铁路旅客票价优化问题的相关模型及算法
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摘要
随着公路与航空运输的迅速崛起，现有铁路旅客票价机制已不能适应市场竞争的要求，为此，需要推进铁路旅客票价改革，建立灵活的、能适应市场竞争的铁路旅客票价机制。
本论文在介绍铁路旅客票价制定的相关理论与方法的基础上，分析了应用双层规划理论来制定铁路旅客票价的优点，指出应用双层规划方法来制定铁路旅客票价的突出优势在于铁路客运部门的票价策略能及时考虑到旅客的选择行为，并介绍了在求解双层规划模型常用到的基于灵敏度分析的启发式算法。
本论文主要考虑铁路旅客运输与其它客运方式竞争的情况下，如何制定合适的旅客票价，使铁路客运企业取得最大的经济效益。围绕铁路旅客票价制定优化问题，本论文主要做了以下几方面的工作：
1、提出了在弹性需求条件下铁路旅客票价合理制定的双层规划模型，并给出了将弹性需求模型转换成固定需求模型后的求解双层规划模型的灵敏度分析方法，用一个算例说明模型的具体求解过程，分析了铁路票价变化对铁路客流及铁路客运企业的经济效益的影响。
2、考虑了不同等级旅客列车的票价制定问题，并用两个相关联双层规划模型来描述铁路与其它客运方式竞争及铁路内部不同等级的列车也存在竞争的票价制定问题，模型以用户盈余最大化为目标来制定铁路客运企业总体票价水平，从企业经济效益最大化的角度来制定不同等级列车的票价。给出了求解这两个相关联的双层规划模型的启发式算法，并用具体的算例论证算法的可行性。
3、考虑了列车提速条件下的铁路旅客票价制定问题，分析了列车提速对客流的影响及相应的票价策略，并考虑了票价制定与线路提速方案的优化问题，给出了在资金约束的条件下线路提速方案及铁路旅客票价制定问题综合优化模型及相应的求解算法，并用具体的算例证明算法是可行的。
4、总结了城市间旅客运输与城市道路交通出行费用的区别，在此基础上，提出一个新广义出行费用函数，并根据这个函数用Logit模型分配客流，设计了一个考虑列车出发时刻的铁路旅客票价优化模型。
5、考虑了铁路客运企业与其它客运方式博弈定价问题，用广义Nash均衡模型来描述不同客运方式的企业为取得本企业最大利润的票价博弈问题，并设计了算例验证算法的可行性，分析了不同运输条件下的博弈效果。
In our country, railway passenger-ticket price is determined by government without regarding to the competitive market. It is no longer suitable for the railway transportation to compete with other traffic modes. With the rapid development of the highway and civil aviation, the railway traffic should reform its pricing mechanism to deal with the competition situation. That is to say that the railway department should have the appropriate right to decide its ticket price and the price should reflect the competition.
Based on the theories and methods about railway passenger ticket pricing, the bi-level programming theory is introduced. According to the advantage of bi-level programming, passengers' choice behavior is considered when determining the ticket price. The heuristic sensitivity-based algorithm is introduced to solve the bi-level programming.
This thesis investigated how to determine the ticket price to maximize the railway revenue with the competition of several transportation modes. The main works of this thesis are as follows:
1. A bi-level programming model is proposed to describe the problem of Passenger-Ticket Pricing under the condition of multi-mode transportation with elastic demands. The elastic bi-level programming is transformed into a fixed one, and then a corresponding heuristic sensitivity-based algorithm is presented. A simple numerical example is given to illustrate the applications of the model and its solution.
2. To describe the relationship of passenger-ticket pricing problem between different kinds of passenger trains, a model composed of two related bi-level programming models is proposed. One bi-level programming model aims to decide railway average ticket price optimizing the consumer surplus. The other is to decide railway ticket prices of different passenger trains. The numerical example illustrates that the model is feasible.
3. A bi-level programming model is presented to optimize the railway passenger-ticket pricing and train speeding-up scheme under the condition of a certain fund. A heuristic sensitivity-based algorithm is given to solve the bi-level problem.
4. By analyzing the characters of the railway passenger transportation, a new method is proposed to measure the generalized travel cost. A Logit model is used to describe the passenger's mode-choice behavior with the competition of different transportation mode. The optimal railway passenger-ticket price model is presented by considering departing time of the trains.
5. A generalized Nash equilibrium model is proposed to describe the game theory problem between the railway department and other transportation corporation. And the price management is analyzed by a numerical example.
本论文在介绍铁路旅客票价制定的相关理论与方法的基础上，分析了应用双层规划理论来制定铁路旅客票价的优点，指出应用双层规划方法来制定铁路旅客票价的突出优势在于铁路客运部门的票价策略能及时考虑到旅客的选择行为，并介绍了在求解双层规划模型常用到的基于灵敏度分析的启发式算法。
本论文主要考虑铁路旅客运输与其它客运方式竞争的情况下，如何制定合适的旅客票价，使铁路客运企业取得最大的经济效益。围绕铁路旅客票价制定优化问题，本论文主要做了以下几方面的工作：
1、提出了在弹性需求条件下铁路旅客票价合理制定的双层规划模型，并给出了将弹性需求模型转换成固定需求模型后的求解双层规划模型的灵敏度分析方法，用一个算例说明模型的具体求解过程，分析了铁路票价变化对铁路客流及铁路客运企业的经济效益的影响。
2、考虑了不同等级旅客列车的票价制定问题，并用两个相关联双层规划模型来描述铁路与其它客运方式竞争及铁路内部不同等级的列车也存在竞争的票价制定问题，模型以用户盈余最大化为目标来制定铁路客运企业总体票价水平，从企业经济效益最大化的角度来制定不同等级列车的票价。给出了求解这两个相关联的双层规划模型的启发式算法，并用具体的算例论证算法的可行性。
3、考虑了列车提速条件下的铁路旅客票价制定问题，分析了列车提速对客流的影响及相应的票价策略，并考虑了票价制定与线路提速方案的优化问题，给出了在资金约束的条件下线路提速方案及铁路旅客票价制定问题综合优化模型及相应的求解算法，并用具体的算例证明算法是可行的。
4、总结了城市间旅客运输与城市道路交通出行费用的区别，在此基础上，提出一个新广义出行费用函数，并根据这个函数用Logit模型分配客流，设计了一个考虑列车出发时刻的铁路旅客票价优化模型。
5、考虑了铁路客运企业与其它客运方式博弈定价问题，用广义Nash均衡模型来描述不同客运方式的企业为取得本企业最大利润的票价博弈问题，并设计了算例验证算法的可行性，分析了不同运输条件下的博弈效果。
In our country, railway passenger-ticket price is determined by government without regarding to the competitive market. It is no longer suitable for the railway transportation to compete with other traffic modes. With the rapid development of the highway and civil aviation, the railway traffic should reform its pricing mechanism to deal with the competition situation. That is to say that the railway department should have the appropriate right to decide its ticket price and the price should reflect the competition.
