模糊数排序及判断矩阵的优先权重
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摘要
模糊数排序是模糊优化中的一个重要问题,从模糊集的定义及性质中可知,模糊数之间的序关系不是通常意义下的全序关系,而是格结构下的半序关系,这使得模糊数的比较和判别成为模糊决策中既重要又艰难的任务之一。
近年来许多模糊数排序方法涌现出来。对这些排序方法进行分析的基础上,本文提出了一种基于左、右拟优于度的区间数排序公式。通过构造一致性正互反判断矩阵得到的排序指标,满足模糊排序合理性的五个公理:序关系的完全性、不相交模糊量的性质、不相关模糊量的独立性、序关系的传递性和对加的相容性。接下来将三角模糊数转换为α截集,利用新提出的区间数排序公式确定各个三角模糊数在α水平下的排序,再通过加权积分最终确定三角模糊数的序关系。这种排序方法考虑到了模糊数的各截集水平对排序结果的影响,还可以通过调整悲观系数λ的取值,使排序问题更加灵活,排序结果更加科学。
作为模糊决策最常见的一种决策方法,层次分析法,其关键是获取决策者完备的偏好信息,构造判断矩阵,再运用适当的技术确定优先权重。在AHP分析方法中,具有一致性是判断矩阵应用的前提。本文提出一种确定正互反判断矩阵优先权重的方法,揭示了一致性正互反判断矩阵元素与优先权重新的逻辑关系。新的逻辑关系不仅能充分利用已知的判断信息,也能很大程度上利用决策者的信息偏好,所确定的优先权重含有参数,决策者可通过参数的选择,达到对优先权重分辨率的要求。
Ranking fuzzy number is an important issue in the process of fuzzy optimization. Considering the definition and quality of fuzzy sets, the sequence of fuzzy numbers is not the entire-sequence but semi-sequence under lattice structures, which makes both the comparison and discrimination of fuzzy numbers become important and hardship tasks in the decision-makers.
In recent years, various methods of ranking fuzzy number have been proposed. A sequence formula of interval numbers which is based on left and right imitated preceding degrees of fuzzy numbers is provided in this paper. This new index, which is obtained by constructing consistent positive reciprocal judgment matrix, satisfies five axioms of reasonable ranking fuzzy number, including the completeness of sequence, the property of uncross fuzzy quantity, the independence of misrelated fuzzy quantity, the transitivity of sequence and the compatibility to addition. Subsequently, triangle fuzzy numbers are converted toα-cut sets, and the sequence of triangle fuzzy numbers underα-level is determined by the new sequence formula of interval numbers. Finally, the sequence of triangle fuzzy numbers is determined by weighted integral. This sort of approach takes account of the impact of fuzzy-numbers' cut-set levels on the outcome of the sequence, and adjusts the value of pessimistic coefficientλto make the sequence more flexible and the outcome more scientific.
The Analytic Hierarchy Process (AHP) is the most common method of decision-making. The key of AHP is to construct the judgment matrices by obtaining decision-makers' complete favorable information, and determine priority vectors with appropriate method. In the AHP method, the consistency of the judgment matrices is the precondition of their application. In this paper, a new approach to determine priority vectors of consistent positive reciprocal judgment matrix is provided. After that, the logic relationship between elements of consistent positive reciprocal judgment matrix with the parameter and the priority is revealed. It could not only take full advantage of the judgment of known information, but also take use of information preference of decision-makers to a large extent. The priority determined by this new logic relationship contains parameters can help the decision-makers choose parameters and meet the requirement of priority resolution.
近年来许多模糊数排序方法涌现出来。对这些排序方法进行分析的基础上,本文提出了一种基于左、右拟优于度的区间数排序公式。通过构造一致性正互反判断矩阵得到的排序指标,满足模糊排序合理性的五个公理:序关系的完全性、不相交模糊量的性质、不相关模糊量的独立性、序关系的传递性和对加的相容性。接下来将三角模糊数转换为α截集,利用新提出的区间数排序公式确定各个三角模糊数在α水平下的排序,再通过加权积分最终确定三角模糊数的序关系。这种排序方法考虑到了模糊数的各截集水平对排序结果的影响,还可以通过调整悲观系数λ的取值,使排序问题更加灵活,排序结果更加科学。
作为模糊决策最常见的一种决策方法,层次分析法,其关键是获取决策者完备的偏好信息,构造判断矩阵,再运用适当的技术确定优先权重。在AHP分析方法中,具有一致性是判断矩阵应用的前提。本文提出一种确定正互反判断矩阵优先权重的方法,揭示了一致性正互反判断矩阵元素与优先权重新的逻辑关系。新的逻辑关系不仅能充分利用已知的判断信息,也能很大程度上利用决策者的信息偏好,所确定的优先权重含有参数,决策者可通过参数的选择,达到对优先权重分辨率的要求。
Ranking fuzzy number is an important issue in the process of fuzzy optimization. Considering the definition and quality of fuzzy sets, the sequence of fuzzy numbers is not the entire-sequence but semi-sequence under lattice structures, which makes both the comparison and discrimination of fuzzy numbers become important and hardship tasks in the decision-makers.
