基于可信性理论的F-GERT网络模型及其应用研究
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摘要
基于经典概率描述的GERT网络理论已经相对比较成熟,其应用范围十分广泛,几乎涵盖了包括资源规划、应急计划、地震分析、产品开发试制及改进计划等许多行业和领域。它是处理经典的概率网络关系的强有力工具。然而,现实世界中除了概率的不确定性以外,还存在着大量的非概率不确定性问题,如模糊不确定性和灰色不确定性等。模糊现象几乎渗透到了所有的科学和技术领域,可信性理论是完整地研究模糊性的理论。因此,有效的把GERT网络技术与可信性理论进行结合,建立基于可信性理论的模糊GERT网络技术(简称F-GERT)既有重要的理论价值,又具有潜在的应用前景。
本文首次将GERT网络研究的范围从传统的随机环境拓展至模糊环境,将可信性理论引入了GERT网络领域。针对含有模糊信息参量的随机网络,提出了一种新的考虑模糊信息的F-GERT网络的建模技术,构造了F-GERT网络的矩、矩母函数,对其4个方面的重要的数学性质做了详细的讨论;在此基础上,利用信号流图原理研究了F-GERT网络的仿真算法。
然后,对含有模糊信息的F-GERT网络要素重复执行次数的模型构建问题进行了研究,定义了反馈网络的基本单元、多重反馈网络、多重反馈递进网络、多重反馈并进网络、多重反馈混合网络以及模糊反馈网络,并对模型网络简化的规则做了研究;在F-GERT网络的重复执行次数的求解思路的基础上,对重复执行次数的F-GERT网络的执行次数算法、条件矩母函数、过程到达时间进行了研究。
最后,用信号流图的拓扑方程确定F-GERT网络的等价模糊传递函数和等价模糊概率,根据WF函数的定义,将等价模糊传递函数转为等价模糊矩母函数。通过等价模糊矩母函数逆向推导F-GERT网络的各项基本参数,基于模糊信息的F-GERT网络模型设计了结合模糊模拟和遗传算法的混合智能算法,并通过数值试验验证了算法的有效性。
The GERT network based on classical probability theory has been relatively mature, and its scope of application is very extensive, including almost resource planning, contingency planning, seismic analysis, product development, trial and improvement plans, and many other industries and fields. However, there is also a large number of non-probabilistic uncertainty outside of the probability of uncertainty, such as the fuzzy uncertainty and grey uncertainty. Fuzziness infiltrate into almost all fields of science and technology, and credibility theory is a complete study of the theory of ambiguity. Thus, it has important theoretical value and potential applications to combine the GERT network technology with the credibility theory effectively and establish credibility based on the theory of GERT network technology (referred to F-GERT) .
It is the first time extending GERT network study range from the traditional random environment to fuzzy environment. In view of random network with the fuzzy information parameter, a new F-GERT network modeling technique which considers the fuzzy information is proposed in this thesis. Moreover, the moments and moment generating function of F-GERT network are constructed and the important mathematical nature of its four aspects is discussed in detail. On this basis, the simulation algorithm of F-GERT network is studied using signal flow graph principle.
Then, it studied the moedl of fuzzy information of the F-GERT network with the number of times repeated questions. The thesis has defined the basic unit of feedback network, multi-feedback network, multi-feedback progressive network, multi-hand feedback network, multi-feedback hybrid networks and fuzzy feedback network, and examined the rules of the network model simplified. On this basis, the number of times to repeat the implementation of ideas for solving based on the number of pairs of repeated F-GERT network, the implementation of the number of algorithms, conditional moment generating function, the process arrival time were studied.
Finally, fuzzy equivalent transfer function and equivalent to fuzzy probabilit were determined by the signal flow graph of the topological equations to F-GERT networks. The transfer function was equivalent to fuzzy moment generating function according to the definition of WF function. It anti-derivated the basic parameters of F-GERT network by the equivalent fuzzy moment generating function and designed hybrid intelligent algorithm based on fuzzy simulation and genetic algorithm of F-GERT network model.Numerical tests were given in detail . Through simulating we know that the effect is good.
