoulos, Intercomparison of numerical urban dispersion models: Part I & II, Water, Air, & Soil Pollution: Focus, 2 (2002), 603-613. Google Scholar

[32]

F. H. Knight, Some fallacies in the interpretation of social cost, The Quarterly Journal of Economics, 38 (1924), 582-606. Google Scholar

[33]

M. Labbe, P. Marcotte and G. Savard, A bilevel model of taxation and its application to optimal highway pricing, Management Science, 44 (1998), 1608-1622. Google Scholar

[34]

S. Lawphongpanich and D. W. Hearn, An MPEC approach to second-best toll pricing, Mathematical Programming, Series B, 101 (2004), 33-55. doi: 10.1007/s10107-004-0536-5.  Google Scholar

[35]

T. Litman, "Well Measured: Developing Indicators for Sustainable and Livable Transport Planning," Tech. Report, Victoria Transport Policy Institute, 2010. Google Scholar

[36]

T. Litman and D. Burwell, Issues in sustainable transportation, International Journal of Global Environmental Issues, 6 (2006), 331-347. Google Scholar

[37]

Z.-Q. Luo, J.-S. Pang, and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, Cambridge, UK, 1996. doi: 10.1017/CBO9780511983658.  Google Scholar

[38]

D. R. Lynam and G. D. Pfeifer, Human health effects of highway-related pollutants, in "Highway Pollution" (eds. R. S. Hamilton and R. M. Harrison), Studies in Environmental Science, 44, Elsevier, Amsterdam, (1991), 259-280. Google Scholar

[39]

T. L. Magnanti and R. T. Wong, Network Design and Transportation Planning: Models and Algorithms, Transportation Science, 18 (1984), 1-55. Google Scholar

[40]

P. Marcotte, Network design problem with congestion effects: A case of bilevel programming, Mathematical Programming, 34 (1986), 142-162. doi: 10.1007/BF01580580.  Google Scholar

[41]

A. Nagurney, Congested urban transportation networks and emission paradoxes, Transportation Research Part D: Transport and Environment, 5 (2000), 145-151. Google Scholar

[42]

A. Nagurney, "Sustainable Transportation Networks," Edward Elgar Publishing, 2000. Google Scholar

[43]

A. Nagurney, Paradoxes in networks with zero emission links: Implications for telecommunications versus transportation, Transportation Research Part D: Transport and Environment, 6 (2001), 283-296. Google Scholar

[44]

M. Patriksson, "The Traffic Assignment Problem: Models and Methods," V.S.P. International Science, Utrecht, The Netherlands, 1994. Google Scholar

[45]

M. Patriksson and R. T. Rockafellar, A mathematical model and descent algorithm for bilevel traffic management, Transportation Science, 36 (2002), 271-291. Google Scholar

[46]

V. Patton, Opportunity NOx, The Environmental Forum, 18 (2001), 30-35. Google Scholar

[47]

A. Pigou, "Wealth and Welfare," Macmillan, London, UK, 1920. Google Scholar

[48]

A. Rahman and R. V. Grol, "SUMMA Final Publishable Report Version 2.0," Tech. Report, RAND Europe, 2005. Google Scholar

[49]

L. Rilett and C. Benedek, Traffic assignment under environmental and equity objectives, Transportation Research Record, 1443 (1994), 92-99. Google Scholar

[50]

J. Rouwendal and E. T. Verhoef, Basic economic principles of road pricing: From theory to applications, Transport Policy, 13 (2006), 106-114. Google Scholar

[51]

Y. Sheffi, "Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods," Prentice Hall, England Cliffs, NJ, 1985. Google Scholar

[52]

S. Sugawara and D. A. Niemeier, How much can vehicle emissions be reduced? Exploratory analysis of an upper boundary using an emissions-optimized trip assignment, Transportation Research Record, 1815 (2002), 29-37. Google Scholar

[53]

L. Sun, Z. Gao and Y. Wang, A Stackelberg game management model of the urban public transport, Journal of Industrial and Management Optimization, 8 (2012), 507-520. doi: 10.3934/jimo.2012.8.507.  Google Scholar

[54]

G. H. Tzeng and C. H. Chen, Multiobjective decision making for traffic assignment, IEEE Transactions on Engineering Management, 40 (1993), 180-187. Google Scholar

[55]

, "World Urbanization Prospects: The 2009 Revision,", Tech. Report, (2009).   Google Scholar

[56]

W. S. Vickrey, Congestion theory and transport investment, American Economic Review, 59 (1969), 251-260. Google Scholar

[57]