Based on the theories and methods about railway passenger ticket pricing, the bi-level programming theory is introduced. According to the advantage of bi-level programming, passengers' choice behavior is considered when determining the ticket price. The heuristic sensitivity-based algorithm is introduced to solve the bi-level programming.
This thesis investigated how to determine the ticket price to maximize the railway revenue with the competition of several transportation modes. The main works of this thesis are as follows:
1. A bi-level programming model is proposed to describe the problem of Passenger-Ticket Pricing under the condition of multi-mode transportation with elastic demands. The elastic bi-level programming is transformed into a fixed one, and then a corresponding heuristic sensitivity-based algorithm is presented. A simple numerical example is given to illustrate the applications of the model and its solution.
2. To describe the relationship of passenger-ticket pricing problem between different kinds of passenger trains, a model composed of two related bi-level programming models is proposed. One bi-level programming model aims to decide railway average ticket price optimizing the consumer surplus. The other is to decide railway ticket prices of different passenger trains. The numerical example illustrates that the model is feasible.
3. A bi-level programming model is presented to optimize the railway passenger-ticket pricing and train speeding-up scheme under the condition of a certain fund. A heuristic sensitivity-based algorithm is given to solve the bi-level problem.
4. By analyzing the characters of the railway passenger transportation, a new method is proposed to measure the generalized travel cost. A Logit model is used to describe the passenger's mode-choice behavior with the competition of different transportation mode. The optimal railway passenger-ticket price model is presented by considering departing time of the trains.
5. A generalized Nash equilibrium model is proposed to describe the game theory problem between the railway department and other transportation corporation. And the price management is analyzed by a numerical example.
引文
[01] 顾海兵．现行铁路春运的标价制度的批判与重建．价格理论与实践，2003，3：41-43
[02] http://news.xinhuanet.com/newscenter/2002-01/12/content_235130.htm
[03] http://politics.people.com.cn/GB/1026/3711716.html
[04] 中华人民共和国铁道部．铁路客运运价规则．北京：中国铁道出版社，1998
[05] 杜五一．完善我国铁路旅客票价体系的构想．铁路运营技术，1999，5：142-148
[06] 赵新刚，郭树东．铁路客运合理运价水平分析．中国物价，2006，5：28-30
[07] 高自友，四兵锋．市场经济条件下铁路旅客票价系统分析——优化模型与求解算法．北京：中国铁道出版社，2002，1-181
[08] 严作人．运输经济学．北京：人民交通出版社，2003，25-36
[09] 何开祥，夏安桃．我国现行铁路客运票价体系的新构想．中国铁路，2001，6：26-28
[10] 张光远．改进铁路客运价格管理设想．价格理论与实践，2005，9：13-15
[11] 张周堂．也谈铁路客运价格的削峰填谷作用．价格理论与实践，2005，5：25-26
[12] 李传翔．关于我国运价问题的思考．铁道运输与经济，2005，27(7)：10-12
[13] 孙林，汪红飞．铁路春运价格调整的法理辨析．铁道经济研究，2005，2：34-38
[14] 张光远，张冬生．市场供求关系与运价管理改革．综合运输，2004，2：28-32
[15] 文力．国家放松对铁路运价管制的可行性分析．中国流通经济，2003，17(10)：33-36
[16] 张丽．我国铁路运价的体制改革．综合运输，2002，5：19-19
[17] 韩潇．铁路客运运价问题研究．铁道运输与经济，2001，23(7)：9-10
[18] 范文，胡建明．浅谈铁路建立三级运价管理体系．中国铁路，2000，9：36-37
[19] 庞宝玉．深化铁路客运体制改革的实践与思考．铁道经济研究，2000，5：25-27
[20] 国建华，文力．关于放松铁路运价管制问题的研究．铁道学报，2000，22(3)：107-111
[21] 薛卫．改革铁路客票管理体制，强化客票营销．铁路运输与经济，1999，10：22-23
[22] 李伟海．关于建立适应市场的客、货营销新机制．铁路运输与经济，1999，9：11-12
[23] 谢晓凌．日本铁路票价政策对我国铁路的启示．铁路运输与经济，1998，7：4-5
[24] 刘重庆．俄罗斯铁路运价政策．铁道经济研究，2002，1：39-41
[25] 刘重庆．俄罗斯铁路改革：政企正式分离．铁道经济研究，2004，2：40-40
[26] 张冬生．美国铁路运价考察．综合运输，2002，6：32-34
[27] 铁道部资金清算中心．北美铁路运价、成本和清算讲座综述．铁道经济研究，2002，6：8-14
[28] 四兵锋，赵小梅，高自友．综合运输体系下的客运流量分离模型及算法研究．铁道学报，2004，26(6)：14-19
[29] 四兵锋，高自友．综合运输体系下铁路客运市场的优化策略模型及算法．铁道 学报，2005，27(5)：6-10
[30] 任民．铁路客票价格分析及客运运价水平的合理制定．铁道运输与经济，1999，6：22-24
[31] 赵新刚，郭树东．中国铁路运价水平动态调节模型研究．北京交通大学学报(社会科学版)，2006，5(1)：18-22
[32] 胡郁葱，勒文舟，徐建闽．一种考虑旅客需求评估客运票价的经验模型．暨南大学学报(自然科学版)，2001，22(3)：27-31
[33] 何伟，匡敏，胡思继，金福才．基于产品差异化的运输企业博弈分析．交通运输系统工程与信息，2004，4(3)：113-117
[34] Ngostino N, Uumberto C, Francesca G. A Behavioural Choice Model for the Evaluation of Railway Supply and Pricing Policies. Transportation Research A, 1999, 34(85): 395-404
[35] 周晶，徐晏．公共交通网络系统的广义Nash经营博弈模型．系统工程学报， 2001，4：261-267
