模糊结构元理论拓展及其决策应用
|
摘要
由于现代决策日趋复杂,模糊不确定性更加突出,模糊决策理论具有重要的应用价值。目前,模糊运算大多是建立Zadeh模糊扩张原理之上的,不过这种运算方法存在运算困难与繁杂的问题。为了解决该问题,郭嗣琮教授提出了模糊结构元理论,该理论思想是将模糊数的运算转换成函数的运算。不过,该理论对一些决策模型,存在无法应用的问题。因此,对结构元理论进行拓展,得到了若干模糊决策模型。
首先,研究了模糊数非单调变换条件下的结构元表示方法。结构元理论主要思想:将任意的有界闭模糊数A用结构元E (即一类特殊的模糊数)和一单调函数来表示,即A = f ( E),进而将模糊数的运算转换成单调函数的运算。本文将变换函数f的限制条件由单调拓展为连续,即A = f ( E)中,若f连续,则A为模糊数。同时,给出了由连续变换函数转换成单调函数的方法。解决了一类模糊值函数无法微积分的问题。研究了模糊限定运算的结构元表示方法。限定运算主要是体现集合间元素的对应关系,不同的限定算子体现了不同的关系。本文利用函数的运算来表示这种关系,得到了模糊限定运算更为一般的表述形式。解决了结构元理论中,由于引人限定算子而导致的应用局限和运算困难。
其次,在以上研究的基础上,进一步对结构元理论进行研究。一是对模糊数加减乘除四则运算进行拓展。拓展后的结果表明,模糊数运算时仅要求单调变换单调就可以运算,将多个运算法则统一成一个基本的运算定理。二是给出了无界模糊数的单调变换函数与其隶属函数之间的转换定理,证明了无界模糊数的加减乘除运算法则。三是给出了含零模糊数运算的机构元表示。
在拓展后结构元理论的基础上,重点研究了模糊数的矩阵应用问题和运筹学中的模糊优化问题。给出了模糊数权重、模糊排队论、模糊最大流与最短路等模型的求解方法。在模糊最短路模型中提出了组合序,在模糊博弈模型总提出了元序,这两个序具有不仅具有良好的数学性质,还能反映决策者的偏好。
As the modern decision is more complicated, fuzzy uncertainty is more prominent, and the fuzzy decision theory is important for application. At present, most fuzzy operation is based on the principle of fuzzy expansion that Zadeh put forward, but this kind of operation method are difficult and complex. In order to solve the problem, the professor GuoSiCong put forward the theory of fuzzy structure element, it converted the operations of fuzzy Numbers into the function operation. However, the theory can't be applied in decision-making model. Therefore, the structure theory is developed, some fuzzy decision model are reached.
First of all, it is researched that how to express the fuzzy number in structure element when it is non-monotone transformation. In the theory of Structure element ,any major thought is : for any bounded closed fuzzy number Aand structured element E (namely : a kind of special fuzzy number), there is always the only monotonic increase (or decrease) function (called the transformed-function), make A = f ( E).and the operations of fuzzy number is converted the operations of monotonic function. in this article, restriction condition of transform function is developed from monotonic to continuous, namely, in A = f ( E),if f is continuous, then A is fuzzy number. And at the same time, the method is gave that the continuous transformation function is converted into monotonic function. Solve the calculus of fuzzy value function. it is researched how to express fuzzy limit operation in structure element .Limit operation mainly reflect the corresponding relationship between collection elements, different limit operator reflects different relationship. In this paper, by using the function operations this relationship is expressed, the more general expression form of the fuzzy limit operation is researched. It is Solved the problem of difficult operation and the application limit as a result of limit operator in structure element theory.
Secondly, on basis of the research in this article, there is further study for the theory of structure element .one : the arithmetic of fuzzy number are developed, and The results showed that after developing, only fuzzy number is monotone and transformed ,the operation can be carried out ,so several algorithms are unified into a basic computing theorems. The other is that The transformation-theorem between monotonic transformed-function of unbounded fuzzy number and membership function is put forward, and the computed-algorithm of unbounded fuzzy number is proved .the last is the operations that the fuzzy number that contain zero are given.
on basis of the theory of structure element that has been developed, application of fuzzy matrix and fuzzy optimization are researched. The model of fuzzy number weight, fuzzy queuing theory and fuzzy maximum flow and the most short circuit of fuzzy number are solved.. In the fuzzy shortest path model, combination sequence is proposed, and the element order in the fuzzy shortest path, the two orders not only have good mathematical property, but also reflect the preferences of decision makers. Therefore, they are more practical.