In recent years, various methods of ranking fuzzy number have been proposed. A sequence formula of interval numbers which is based on left and right imitated preceding degrees of fuzzy numbers is provided in this paper. This new index, which is obtained by constructing consistent positive reciprocal judgment matrix, satisfies five axioms of reasonable ranking fuzzy number, including the completeness of sequence, the property of uncross fuzzy quantity, the independence of misrelated fuzzy quantity, the transitivity of sequence and the compatibility to addition. Subsequently, triangle fuzzy numbers are converted toα-cut sets, and the sequence of triangle fuzzy numbers underα-level is determined by the new sequence formula of interval numbers. Finally, the sequence of triangle fuzzy numbers is determined by weighted integral. This sort of approach takes account of the impact of fuzzy-numbers' cut-set levels on the outcome of the sequence, and adjusts the value of pessimistic coefficientλto make the sequence more flexible and the outcome more scientific.
The Analytic Hierarchy Process (AHP) is the most common method of decision-making. The key of AHP is to construct the judgment matrices by obtaining decision-makers' complete favorable information, and determine priority vectors with appropriate method. In the AHP method, the consistency of the judgment matrices is the precondition of their application. In this paper, a new approach to determine priority vectors of consistent positive reciprocal judgment matrix is provided. After that, the logic relationship between elements of consistent positive reciprocal judgment matrix with the parameter and the priority is revealed. It could not only take full advantage of the judgment of known information, but also take use of information preference of decision-makers to a large extent. The priority determined by this new logic relationship contains parameters can help the decision-makers choose parameters and meet the requirement of priority resolution.
引文
[1]李荣均.模糊多准则决策理论与应用[M].北京:科学出版社,2002
[2]Ying-Ming Wang,Jian-Bo Yang,Dong-Ling Xu,Kwai-Sang Chin.On the centroids of fuzzy numbers.Fuzzy Sets and Systems 157(2006)919-926
[3]L.-H.CHEN,H.-W.LU.An Approximate Approach for Ranking Fuzzy Numbers Based on Left and Right Dominance.Computers and Mathematics with Applications 41(2001)1589-1602
[4]姜艳萍,樊治平.一种三角模糊数互补判断矩阵的排序方法[J].系统工程与电子技术,2002,Vol.24,No.17
[5]李宏艳,刘海娟.模糊数的一种类序和类距离的定义[J].辽宁工程技术大学学报,2003,Vol.22,No.5:704-706
[6]魏毅强,刘进生,王绪柱.不确定型AHP中判断矩阵的一致性概念及权重[J].系统工程与理论实践,1994,7(4):16-22
[7]韦兰用,韦振中.区间数判断矩阵中区间数的运算[J].数学的实践与认识,2003,33(9):75-79
[8]王绪柱,单静.模糊量排序综述[J].模糊系统与数学,2002,Vol.16,No.4:28-34
[9]Xuzhu Wang,Etienne E.Kerre.Reasonable properties for the ordering of fuzzy quantities(Ⅰ)[J]Fuzzy Sets and Systems.118(2001):375-385
[10]Xuzhu Wang,Etienne E.Kerre.Reasonable properties for the ordering of fuzzy quantities(Ⅱ)[J]Fuzzy Sets and Systems.118(2001):387-405
[11]Xin-Wang Liu,Shi-Lian Han.Ranking Fuzzy Numbers with Preference Weighting Function Expectations.Computers and Mathematics with Applications 49(2005)1731-1753