[36] Ferrari P. Road Pricing and Network Equilibrium. Transportation Research B,1995, 29(5): 357-372
[37] Yang H, Bell M G H. Traffic Restraint, Road Pricing and Network Equilibrium. Transportation Research B, 1997, 31(4): 303-314
[38] 肖伯春，彭杰．中国民航客运机票定价策略研究．四川大学学报，2000，6：10-15
[39] 杨淑伶，张秀昇．价格市场化的博弈分析．商业研究，2002，25(9)：10-13
[40] 四兵锋，高自友．铁路客票价格与客流量之间的灵敏度分析．铁道学报，1999，21(4)：13-16
[41] 四兵锋，高自友．城市间旅客列车的票价与流量的灵敏度分析．北方交通大学学报，2001，25(2)：49-53
[42] 四兵锋，高自友．城市间公路客运的客票票价与其客流量之间的灵敏度分析．中国公路学报，2000，13(2)：91-95
[43] 四兵锋，高自友．合理制定铁路客票价格的优化模型及算法．管理科学学报，2001，4(2)：45-51
[44] 高自友，四兵锋．市场竞争条件下铁路旅客票价制定的模型与算法．交通运输系统工程与信息，2001，1(1)：50-55
[45] 四兵锋，高自友．多模式条件下合理制定旅客票价制定的模型及算法．中国管理科学，2000，8(4)：55-62
[46] 四兵锋．双层规划在交通运输问题中的应用——铁路旅客票价制定问题及O-D需求估计问题的优化模型计算法的研究．[学位论文]．北京：北方交通大学，2001
[47] 四兵锋，高自友．市场竞争条件下的客运价格优化策略模型及算法．交通运输系统工程与信息，2007，7(1)：73-78
[48] 盛昭瀚．主从递阶决策论—Stackelberg问题．北京：科学出版社，1998，23-56
[49] 滕春贤，李智慧．二层规划的理论与应用．北京：科学出版社，2002，84-92
[50] Yang H, Lam W H K. Optimal Road Tolls under Conditions of Queuing and Congestion. Transportation Research B, 1996, 30(5): 319-331
[51] Marcotte P, Marquis G. Efficient Implementation of Heuristics for the Continuous Network Design Problem. Annals of Operations Research, 1992, 34:163-176
[52] 李志纯，谷强，史峰．弹性需求拥挤道路收费的模型和算法研究．交通运输工程学报，2001，1(3)：81-85
[53] Yang H, Yagar S, lida Y, et al. An Algorithm for the Inflow Control Problem on Urban Freeway Networks with User-optimal Flows. Transportation Research B, 1994, 28(2): 123-139
[54] Yang H, Yagar S. Traffic Assignment and Signal Control in Saturated Road Network. Transportation Research A, 1995, 29(2): 125-139
[55] Yang H, Bell M G H. Transport Bilevel Programming Problems: Recent Methodological Advances. Transportation Research B, 2001, 35(1): 1-4
[56] Ben-Ayed O, Boyce D E, Blair C E. A General Bilevel Linear Programming Formulation of The Network Design Problem. Transportation Research B, 1988, 22(4): 311-318
[57] Jeroslow R G. The Polynomial Hierarchy and a Simple Model for Competitive Analysis. Mathematical Programming, 1985, 32(2): 146-164
[58] Ying J Q, Yang H. Sensitivity Analysis of Stochastic User Equilibrium Flows in a Bi-modal Network with Application to Optimal Pricing. Transportation Research B, 2005, 39(9): 769-795
[59] Abdulaal M, LeBlanc L J. Methods for combining modal split and equilibrium assignment models. Transportation Science, 1979, 13(4): 292-314
[60] Chen Y, Florian M. The Nonlinear Bilevel Programming Problem: Formulation, Regularity and Optimality Conditions. Optimization, 1995, 32(1): 193-209
[61] Yang H. Sensitivity Analysis for the Elastic Demand Network Equilibrium Problem with Applications. Transportation Research B, 1997, 31(2): 55-70
[62] Tobin R L, Friesz T L. Sensitivity Analysis for Equilibrium Network Flow. Transportation Science, 1988, 22(4): 242-250
[63] Taplin J H E, Hensher D A, Smith B. Preserving the Symmetry of Estimated Commuter Travel Elasticities. Transportation Research B, 1999, 33(3): 215-232
[64] Bellei G, Gentile G, Papola N. Network Pricing Optimization in Multi-user and Multimodel Context with Elastic Demand. Transportation Research B, 2002, 36(9):779-798
[65] 高自友，宋一凡，四兵锋．城市交通连续平衡网络设计—理论与方法．北京：中国铁道出版社，2000，93-106
[66] Tobin R L, Friesz T L. Sensitivity Analysis for Equilibrium Network Flow. Transportation Science, 1988, 22(4): 242-250
[67] 李观军，帅斌．铁路客运专线成本曲线分析．铁道运输与经济，2006，28(5)：1-3
[68] 程琳，王炜，邵昀泓．社会剩余最大化条件下的道路拥挤收费研究．交通运输系统工程与信息，2003，3：47-50
[69] 刘南．城市交通拥挤定价的经济学原理与应用．土木工程学报(交通工程分册)，2002，2：83-87
[70] Hamdouch Y, Florian M, Hearn D W, et al. Congestion Pricing for Multi-modal Transportation Systems. Transportation Research B, 2007, 41(3): 275-291
[71] 黄海军．拥挤道路使用收费的研究进展和实践难题．中国科学基会，2003，4：198-203
[72] 黄海军．城市交通网络平衡分析—理论与分析．北京：人民交通出版社，1994，13-36
[73] Mardman M. Demand for Rail Travel and the Effects of External Factors. Transportation Research E, 2006, 42(3): 129-148
[74] Yildirim M B, Hearn D W. A First Best Toll Pricing Framework for Variable Demand Traffic Assignment Problems. Transportation Research B, 2005, 39(8): 659-678
[75] Brons M, Pels E, Nijkamp P, et al. Price Elasticities of Demand for Passenger Air Travel: A Meta-Analysis. Journal of Air Transport Management, 2002, 8(3):165-175