首先,研究了模糊数非单调变换条件下的结构元表示方法。结构元理论主要思想:将任意的有界闭模糊数A用结构元E (即一类特殊的模糊数)和一单调函数来表示,即A = f ( E),进而将模糊数的运算转换成单调函数的运算。本文将变换函数f的限制条件由单调拓展为连续,即A = f ( E)中,若f连续,则A为模糊数。同时,给出了由连续变换函数转换成单调函数的方法。解决了一类模糊值函数无法微积分的问题。研究了模糊限定运算的结构元表示方法。限定运算主要是体现集合间元素的对应关系,不同的限定算子体现了不同的关系。本文利用函数的运算来表示这种关系,得到了模糊限定运算更为一般的表述形式。解决了结构元理论中,由于引人限定算子而导致的应用局限和运算困难。
其次,在以上研究的基础上,进一步对结构元理论进行研究。一是对模糊数加减乘除四则运算进行拓展。拓展后的结果表明,模糊数运算时仅要求单调变换单调就可以运算,将多个运算法则统一成一个基本的运算定理。二是给出了无界模糊数的单调变换函数与其隶属函数之间的转换定理,证明了无界模糊数的加减乘除运算法则。三是给出了含零模糊数运算的机构元表示。
在拓展后结构元理论的基础上,重点研究了模糊数的矩阵应用问题和运筹学中的模糊优化问题。给出了模糊数权重、模糊排队论、模糊最大流与最短路等模型的求解方法。在模糊最短路模型中提出了组合序,在模糊博弈模型总提出了元序,这两个序具有不仅具有良好的数学性质,还能反映决策者的偏好。
As the modern decision is more complicated, fuzzy uncertainty is more prominent, and the fuzzy decision theory is important for application. At present, most fuzzy operation is based on the principle of fuzzy expansion that Zadeh put forward, but this kind of operation method are difficult and complex. In order to solve the problem, the professor GuoSiCong put forward the theory of fuzzy structure element, it converted the operations of fuzzy Numbers into the function operation. However, the theory can't be applied in decision-making model. Therefore, the structure theory is developed, some fuzzy decision model are reached.
First of all, it is researched that how to express the fuzzy number in structure element when it is non-monotone transformation. In the theory of Structure element ,any major thought is : for any bounded closed fuzzy number Aand structured element E (namely : a kind of special fuzzy number), there is always the only monotonic increase (or decrease) function (called the transformed-function), make A = f ( E).and the operations of fuzzy number is converted the operations of monotonic function. in this article, restriction condition of transform function is developed from monotonic to continuous, namely, in A = f ( E),if f is continuous, then A is fuzzy number. And at the same time, the method is gave that the continuous transformation function is converted into monotonic function. Solve the calculus of fuzzy value function. it is researched how to express fuzzy limit operation in structure element .Limit operation mainly reflect the corresponding relationship between collection elements, different limit operator reflects different relationship. In this paper, by using the function operations this relationship is expressed, the more general expression form of the fuzzy limit operation is researched. It is Solved the problem of difficult operation and the application limit as a result of limit operator in structure element theory.
Secondly, on basis of the research in this article, there is further study for the theory of structure element .one : the arithmetic of fuzzy number are developed, and The results showed that after developing, only fuzzy number is monotone and transformed ,the operation can be carried out ,so several algorithms are unified into a basic computing theorems. The other is that The transformation-theorem between monotonic transformed-function of unbounded fuzzy number and membership function is put forward, and the computed-algorithm of unbounded fuzzy number is proved .the last is the operations that the fuzzy number that contain zero are given.
on basis of the theory of structure element that has been developed, application of fuzzy matrix and fuzzy optimization are researched. The model of fuzzy number weight, fuzzy queuing theory and fuzzy maximum flow and the most short circuit of fuzzy number are solved.. In the fuzzy shortest path model, combination sequence is proposed, and the element order in the fuzzy shortest path, the two orders not only have good mathematical property, but also reflect the preferences of decision makers. Therefore, they are more practical.
引文
[1] Knight F.H.Risk, Uncertainty, and Probability. University of Chicago Press, Chicago.Keynes, J.M.A Treatise on Probability, London, Macmillan.1921.
[2] Langer, Ellen J.“The Illusion of Control”. Journal of Personality and Social Psychology 1975, 32, 311-328.
[3]郭嗣琮.模糊分析中的结构元方法(Ⅰ)、(Ⅱ)[J].辽宁工程技术大学学报,2002, 21(5):670-677.
[4]郭嗣琮、苏志雄、王磊.模糊分析计算中的结构元方法[J].模糊系统与数学,2004,18(3):68-75.
[5]郭嗣琮.模糊实数空间与[-1,1]上同序单调函数类的同胚[J].自然科学进展,2004,14(10).
[6]郭嗣琮.[-1,1]上同序单调函数的同序变换群与模糊数运算[J].模糊系统与数学,2005, 19(3).105-110.
[7]郭嗣琮.基于结构元方法的模糊值函数分析学表述理论[J].自然科学进展, 2005, 15(5):547-552.
[8]郭嗣琮.基于结构元的模糊值函数的一般表示方法[J].模糊系统与数学2005, 1 (2) : 82-86.
[9] Zadeh L A. Probability measures of fuzzy events[J]. J.Math.Anal.Appl,23 (1968): 421-427.
[10] Bortolan G, Degain R. A review of some methods for ranking fuzzy subsets[J]. Fuzzy Sets and Systems,1985,15:1~21.