J. G. Wardrop, Some theoretical aspects of road traffic research, ICE Proceedings: Engineering Divisions, 1 (1952), 325-378. Google Scholar

[58]

H. Yang and M. G. H. Bell, Models and algorithms for road network design: A review and some new developments, Transport Reviews: A Transnational Transdisciplinary Journal, 18 (1998), 257-278. Google Scholar

[59]

H. Yang and Q. Meng, Highway pricing and capacity choice in a road network under a build-operate-transfer scheme, Transportation Research Part A: Policy and Practice, 34 (2000), 207-222. Google Scholar

[60]

H. Yang and Q. Meng, A note on highway pricing and capacity choice in a road network under a build-operate-transfer scheme, Transportation Research Part A: Policy and Practice, 36 (2002), 659-663. Google Scholar

[61]

Y. Yin and S. Lawphongpanich, Internalizing emission externality on road networks, Transportation Research Part D: Transport and Environment, 11 (2006), 292-301. Google Scholar

[62]

H. Zhang and Z. Gao, Bilevel programming model and solution method for mixed transportation network design problem, Journal of Systems Science and Complexity, 22 (2009), 446-459. doi: 10.1007/s11424-009-9177-3.  Google Scholar

show all references

References:
[1]

M. Abdulaal and L. J. LeBlanc, Continuous equilibrium network design models, Transportation Research Part B: Methodological, 13 (1979), 19-32. Google Scholar

[2]

R. Akçelik, "Speed-Flow Models for Uninterrupted Traffic Facilities," Tech. Report, Akcelik Associates Pty Ltd, 2003. Google Scholar

[3]

R. Akçelik, "Speed-Flow and Bunching Relationships for Uninterrupted Flows," Proceeding of the 5th International Symposium on Highway Capacity and Quality of Service, Yokohama, Japan, 2006. Google Scholar

[4]

R. Arnott and K. Small, "The Economics of Traffic Congestion," Boston College working papers in economics, Boston College Department of Economics, 1993. Google Scholar

[5]

F. Babonneau and J.-P. Vial, An efficient method to compute traffic assignment problems with elastic demands, Transportation Science, 42 (2008), 249-260. Google Scholar

[6]

C. Benedek and L. Rilett, Equitable traffic assignment with environmental cost functions, Journal of Transportation Engineering, 124 (1998), 16-22. Google Scholar

[7]

Ş. I. Birbil, G. Gürkan and O. Listeş, Solving stochastic mathematical programs with complementarity constraints using simulation, Mathematics of Operations Research, 31 (2006), 739-760. doi: 10.1287/moor.1060.0215.  Google Scholar

[8]

, "Traffic Assignment Manual for Application with a Large, High Speed Computer,", U.S. Dept. of Commerce, (1964).   Google Scholar

[9]

L. Brotcorne, M. Labbe, P. Marcotte and G. Savard, A bilevel model for toll optimization on a multicommodity transportation network, Transportation Science, 35 (2001), 345-358. Google Scholar

[10]

, "Methods to Find the Cost-Effectiveness of Funding Air Quality Projects,", Tech. Report, (2010).   Google Scholar

[11]

A. Chen and K. Subprasom, Analysis of regulation and policy of private toll roads in a build-operate-transfer scheme under demand uncertainty, Transportation Research Part A: Policy and Practice, 41 (2007), 537-558. Google Scholar

[12]

S.-W. Chiou, Bilevel programming for the continuous transport network design problem, Transportation Research Part B: Methodological, 39 (2005), 361-383. Google Scholar

[13]

J. Dargay, D. Gately and M. Sommer, Vehicle ownership and income growth, worldwide: 1960-2030, The Energy Journal, 28 (2007), 163-190. Google Scholar

[14]

G. de Ceuster, B. van Herbruggen, O. Ivanova, K. Carlier, A. Martino and D. Fiorello, "TREMOVE - Final Report," Tech. Report, Transport Mobility Leuven, 2007. Google Scholar

[15]

E. Deakin, "Sustainable Development and Sustainable Transportation: Strategies for Economic Prosperity Environmental Quality and Equity," Tech. Report 2001'03, Institute of Urban and Regional Development, University of California, Berkeley, 2001. Google Scholar

[16]

S. P. Dirkse and M. C. Ferris, Traffic modeling and variational inequalities using GAMS, in "Operations Research and Decision Aid Methodologies in Traffic and Transportation Management," NATO ASI Series F, Springer-Verlag, (1997), 136-163. Google Scholar

