区间不确定下的交通需求预测
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摘要
交通需求预测是进行交通规划的基础,其预测结果,直接影响着决策者对规划方案的制定。传统交通需求预测方法,往往忽视不确定性,采用确定性的方法进行“点”预测,存在巨大的决策风险。为了提高预测结果的可靠度,降低决策风险,需要考虑交通需求预测中的不确定性。
本论文主要是针对区间不确定下的交通需求预测模型与算法进行研究。首先,对交通需求预测中的不确定性进行了定性分析,明确了主要影响因素。通过对当前主要四种不确定性度量方法的对比分析,论证了采用区间数度量不确定性交通需求的优越性。其次,考虑需求的区间不确定性,构建了区间需求下的交通分配模型。运用区间分析方法,设计了基于minmax后悔法的鲁棒最短路算法,在此基础上构造了区间连续平均法进行模型求解。再次,考虑路段流量的区间不确定性,构建了区间OD反推双层规划模型。将区间分析方法与遗传算法相结合,通过构造能够进行区间运算的遗传算子,设计了区间遗传算法。最后,基于MATLAB平台编制算法程序,利用算例对区间阻抗下的鲁棒最短路算法、区间交通分配法和区间OD反推法进行了分析验证。
本论文通过利用区间分析的方法,对传统交通需求预测模型进行改进,综合考虑了需求的不确定性,其预测的结果将会是一个包含多种情况的区间值,改变过去传统“点”预测存在的巨大决策风险性,增强了预测结果的可靠性,为决策者进行决策提供了更合理的依据。
The travel demand forecast is the base of transportation planning, which directly affects the decision of policy-maker. However, in the traditional travel demand forecasts, uncertainty is usually ignored and deterministic point prediction methods are used which bring forth massive decision risks. In order to raise the reliability of forecasting result, the uncertainty should be taken into account in travel demand forecast.
In this thesis, the problems about travel demand forecast models and algorithms under interval uncertainty are studied. Firstly, the uncertainty of travel demand is qualitatively analyzed to point out the main influence factors clearly. By contrast with four different kinds of uncertainty measurement, the advantage of interval number measuring the travel demand uncertainty is demonstrated. Secondly, a traffic assignment model under interval uncertain demand is established. Using interval analysis method, a robust shortest path algorithm is designed based on minmax regret principle and an interval successive average method is put forward to solve the traffic assignment model. Thirdly, considering uncertainty of interval link flow, an interval bi-level model is established to estimate interval OD matrix. Combining interval analysis with genetic algorithm, an interval genetic algorithm is designed by constructing new genetic operators which can dispose interval operations. Finally, the MATLAB program is developed to solve the models, and a simple example is used to test the above models and algorithms, including robust shortest path algorithm under interval impedance, method of interval traffic assignment and interval OD matrix estimation.
The traditional forecast model of travel demand is improved by using interval analysis and considering demand uncertainty. The forecast result is a interval value which implies a variety of uncertainty and avoids decision-making risk in traditional point prediction ,thus enhancing the reliability of prediction. The policy-maker can make a more reasonable plan based on the results forecasted.
本论文主要是针对区间不确定下的交通需求预测模型与算法进行研究。首先,对交通需求预测中的不确定性进行了定性分析,明确了主要影响因素。通过对当前主要四种不确定性度量方法的对比分析,论证了采用区间数度量不确定性交通需求的优越性。其次,考虑需求的区间不确定性,构建了区间需求下的交通分配模型。运用区间分析方法,设计了基于minmax后悔法的鲁棒最短路算法,在此基础上构造了区间连续平均法进行模型求解。再次,考虑路段流量的区间不确定性,构建了区间OD反推双层规划模型。将区间分析方法与遗传算法相结合,通过构造能够进行区间运算的遗传算子,设计了区间遗传算法。最后,基于MATLAB平台编制算法程序,利用算例对区间阻抗下的鲁棒最短路算法、区间交通分配法和区间OD反推法进行了分析验证。
本论文通过利用区间分析的方法,对传统交通需求预测模型进行改进,综合考虑了需求的不确定性,其预测的结果将会是一个包含多种情况的区间值,改变过去传统“点”预测存在的巨大决策风险性,增强了预测结果的可靠性,为决策者进行决策提供了更合理的依据。
The travel demand forecast is the base of transportation planning, which directly affects the decision of policy-maker. However, in the traditional travel demand forecasts, uncertainty is usually ignored and deterministic point prediction methods are used which bring forth massive decision risks. In order to raise the reliability of forecasting result, the uncertainty should be taken into account in travel demand forecast.