[11] Nakamura K. Preference relations on a set of fuzzy utilities as a basis for decision making[J]. Fuzzy sets and Systems,1986,20:147-162.
[12] Lee E S, Li R J. Comparison of fuzzy numbers based on the probability measure of fuzzy events[J]. Comp.Math.Applic,1988,15:887-896.
[13] Li R J, Lee E S. Ranking fuzzy number - A comparison[C]//Proc NAFIPS Workshop,West Lafayette,IL,1987:169-204.
[14] Dubois D, Prade H. Ranking of fuzzy numbers in the setting of possibility theory[J] Information Seienees,1983,30:183-224.
[15] Bortolan G, Degani R. A review of some methods for ranking fuzzy subsets[J].Fuzzy Sets and Systems,1985,15:1-19.
[16] Kim K, Park K S. Ranking fuzzy numbers with index of optimism[J].Fuzzy Sets and systems,1990,35:143-150.
[17] Choobineh F, Li H. An index for ordering fuzzy numbers[J]Fuzzy Sets and Systems,1993,54:287-294.
[18]郭嗣琮.模糊数比较与排序的结构元方法[J].系统工程理论与实践,2009,6(6):94-100.
[19]刘海涛,郭嗣琮.基于模糊结构元表述的模糊数排序[J].模糊系统与数学,2010,24(5):61-67.
[20]孙旭东,郭嗣琮.幂模糊数方程和二次模糊方程的结构元求解方法[J].模糊系统与数学,2009,23(5):79-85.
[21]孙旭东,郭嗣琮.求解模糊结构元线性生成的模糊线性系统[J].海南师范大学学报(自然科学版),2008,21(4):407-411.
[22] Friedman M, Ma M, Kandel A. Fhzzy linear systems[J]. Fhzzy Sets and Systems, 1998(96): 201-209.
[23] Ma M, Friedman M, Kandel A. Duality in fuzzy linear systems[J]. Fhzzy Sets and Systems, 2000(109): 55-58.
[24] Asady B, Abbasbandy S, Alavi M. Solution of a fuzzy system of linear equation(J]. Applied Mathematics and Computation, 2006(175): 519-531.
[25] Asady B, Abbasbandy S, Alavi M. Fhzzy general linear systems(J]. Applied Mathematics and Computation,2005(169): 34-40.
[26] Zheng B, Wang K. General fuzzy linear systems(J]. Applied Mathematics and Computation, 2006(181): 1276-1286.
[27] Abbasbandy S, Otadi M, Mosleh M. Minimal solution of general dual fuzzy linear systems[J]. Chaos, Solitons&Factals, 2008(37): 1113-1124.
[28]彭晓华,屠兴汉,郭嗣琮.用QR分解法求解完全模糊线性系统[J].系统工程理论与实践,2010,29(5):778-781.
[29] Dehghan M,Hashemi B,Ghatee M.Computational meth-ods for solving fully fuzzy linear systems[J].Applied Mathematics and Computation,2006,179:328-343.
[30] Dehghan M,Hashemi B.Iterative solution of fuzzy linear systems[J].Applied Mathematics and Computation,2006,175:645-674.
[31] Mosleh M,Otadi M,Khanmirzaie A.Decomposition method for solving fully fuzzy linear systems[J].IranianJournal of Optimization, 2009,1:188-198.
[32] Nasseri S H,Sohrab M,Ardil E.Solving fully fuzzy linear systems by use of a certain decomposition of the coefficient matrix[J].International Journal of Computa-tional and Mathematical Sciences,2008,2:140-142.
[33] G.J.Klir. Fuzzy arithmetic with requisite constraints [J]. Fuzzy Sets and Systems, Vol. 91(1997):165-175
[34]郭嗣琮,刘海涛.模糊数的相等、同一与等式限定运算[J].模糊系统与数学,2008, 22(6):76-82.
[35]刘海涛,郭嗣琮.基于结构元方法的变量模糊的线性规划[J].系统工程理论与实践,2008,6(6):95-99.
[36]赵海坤,郭嗣琮.全系数模糊两层线性规划[J].模糊系统与数学,2010,24(3):98-106.
[37]付永红,杜纲.具有模糊系数的两层线性规划[J].管理科学学报,1999,3:42-49.
[38]张艳菊,刘佳旭.基于结构元理论的模糊多服务台排队模型[J]. .系统工程理论与实践,2010,30(10):1816-1821.
[39]闫艳,赵宝福,岳立柱.基于结构元理论的模糊最大流算法研究[J].运筹与管理,2010,19 (4):26-30.
[40]岳立柱,仲维清,闫艳.基于结构元方法的模糊矩阵博弈求解[J].系统工程理论与实践,2010, 30(2): 272-276.
[41]闫艳,仲维清,岳立柱.具有风险偏好的模糊网络最短路算法[J].数学的实践与认识,2010,24 (4):26-30.