[12]T.C.Chu,C.T.Tsao.Ranking fuzzy numbers with an area between the centroid point and original point.Computers and Mathematics with Applications.2002,43:111-117
[13]D.YONG,Z.F ZHU,Q.LIU.Ranking Fuzzy Numbers with an Area Method using Radius of Gyration.Computers and Mathematics with Applications 51(2006)1127-1136
[14]B.Asady,A.Zendehnam,Ranking fuzzy numbers by distance minimization,Appl.Math. Modell.(2006),doi:10.1016/j.apm.2006.10.018
[15]P.Anand Raj,D.Nagesh Kumar.Ranking alternatives with fuzzy weights using maximizing set and minimizing set.Fuzzy Sets and Systems 105(1999)365-375
[16]Liem Tran,Lucien Duckstein.Comparison of fuzzy numbers using a fuzzy distance measure.Fuzzy Sets and Systems 130(2002)331-341
[17]Jing-Shing Yao,Kweimei Wu.Ranking fuzzy numbers based on decomposition principle and signed distance.Fuzzy Sets and Systems 116(2000)275-288
[18]E.S.Lee and R.J.Li.Comparison of fuzzy numbers based on the probability measure of fuzzy events,Computers and Mathematics with Applications 15(1988),887-896
[19]H.-C.TANG.Inconsistent Property of Lee and Li Fuzzy Ranking Method,Computers and Mathematics with Applications 45(2003)709-713
[20]H.B.Mitchell,P.A Schaefer,On ordering fuzzy numbers,International Journal of Intelligent Systems15(2000),981-993
[21]R.R.Yager,Ranking fuzzy subsets over the unit interval,Prc.1978 CDC(1978),1435-1437
[22]R.R.Yager,A procedure for ordering fuzzy subsets of the unit interval,Inform.Scie.24(1981),143-161
[23]R.C.Hibbeler,Mechanics of Materials,Prentice Hall,New Jersey,(2000).
[24]麻晓波,王绪柱,张会娟.模糊最大集与极大集的关系研究[J].中北大学学报(自然科学版),2006,Vol.27,No.2:144-146
[25]王绪柱.以参考集为基础的排序指标之间的关系[J].华北工学院学报,2001,Vol.22,No.5:323-326
[26]李荣钧.模糊决策的基础-模糊集的比较与排序[J].控制与决策,2003(3),Vol.18,No.2,221-224
[27]R.Goetschel,W.Voxman,Elementary calculus,Fuzzy Sets Syst.18(1986)31-43.
[28]P.Grzegorzewski,Nearest interval approximation of a fuzzy number,Fuzzy Sets Syst.130(2002)321-330.
[29]Mohammad Modarres,Soheil Sadi-Nezhad.Ranking fuzzy numbers by preference ratio.Fuzzy Sets and Systems 118(2001)429-436
[30]郭志林,薛明志.Fuzzy区间数的一种排序方法及综合评判模型[J].数学的实践与认识,2005(7)Vol.35,No.7:244-247
[31]牟琼,吴庆军,鲁成国.一种考虑多个指标的模糊数排序方法[J].广西师范学院学报(自然科学版),2004,Vol.21,NO-2:17-21
[32]宋志华,张多林,龚立占.一种相对隶属度新概念下的模糊数排序方法[J].空军工程大学学报(自然科学版),2006,Vol.7,No.6:81-83
[33]鞠红梅,包红.运用期望值比较模糊数[J].辽宁师范大学学报(自然科学版),2005,Vol.28,NO.1:32-35
[34]李引珍.模糊最短路的一种算法[J].运筹与管理,2004,Vol.13,No.5:7-11
[35]孔峰.华北电力大学管理学博士学位论文.2005年8月
[36]罗承忠.模糊集引论(上册)[M].北京:北京师范大学出版社,1989
[37]侯福均,吴祈宗.基于Hausdorff距离的模糊数互补判断矩阵排序研究[J].2005(6)模糊系统与数学,Vol.19,No.2:110-115
[38]刘华文.基于距离测度的模糊数排序[J].山东大学学报(理学版),2004(4)Vol.39No.2:30-36
[39]S.Abbasbandy,B.Asady.Ranking of fuzzy numbers by sign distance.Information Sciences 176(2006)2405-2416
[40]L.-H.CHEN,H.-W.LU.The Preference Order of Fuzzy Numbers.Computers and Mathematics with Applications 44(2002)1455-1465
[41]曾文艺,李洪兴,谷云东.模糊数的排序方法[J].北京师范大学学报(自然科学版),2001(12)Vol.37 NO.6:711-714
[42]刘自新,张成,刚家泰.关于模糊数的顺序[J].辽宁师范大学学报(自然科学版),2002(12)Vol.25,No.4:362-365
[43]孔峰,刘鸿雁.基于决策者不确定偏好的模糊多属性决策方法[J].华北电力大学学报,2005(7)Vol.32,No.4:52-55