In this thesis, the problems about travel demand forecast models and algorithms under interval uncertainty are studied. Firstly, the uncertainty of travel demand is qualitatively analyzed to point out the main influence factors clearly. By contrast with four different kinds of uncertainty measurement, the advantage of interval number measuring the travel demand uncertainty is demonstrated. Secondly, a traffic assignment model under interval uncertain demand is established. Using interval analysis method, a robust shortest path algorithm is designed based on minmax regret principle and an interval successive average method is put forward to solve the traffic assignment model. Thirdly, considering uncertainty of interval link flow, an interval bi-level model is established to estimate interval OD matrix. Combining interval analysis with genetic algorithm, an interval genetic algorithm is designed by constructing new genetic operators which can dispose interval operations. Finally, the MATLAB program is developed to solve the models, and a simple example is used to test the above models and algorithms, including robust shortest path algorithm under interval impedance, method of interval traffic assignment and interval OD matrix estimation.
The traditional forecast model of travel demand is improved by using interval analysis and considering demand uncertainty. The forecast result is a interval value which implies a variety of uncertainty and avoids decision-making risk in traditional point prediction ,thus enhancing the reliability of prediction. The policy-maker can make a more reasonable plan based on the results forecasted.
引文
[1] Pollack H N. Uncertain Science, Uncertain World[M]. Cambridge University Press, 2003.
[2] Hunt J D, Johnston R, Abraham J E, et al. Comparisons From Sacramento Model Test Bed[J]. Journal of Transportation Research Record. 2001, 1780: 53-63.
[3] Metropolitan Travel Forecasting Current Practice and Future Direction– Special Report 288[R]. Transportation research board, 2007.
[4] C R Rao. Statistics and truth– putting chance to work[M]. World Scientific Publishing. 1999.
[5] Moore R E Interval Analysis[M]. Englewood: Prentice-Hall, 1966.
[6] Moore R E, Kearfott R B, Cloud M J. Introduction to Interval Analysis[M]. Society for Industrial and Applied Mathematics, 2009.
[7] E. R. Hansen. Global optimization using interval analysis: The one-dimensional case[J]. Journal of Optimization Theory and Applications, 1979, 29(3): 331-344
[8] Eldon Hansen. Global optimization using interval analysis the multi-dimensional Case[J]. Numerische Mathematik, 1980 34: 247-270.
[9] Eldon Hansen, Saumyendra Sengupta. Bounding solutions of systems of equations using interval analysis[J]. BIT, 1981, 21: 203-211.
[10] A. Fusiello, A. Benedetti, A. Busti, et al. Globally convergent autocalibration using interval analysis[J]. Universita Di Verona-Dipartimento Di Informatica. 2003.
[11]李政文.区间数不确定多属性决策方法研究:[西南交通大学硕士学位论文].成都:西南交通大学,2010.
[12] Ramon Moore, Weldon Lodwick. Interval analysis and fuzzy set theory[J]. Fuzzy Sets and Systems. 2003, 135: 5-9.
[13] Horner P. Planning under uncertainty[J]. OR/MS Today. 1999, 26: 26-30.
[14] Glynn P, Robinson S. Introduction to Stochastic Programming[M].Springer-Verlag, 1997.
[15] Infanger G. Stochastic Programming– The State of the Art In Honor of George B. Dantzig[M]. Springer Science+Business Media, LLC, 2011.