[42]王凯兴,郭嗣琮.模糊需求下的库存风险及最优库存决策[J].模糊系统与数学,2010,24(1):98-102.
[43]曾繁慧.模糊线性回归分析的结构元理论[J].科学技术与工程,2005,5(10):635-639.
[44]孙旭东,郭嗣琮.O型模糊数及其结构元表示方法[J].模糊系统与数学,2009,23(3):61-67.
[45]张艳菊、郭嗣琮.基于结构元的复模糊数四则运算[J].模糊系统与数学, 2008, 22(4):46-49.
[46]刘海涛,郭嗣琮.基于结构元理论的折衷型决策方法[J].辽宁工程技术大学学报(社会科学版), 2008,.10(1):52-56.
[47]岳立柱,仲维清,闫艳.基于模糊数的模糊事件概率的结构元表示[J].模糊系统与数学, 2008, 22(4):46-49.
[48]钟佑明,吕恩琳,王应芳.区间概率随机变量及其数字特征[J].重庆大学学报,2001,24(1):24-27.
[49]吕恩琳,钟佑明.模糊概率随机变量[J].应用数学和力学,2003,23(4):434-439.
[50]王明文.基于概率区间的Bayes决策方法[J].系统工程理论与实践,1997,17(11):79-80.
[51]韩立岩,汪培庄.应用模糊数学[M].北京.首都经济贸易大学出版社,1998: 66-68.
[52]郭嗣琮.扩张原理的一种新的表现形式[J].模糊系统与数学,17(2003) : 43-47
[53] G.J.Klir. Fuzzy arithmetic with requisite constraints [J]. Fuzzy Sets and Systems, Vol. 91(1997):165-175.
[54]郭嗣琮.多种模糊值函数微分定义的统一表述[J].模糊系统与数学3 (2008) : 116-121.
[55]郭嗣琮.基于模糊结构元理论的模糊分析数学原理[M].沈阳.东北大学出版社,2004:873; 21(6):808-810.
[56] Orlovsky S A. Decision-making with a Fuzzy Preference Relation[J]. Fuzzy Sets and Systems, 1978(1):155-167.
[57]杜栋. AHP判断矩阵一致性问题的数学变换解决方法[A].决策科学及其应用[C].北京:海洋出版社, 1996.
[58]林钧昌,徐泽水.模糊AHP中一种新的标度法[J].运筹与管理, 1998, 7(2): 37-40.
[59]樊治平,姜艳萍.互补判断矩阵一致性改进方法[J].东北大学学报(自然科学版),2003,24(1):98-101.
[60]徐泽水.三角模糊数互补判断矩阵的一种排序方法[J].模糊系统与数学,2002,16(1):47-50.
[61]姜艳萍,樊治平.三角模糊数互补判断矩阵排序的一种实用方法[J].系统工程,2002,20(2):89-92.
[62]侯福均,吴祈宗.模糊数互补判断矩阵的加性一致性[J].北京理工大学学报,2004,24(4):367-372.
[63]徐泽水.广义模糊一致性矩阵及其排序方法[J].解放军理工大学学报,2000,1 (6):97-99.
[64]徐泽水.模糊互补判断矩阵排序一种算法[J].系统工程学报,2001,16(4):311-314.
[65]姚敏.模糊决策方法研究[J].系统工程理论与实践,1999,(11),61-70.
[66]高虹霓.基于模糊AHP的道路选优评价方法研究[J].空军工程大学学报,2001,2(2):82-84.
[67]徐泽水.AHP中两类标度的关系研究[J].系统工程理论与实践,1999,(7):97-101.
[68]吕跃进.基于模糊一致矩阵的模糊层次分析法的排序[J].模糊系统与数学,2002,16(2):79-85.
[69]王应明.判断矩阵排序方法综述[J].决策与决策支持系统,1995,(3):101-114.
[70]李梅霞. AHP中判断矩阵一致性改进的一种新方法研究究[J].系统工程理论与实践,2000,2:122-125.
[71]陈永义.模糊控制技术及应用实例[M].北京师范大学出版社,1993.
[72]王国俊.中外模糊系统研究比较[J].国际学术动态,1994(4):48-49.
[73] Zadeh L A. Outline of a new approach to the analysis of complex systems and decision processes[J]. IEEE Trans. Systems Man Cybernet,1973(3):28-44.
[74] Mamdani E H. Application of fuzzy logic to approximate reasoning using lingustic systems[J].IEEE Trans. Comput.,1977(26):1182-1191.
[75] Zadeh L A. The concept of lingustic variable and its application to approximate reasoning,part 1-3[J]. Infrom. Sci.,1975(8):199-249;301-357;1975(9):43-80.
[76]吴望名.模糊推理的原理和方法[M].贵州科技出版社,1994.
[77]吴望名.关于模糊逻辑的一场争论[J].模糊系统与数学, 1995;9(2): 1-9.
[78]王国俊.模糊逻辑与模糊推理[C].全国第七届多值逻辑与模糊逻辑学术会议论文集,西安, 1996:82-96.