[76] Bel G. Changes in Travel Time Across Modes and Its Impact on the Demand for Inter-Urban Rail Travel. Transportation Research E, 1997, 33(1): 43-52
[77] Hamerslag R, Immers B H. Estimation of Trip Matrices: Shortcomings and Possibilities for Improvement. Transportation Research Record, 1988, 12(3): 27-39
[78] Norbert O. Urban Travel Demand Modeling: From Individual Choice to General Equilibrium. A Wiley-lnterscience Publication, John Wiley & Sons, INC, 1994,112-203
[79] Hallefjord A, Jornsten K, Storoy S. Traffic Equilibrium Paradoxes when Travel Demand is Elastic. Asia-Pacific Journal of Operations Research, 1994, 11(1): 41-50
[80] Cree N D, Maher M J. The Continuous Equilibrium Optimal Network Design Problem: A Genetic Approach, In (ed.) Bell M G H. Transportation Network Recent Methodological Advances, Netherlands: Elsevier Ltd, 1998, 163-174
[81] Sheffi Y. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Englewood Cliffs, New Jersey: Prentice-Hall, 1985, 56-131
[82] Dafermos S C. Traffic Equilibrium and Variational Inequalities. Transportation Science, 1980, 14(1): 42-54
[83] Tobin R L, Fiacco A V. Sensitivity Analysis for Variational Inequalities. Journal of Optimization Theory and Application, 1986, 48(1): 191-204
[84] Bard J F. An Algorithm for Solving the General Bi-Level Programming Problem. Mathematics of Operations Research, 1983, 8(2): 260-272
[85] Friesz T L, Tobin R L, Chao H J, et al. Sensitivity Analysis Based Heuristic Algorithms for Mathematical Programs with Variational Inequality Constraints. Mathematical Programming, 1990, 48(2): 265-284
[86] Fiacco A V. Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. New York: Academic Press, 1983, 33-64
[87] LeBlanc L J, Boycc D E. A Bilcvel Programming Algorithm for Exact Solutoion of the Network Design Problem with User-Optimal Flows. Transportation Research B, 1986, 20(3): 259-265
[88] Magnanti T L, Wong R T. Network Design and Transportation Planning: Models and Algorithms. Transportation Science, 1984, 18(1): 1-55
[89] Marcottc P. A Note on a Bilcvel Programming Algorithm by Lcblanc and Boyce. Transportation Research B, 1988, 22(3): 233-236
[90] Yang H. Heuristic Algorithm for the Bilevel Origin-Destination Matrix Estimation Problem. Transportation Research B, 1995, 29(4): 231-242
[91] Yang H, Bell M G H. Models and Algorithms for Road Network Design: A Review and Some New Development. Transport Reviews, 1998, 18(3): 257-278
[92] Hyman G, Mayhew L. Optimizing the Benefits of Urban Road User Charging. Transport Policy, 2002, 9(3): 189-207
[93] Milan J. The European Railway Network. Transport Policy, 2002(9): 179-188
[94] Daniel V. Optimal Pricing in Railway Passenger Transport: Theory and Practice in the Netherlands. Transport Policy, 2002, 9(2): 95-106
[95] Nic Y, Zhang H M, Lee D H. Models and Algorithms for the Traffic Assignment Problem with Link Capacity Constraints. Transportation Research B, 2004, 38(4):285-312
[96] Fcrrari P. Capacity Constraints in Urban Transport Networks. Transportation Research B, 1997, 31(4): 291-300
[97] 程琳，王炜，王欣等．一种求解容量制约下交通网络流模型的新梯度方法．同济大学学报(自然科学版)，2006，34(3)：345-349
[98] 李志纯，谷强，史峰．弹性需求下拥挤道路收费的模型与算法研究．交通运输工程学报，2001，1(3)：81-85
[99] Yang H, Huang H J. Principle of marginal-cost pricing: how does it work in a general road network? Transportation Research A, 1998, 32(1): 45-54
[100] 付连军．铁路提速对民航客运影响的实证分析．郑州航空工业管理学院学报(管理科学版)，2005，23(1)：29-32
[101] Leblanc L J. An Algorithm for the Discrete Network Design Problem. Transportation Science, 1975, 9(3): 183-199
[102] Stairs S. Selecting An Optimal Traffic Network. Journal of Transport Economics and Policy, 1968, 2(2): 218-231
[103] 赵彤．城市交通离散网络设计及相关问题研究．[学位论文]．北京：北方交通大学，2004
[104] 南敬林．京沪通道旅客行为时间价值研究．铁道经济研究，2002，3：36-38
[105] 南敬林．旅客旅行舒适度的需求分析．铁道运输与经济，2005，27(6)：79-81
[106] 李友好，刘晓佳，施其洲．基于网络的运输通道客运流量分配模型．交通运输工程与信息学报，2005，3(2)：57-62
[107] Winston C, Waheshri V. On the Social Desirability of Urban Rail Transit Systems. Journal of Urban Economics. 2007, 61(2): 299-312
[108] Crisalli U. User's Behaviour Simulation of Intercity Rail Service Choices. Simulation Practice and Theory, 1999, 7(3): 233-249