[16] Liu B. Uncertainty Theory, 3rd Edition[Z]. UTLAB, 2009.
[17]陈洋洋.基于鲁棒优化的电量分配模型的研究:[长沙理工大学硕士学位论文].长沙:长沙理工大学,2010.
[18] Kouvelis P, Yu G Robust Discrete Optimization and Its Applications[M]. KiuwerAcldemic Publishers, 1997.
[19] Marorell S, Carlos S, Sanchez A, et al. Constrained optimization of test intervals using a steady-state genetic algorithm[J]. Reliability Engineering & System Safety. 2000, 67(3): 215-232.
[20] Chen S H, Wu J, Chen Y D Y D. Interval optimization for uncertain structures[J]. Finite Elements in Analysis and Design. 2004, 40(11): 1379-1398.
[21] Averbakh I, Lebedev V. Interval data minmax regret network optimization problems[J]. Discrete Applied Mathematics. 2004, 138(3): 289-301.
[22] Kasperski A, Zielinski P. An approximation algorithm for interval data minmax regret combinatorial optimization problems[J]. Information Processing Letters. 2006, 97(5): 177-180.
[23] Jiang C, Han X, Liu G R. Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval[J]. Computer Methods in Applied Mechanics and Engineering. 2007, 196(49-52): 4791-4800.
[24] Jiang C, Han X, Liu G R, et al. A nonlinear interval number programming method for uncertain optimization problems[J]. European Journal of Operational Research. 2008, 188(1): 1-13.
[25] Kasperki A. Discrete Optimization with Interval Data: Minmax Regret and Fuzzy Approach[M]. Springer-Verlag, 2008.
[26] Harvey G, Deakin E. A Manual of Regional Transportation Modeling Practice for Air Quality Analysis[R]. National Association of Regional Councils, Washington, D.C, 1993.
[27] Zhao Y, Kockelman K M. The propagation of uncertainty through travel demand models: An exploratory analysis[J]. The Annals of Regional Science. 2002(36): 145-163.
[28]王维凤,陈小鸿,林航飞等.高速公路流量预测风险分析[J].长安大学学报(自然科学版). 2003(06).
[29]王维凤,陈小鸿,林航飞等.崇明越江通道流量预测风险分析[J].上海公路. 2003(01).
[30]王正武,罗大庸,谢永彰.交通需求预测中不确定性的传播分析[J].系统工程.2005(07).
[31] Zhou H, Liu W, Li L. Estimation of origin-destination matrix from uncertain link counts using mixed intelligent algorithm[C]. 2008. International conference on intelligent computation technology and automation, ICICTA, 2008: 368-372.
[32]周和平,胡列格,晏克非.基于模糊路段流量的OD反推的不确定规划模型与算法研究[J].铁道科学与工程学报. 2005(05).
[33]周和平,晏克非,胡列格.基于随机模拟的遗传算法在OD反推中的应用研究[J].系统工程. 2003(05).
[34] Waller S T. Evaluation with traffic assignment under demand uncertainty[J]. Transportation Research Record. 2011(1771): 69-74.
[35] Zhang C, Chen X, Sumalee A. Robust Wardrop’s user equilibrium assignment under stochastic demand and supply: Expected residual minimization approach[J]. Transportation Research Part B: Methodological. 2011, 45(3): 534-552.
[36] Ghatee M, Hashemi S M. Traffic assignment model with fuzzy level of travel demand: An efficient algorithm based on quasi-Logit formulas[J]. 2009, 194(2):432-451.
[37]罗京.四阶段交通需求预测模型不确定性传递分析:[长安大学硕士学位论文].西安:长安大学,2009,1-8.
[38]潘艳荣.基于不确定性分析的交通网络设计问题研究:[东南大学博士学位论文].南京:东南大学,2008.
[39]刘宝碇,赵瑞清,王纲.不确定规划及应用[M].清华大学出版社. 2003.