[79]王国俊.模糊推理的一个新方法[J].模糊系统与数学, 1999;13(3):1-9.
[80]王国俊. Fuzzy命题演算的一种形式演绎系统[J].科学通报, 1997;42(10):1041-1045.
[81] Dubois D,Prade H. Fuzzy sets and systems theory and applications[M]. New York:Academic Press,1980.
[82] Gorzalczany M B. An interval-valued fuzzy inference method: some basic properties[J]. Fuzzy Sets and Systems,1989,31:243-251.
[83] Turksen I B,Zhong Z. An approximate analogical reasoning scheme based on similarity measures and interval valued fuzzy sets [J]. Fuzzy Sets and Systems,1990,34:323~346.
[84] Atanassov K. Intuitionstic fuzzy sets: theory and applications[M]. Heidelberg,New York:Physica-Verlag,1999.
[85] Ren P. Generalized fuzzy sets and representation of incomplete knowledge[J]. Fuzzy Sets and Systems,1990,36:91-96.
[86]曾文艺,李洪兴,施煜.区间值模糊集合的分解定理[J].北京师范大学学报(自然科学版),2003,39:171~177.
[87]曾文艺,于福生,李洪兴.区间值模糊推理[J].模糊系统与数学, 2007;21(1):68-74
[88]汪培庄.模糊集合论及其应用.上海:上海科技出版社,1983.
[89]王涛,王艳平,唐剑涛.模糊数学及其应用[M].沈阳:东北大学出版社2005.6:48-51.
[90]刘兆君.模糊集贴近度的一般表示形式[J].数学的实践与认识,2009,39(18):163-169.
[91]王贵君,李晓萍.基于IVFS的IV—模糊度与IV—贴近度及积分表示[J].工程数学学报,2004, 21 (2):195-201.
[92]范九伦.Vague值与Vague集上的贴近度[J].系统工程理论与实践, 2006,8: 95-100
[93]郭嗣琮,陈刚.信息科学中的软计算方法[M].沈阳:东北大学出版社2001:48-51.
[94] Stanford R E.The set of limiting distributions for a Markov chain with fuzzy transition probabilities[J].Fuzzy Sets and Systems,1982,7(1):71-78.
[95] Li R J,Lee E S.Analysis of fuzzy queues[J].Computers & Mathematics with Applications,1989,17(7):1143-1147.
[96] Negi D S,Lee E S.Analysis and simulation of fuzzy queues[J].Fuzzy Sets and Systems,1992,46(3):321-330.
[97] Chanas S,Nowakowski M.Single value simulation of fuzzy variable[J].Fuzzy Sets and Systems,1988,25(1):43-57.
[98] Chanas S,Heilpern S.Single value simulation of fuzzy variable some further results[J].Fuzzy Sets and Systems,1989,33(1):29-36.
[99] Hansson A.A stochastic interpretation of membership functions[J]. Automatica,1994,30(3):551-553.
[100] Kao C,Li C C,Chen S P.Parametric programming to the analysis of fuzzy queues[J].Fuzzy Sets and Systems,1999,107(1):93-100.
[101]郭嗣琮,王世辉.二列模糊排队的主要指标求解的结构元方法[J].辽宁工程技术大学学报(自然科学版),2010,29(5):827-830.
[102] Dubois D, Prade H.Systems of linear fuzzy constraints[J]. Fuzzy Sets and Systems, 1980,17(3):37-48.
[103] Osaka S.Gen M.,Fuzzy Shortest Paths Problem, Computers Industrial Engineering, 1994, 27(4):465-468.
[104] Osaka S.,Gen,M.Order Relation between intervals and its application to shortest path problem, Computers Industrial Engineering, 1994,25(1):147-150.
[105] Liu C.,He J.,Shi J.New methods to solve fuzzy shortest path problems, Jornal of Southeast University, 2001,17(1):18-21.
[106]李引珍.模糊最短路的一种算法[J].运筹与管理,2004,13(5):7-11.
[107] Copeland T E, Weston J F. Financial theory and corporate policy [M]. Addison-Wesley Publishing Company, 1992.85-99.
[108] Mas-Colell, Whinston A M, Green J. Microe-conomic theory[M]. New York: Oxford University Press, 1995.167-205.
[109] Ford-Fulkerson. Operations on fuzzy numbers[J]. International Journal for Systems Sciences,1978,9(7):613-626.
[110]胡劲松、吴斐、钟永光.模糊网络最大流算法研究[J].数学的实践与认识,2006,36(8):293-299.
[111] Nishizaki I,Sakawa M.Fuzzy and multiobjective game for conflict resolution [M].Heidelberg,Germany:Physica-verleg,2001.
[112] Bector C R,Chandra S ,Vijay V. Duality in linear programming with fuzzy prrameters and matrix games with fuzzy pay-offs [J].Fuzzy Sets and Systems,2004,146(2):253-269.
[113]胡劲松、金毅、达庆丽.模糊矩阵对策问题及求解方法[J].管理工程学报,1998,12(2):30-34.