本文首次将GERT网络研究的范围从传统的随机环境拓展至模糊环境,将可信性理论引入了GERT网络领域。针对含有模糊信息参量的随机网络,提出了一种新的考虑模糊信息的F-GERT网络的建模技术,构造了F-GERT网络的矩、矩母函数,对其4个方面的重要的数学性质做了详细的讨论;在此基础上,利用信号流图原理研究了F-GERT网络的仿真算法。
然后,对含有模糊信息的F-GERT网络要素重复执行次数的模型构建问题进行了研究,定义了反馈网络的基本单元、多重反馈网络、多重反馈递进网络、多重反馈并进网络、多重反馈混合网络以及模糊反馈网络,并对模型网络简化的规则做了研究;在F-GERT网络的重复执行次数的求解思路的基础上,对重复执行次数的F-GERT网络的执行次数算法、条件矩母函数、过程到达时间进行了研究。
最后,用信号流图的拓扑方程确定F-GERT网络的等价模糊传递函数和等价模糊概率,根据WF函数的定义,将等价模糊传递函数转为等价模糊矩母函数。通过等价模糊矩母函数逆向推导F-GERT网络的各项基本参数,基于模糊信息的F-GERT网络模型设计了结合模糊模拟和遗传算法的混合智能算法,并通过数值试验验证了算法的有效性。
The GERT network based on classical probability theory has been relatively mature, and its scope of application is very extensive, including almost resource planning, contingency planning, seismic analysis, product development, trial and improvement plans, and many other industries and fields. However, there is also a large number of non-probabilistic uncertainty outside of the probability of uncertainty, such as the fuzzy uncertainty and grey uncertainty. Fuzziness infiltrate into almost all fields of science and technology, and credibility theory is a complete study of the theory of ambiguity. Thus, it has important theoretical value and potential applications to combine the GERT network technology with the credibility theory effectively and establish credibility based on the theory of GERT network technology (referred to F-GERT) .
It is the first time extending GERT network study range from the traditional random environment to fuzzy environment. In view of random network with the fuzzy information parameter, a new F-GERT network modeling technique which considers the fuzzy information is proposed in this thesis. Moreover, the moments and moment generating function of F-GERT network are constructed and the important mathematical nature of its four aspects is discussed in detail. On this basis, the simulation algorithm of F-GERT network is studied using signal flow graph principle.
Then, it studied the moedl of fuzzy information of the F-GERT network with the number of times repeated questions. The thesis has defined the basic unit of feedback network, multi-feedback network, multi-feedback progressive network, multi-hand feedback network, multi-feedback hybrid networks and fuzzy feedback network, and examined the rules of the network model simplified. On this basis, the number of times to repeat the implementation of ideas for solving based on the number of pairs of repeated F-GERT network, the implementation of the number of algorithms, conditional moment generating function, the process arrival time were studied.
Finally, fuzzy equivalent transfer function and equivalent to fuzzy probabilit were determined by the signal flow graph of the topological equations to F-GERT networks. The transfer function was equivalent to fuzzy moment generating function according to the definition of WF function. It anti-derivated the basic parameters of F-GERT network by the equivalent fuzzy moment generating function and designed hybrid intelligent algorithm based on fuzzy simulation and genetic algorithm of F-GERT network model.Numerical tests were given in detail . Through simulating we know that the effect is good.
引文
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[3]方志耕,杨保华,陆志鹏等.基于Bayes推理的灾害演化GERT网络模型研究[J].中国管理科学,2009,17(2):102~107.
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[5]何正文,徐渝,朱少英等.基于GERT模型的新产品研发项目周期仿真分析[J].系统工程,2003,21(2):92~97.
[6]符志民.基于随机网络模拟的航天项目风险分析和评估[J].系统工程与电子技术,2007,29(3):376~377.
[7]乔立红,郭建飞.并行产品开发随机网络计划与仿真方法[J].计算机集成制造系统,2008,14(12):2354~2360.