[17]

A. Drud, CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems, Mathematical Programming, 31 (1985), 153-191. doi: 10.1007/BF02591747.  Google Scholar

[18]

, "Major Pollutants from Petrol and Diesel Engines,", Tech. Report, (2010).   Google Scholar

[19]

, "User's Guide to Mobile 6.1 and Mobile 6.2 - Mobile Source Emission Factor Model,", Environmental Protection Agency, (2003).   Google Scholar

[20]

M. C. Ferris, S. P. Dirkse and A. Meeraus, Mathematical programs with equilibrium constraints: Automatic reformulation and solution via constrained optimization, in "Frontiers in Applied General Equilibrium Modeling" (eds. T. J.Kehoe, T. N. Srinivasan and J. Whalley), Cambridge University Press, (2005), 67-93. Google Scholar

[21]

T. L. Friesz, R. L. Tobin, H.-J. Cho and N. J. Mehta, Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints, Mathematical Programming, 48 (1990), 265-284. doi: 10.1007/BF01582259.  Google Scholar

[22]

, "GAMS-A User's Guide,", General Algebraic Modeling System, (2010).   Google Scholar

[23]

Z. Gao and Y. Song, A reserve capacity model of optimal signal control with user-equilibrium route choice, Transportation Research: Part B, 36 (2002), 313-323. Google Scholar

[24]

D. Gkatzoflias, C. Kouridis, L. Ntziachristos and Z. Samaras, "Copert 5 - Computer Programme to Calculate Emissions from Road Transport, User Manual, Version 5.0," European Environment Agency, 2007. Google Scholar

[25]

S. Gokhale and M. Khare, A review of deterministic, stochastic and hybrid vehicular exhaust emission models, International Journal of Transport Management, 2 (2004), 59-74. Google Scholar

[26]

R. L. Guensler and D. Sperling, Congestion pricing and motor vehicle emissions: An initial review, Transportation Research Board Special Report, 242 (1994), 356-379. Google Scholar

[27]

E. Gutt, V. Patton and N. Spencer, "Building on 30 Years of Clean Air Act Success: The Case for Reducing NOx Air Pollution," Tech. Report, Environmental Defense Fund, 2000. Google Scholar

[28]

D. W. Hearn and M. V. Ramana, Solving congestion toll pricing models, in "Equilibrium and Advanced Transportation Modeling" (eds. P. Marcotte and S. Nguyen), Kluwer Academic Publishers, (1998), 109-124. Google Scholar

[29]

C. M. Jeon and A. Amekudzi, Addressing sustainability in transportation systems: Definitions, indicators, and metrics, Journal of Infrastructure Systems, 11 (2005), 31-50. Google Scholar

[30]

O. Johansson-Stenman and T. Sterner, What is the scope for environmental road pricing?, in "Road Pricing, Traffic Congestion and the Environment: Issues of Efficiency and Social Feasibility" (eds. K. J. Button and E. T. Verhoef), Edward Elgar Pub, London, UK, (1998), 150-167. Google Scholar

[31]

M. Ketzel, P. Louka, P. Sahm, E. Guilloteau, J.-F. Sini and N. Moussiopoulos, Intercomparison of numerical urban dispersion models: Part I & II, Water, Air, & Soil Pollution: Focus, 2 (2002), 603-613. Google Scholar

[32]

F. H. Knight, Some fallacies in the interpretation of social cost, The Quarterly Journal of Economics, 38 (1924), 582-606. Google Scholar

[33]

M. Labbe, P. Marcotte and G. Savard, A bilevel model of taxation and its application to optimal highway pricing, Management Science, 44 (1998), 1608-1622. Google Scholar

[34]

S. Lawphongpanich and D. W. Hearn, An MPEC approach to second-best toll pricing, Mathematical Programming, Series B, 101 (2004), 33-55. doi: 10.1007/s10107-004-0536-5.  Google Scholar

[35]

T. Litman, "Well Measured: Developing Indicators for Sustainable and Livable Transport Planning," Tech. Report, Victoria Transport Policy Institute, 2010. Google Scholar

[36]

T. Litman and D. Burwell, Issues in sustainable transportation, International Journal of Global Environmental Issues, 6 (2006), 331-347. Google Scholar

[37]

Z.-Q. Luo, J.-S. Pang, and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, Cambridge, UK, 1996. doi: 10.1017/CBO9780511983658.  Google Scholar

[38]