[44]周宏安,刘三阳,李炳杰.基于目标规划和相对优势度的区间数互反判断矩阵排序法[J].数学的实践与认识,2006(6)Vol.36,No.6:63-67
[45]郭元春,魏毅强,王绪柱.区间数代数[J].华北工学院学报,2004,Vol.25,No.6:441-445
[46]刘进生,王绪柱,张宝玉.区间数排序[J].工程数学学报,2001,Vol.18,No.4:103-109
[47]徐泽水,达庆利.区间数排序的可能度法及其应用[J].系统工程学报,2003,Vol.18,No.1:67-70
[48]兰继斌,王中兴,霍良安.不确定性多属性决策的新方法[J].运筹与管理, 2006(12),Vol.15,No.6:48-53
[49]王莲芬,许树柏.层次分析引论[M].北京:中国人民大学出版社,1990
[50]章志敏,徐梅芳,李梅霞.AHP中新元素导入的保序性条件[J].1998(1),Vol.24,No.1:9-13
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[2]Ying-Ming Wang,Jian-Bo Yang,Dong-Ling Xu,Kwai-Sang Chin.On the centroids of fuzzy numbers.Fuzzy Sets and Systems 157(2006)919-926
[3]L.-H.CHEN,H.-W.LU.An Approximate Approach for Ranking Fuzzy Numbers Based on Left and Right Dominance.Computers and Mathematics with Applications 41(2001)1589-1602
[4]姜艳萍,樊治平.一种三角模糊数互补判断矩阵的排序方法[J].系统工程与电子技术,2002,Vol.24,No.17
[5]李宏艳,刘海娟.模糊数的一种类序和类距离的定义[J].辽宁工程技术大学学报,2003,Vol.22,No.5:704-706
[6]魏毅强,刘进生,王绪柱.不确定型AHP中判断矩阵的一致性概念及权重[J].系统工程与理论实践,1994,7(4):16-22
[7]韦兰用,韦振中.区间数判断矩阵中区间数的运算[J].数学的实践与认识,2003,33(9):75-79
[8]王绪柱,单静.模糊量排序综述[J].模糊系统与数学,2002,Vol.16,No.4:28-34
[9]Xuzhu Wang,Etienne E.Kerre.Reasonable properties for the ordering of fuzzy quantities(Ⅰ)[J]Fuzzy Sets and Systems.118(2001):375-385
[10]Xuzhu Wang,Etienne E.Kerre.Reasonable properties for the ordering of fuzzy quantities(Ⅱ)[J]Fuzzy Sets and Systems.118(2001):387-405
[11]Xin-Wang Liu,Shi-Lian Han.Ranking Fuzzy Numbers with Preference Weighting Function Expectations.Computers and Mathematics with Applications 49(2005)1731-1753
[12]T.C.Chu,C.T.Tsao.Ranking fuzzy numbers with an area between the centroid point and original point.Computers and Mathematics with Applications.2002,43:111-117
[13]D.YONG,Z.F ZHU,Q.LIU.Ranking Fuzzy Numbers with an Area Method using Radius of Gyration.Computers and Mathematics with Applications 51(2006)1127-1136
[14]B.Asady,A.Zendehnam,Ranking fuzzy numbers by distance minimization,Appl.Math. Modell.(2006),doi:10.1016/j.apm.2006.10.018
[15]P.Anand Raj,D.Nagesh Kumar.Ranking alternatives with fuzzy weights using maximizing set and minimizing set.Fuzzy Sets and Systems 105(1999)365-375
[16]Liem Tran,Lucien Duckstein.Comparison of fuzzy numbers using a fuzzy distance measure.Fuzzy Sets and Systems 130(2002)331-341
[17]Jing-Shing Yao,Kweimei Wu.Ranking fuzzy numbers based on decomposition principle and signed distance.Fuzzy Sets and Systems 116(2000)275-288
[18]E.S.Lee and R.J.Li.Comparison of fuzzy numbers based on the probability measure of fuzzy events,Computers and Mathematics with Applications 15(1988),887-896
[19]H.-C.TANG.Inconsistent Property of Lee and Li Fuzzy Ranking Method,Computers and Mathematics with Applications 45(2003)709-713
[20]H.B.Mitchell,P.A Schaefer,On ordering fuzzy numbers,International Journal of Intelligent Systems15(2000),981-993
[21]R.R.Yager,Ranking fuzzy subsets over the unit interval,Prc.1978 CDC(1978),1435-1437
[22]R.R.Yager,A procedure for ordering fuzzy subsets of the unit interval,Inform.Scie.24(1981),143-161
[23]R.C.Hibbeler,Mechanics of Materials,Prentice Hall,New Jersey,(2000).