[8]徐哲,吴瑾瑾,姚李刚.基于GERT仿真的武器装备技术风险量化评估模型[J].系统仿真学报,2008,20(7):1655~1644.
[9]郑轶松,齐二石,郑晓东.基于图示评审技术GERT的高科技产品开发研究[J].系统工程,2005,23(11):112~115.
[10]方志耕,龚正,黄西林.公路军事交通运输勤务综合演习项目GERT网络模型研究与分析[J].系统工程理论与实践,2000,4(4):132~135.
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[12]邱万国,徐国英,邢俊文.进度风险的随机网络建模与仿真分析[J].装甲兵工程学院学报,2005,19(1):48~51.
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[14]谢家平,任毅,赵忠.装配式产品拆卸的随机网络模型研究[J].管理学报,2007,4(2):174~179.
[15]张建强,张涛,郭波.基于嵌入Petri网的GERTS维修保障流程仿真模型研究[J].系统仿真学报,2004,16(8):1641~1644.
[16]邓堃，吴静，柳世．基于GSPN的GERTS仿真模型研究[J].计算机仿真,2007,24(12):32~36.
[17]阮爱清,徐霄峰.区域产业集群形成与发展的GERT网络研究[J].科技进步与对策,2009,26(5):31~34.
[18]李强,张静,张耀坦.蒙特卡洛仿真技术在工程进度计划中的应用[J].长江大学学报(自科版)理工卷,2007,4(2):62~65.
[19]阮爱清,刘思峰.灰色GERT网络及基于顾客需求的灰数估计精度[J].系统工程,2007,25(12):100~104.
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[23]Phillips, D.T. .Fundamentals of Network Analysis[M].Prentice-Hall, inc., 1981.
[24]Banks, J.and Carson, J.S. .Discrete-Event System Simulation[M].Prentice-Hall, inc., 1984.
[25]Law, A.M.and Kelton, W.D. .Simulation Modeling and Analysis[M].McGraw-Hill Book Co., 1982.
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[27]Pritsker, A.A.B.and Whitehouse, G.E. .GERT:Graphical Evaluation and Review Technique, Part 2[J].Probabilistic and Industuial Engineering Applications,The Journal of Industrial Engineering, Vol.17, No.6, 1996.
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[29] L A.Zadeh.Fuzzy sets[J].Information and Control,Vol.8, 1965, 338~353.
[30] A.Kaufmann.Introduction to the Theory of Fuzzy Subsets[M] .Academic Press, New York, VoI.I, 1975.
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[32] D.Dubois and H.Prade.Possibility Theory: An Approach to Computerized Processing of Uncertainty[M].Plenum, New York, 1988.
[33] Liu B, Liu Y.Expected value of fuzzy variable and fuzzy expected value models[J].IEEE Transactions on Fuzzy Systems, 2002, 10(4):445~450.
[34] Li X, Liu B.A.sufficient and necessary condition for credibility measures[J].International Journalof Uncertainty, Fuzziness&Knowledge-Based Systems, 2006, 14(5):527~535.
[35] Liu B.Theory and Practice of Uncertain Programming[M].Heidelberg:Physica-Verlag, 2002.
[36] Li X, Liu B.Moment estimation theorems for various types of uncertain variable[M].Technical Report, 2008.
[37]Li X, Liu B.On distance between fuzzy variables[J].Journal of Intelligent and Fuzzy Systems, In press.
[38]Li P, Liu B.Entropy of credibility distributions for fuzzy variables[J].IEEE Transactions on Fuzzy Systems, 2008, 16(1):123~129.
[39]Li X, Liu B.Maximum entropy principle for fuzzy variables[J].International Journal of Uncertainty, Fuzziness&Knowledge-Based Systems, 2007, 15(supp 2):43~52.
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[43]Puri M L, Ralescu D A.Fuzzy random variables[J].Journal of Mathematical Analysis and Applications, 1986, 114:409~422.