D. R. Lynam and G. D. Pfeifer, Human health effects of highway-related pollutants, in "Highway Pollution" (eds. R. S. Hamilton and R. M. Harrison), Studies in Environmental Science, 44, Elsevier, Amsterdam, (1991), 259-280. Google Scholar

[39]

T. L. Magnanti and R. T. Wong, Network Design and Transportation Planning: Models and Algorithms, Transportation Science, 18 (1984), 1-55. Google Scholar

[40]

P. Marcotte, Network design problem with congestion effects: A case of bilevel programming, Mathematical Programming, 34 (1986), 142-162. doi: 10.1007/BF01580580.  Google Scholar

[41]

A. Nagurney, Congested urban transportation networks and emission paradoxes, Transportation Research Part D: Transport and Environment, 5 (2000), 145-151. Google Scholar

[42]

A. Nagurney, "Sustainable Transportation Networks," Edward Elgar Publishing, 2000. Google Scholar

[43]

A. Nagurney, Paradoxes in networks with zero emission links: Implications for telecommunications versus transportation, Transportation Research Part D: Transport and Environment, 6 (2001), 283-296. Google Scholar

[44]

M. Patriksson, "The Traffic Assignment Problem: Models and Methods," V.S.P. International Science, Utrecht, The Netherlands, 1994. Google Scholar

[45]

M. Patriksson and R. T. Rockafellar, A mathematical model and descent algorithm for bilevel traffic management, Transportation Science, 36 (2002), 271-291. Google Scholar

[46]

V. Patton, Opportunity NOx, The Environmental Forum, 18 (2001), 30-35. Google Scholar

[47]

A. Pigou, "Wealth and Welfare," Macmillan, London, UK, 1920. Google Scholar

[48]

A. Rahman and R. V. Grol, "SUMMA Final Publishable Report Version 2.0," Tech. Report, RAND Europe, 2005. Google Scholar

[49]

L. Rilett and C. Benedek, Traffic assignment under environmental and equity objectives, Transportation Research Record, 1443 (1994), 92-99. Google Scholar

[50]

J. Rouwendal and E. T. Verhoef, Basic economic principles of road pricing: From theory to applications, Transport Policy, 13 (2006), 106-114. Google Scholar

[51]

Y. Sheffi, "Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods," Prentice Hall, England Cliffs, NJ, 1985. Google Scholar

[52]

S. Sugawara and D. A. Niemeier, How much can vehicle emissions be reduced? Exploratory analysis of an upper boundary using an emissions-optimized trip assignment, Transportation Research Record, 1815 (2002), 29-37. Google Scholar

[53]

L. Sun, Z. Gao and Y. Wang, A Stackelberg game management model of the urban public transport, Journal of Industrial and Management Optimization, 8 (2012), 507-520. doi: 10.3934/jimo.2012.8.507.  Google Scholar

[54]

G. H. Tzeng and C. H. Chen, Multiobjective decision making for traffic assignment, IEEE Transactions on Engineering Management, 40 (1993), 180-187. Google Scholar

[55]

, "World Urbanization Prospects: The 2009 Revision,", Tech. Report, (2009).   Google Scholar

[56]

W. S. Vickrey, Congestion theory and transport investment, American Economic Review, 59 (1969), 251-260. Google Scholar

[57]

J. G. Wardrop, Some theoretical aspects of road traffic research, ICE Proceedings: Engineering Divisions, 1 (1952), 325-378. Google Scholar

[58]

H. Yang and M. G. H. Bell, Models and algorithms for road network design: A review and some new developments, Transport Reviews: A Transnational Transdisciplinary Journal, 18 (1998), 257-278. Google Scholar

[59]

H. Yang and Q. Meng, Highway pricing and capacity choice in a road network under a build-operate-transfer scheme, Transportation Research Part A: Policy and Practice, 34 (2000), 207-222. Google Scholar

[60]

H. Yang and Q. Meng, A note on highway pricing and capacity choice in a road network under a build-operate-transfer scheme, Transportation Research Part A: Policy and Practice, 36 (2002), 659-663. Google Scholar

[61]

Y. Yin and S. Lawphongpanich, Internalizing emission externality on road networks, Transportation Research Part D: Transport and Environment, 11 (2006), 292-301. Google Scholar

[62]

H. Zhang and Z. Gao, Bilevel programming model and solution method for mixed transportation network design problem, Journal of Systems Science and Complexity, 22 (2009), 446-459. doi: 10.1007/s11424-009-9177-3.  Google Scholar

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