[24]麻晓波,王绪柱,张会娟.模糊最大集与极大集的关系研究[J].中北大学学报(自然科学版),2006,Vol.27,No.2:144-146
[25]王绪柱.以参考集为基础的排序指标之间的关系[J].华北工学院学报,2001,Vol.22,No.5:323-326
[26]李荣钧.模糊决策的基础-模糊集的比较与排序[J].控制与决策,2003(3),Vol.18,No.2,221-224
[27]R.Goetschel,W.Voxman,Elementary calculus,Fuzzy Sets Syst.18(1986)31-43.
[28]P.Grzegorzewski,Nearest interval approximation of a fuzzy number,Fuzzy Sets Syst.130(2002)321-330.
[29]Mohammad Modarres,Soheil Sadi-Nezhad.Ranking fuzzy numbers by preference ratio.Fuzzy Sets and Systems 118(2001)429-436
[30]郭志林,薛明志.Fuzzy区间数的一种排序方法及综合评判模型[J].数学的实践与认识,2005(7)Vol.35,No.7:244-247
[31]牟琼,吴庆军,鲁成国.一种考虑多个指标的模糊数排序方法[J].广西师范学院学报(自然科学版),2004,Vol.21,NO-2:17-21
[32]宋志华,张多林,龚立占.一种相对隶属度新概念下的模糊数排序方法[J].空军工程大学学报(自然科学版),2006,Vol.7,No.6:81-83
[33]鞠红梅,包红.运用期望值比较模糊数[J].辽宁师范大学学报(自然科学版),2005,Vol.28,NO.1:32-35
[34]李引珍.模糊最短路的一种算法[J].运筹与管理,2004,Vol.13,No.5:7-11
[35]孔峰.华北电力大学管理学博士学位论文.2005年8月
[36]罗承忠.模糊集引论(上册)[M].北京:北京师范大学出版社,1989
[37]侯福均,吴祈宗.基于Hausdorff距离的模糊数互补判断矩阵排序研究[J].2005(6)模糊系统与数学,Vol.19,No.2:110-115
[38]刘华文.基于距离测度的模糊数排序[J].山东大学学报(理学版),2004(4)Vol.39No.2:30-36
[39]S.Abbasbandy,B.Asady.Ranking of fuzzy numbers by sign distance.Information Sciences 176(2006)2405-2416
[40]L.-H.CHEN,H.-W.LU.The Preference Order of Fuzzy Numbers.Computers and Mathematics with Applications 44(2002)1455-1465
[41]曾文艺,李洪兴,谷云东.模糊数的排序方法[J].北京师范大学学报(自然科学版),2001(12)Vol.37 NO.6:711-714
[42]刘自新,张成,刚家泰.关于模糊数的顺序[J].辽宁师范大学学报(自然科学版),2002(12)Vol.25,No.4:362-365
[43]孔峰,刘鸿雁.基于决策者不确定偏好的模糊多属性决策方法[J].华北电力大学学报,2005(7)Vol.32,No.4:52-55
[44]周宏安,刘三阳,李炳杰.基于目标规划和相对优势度的区间数互反判断矩阵排序法[J].数学的实践与认识,2006(6)Vol.36,No.6:63-67
[45]郭元春,魏毅强,王绪柱.区间数代数[J].华北工学院学报,2004,Vol.25,No.6:441-445
[46]刘进生,王绪柱,张宝玉.区间数排序[J].工程数学学报,2001,Vol.18,No.4:103-109
[47]徐泽水,达庆利.区间数排序的可能度法及其应用[J].系统工程学报,2003,Vol.18,No.1:67-70
[48]兰继斌,王中兴,霍良安.不确定性多属性决策的新方法[J].运筹与管理, 2006(12),Vol.15,No.6:48-53
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