[44]Kruse R, Meyer K D.Statistics with Vague Data[M].Dordrecht:D.Reidel Publishing Company, 1987.
[45]Liu Y, Liu B.Fuzzy random variables:A scalar expected value operator[J].Fuzzy Optimization and Decision Making, 2003, 2(2):143~160.
[46]Liu B.Fuzzy random chance-constrained programming[J].IEEE Transactions on Fuzzy Systems, 2001, 9(5):713~720.
[47]Liu B.Fuzzy random dependent-chance programming[J].IEEE Transactions on Fuzzy Systems, 2001, 9(5):721~726.
[48]Liu Y, Liu B.On minimum-risk problems in fuzzy random decision systems[J].Computers&Operations Research, 2005, 32(2):257~283.
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[2]刘思峰,万寿庆,陆志鹏.复杂交通网络中救援点与事故点间的路段重要性评价模型研究[J].中国管理科学,2009,17(1):119~2009.
[3]方志耕,杨保华,陆志鹏等.基于Bayes推理的灾害演化GERT网络模型研究[J].中国管理科学,2009,17(2):102~107.
[4]张杨.不确定环境下城市交通中车辆路径选择研究,[博士学位论文].四川:西南交通大学,2003.
[5]何正文,徐渝,朱少英等.基于GERT模型的新产品研发项目周期仿真分析[J].系统工程,2003,21(2):92~97.
[6]符志民.基于随机网络模拟的航天项目风险分析和评估[J].系统工程与电子技术,2007,29(3):376~377.
[7]乔立红,郭建飞.并行产品开发随机网络计划与仿真方法[J].计算机集成制造系统,2008,14(12):2354~2360.
[8]徐哲,吴瑾瑾,姚李刚.基于GERT仿真的武器装备技术风险量化评估模型[J].系统仿真学报,2008,20(7):1655~1644.
[9]郑轶松,齐二石,郑晓东.基于图示评审技术GERT的高科技产品开发研究[J].系统工程,2005,23(11):112~115.
[10]方志耕,龚正,黄西林.公路军事交通运输勤务综合演习项目GERT网络模型研究与分析[J].系统工程理论与实践,2000,4(4):132~135.
[11]李存斌,王恪铖.网络计划项目风险元传递解析模型研究[J].中国管理科学,2007,15(3):108~113.
[12]邱万国,徐国英,邢俊文.进度风险的随机网络建模与仿真分析[J].装甲兵工程学院学报,2005,19(1):48~51.
[13]吴艳霞,胡海青,申团营.企业技术创新项目群风险与周期评估的组合GERT网络模型研究[J].管理工程学报,2007,21(2):55~57.
[14]谢家平,任毅,赵忠.装配式产品拆卸的随机网络模型研究[J].管理学报,2007,4(2):174~179.
[15]张建强,张涛,郭波.基于嵌入Petri网的GERTS维修保障流程仿真模型研究[J].系统仿真学报,2004,16(8):1641~1644.
[16]邓堃，吴静，柳世．基于GSPN的GERTS仿真模型研究[J].计算机仿真,2007,24(12):32~36.
[17]阮爱清,徐霄峰.区域产业集群形成与发展的GERT网络研究[J].科技进步与对策,2009,26(5):31~34.
[18]李强,张静,张耀坦.蒙特卡洛仿真技术在工程进度计划中的应用[J].长江大学学报(自科版)理工卷,2007,4(2):62~65.
[19]阮爱清,刘思峰.灰色GERT网络及基于顾客需求的灰数估计精度[J].系统工程,2007,25(12):100~104.
[20] Whitehouse, G.E.Systems Analysis and Design Using Network Techniques[M].Prentice-Hall, inc. , 1973.
[21]Moore, L.J and Clayton, E.R.GERT Modeling and Simulation fundamentals and Applications[M].Petrocelli/Charter, 1976.
[22]Pritsker, A.A.B. .Modeling and Analysis Using Q-GERT[M] .Nerwork John Wiley&Sons, 1977.