[109] 金一兵．模糊数学在预测车站最高聚集人数中的应用．铁路运输与经济，2004，26(2)：68-70
[110] 何宇强，毛保华，陈绍宽等．铁路客运站旅客最高聚集人数计算方法研究．铁道学报，2006，28(1)：6-11
[111] Bassanini A, La B A, Nastasi A. Allocation of Railroad Capacity under Competition: A Game Theoretic Approach to Track Time Pricing, in (ed.) Gendreau M, Marcotte P. Transportation and Network Analysis: Current Trends. Netherlands: Kluwer Academic Publishers, 2002,1-17
[112] Luce R D. Individual Choice Behavior: A Theoretical Analysis. New York: Wiley and Sons, 1959, 75-90
[113] McFadden D. Conditional Logit Analysis of Qualitative Choice Behavior, in (ed.) Zarembka P. Frontiers in Econometrics, New York: Academic Press, 1974,105-142
[114] 王树盛．Probit模型及其在交通方式分担中的应用研究．ITS通讯，2005，7(4)：48-50
[115] Harker P T, Hong S. Pricing of Track Time in Railroad Operations: An Internal Market Approach. Transportation Research B, 1994, 28(3): 197-212
[116] Sun L J, Gao Z Y. An Equilibrium Model for Urban Transit Assignment Based on Game Theory. European Journal of Operations Research, 2007, 181(1): 305-314
[117] Nagurney A. A Multiclass, Multicriteria Traffic Network Equilibrium Model. Mathematical and Computer Modelling, 2000, 32(3): 393-411
[118] Nagurney A, Dong J. A Multiclass, Multicriteria Traffic Network Equilibrium Model with Elastic Denamd. Transportation Research B, 2002, 36(5): 445-469
[119] Zhou J, Lain W H K, Heydecher B G. The Generalized Nash Equilibrium Model for Oligopolistic Transit Market with Elastic Demand. Transportation Research B, 2005, 39(6): 519-544
[120] 黄园高，周晶．收费公路和公共交通之间的定价博弈分析．东南大学学报(自然科学版)，2004，34(2)：268-273
[121] 李静，范炳全．公交运营规模的博弈分析．城市公共交通，2003，2：9-10
[122] 张维迎．博弈论和信息经济学．上海：上海三联书店和上海人民出版社，1996，135-233
[123] Samuelson L. Evolutionary Games and Equilibrium Selection, Cambridge: The MIT Press, 1997, 26-45
[124] 黄涛．博弈论教程—理论与应用．北京：首都经济贸易大学出版社，2004，1-66
[125] Nash J F. Noncooperative Games. Annals of Mathematics, 1951, 54(2): 286-298
[126] Nash J F. Equilibrium Points in N-Person Games. Proceeding of the National Academy of Sciences, 1950, 36:48-49
[127] Friedman J. A Noncooperative Equilibrium for Supergame. Review of Economic Studies, 1971, 38:1-12
[128] Harker P T, Hong S. Generalized Nash Games and Qusi-Variational Inequalities. European Journal of Operations Research, 1991, 54(1): 81-94
[129]Harker P T. Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: a Survey of Theory, Algorithms and Applications. Mathematical Programming, 1990,48(2): 161-220
[130] Roy A, Dominique M. Competitive Pricing by A Price Leader. Management Science, 1994, 20(7): 809-823
[131]Ichiishi T. Game Theory for Economic Analysis. New York: Academic Press, 1983, 90-150
[132]Pang J S, Yang J M. Parallel Newton's Method for Computing the Nonlinear Complementarity Problems. Mathematical Programming, 1998,42 (3): 407-420
[133] Yao J C. The Generalized Quasi-Variational Inequality Problem with Applications. Journal of Mathematical Analysis and Applications, 1991,158(1): 139-160
[134]Daniele P, Maugeri A. Variational Inequalities and Discrete and Continuum Models of Network Equilibrium Problems. Mathematical and Computer Modelling, 2002, 35(5): 689-708
[02] http://news.xinhuanet.com/newscenter/2002-01/12/content_235130.htm
[03] http://politics.people.com.cn/GB/1026/3711716.html
[04] 中华人民共和国铁道部．铁路客运运价规则．北京：中国铁道出版社，1998
[05] 杜五一．完善我国铁路旅客票价体系的构想．铁路运营技术，1999，5：142-148
[06] 赵新刚，郭树东．铁路客运合理运价水平分析．中国物价，2006，5：28-30
[07] 高自友，四兵锋．市场经济条件下铁路旅客票价系统分析——优化模型与求解算法．北京：中国铁道出版社，2002，1-181
[08] 严作人．运输经济学．北京：人民交通出版社，2003，25-36
[09] 何开祥，夏安桃．我国现行铁路客运票价体系的新构想．中国铁路，2001，6：26-28
[10] 张光远．改进铁路客运价格管理设想．价格理论与实践，2005，9：13-15
[11] 张周堂．也谈铁路客运价格的削峰填谷作用．价格理论与实践，2005，5：25-26
[12] 李传翔．关于我国运价问题的思考．铁道运输与经济，2005，27(7)：10-12
[13] 孙林，汪红飞．铁路春运价格调整的法理辨析．铁道经济研究，2005，2：34-38
[14] 张光远，张冬生．市场供求关系与运价管理改革．综合运输，2004，2：28-32
[15] 文力．国家放松对铁路运价管制的可行性分析．中国流通经济，2003，17(10)：33-36
[16] 张丽．我国铁路运价的体制改革．综合运输，2002，5：19-19
[17] 韩潇．铁路客运运价问题研究．铁道运输与经济，2001，23(7)：9-10
[18] 范文，胡建明．浅谈铁路建立三级运价管理体系．中国铁路，2000，9：36-37
[19] 庞宝玉．深化铁路客运体制改革的实践与思考．铁道经济研究，2000，5：25-27
[20] 国建华，文力．关于放松铁路运价管制问题的研究．铁道学报，2000，22(3)：107-111