[2] Langer, Ellen J.“The Illusion of Control”. Journal of Personality and Social Psychology 1975, 32, 311-328.
[3]郭嗣琮.模糊分析中的结构元方法(Ⅰ)、(Ⅱ)[J].辽宁工程技术大学学报,2002, 21(5):670-677.
[4]郭嗣琮、苏志雄、王磊.模糊分析计算中的结构元方法[J].模糊系统与数学,2004,18(3):68-75.
[5]郭嗣琮.模糊实数空间与[-1,1]上同序单调函数类的同胚[J].自然科学进展,2004,14(10).
[6]郭嗣琮.[-1,1]上同序单调函数的同序变换群与模糊数运算[J].模糊系统与数学,2005, 19(3).105-110.
[7]郭嗣琮.基于结构元方法的模糊值函数分析学表述理论[J].自然科学进展, 2005, 15(5):547-552.
[8]郭嗣琮.基于结构元的模糊值函数的一般表示方法[J].模糊系统与数学2005, 1 (2) : 82-86.
[9] Zadeh L A. Probability measures of fuzzy events[J]. J.Math.Anal.Appl,23 (1968): 421-427.
[10] Bortolan G, Degain R. A review of some methods for ranking fuzzy subsets[J]. Fuzzy Sets and Systems,1985,15:1~21.
[11] Nakamura K. Preference relations on a set of fuzzy utilities as a basis for decision making[J]. Fuzzy sets and Systems,1986,20:147-162.
[12] Lee E S, Li R J. Comparison of fuzzy numbers based on the probability measure of fuzzy events[J]. Comp.Math.Applic,1988,15:887-896.
[13] Li R J, Lee E S. Ranking fuzzy number - A comparison[C]//Proc NAFIPS Workshop,West Lafayette,IL,1987:169-204.
[14] Dubois D, Prade H. Ranking of fuzzy numbers in the setting of possibility theory[J] Information Seienees,1983,30:183-224.
[15] Bortolan G, Degani R. A review of some methods for ranking fuzzy subsets[J].Fuzzy Sets and Systems,1985,15:1-19.
[16] Kim K, Park K S. Ranking fuzzy numbers with index of optimism[J].Fuzzy Sets and systems,1990,35:143-150.
[17] Choobineh F, Li H. An index for ordering fuzzy numbers[J]Fuzzy Sets and Systems,1993,54:287-294.
[18]郭嗣琮.模糊数比较与排序的结构元方法[J].系统工程理论与实践,2009,6(6):94-100.
[19]刘海涛,郭嗣琮.基于模糊结构元表述的模糊数排序[J].模糊系统与数学,2010,24(5):61-67.
[20]孙旭东,郭嗣琮.幂模糊数方程和二次模糊方程的结构元求解方法[J].模糊系统与数学,2009,23(5):79-85.
[21]孙旭东,郭嗣琮.求解模糊结构元线性生成的模糊线性系统[J].海南师范大学学报(自然科学版),2008,21(4):407-411.
[22] Friedman M, Ma M, Kandel A. Fhzzy linear systems[J]. Fhzzy Sets and Systems, 1998(96): 201-209.
[23] Ma M, Friedman M, Kandel A. Duality in fuzzy linear systems[J]. Fhzzy Sets and Systems, 2000(109): 55-58.
[24] Asady B, Abbasbandy S, Alavi M. Solution of a fuzzy system of linear equation(J]. Applied Mathematics and Computation, 2006(175): 519-531.
[25] Asady B, Abbasbandy S, Alavi M. Fhzzy general linear systems(J]. Applied Mathematics and Computation,2005(169): 34-40.
[26] Zheng B, Wang K. General fuzzy linear systems(J]. Applied Mathematics and Computation, 2006(181): 1276-1286.
[27] Abbasbandy S, Otadi M, Mosleh M. Minimal solution of general dual fuzzy linear systems[J]. Chaos, Solitons&Factals, 2008(37): 1113-1124.
[28]彭晓华,屠兴汉,郭嗣琮.用QR分解法求解完全模糊线性系统[J].系统工程理论与实践,2010,29(5):778-781.
[29] Dehghan M,Hashemi B,Ghatee M.Computational meth-ods for solving fully fuzzy linear systems[J].Applied Mathematics and Computation,2006,179:328-343.
[30] Dehghan M,Hashemi B.Iterative solution of fuzzy linear systems[J].Applied Mathematics and Computation,2006,175:645-674.
[31] Mosleh M,Otadi M,Khanmirzaie A.Decomposition method for solving fully fuzzy linear systems[J].IranianJournal of Optimization, 2009,1:188-198.
[32] Nasseri S H,Sohrab M,Ardil E.Solving fully fuzzy linear systems by use of a certain decomposition of the coefficient matrix[J].International Journal of Computa-tional and Mathematical Sciences,2008,2:140-142.