[40]刘宝碇,彭锦.不确定理论研究:回顾与展望[C].南京:Global-Link Publisher. 2005.
[41]张文修,吴伟志.粗糙集理论介绍和研究综述[J].模糊系统与数学. 2000, 14(4).
[42] Schjaer-Jacobsen H. Representation and calculation of economic uncertainties: Intervals, fuzzy numbers, and probabilities[J]. 2002, 78(1): 91-98.
[43] Sengupta A, Pal T K. Fuzzy Preference Ordering of Interval Numbers in Decision Problem[M]. Springer, 2009.
[44]陆华普.交通规划理论与方法[M],北京:清华大学出版社. 2006.
[45]潘艳荣,邓卫.不确定需求条件下的交通平衡问题分析方法研究[J].公路交通科技. 2009,111-115.
[46]路启.旅行时间不确定条件下的交通网络配流分析:[东南大学硕士学位论文].南京:东南大学,2007.
[47] Mudchanatongsuk S, Ordonez F, Liu J. Robust Solutions for Network Design under Transportation Cost and Demand Uncertainty[J]. Journal of the Operational Research Society. 2008, 59(5): 652-662.
[48]邵春福.交通规划原理[M].北京:中国铁道出版社. 2004.
[49]李青.交通分析中的OD特性研究:[同济大学博士学位论文].上海:同济大学,2008,12-18.
[1] Pollack H N. Uncertain Science, Uncertain World[M]. Cambridge University Press, 2003.
[2] Hunt J D, Johnston R, Abraham J E, et al. Comparisons From Sacramento Model Test Bed[J]. Journal of Transportation Research Record. 2001, 1780: 53-63.
[3] Metropolitan Travel Forecasting Current Practice and Future Direction– Special Report 288[R]. Transportation research board, 2007.
[4] C R Rao. Statistics and truth– putting chance to work[M]. World Scientific Publishing. 1999.
[5] Moore R E Interval Analysis[M]. Englewood: Prentice-Hall, 1966.
[6] Moore R E, Kearfott R B, Cloud M J. Introduction to Interval Analysis[M]. Society for Industrial and Applied Mathematics, 2009.
[7] E. R. Hansen. Global optimization using interval analysis: The one-dimensional case[J]. Journal of Optimization Theory and Applications, 1979, 29(3): 331-344
[8] Eldon Hansen. Global optimization using interval analysis the multi-dimensional Case[J]. Numerische Mathematik, 1980 34: 247-270.
[9] Eldon Hansen, Saumyendra Sengupta. Bounding solutions of systems of equations using interval analysis[J]. BIT, 1981, 21: 203-211.
[10] A. Fusiello, A. Benedetti, A. Busti, et al. Globally convergent autocalibration using interval analysis[J]. Universita Di Verona-Dipartimento Di Informatica. 2003.
[11]李政文.区间数不确定多属性决策方法研究:[西南交通大学硕士学位论文].成都:西南交通大学,2010.
[12] Ramon Moore, Weldon Lodwick. Interval analysis and fuzzy set theory[J]. Fuzzy Sets and Systems. 2003, 135: 5-9.
[13] Horner P. Planning under uncertainty[J]. OR/MS Today. 1999, 26: 26-30.
[14] Glynn P, Robinson S. Introduction to Stochastic Programming[M]. Springer-Verlag, 1997.
[15] Infanger G. Stochastic Programming– The State of the Art In Honor of George B. Dantzig[M]. Springer Science+Business Media, LLC, 2011.
[16] Liu B. Uncertainty Theory, 3rd Edition[Z]. UTLAB, 2009.
[17]陈洋洋.基于鲁棒优化的电量分配模型的研究:[长沙理工大学硕士学位论文].长沙:长沙理工大学,2010.
[18] Kouvelis P, Yu G Robust Discrete Optimization and Its Applications[M]. Kiuwer Acldemic Publishers, 1997.
[19] Marorell S, Carlos S, Sanchez A, et al. Constrained optimization of test intervals using a steady-state genetic algorithm[J]. Reliability Engineering & System Safety. 2000, 67(3): 215-232.