[21] 薛卫．改革铁路客票管理体制，强化客票营销．铁路运输与经济，1999，10：22-23
[22] 李伟海．关于建立适应市场的客、货营销新机制．铁路运输与经济，1999，9：11-12
[23] 谢晓凌．日本铁路票价政策对我国铁路的启示．铁路运输与经济，1998，7：4-5
[24] 刘重庆．俄罗斯铁路运价政策．铁道经济研究，2002，1：39-41
[25] 刘重庆．俄罗斯铁路改革：政企正式分离．铁道经济研究，2004，2：40-40
[26] 张冬生．美国铁路运价考察．综合运输，2002，6：32-34
[27] 铁道部资金清算中心．北美铁路运价、成本和清算讲座综述．铁道经济研究，2002，6：8-14
[28] 四兵锋，赵小梅，高自友．综合运输体系下的客运流量分离模型及算法研究．铁道学报，2004，26(6)：14-19
[29] 四兵锋，高自友．综合运输体系下铁路客运市场的优化策略模型及算法．铁道 学报，2005，27(5)：6-10
[30] 任民．铁路客票价格分析及客运运价水平的合理制定．铁道运输与经济，1999，6：22-24
[31] 赵新刚，郭树东．中国铁路运价水平动态调节模型研究．北京交通大学学报(社会科学版)，2006，5(1)：18-22
[32] 胡郁葱，勒文舟，徐建闽．一种考虑旅客需求评估客运票价的经验模型．暨南大学学报(自然科学版)，2001，22(3)：27-31
[33] 何伟，匡敏，胡思继，金福才．基于产品差异化的运输企业博弈分析．交通运输系统工程与信息，2004，4(3)：113-117
[34] Ngostino N, Uumberto C, Francesca G. A Behavioural Choice Model for the Evaluation of Railway Supply and Pricing Policies. Transportation Research A, 1999, 34(85): 395-404
[35] 周晶，徐晏．公共交通网络系统的广义Nash经营博弈模型．系统工程学报， 2001，4：261-267
[36] Ferrari P. Road Pricing and Network Equilibrium. Transportation Research B,1995, 29(5): 357-372
[37] Yang H, Bell M G H. Traffic Restraint, Road Pricing and Network Equilibrium. Transportation Research B, 1997, 31(4): 303-314
[38] 肖伯春，彭杰．中国民航客运机票定价策略研究．四川大学学报，2000，6：10-15
[39] 杨淑伶，张秀昇．价格市场化的博弈分析．商业研究，2002，25(9)：10-13
[40] 四兵锋，高自友．铁路客票价格与客流量之间的灵敏度分析．铁道学报，1999，21(4)：13-16
[41] 四兵锋，高自友．城市间旅客列车的票价与流量的灵敏度分析．北方交通大学学报，2001，25(2)：49-53
[42] 四兵锋，高自友．城市间公路客运的客票票价与其客流量之间的灵敏度分析．中国公路学报，2000，13(2)：91-95
[43] 四兵锋，高自友．合理制定铁路客票价格的优化模型及算法．管理科学学报，2001，4(2)：45-51
[44] 高自友，四兵锋．市场竞争条件下铁路旅客票价制定的模型与算法．交通运输系统工程与信息，2001，1(1)：50-55
[45] 四兵锋，高自友．多模式条件下合理制定旅客票价制定的模型及算法．中国管理科学，2000，8(4)：55-62
[46] 四兵锋．双层规划在交通运输问题中的应用——铁路旅客票价制定问题及O-D需求估计问题的优化模型计算法的研究．[学位论文]．北京：北方交通大学，2001
[47] 四兵锋，高自友．市场竞争条件下的客运价格优化策略模型及算法．交通运输系统工程与信息，2007，7(1)：73-78
[48] 盛昭瀚．主从递阶决策论—Stackelberg问题．北京：科学出版社，1998，23-56
[49] 滕春贤，李智慧．二层规划的理论与应用．北京：科学出版社，2002，84-92
[50] Yang H, Lam W H K. Optimal Road Tolls under Conditions of Queuing and Congestion. Transportation Research B, 1996, 30(5): 319-331
[51] Marcotte P, Marquis G. Efficient Implementation of Heuristics for the Continuous Network Design Problem. Annals of Operations Research, 1992, 34:163-176
[52] 李志纯，谷强，史峰．弹性需求拥挤道路收费的模型和算法研究．交通运输工程学报，2001，1(3)：81-85
[53] Yang H, Yagar S, lida Y, et al. An Algorithm for the Inflow Control Problem on Urban Freeway Networks with User-optimal Flows. Transportation Research B, 1994, 28(2): 123-139
[54] Yang H, Yagar S. Traffic Assignment and Signal Control in Saturated Road Network. Transportation Research A, 1995, 29(2): 125-139
[55] Yang H, Bell M G H. Transport Bilevel Programming Problems: Recent Methodological Advances. Transportation Research B, 2001, 35(1): 1-4
[56] Ben-Ayed O, Boyce D E, Blair C E. A General Bilevel Linear Programming Formulation of The Network Design Problem. Transportation Research B, 1988, 22(4): 311-318
[57] Jeroslow R G. The Polynomial Hierarchy and a Simple Model for Competitive Analysis. Mathematical Programming, 1985, 32(2): 146-164
[58] Ying J Q, Yang H. Sensitivity Analysis of Stochastic User Equilibrium Flows in a Bi-modal Network with Application to Optimal Pricing. Transportation Research B, 2005, 39(9): 769-795
[59] Abdulaal M, LeBlanc L J. Methods for combining modal split and equilibrium assignment models. Transportation Science, 1979, 13(4): 292-314
[60] Chen Y, Florian M. The Nonlinear Bilevel Programming Problem: Formulation, Regularity and Optimality Conditions. Optimization, 1995, 32(1): 193-209
[61] Yang H. Sensitivity Analysis for the Elastic Demand Network Equilibrium Problem with Applications. Transportation Research B, 1997, 31(2): 55-70
[62] Tobin R L, Friesz T L. Sensitivity Analysis for Equilibrium Network Flow. Transportation Science, 1988, 22(4): 242-250
[63] Taplin J H E, Hensher D A, Smith B. Preserving the Symmetry of Estimated Commuter Travel Elasticities. Transportation Research B, 1999, 33(3): 215-232
[64] Bellei G, Gentile G, Papola N. Network Pricing Optimization in Multi-user and Multimodel Context with Elastic Demand. Transportation Research B, 2002, 36(9):779-798
[65] 高自友，宋一凡，四兵锋．城市交通连续平衡网络设计—理论与方法．北京：中国铁道出版社，2000，93-106
