A Stackelberg game management model of the urban public transport

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April 2012, 8(2): 507-520. doi: 10.3934/jimo.2012.8.507

A Stackelberg game management model of the urban public transport

Lianju Sun ^1, , Ziyou Gao ^2, and Yiju Wang ^3,

1.	School of Mathematical Sciences, Qufu Normal University, Qufu Shandong, 273165, China
2.	School of Traffic and Transportation, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China
3.	School of Management Science, Qufu Normal University, Rizhao Shandong, 276800

Received January 2011 Revised January 2012 Published April 2012

Abstract
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For the urban public transport management problem, based on analysis of the competition and cooperation relationship among operators on a common network, we establish a generalized Nash equilibrium game first; and then by analyzing the dynamic interaction between manager and operators, we propose a non-cooperative Stackelberg game model in which the manager is the leader and the operators are the followers. To solve the model, we transform it into a variational inequality problem, and the gap function method and the augmented Lagrange algorithm are applied. The given numerical experiments show the efficiency of the proposed model and algorithms.

Keywords: Urban public transport, Nash equilibrium, game theory, augmented Lagrange algorithm., variational inequality.

Mathematics Subject Classification: Primary: 90B20, 91A80; Secondary: 91B5.

Citation: Lianju Sun, Ziyou Gao, Yiju Wang. A Stackelberg game management model of the urban public transport. Journal of Industrial & Management Optimization, 2012, 8 (2) : 507-520. doi: 10.3934/jimo.2012.8.507

References:

[1]	M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Second edition, "Nonlinear Programming: Theory and Algorithms,", Wiley, (1993).
[2]	A. Bensoussan, Points de Nash dans le cas de fonctionnelles quadratiques et jeux différentials linéaires à $N$ personnes,, SIAM Journal Control, 12 (1974), 460. doi: 10.1137/0312037.
[3]	D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods,", Computer Science and Applied Mathematics, (1982).
[4]	T. Boulogne, E. Altman, H. Kameda and O. Pourtallier, Mixed equilibrium (ME) for multi-class routing games,, IEEE Transactions on Automatic Control, 47 (2002), 903. doi: 10.1109/TAC.2002.1008357.
[5]	W.-K. Ching, S.-M. Choi and M. Huang, Optimal service capacity in a multiple-server queueing system: A game theory approach,, Journal of Industrial and Management Optimization, 6 (2010), 73.
[6]	M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,, Mathematical Programming, 53 (1992), 99. doi: 10.1007/BF01585696.
[7]	P. T. Harker, Generalized Nash games and quasi-variational inequalities,, European Journal of Operational Research, 54 (1991), 81. doi: 10.1016/0377-2217(91)90325-P.
[8]	P. T. Harker, Multiple equilibrium behaviors on networks,, Transportation Science, 22 (1988), 39. doi: 10.1287/trsc.22.1.39.
[9]	B.-S. He, H. Yang and C.-S. Zhang, A modified augmented Lagrangian method for a class of monotone variational inequalities,, European Journal of Operational Research, 159 (2004), 35. doi: 10.1016/S0377-2217(03)00385-0.
[10]	D. W. Hearn and M. V. Ramana, Solving congestion toll pricing models,, in, (1998), 109.
[11]	O. Kofi and I. Ugboro, Effective strategic planning in public transit systems,, Transportation Research Part E, 44 (2008), 420. doi: 10.1016/j.tre.2006.10.008.
[12]	R. J. La and V. Anantharam, Optimal routing control: Repeated game approach,, IEEE Transactions on Automatic Control, 47 (2002), 437. doi: 10.1109/9.989076.
[13]	G. Laporte, J. A. Mesa and P. Federico, A game theoretic frame work for the robust railway transit network design problem,, Transportation Research Part B, 44 (2010), 447. doi: 10.1016/j.trb.2009.08.004.
[14]	Z.-Q. Luo, J.-S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints,", Cambridge University Press, (1996). doi: 10.1017/CBO9780511983658.
[15]	Q. Meng, H. Yang and M. G. H. Bell, An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem,, Transportation Research Part B, 35 (2001), 83. doi: 10.1016/S0191-2615(00)00016-3.
[16]	Y. Nie, H. M. Zhang and D. H. Lee, Models and algorithms for the traffic assignment problem with link capacity constraints,, Transportation Research Part B, 38 (2004), 285. doi: 10.1016/S0191-2615(03)00010-9.
[17]	J. S. Pang, The generalized quasi-variational inequality problem,, Mathematics of Operations Research, 7 (1982), 211. doi: 10.1287/moor.7.2.211.
[18]	M. Patriksson and R. T. Rockafellar, A mathematical model and descent algorithm for bilevel traffic management,, Transportation Science, 36 (2002), 271. doi: 10.1287/trsc.36.3.271.7826.
[19]	K. Shimizu, Y. Ishizuka and J. F. Bard, "Non-Differentiable and Two-Level Mathematical Programming,", Kluwer Academic Publishers, (1997).
[20]	L. J. Sun and Z. Y. Gao, An equilibrium model for urban transit assignment based on game theory,, European Journal of Operational Research, 181 (2007), 305. doi: 10.1016/j.ejor.2006.05.028.
[21]	T. Van Vuren, D. Van Vliet and M. Smith, Combined equilibrium in a network with partial route guidance,, in, (1990), 375.
[22]	T. Van Vuren and D. Watling, A multiple user class assignment model for route guidance,, Transportation Research Record, 1306 (1991), 22.
[23]	Q.-Y. Yan, J.-B. Li and J.-L. Zhang, Licensing schemes in Stackelberg model under asymmetric information of product costs,, Journal of Industrial and Management Optimization, 3 (2007), 763. doi: 10.3934/jimo.2007.3.763.
[24]	H. Yang, X. N. Zhang and Q. Meng, Modeling private highways in networks with entry-exit based toll charges,, Transportation Research Part B, 38 (2004), 191. doi: 10.1016/S0191-2615(03)00050-X.
[25]	H. Yang, X. N. Zhang and Q. Meng, Stackelberg games and multiple equilibrium behaviors on networks,, Transportation Research Part B, 41 (2007), 841. doi: 10.1016/j.trb.2007.03.002.
[26]	J. Zhou, W. H. K. Lam and B. Heydecker, The generalized Nash equilibrium model for oligopolistic transit market with elastic demand,, Transportation Research Part B, 39 (2005), 519. doi: 10.1016/j.trb.2004.07.003.
[27]	D. Zhu and P. Marcotte, Modified descent methods for solving the monotone variational inequality problem,, Operations Research Letters, 14 (1993), 111. doi: 10.1016/0167-6377(93)90103-N.

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References:

[1]	M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Second edition, "Nonlinear Programming: Theory and Algorithms,", Wiley, (1993).
[2]	A. Bensoussan, Points de Nash dans le cas de fonctionnelles quadratiques et jeux différentials linéaires à $N$ personnes,, SIAM Journal Control, 12 (1974), 460. doi: 10.1137/0312037.
[3]	D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods,", Computer Science and Applied Mathematics, (1982).
[4]	T. Boulogne, E. Altman, H. Kameda and O. Pourtallier, Mixed equilibrium (ME) for multi-class routing games,, IEEE Transactions on Automatic Control, 47 (2002), 903. doi: 10.1109/TAC.2002.1008357.
[5]	W.-K. Ching, S.-M. Choi and M. Huang, Optimal service capacity in a multiple-server queueing system: A game theory approach,, Journal of Industrial and Management Optimization, 6 (2010), 73.
[6]	M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,, Mathematical Programming, 53 (1992), 99. doi: 10.1007/BF01585696.
[7]	P. T. Harker, Generalized Nash games and quasi-variational inequalities,, European Journal of Operational Research, 54 (1991), 81. doi: 10.1016/0377-2217(91)90325-P.
[8]	P. T. Harker, Multiple equilibrium behaviors on networks,, Transportation Science, 22 (1988), 39. doi: 10.1287/trsc.22.1.39.
[9]	B.-S. He, H. Yang and C.-S. Zhang, A modified augmented Lagrangian method for a class of monotone variational inequalities,, European Journal of Operational Research, 159 (2004), 35. doi: 10.1016/S0377-2217(03)00385-0.
[10]	D. W. Hearn and M. V. Ramana, Solving congestion toll pricing models,, in, (1998), 109.
[11]	O. Kofi and I. Ugboro, Effective strategic planning in public transit systems,, Transportation Research Part E, 44 (2008), 420. doi: 10.1016/j.tre.2006.10.008.
[12]	R. J. La and V. Anantharam, Optimal routing control: Repeated game approach,, IEEE Transactions on Automatic Control, 47 (2002), 437. doi: 10.1109/9.989076.
[13]	G. Laporte, J. A. Mesa and P. Federico, A game theoretic frame work for the robust railway transit network design problem,, Transportation Research Part B, 44 (2010), 447. doi: 10.1016/j.trb.2009.08.004.
[14]	Z.-Q. Luo, J.-S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints,", Cambridge University Press, (1996). doi: 10.1017/CBO9780511983658.
[15]	Q. Meng, H. Yang and M. G. H. Bell, An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem,, Transportation Research Part B, 35 (2001), 83. doi: 10.1016/S0191-2615(00)00016-3.
[16]	Y. Nie, H. M. Zhang and D. H. Lee, Models and algorithms for the traffic assignment problem with link capacity constraints,, Transportation Research Part B, 38 (2004), 285. doi: 10.1016/S0191-2615(03)00010-9.
[17]	J. S. Pang, The generalized quasi-variational inequality problem,, Mathematics of Operations Research, 7 (1982), 211. doi: 10.1287/moor.7.2.211.
[18]	M. Patriksson and R. T. Rockafellar, A mathematical model and descent algorithm for bilevel traffic management,, Transportation Science, 36 (2002), 271. doi: 10.1287/trsc.36.3.271.7826.
[19]	K. Shimizu, Y. Ishizuka and J. F. Bard, "Non-Differentiable and Two-Level Mathematical Programming,", Kluwer Academic Publishers, (1997).
[20]	L. J. Sun and Z. Y. Gao, An equilibrium model for urban transit assignment based on game theory,, European Journal of Operational Research, 181 (2007), 305. doi: 10.1016/j.ejor.2006.05.028.
[21]	T. Van Vuren, D. Van Vliet and M. Smith, Combined equilibrium in a network with partial route guidance,, in, (1990), 375.
[22]	T. Van Vuren and D. Watling, A multiple user class assignment model for route guidance,, Transportation Research Record, 1306 (1991), 22.
[23]	Q.-Y. Yan, J.-B. Li and J.-L. Zhang, Licensing schemes in Stackelberg model under asymmetric information of product costs,, Journal of Industrial and Management Optimization, 3 (2007), 763. doi: 10.3934/jimo.2007.3.763.
[24]	H. Yang, X. N. Zhang and Q. Meng, Modeling private highways in networks with entry-exit based toll charges,, Transportation Research Part B, 38 (2004), 191. doi: 10.1016/S0191-2615(03)00050-X.
[25]	H. Yang, X. N. Zhang and Q. Meng, Stackelberg games and multiple equilibrium behaviors on networks,, Transportation Research Part B, 41 (2007), 841. doi: 10.1016/j.trb.2007.03.002.
[26]	J. Zhou, W. H. K. Lam and B. Heydecker, The generalized Nash equilibrium model for oligopolistic transit market with elastic demand,, Transportation Research Part B, 39 (2005), 519. doi: 10.1016/j.trb.2004.07.003.
[27]	D. Zhu and P. Marcotte, Modified descent methods for solving the monotone variational inequality problem,, Operations Research Letters, 14 (1993), 111. doi: 10.1016/0167-6377(93)90103-N.

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