[20] Chen S H, Wu J, Chen Y D Y D. Interval optimization for uncertain structures[J]. Finite Elements in Analysis and Design. 2004, 40(11): 1379-1398.
[21] Averbakh I, Lebedev V. Interval data minmax regret network optimization problems[J]. Discrete Applied Mathematics. 2004, 138(3): 289-301.
[22] Kasperski A, Zielinski P. An approximation algorithm for interval data minmax regret combinatorial optimization problems[J]. Information Processing Letters. 2006, 97(5): 177-180.
[23] Jiang C, Han X, Liu G R. Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval[J]. Computer Methods in Applied Mechanics and Engineering. 2007, 196(49-52): 4791-4800.
[24] Jiang C, Han X, Liu G R, et al. A nonlinear interval number programming method for uncertain optimization problems[J]. European Journal of Operational Research. 2008, 188(1): 1-13.
[25] Kasperki A. Discrete Optimization with Interval Data: Minmax Regret and Fuzzy Approach[M]. Springer-Verlag, 2008.
[26] Harvey G, Deakin E. A Manual of Regional Transportation Modeling Practice for Air Quality Analysis[R]. National Association of Regional Councils, Washington, D.C, 1993.
[27] Zhao Y, Kockelman K M. The propagation of uncertainty through travel demand models: An exploratory analysis[J]. The Annals of Regional Science. 2002(36): 145-163.
[28]王维凤,陈小鸿,林航飞等.高速公路流量预测风险分析[J].长安大学学报(自然科学版). 2003(06).
[29]王维凤,陈小鸿,林航飞等.崇明越江通道流量预测风险分析[J].上海公路. 2003(01).
[30]王正武,罗大庸,谢永彰.交通需求预测中不确定性的传播分析[J].系统工程.2005(07).
[31] Zhou H, Liu W, Li L. Estimation of origin-destination matrix from uncertain link counts using mixed intelligent algorithm[C]. 2008. International conference on intelligent computation technology and automation, ICICTA, 2008: 368-372.
[32]周和平,胡列格,晏克非.基于模糊路段流量的OD反推的不确定规划模型与算法研究[J].铁道科学与工程学报. 2005(05).
[33]周和平,晏克非,胡列格.基于随机模拟的遗传算法在OD反推中的应用研究[J].系统工程. 2003(05).
[34] Waller S T. Evaluation with traffic assignment under demand uncertainty[J]. Transportation Research Record. 2011(1771): 69-74.
[35] Zhang C, Chen X, Sumalee A. Robust Wardrop’s user equilibrium assignment under stochastic demand and supply: Expected residual minimization approach[J]. Transportation Research Part B: Methodological. 2011, 45(3): 534-552.
[36] Ghatee M, Hashemi S M. Traffic assignment model with fuzzy level of travel demand: An efficient algorithm based on quasi-Logit formulas[J]. 2009, 194(2):432-451.
[2] Hunt J D, Johnston R, Abraham J E, et al. Comparisons From Sacramento Model Test Bed[J]. Journal of Transportation Research Record. 2001, 1780: 53-63.
[3] Metropolitan Travel Forecasting Current Practice and Future Direction– Special Report 288[R]. Transportation research board, 2007.
[4] C R Rao. Statistics and truth– putting chance to work[M]. World Scientific Publishing. 1999.
[5] Moore R E Interval Analysis[M]. Englewood: Prentice-Hall, 1966.
[6] Moore R E, Kearfott R B, Cloud M J. Introduction to Interval Analysis[M]. Society for Industrial and Applied Mathematics, 2009.
[7] E. R. Hansen. Global optimization using interval analysis: The one-dimensional case[J]. Journal of Optimization Theory and Applications, 1979, 29(3): 331-344
[8] Eldon Hansen. Global optimization using interval analysis the multi-dimensional Case[J]. Numerische Mathematik, 1980 34: 247-270.