[66] Tobin R L, Friesz T L. Sensitivity Analysis for Equilibrium Network Flow. Transportation Science, 1988, 22(4): 242-250
[67] 李观军，帅斌．铁路客运专线成本曲线分析．铁道运输与经济，2006，28(5)：1-3
[68] 程琳，王炜，邵昀泓．社会剩余最大化条件下的道路拥挤收费研究．交通运输系统工程与信息，2003，3：47-50
[69] 刘南．城市交通拥挤定价的经济学原理与应用．土木工程学报(交通工程分册)，2002，2：83-87
[70] Hamdouch Y, Florian M, Hearn D W, et al. Congestion Pricing for Multi-modal Transportation Systems. Transportation Research B, 2007, 41(3): 275-291
[71] 黄海军．拥挤道路使用收费的研究进展和实践难题．中国科学基会，2003，4：198-203
[72] 黄海军．城市交通网络平衡分析—理论与分析．北京：人民交通出版社，1994，13-36
[73] Mardman M. Demand for Rail Travel and the Effects of External Factors. Transportation Research E, 2006, 42(3): 129-148
[74] Yildirim M B, Hearn D W. A First Best Toll Pricing Framework for Variable Demand Traffic Assignment Problems. Transportation Research B, 2005, 39(8): 659-678
[75] Brons M, Pels E, Nijkamp P, et al. Price Elasticities of Demand for Passenger Air Travel: A Meta-Analysis. Journal of Air Transport Management, 2002, 8(3):165-175
[76] Bel G. Changes in Travel Time Across Modes and Its Impact on the Demand for Inter-Urban Rail Travel. Transportation Research E, 1997, 33(1): 43-52
[77] Hamerslag R, Immers B H. Estimation of Trip Matrices: Shortcomings and Possibilities for Improvement. Transportation Research Record, 1988, 12(3): 27-39
[78] Norbert O. Urban Travel Demand Modeling: From Individual Choice to General Equilibrium. A Wiley-lnterscience Publication, John Wiley & Sons, INC, 1994,112-203
[79] Hallefjord A, Jornsten K, Storoy S. Traffic Equilibrium Paradoxes when Travel Demand is Elastic. Asia-Pacific Journal of Operations Research, 1994, 11(1): 41-50
[80] Cree N D, Maher M J. The Continuous Equilibrium Optimal Network Design Problem: A Genetic Approach, In (ed.) Bell M G H. Transportation Network Recent Methodological Advances, Netherlands: Elsevier Ltd, 1998, 163-174
[81] Sheffi Y. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Englewood Cliffs, New Jersey: Prentice-Hall, 1985, 56-131
[82] Dafermos S C. Traffic Equilibrium and Variational Inequalities. Transportation Science, 1980, 14(1): 42-54
[83] Tobin R L, Fiacco A V. Sensitivity Analysis for Variational Inequalities. Journal of Optimization Theory and Application, 1986, 48(1): 191-204
[84] Bard J F. An Algorithm for Solving the General Bi-Level Programming Problem. Mathematics of Operations Research, 1983, 8(2): 260-272
[85] Friesz T L, Tobin R L, Chao H J, et al. Sensitivity Analysis Based Heuristic Algorithms for Mathematical Programs with Variational Inequality Constraints. Mathematical Programming, 1990, 48(2): 265-284
[86] Fiacco A V. Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. New York: Academic Press, 1983, 33-64
[87] LeBlanc L J, Boycc D E. A Bilcvel Programming Algorithm for Exact Solutoion of the Network Design Problem with User-Optimal Flows. Transportation Research B, 1986, 20(3): 259-265
[88] Magnanti T L, Wong R T. Network Design and Transportation Planning: Models and Algorithms. Transportation Science, 1984, 18(1): 1-55
[89] Marcottc P. A Note on a Bilcvel Programming Algorithm by Lcblanc and Boyce. Transportation Research B, 1988, 22(3): 233-236
[90] Yang H. Heuristic Algorithm for the Bilevel Origin-Destination Matrix Estimation Problem. Transportation Research B, 1995, 29(4): 231-242
[91] Yang H, Bell M G H. Models and Algorithms for Road Network Design: A Review and Some New Development. Transport Reviews, 1998, 18(3): 257-278
[92] Hyman G, Mayhew L. Optimizing the Benefits of Urban Road User Charging. Transport Policy, 2002, 9(3): 189-207
[93] Milan J. The European Railway Network. Transport Policy, 2002(9): 179-188
[94] Daniel V. Optimal Pricing in Railway Passenger Transport: Theory and Practice in the Netherlands. Transport Policy, 2002, 9(2): 95-106
[95] Nic Y, Zhang H M, Lee D H. Models and Algorithms for the Traffic Assignment Problem with Link Capacity Constraints. Transportation Research B, 2004, 38(4):285-312
[96] Fcrrari P. Capacity Constraints in Urban Transport Networks. Transportation Research B, 1997, 31(4): 291-300
[97] 程琳，王炜，王欣等．一种求解容量制约下交通网络流模型的新梯度方法．同济大学学报(自然科学版)，2006，34(3)：345-349