[23]Phillips, D.T. .Fundamentals of Network Analysis[M].Prentice-Hall, inc., 1981.
[24]Banks, J.and Carson, J.S. .Discrete-Event System Simulation[M].Prentice-Hall, inc., 1984.
[25]Law, A.M.and Kelton, W.D. .Simulation Modeling and Analysis[M].McGraw-Hill Book Co., 1982.
[26]Pritsker, A.A.B., and Happ, W.W. .GERT:Graphical Evaluation and Review Technique, Part 1, Fundmentals[J].The Journal of Industrial Engineering, Vol.17, No.5, 1966.
[27]Pritsker, A.A.B.and Whitehouse, G.E. .GERT:Graphical Evaluation and Review Technique, Part 2[J].Probabilistic and Industuial Engineering Applications,The Journal of Industrial Engineering, Vol.17, No.6, 1996.
[28]Whitehouse, G.E.and Pritsker, A.A.B. .GERT:Part 3-Further Statistical Results,Counters, Renewal Times, and Correlations[J].AIIE Transactrons, March 1969.
[29] L A.Zadeh.Fuzzy sets[J].Information and Control,Vol.8, 1965, 338~353.
[30] A.Kaufmann.Introduction to the Theory of Fuzzy Subsets[M] .Academic Press, New York, VoI.I, 1975.
[31] LA. Zadeh.Fuzzy sets as a basis for a theory of possibility[J]. Fuzzy Sets and Systems, Vol.1, 1978, 3~28.
[32] D.Dubois and H.Prade.Possibility Theory: An Approach to Computerized Processing of Uncertainty[M].Plenum, New York, 1988.
[33] Liu B, Liu Y.Expected value of fuzzy variable and fuzzy expected value models[J].IEEE Transactions on Fuzzy Systems, 2002, 10(4):445~450.
[34] Li X, Liu B.A.sufficient and necessary condition for credibility measures[J].International Journalof Uncertainty, Fuzziness&Knowledge-Based Systems, 2006, 14(5):527~535.
[35] Liu B.Theory and Practice of Uncertain Programming[M].Heidelberg:Physica-Verlag, 2002.
[36] Li X, Liu B.Moment estimation theorems for various types of uncertain variable[M].Technical Report, 2008.
[37]Li X, Liu B.On distance between fuzzy variables[J].Journal of Intelligent and Fuzzy Systems, In press.
[38]Li P, Liu B.Entropy of credibility distributions for fuzzy variables[J].IEEE Transactions on Fuzzy Systems, 2008, 16(1):123~129.
[39]Li X, Liu B.Maximum entropy principle for fuzzy variables[J].International Journal of Uncertainty, Fuzziness&Knowledge-Based Systems, 2007, 15(supp 2):43~52.
[40]Liu B.Uncertainty Theory, 2nd[M].Berlin:Springer-Verlag, 2007.
[41]Kwakernaak H.Fuzzy random variables I[J].Information Sciences, 1978, 15:1~29.
[42]Kwakernaak H.Fuzzy random variables II[J].Information Sciences, 1979, 17:253~278.
[43]Puri M L, Ralescu D A.Fuzzy random variables[J].Journal of Mathematical Analysis and Applications, 1986, 114:409~422.
[44]Kruse R, Meyer K D.Statistics with Vague Data[M].Dordrecht:D.Reidel Publishing Company, 1987.
[45]Liu Y, Liu B.Fuzzy random variables:A scalar expected value operator[J].Fuzzy Optimization and Decision Making, 2003, 2(2):143~160.
[46]Liu B.Fuzzy random chance-constrained programming[J].IEEE Transactions on Fuzzy Systems, 2001, 9(5):713~720.
[47]Liu B.Fuzzy random dependent-chance programming[J].IEEE Transactions on Fuzzy Systems, 2001, 9(5):721~726.
[48]Liu Y, Liu B.On minimum-risk problems in fuzzy random decision systems[J].Computers&Operations Research, 2005, 32(2):257~283.
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