[9] Eldon Hansen, Saumyendra Sengupta. Bounding solutions of systems of equations using interval analysis[J]. BIT, 1981, 21: 203-211.
[10] A. Fusiello, A. Benedetti, A. Busti, et al. Globally convergent autocalibration using interval analysis[J]. Universita Di Verona-Dipartimento Di Informatica. 2003.
[11]李政文.区间数不确定多属性决策方法研究:[西南交通大学硕士学位论文].成都:西南交通大学,2010.
[12] Ramon Moore, Weldon Lodwick. Interval analysis and fuzzy set theory[J]. Fuzzy Sets and Systems. 2003, 135: 5-9.
[13] Horner P. Planning under uncertainty[J]. OR/MS Today. 1999, 26: 26-30.
[14] Glynn P, Robinson S. Introduction to Stochastic Programming[M].Springer-Verlag, 1997.
[15] Infanger G. Stochastic Programming– The State of the Art In Honor of George B. Dantzig[M]. Springer Science+Business Media, LLC, 2011.
[16] Liu B. Uncertainty Theory, 3rd Edition[Z]. UTLAB, 2009.
[17]陈洋洋.基于鲁棒优化的电量分配模型的研究:[长沙理工大学硕士学位论文].长沙:长沙理工大学,2010.
[18] Kouvelis P, Yu G Robust Discrete Optimization and Its Applications[M]. KiuwerAcldemic Publishers, 1997.
[19] Marorell S, Carlos S, Sanchez A, et al. Constrained optimization of test intervals using a steady-state genetic algorithm[J]. Reliability Engineering & System Safety. 2000, 67(3): 215-232.
[20] Chen S H, Wu J, Chen Y D Y D. Interval optimization for uncertain structures[J]. Finite Elements in Analysis and Design. 2004, 40(11): 1379-1398.
[21] Averbakh I, Lebedev V. Interval data minmax regret network optimization problems[J]. Discrete Applied Mathematics. 2004, 138(3): 289-301.
[22] Kasperski A, Zielinski P. An approximation algorithm for interval data minmax regret combinatorial optimization problems[J]. Information Processing Letters. 2006, 97(5): 177-180.
[23] Jiang C, Han X, Liu G R. Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval[J]. Computer Methods in Applied Mechanics and Engineering. 2007, 196(49-52): 4791-4800.
[24] Jiang C, Han X, Liu G R, et al. A nonlinear interval number programming method for uncertain optimization problems[J]. European Journal of Operational Research. 2008, 188(1): 1-13.
[25] Kasperki A. Discrete Optimization with Interval Data: Minmax Regret and Fuzzy Approach[M]. Springer-Verlag, 2008.
[26] Harvey G, Deakin E. A Manual of Regional Transportation Modeling Practice for Air Quality Analysis[R]. National Association of Regional Councils, Washington, D.C, 1993.
[27] Zhao Y, Kockelman K M. The propagation of uncertainty through travel demand models: An exploratory analysis[J]. The Annals of Regional Science. 2002(36): 145-163.
[28]王维凤,陈小鸿,林航飞等.高速公路流量预测风险分析[J].长安大学学报(自然科学版). 2003(06).
[29]王维凤,陈小鸿,林航飞等.崇明越江通道流量预测风险分析[J].上海公路. 2003(01).
[30]王正武,罗大庸,谢永彰.交通需求预测中不确定性的传播分析[J].系统工程.2005(07).
[31] Zhou H, Liu W, Li L. Estimation of origin-destination matrix from uncertain link counts using mixed intelligent algorithm[C]. 2008. International conference on intelligent computation technology and automation, ICICTA, 2008: 368-372.
[32]周和平,胡列格,晏克非.基于模糊路段流量的OD反推的不确定规划模型与算法研究[J].铁道科学与工程学报. 2005(05).
[33]周和平,晏克非,胡列格.基于随机模拟的遗传算法在OD反推中的应用研究[J].系统工程. 2003(05).