[33] G.J.Klir. Fuzzy arithmetic with requisite constraints [J]. Fuzzy Sets and Systems, Vol. 91(1997):165-175
[34]郭嗣琮,刘海涛.模糊数的相等、同一与等式限定运算[J].模糊系统与数学,2008, 22(6):76-82.
[35]刘海涛,郭嗣琮.基于结构元方法的变量模糊的线性规划[J].系统工程理论与实践,2008,6(6):95-99.
[36]赵海坤,郭嗣琮.全系数模糊两层线性规划[J].模糊系统与数学,2010,24(3):98-106.
[37]付永红,杜纲.具有模糊系数的两层线性规划[J].管理科学学报,1999,3:42-49.
[38]张艳菊,刘佳旭.基于结构元理论的模糊多服务台排队模型[J]. .系统工程理论与实践,2010,30(10):1816-1821.
[39]闫艳,赵宝福,岳立柱.基于结构元理论的模糊最大流算法研究[J].运筹与管理,2010,19 (4):26-30.
[40]岳立柱,仲维清,闫艳.基于结构元方法的模糊矩阵博弈求解[J].系统工程理论与实践,2010, 30(2): 272-276.
[41]闫艳,仲维清,岳立柱.具有风险偏好的模糊网络最短路算法[J].数学的实践与认识,2010,24 (4):26-30.
[42]王凯兴,郭嗣琮.模糊需求下的库存风险及最优库存决策[J].模糊系统与数学,2010,24(1):98-102.
[43]曾繁慧.模糊线性回归分析的结构元理论[J].科学技术与工程,2005,5(10):635-639.
[44]孙旭东,郭嗣琮.O型模糊数及其结构元表示方法[J].模糊系统与数学,2009,23(3):61-67.
[45]张艳菊、郭嗣琮.基于结构元的复模糊数四则运算[J].模糊系统与数学, 2008, 22(4):46-49.
[46]刘海涛,郭嗣琮.基于结构元理论的折衷型决策方法[J].辽宁工程技术大学学报(社会科学版), 2008,.10(1):52-56.
[47]岳立柱,仲维清,闫艳.基于模糊数的模糊事件概率的结构元表示[J].模糊系统与数学, 2008, 22(4):46-49.
[48]钟佑明,吕恩琳,王应芳.区间概率随机变量及其数字特征[J].重庆大学学报,2001,24(1):24-27.
[49]吕恩琳,钟佑明.模糊概率随机变量[J].应用数学和力学,2003,23(4):434-439.
[50]王明文.基于概率区间的Bayes决策方法[J].系统工程理论与实践,1997,17(11):79-80.
[51]韩立岩,汪培庄.应用模糊数学[M].北京.首都经济贸易大学出版社,1998: 66-68.
[52]郭嗣琮.扩张原理的一种新的表现形式[J].模糊系统与数学,17(2003) : 43-47
[53] G.J.Klir. Fuzzy arithmetic with requisite constraints [J]. Fuzzy Sets and Systems, Vol. 91(1997):165-175.
[54]郭嗣琮.多种模糊值函数微分定义的统一表述[J].模糊系统与数学3 (2008) : 116-121.
[55]郭嗣琮.基于模糊结构元理论的模糊分析数学原理[M].沈阳.东北大学出版社,2004:873; 21(6):808-810.
[56] Orlovsky S A. Decision-making with a Fuzzy Preference Relation[J]. Fuzzy Sets and Systems, 1978(1):155-167.
[57]杜栋. AHP判断矩阵一致性问题的数学变换解决方法[A].决策科学及其应用[C].北京:海洋出版社, 1996.
[58]林钧昌,徐泽水.模糊AHP中一种新的标度法[J].运筹与管理, 1998, 7(2): 37-40.
[59]樊治平,姜艳萍.互补判断矩阵一致性改进方法[J].东北大学学报(自然科学版),2003,24(1):98-101.
[60]徐泽水.三角模糊数互补判断矩阵的一种排序方法[J].模糊系统与数学,2002,16(1):47-50.
[61]姜艳萍,樊治平.三角模糊数互补判断矩阵排序的一种实用方法[J].系统工程,2002,20(2):89-92.
[62]侯福均,吴祈宗.模糊数互补判断矩阵的加性一致性[J].北京理工大学学报,2004,24(4):367-372.
[63]徐泽水.广义模糊一致性矩阵及其排序方法[J].解放军理工大学学报,2000,1 (6):97-99.
[64]徐泽水.模糊互补判断矩阵排序一种算法[J].系统工程学报,2001,16(4):311-314.
[65]姚敏.模糊决策方法研究[J].系统工程理论与实践,1999,(11),61-70.
[66]高虹霓.基于模糊AHP的道路选优评价方法研究[J].空军工程大学学报,2001,2(2):82-84.
[67]徐泽水.AHP中两类标度的关系研究[J].系统工程理论与实践,1999,(7):97-101.
[68]吕跃进.基于模糊一致矩阵的模糊层次分析法的排序[J].模糊系统与数学,2002,16(2):79-85.
[69]王应明.判断矩阵排序方法综述[J].决策与决策支持系统,1995,(3):101-114.