[98] 李志纯，谷强，史峰．弹性需求下拥挤道路收费的模型与算法研究．交通运输工程学报，2001，1(3)：81-85
[99] Yang H, Huang H J. Principle of marginal-cost pricing: how does it work in a general road network? Transportation Research A, 1998, 32(1): 45-54
[100] 付连军．铁路提速对民航客运影响的实证分析．郑州航空工业管理学院学报(管理科学版)，2005，23(1)：29-32
[101] Leblanc L J. An Algorithm for the Discrete Network Design Problem. Transportation Science, 1975, 9(3): 183-199
[102] Stairs S. Selecting An Optimal Traffic Network. Journal of Transport Economics and Policy, 1968, 2(2): 218-231
[103] 赵彤．城市交通离散网络设计及相关问题研究．[学位论文]．北京：北方交通大学，2004
[104] 南敬林．京沪通道旅客行为时间价值研究．铁道经济研究，2002，3：36-38
[105] 南敬林．旅客旅行舒适度的需求分析．铁道运输与经济，2005，27(6)：79-81
[106] 李友好，刘晓佳，施其洲．基于网络的运输通道客运流量分配模型．交通运输工程与信息学报，2005，3(2)：57-62
[107] Winston C, Waheshri V. On the Social Desirability of Urban Rail Transit Systems. Journal of Urban Economics. 2007, 61(2): 299-312
[108] Crisalli U. User's Behaviour Simulation of Intercity Rail Service Choices. Simulation Practice and Theory, 1999, 7(3): 233-249
[109] 金一兵．模糊数学在预测车站最高聚集人数中的应用．铁路运输与经济，2004，26(2)：68-70
[110] 何宇强，毛保华，陈绍宽等．铁路客运站旅客最高聚集人数计算方法研究．铁道学报，2006，28(1)：6-11
[111] Bassanini A, La B A, Nastasi A. Allocation of Railroad Capacity under Competition: A Game Theoretic Approach to Track Time Pricing, in (ed.) Gendreau M, Marcotte P. Transportation and Network Analysis: Current Trends. Netherlands: Kluwer Academic Publishers, 2002,1-17
[112] Luce R D. Individual Choice Behavior: A Theoretical Analysis. New York: Wiley and Sons, 1959, 75-90
[113] McFadden D. Conditional Logit Analysis of Qualitative Choice Behavior, in (ed.) Zarembka P. Frontiers in Econometrics, New York: Academic Press, 1974,105-142
[114] 王树盛．Probit模型及其在交通方式分担中的应用研究．ITS通讯，2005，7(4)：48-50
[115] Harker P T, Hong S. Pricing of Track Time in Railroad Operations: An Internal Market Approach. Transportation Research B, 1994, 28(3): 197-212
[116] Sun L J, Gao Z Y. An Equilibrium Model for Urban Transit Assignment Based on Game Theory. European Journal of Operations Research, 2007, 181(1): 305-314
[117] Nagurney A. A Multiclass, Multicriteria Traffic Network Equilibrium Model. Mathematical and Computer Modelling, 2000, 32(3): 393-411
[118] Nagurney A, Dong J. A Multiclass, Multicriteria Traffic Network Equilibrium Model with Elastic Denamd. Transportation Research B, 2002, 36(5): 445-469
[119] Zhou J, Lain W H K, Heydecher B G. The Generalized Nash Equilibrium Model for Oligopolistic Transit Market with Elastic Demand. Transportation Research B, 2005, 39(6): 519-544
[120] 黄园高，周晶．收费公路和公共交通之间的定价博弈分析．东南大学学报(自然科学版)，2004，34(2)：268-273
[121] 李静，范炳全．公交运营规模的博弈分析．城市公共交通，2003，2：9-10
[122] 张维迎．博弈论和信息经济学．上海：上海三联书店和上海人民出版社，1996，135-233
[123] Samuelson L. Evolutionary Games and Equilibrium Selection, Cambridge: The MIT Press, 1997, 26-45
[124] 黄涛．博弈论教程—理论与应用．北京：首都经济贸易大学出版社，2004，1-66
[125] Nash J F. Noncooperative Games. Annals of Mathematics, 1951, 54(2): 286-298
[126] Nash J F. Equilibrium Points in N-Person Games. Proceeding of the National Academy of Sciences, 1950, 36:48-49
[127] Friedman J. A Noncooperative Equilibrium for Supergame. Review of Economic Studies, 1971, 38:1-12
[128] Harker P T, Hong S. Generalized Nash Games and Qusi-Variational Inequalities. European Journal of Operations Research, 1991, 54(1): 81-94
[129]Harker P T. Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: a Survey of Theory, Algorithms and Applications. Mathematical Programming, 1990,48(2): 161-220
[130] Roy A, Dominique M. Competitive Pricing by A Price Leader. Management Science, 1994, 20(7): 809-823
[131]Ichiishi T. Game Theory for Economic Analysis. New York: Academic Press, 1983, 90-150
[132]Pang J S, Yang J M. Parallel Newton's Method for Computing the Nonlinear Complementarity Problems. Mathematical Programming, 1998,42 (3): 407-420
[133] Yao J C. The Generalized Quasi-Variational Inequality Problem with Applications. Journal of Mathematical Analysis and Applications, 1991,158(1): 139-160
[134]Daniele P, Maugeri A. Variational Inequalities and Discrete and Continuum Models of Network Equilibrium Problems. Mathematical and Computer Modelling, 2002, 35(5): 689-708