[34] Waller S T. Evaluation with traffic assignment under demand uncertainty[J]. Transportation Research Record. 2011(1771): 69-74.
[35] Zhang C, Chen X, Sumalee A. Robust Wardrop’s user equilibrium assignment under stochastic demand and supply: Expected residual minimization approach[J]. Transportation Research Part B: Methodological. 2011, 45(3): 534-552.
[36] Ghatee M, Hashemi S M. Traffic assignment model with fuzzy level of travel demand: An efficient algorithm based on quasi-Logit formulas[J]. 2009, 194(2):432-451.
[37]罗京.四阶段交通需求预测模型不确定性传递分析:[长安大学硕士学位论文].西安:长安大学,2009,1-8.
[38]潘艳荣.基于不确定性分析的交通网络设计问题研究:[东南大学博士学位论文].南京:东南大学,2008.
[39]刘宝碇,赵瑞清,王纲.不确定规划及应用[M].清华大学出版社. 2003.
[40]刘宝碇,彭锦.不确定理论研究:回顾与展望[C].南京:Global-Link Publisher. 2005.
[41]张文修,吴伟志.粗糙集理论介绍和研究综述[J].模糊系统与数学. 2000, 14(4).
[42] Schjaer-Jacobsen H. Representation and calculation of economic uncertainties: Intervals, fuzzy numbers, and probabilities[J]. 2002, 78(1): 91-98.
[43] Sengupta A, Pal T K. Fuzzy Preference Ordering of Interval Numbers in Decision Problem[M]. Springer, 2009.
[44]陆华普.交通规划理论与方法[M],北京:清华大学出版社. 2006.
[45]潘艳荣,邓卫.不确定需求条件下的交通平衡问题分析方法研究[J].公路交通科技. 2009,111-115.
[46]路启.旅行时间不确定条件下的交通网络配流分析:[东南大学硕士学位论文].南京:东南大学,2007.
[47] Mudchanatongsuk S, Ordonez F, Liu J. Robust Solutions for Network Design under Transportation Cost and Demand Uncertainty[J]. Journal of the Operational Research Society. 2008, 59(5): 652-662.
[48]邵春福.交通规划原理[M].北京:中国铁道出版社. 2004.
[49]李青.交通分析中的OD特性研究:[同济大学博士学位论文].上海:同济大学,2008,12-18.
[1] Pollack H N. Uncertain Science, Uncertain World[M]. Cambridge University Press, 2003.
[2] Hunt J D, Johnston R, Abraham J E, et al. Comparisons From Sacramento Model Test Bed[J]. Journal of Transportation Research Record. 2001, 1780: 53-63.
[3] Metropolitan Travel Forecasting Current Practice and Future Direction– Special Report 288[R]. Transportation research board, 2007.
[4] C R Rao. Statistics and truth– putting chance to work[M]. World Scientific Publishing. 1999.
[5] Moore R E Interval Analysis[M]. Englewood: Prentice-Hall, 1966.
[6] Moore R E, Kearfott R B, Cloud M J. Introduction to Interval Analysis[M]. Society for Industrial and Applied Mathematics, 2009.
[7] E. R. Hansen. Global optimization using interval analysis: The one-dimensional case[J]. Journal of Optimization Theory and Applications, 1979, 29(3): 331-344
[8] Eldon Hansen. Global optimization using interval analysis the multi-dimensional Case[J]. Numerische Mathematik, 1980 34: 247-270.
[9] Eldon Hansen, Saumyendra Sengupta. Bounding solutions of systems of equations using interval analysis[J]. BIT, 1981, 21: 203-211.
[10] A. Fusiello, A. Benedetti, A. Busti, et al. Globally convergent autocalibration using interval analysis[J]. Universita Di Verona-Dipartimento Di Informatica. 2003.
[11]李政文.区间数不确定多属性决策方法研究:[西南交通大学硕士学位论文].成都:西南交通大学,2010.
[12] Ramon Moore, Weldon Lodwick. Interval analysis and fuzzy set theory[J]. Fuzzy Sets and Systems. 2003, 135: 5-9.
[13] Horner P. Planning under uncertainty[J]. OR/MS Today. 1999, 26: 26-30.
[14] Glynn P, Robinson S. Introduction to Stochastic Programming[M]. Springer-Verlag, 1997.
[15] Infanger G. Stochastic Programming– The State of the Art In Honor of George B. Dantzig[M]. Springer Science+Business Media, LLC, 2011.
[16] Liu B. Uncertainty Theory, 3rd Edition[Z]. UTLAB, 2009.
[17]陈洋洋.基于鲁棒优化的电量分配模型的研究:[长沙理工大学硕士学位论文].长沙:长沙理工大学,2010.
[18] Kouvelis P, Yu G Robust Discrete Optimization and Its Applications[M]. Kiuwer Acldemic Publishers, 1997.
[19] Marorell S, Carlos S, Sanchez A, et al. Constrained optimization of test intervals using a steady-state genetic algorithm[J]. Reliability Engineering & System Safety. 2000, 67(3): 215-232.
[20] Chen S H, Wu J, Chen Y D Y D. Interval optimization for uncertain structures[J]. Finite Elements in Analysis and Design. 2004, 40(11): 1379-1398.
[21] Averbakh I, Lebedev V. Interval data minmax regret network optimization problems[J]. Discrete Applied Mathematics. 2004, 138(3): 289-301.
[22] Kasperski A, Zielinski P. An approximation algorithm for interval data minmax regret combinatorial optimization problems[J]. Information Processing Letters. 2006, 97(5): 177-180.
[23] Jiang C, Han X, Liu G R. Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval[J]. Computer Methods in Applied Mechanics and Engineering. 2007, 196(49-52): 4791-4800.
[24] Jiang C, Han X, Liu G R, et al. A nonlinear interval number programming method for uncertain optimization problems[J]. European Journal of Operational Research. 2008, 188(1): 1-13.
[25] Kasperki A. Discrete Optimization with Interval Data: Minmax Regret and Fuzzy Approach[M]. Springer-Verlag, 2008.
[26] Harvey G, Deakin E. A Manual of Regional Transportation Modeling Practice for Air Quality Analysis[R]. National Association of Regional Councils, Washington, D.C, 1993.
[27] Zhao Y, Kockelman K M. The propagation of uncertainty through travel demand models: An exploratory analysis[J]. The Annals of Regional Science. 2002(36): 145-163.
[28]王维凤,陈小鸿,林航飞等.高速公路流量预测风险分析[J].长安大学学报(自然科学版). 2003(06).
[29]王维凤,陈小鸿,林航飞等.崇明越江通道流量预测风险分析[J].上海公路. 2003(01).
[30]王正武,罗大庸,谢永彰.交通需求预测中不确定性的传播分析[J].系统工程.2005(07).
[31] Zhou H, Liu W, Li L. Estimation of origin-destination matrix from uncertain link counts using mixed intelligent algorithm[C]. 2008. International conference on intelligent computation technology and automation, ICICTA, 2008: 368-372.
[32]周和平,胡列格,晏克非.基于模糊路段流量的OD反推的不确定规划模型与算法研究[J].铁道科学与工程学报. 2005(05).
[33]周和平,晏克非,胡列格.基于随机模拟的遗传算法在OD反推中的应用研究[J].系统工程. 2003(05).
[34] Waller S T. Evaluation with traffic assignment under demand uncertainty[J]. Transportation Research Record. 2011(1771): 69-74.
[35] Zhang C, Chen X, Sumalee A. Robust Wardrop’s user equilibrium assignment under stochastic demand and supply: Expected residual minimization approach[J]. Transportation Research Part B: Methodological. 2011, 45(3): 534-552.
[36] Ghatee M, Hashemi S M. Traffic assignment model with fuzzy level of travel demand: An efficient algorithm based on quasi-Logit formulas[J]. 2009, 194(2):432-451.