[70]李梅霞. AHP中判断矩阵一致性改进的一种新方法研究究[J].系统工程理论与实践,2000,2:122-125.
[71]陈永义.模糊控制技术及应用实例[M].北京师范大学出版社,1993.
[72]王国俊.中外模糊系统研究比较[J].国际学术动态,1994(4):48-49.
[73] Zadeh L A. Outline of a new approach to the analysis of complex systems and decision processes[J]. IEEE Trans. Systems Man Cybernet,1973(3):28-44.
[74] Mamdani E H. Application of fuzzy logic to approximate reasoning using lingustic systems[J].IEEE Trans. Comput.,1977(26):1182-1191.
[75] Zadeh L A. The concept of lingustic variable and its application to approximate reasoning,part 1-3[J]. Infrom. Sci.,1975(8):199-249;301-357;1975(9):43-80.
[76]吴望名.模糊推理的原理和方法[M].贵州科技出版社,1994.
[77]吴望名.关于模糊逻辑的一场争论[J].模糊系统与数学, 1995;9(2): 1-9.
[78]王国俊.模糊逻辑与模糊推理[C].全国第七届多值逻辑与模糊逻辑学术会议论文集,西安, 1996:82-96.
[79]王国俊.模糊推理的一个新方法[J].模糊系统与数学, 1999;13(3):1-9.
[80]王国俊. Fuzzy命题演算的一种形式演绎系统[J].科学通报, 1997;42(10):1041-1045.
[81] Dubois D,Prade H. Fuzzy sets and systems theory and applications[M]. New York:Academic Press,1980.
[82] Gorzalczany M B. An interval-valued fuzzy inference method: some basic properties[J]. Fuzzy Sets and Systems,1989,31:243-251.
[83] Turksen I B,Zhong Z. An approximate analogical reasoning scheme based on similarity measures and interval valued fuzzy sets [J]. Fuzzy Sets and Systems,1990,34:323~346.
[84] Atanassov K. Intuitionstic fuzzy sets: theory and applications[M]. Heidelberg,New York:Physica-Verlag,1999.
[85] Ren P. Generalized fuzzy sets and representation of incomplete knowledge[J]. Fuzzy Sets and Systems,1990,36:91-96.
[86]曾文艺,李洪兴,施煜.区间值模糊集合的分解定理[J].北京师范大学学报(自然科学版),2003,39:171~177.
[87]曾文艺,于福生,李洪兴.区间值模糊推理[J].模糊系统与数学, 2007;21(1):68-74
[88]汪培庄.模糊集合论及其应用.上海:上海科技出版社,1983.
[89]王涛,王艳平,唐剑涛.模糊数学及其应用[M].沈阳:东北大学出版社2005.6:48-51.
[90]刘兆君.模糊集贴近度的一般表示形式[J].数学的实践与认识,2009,39(18):163-169.
[91]王贵君,李晓萍.基于IVFS的IV—模糊度与IV—贴近度及积分表示[J].工程数学学报,2004, 21 (2):195-201.
[92]范九伦.Vague值与Vague集上的贴近度[J].系统工程理论与实践, 2006,8: 95-100
[93]郭嗣琮,陈刚.信息科学中的软计算方法[M].沈阳:东北大学出版社2001:48-51.
[94] Stanford R E.The set of limiting distributions for a Markov chain with fuzzy transition probabilities[J].Fuzzy Sets and Systems,1982,7(1):71-78.
[95] Li R J,Lee E S.Analysis of fuzzy queues[J].Computers & Mathematics with Applications,1989,17(7):1143-1147.
[96] Negi D S,Lee E S.Analysis and simulation of fuzzy queues[J].Fuzzy Sets and Systems,1992,46(3):321-330.
[97] Chanas S,Nowakowski M.Single value simulation of fuzzy variable[J].Fuzzy Sets and Systems,1988,25(1):43-57.
[98] Chanas S,Heilpern S.Single value simulation of fuzzy variable some further results[J].Fuzzy Sets and Systems,1989,33(1):29-36.
[99] Hansson A.A stochastic interpretation of membership functions[J]. Automatica,1994,30(3):551-553.
[100] Kao C,Li C C,Chen S P.Parametric programming to the analysis of fuzzy queues[J].Fuzzy Sets and Systems,1999,107(1):93-100.
[101]郭嗣琮,王世辉.二列模糊排队的主要指标求解的结构元方法[J].辽宁工程技术大学学报(自然科学版),2010,29(5):827-830.
[102] Dubois D, Prade H.Systems of linear fuzzy constraints[J]. Fuzzy Sets and Systems, 1980,17(3):37-48.
[103] Osaka S.Gen M.,Fuzzy Shortest Paths Problem, Computers Industrial Engineering, 1994, 27(4):465-468.
[104] Osaka S.,Gen,M.Order Relation between intervals and its application to shortest path problem, Computers Industrial Engineering, 1994,25(1):147-150.
[105] Liu C.,He J.,Shi J.New methods to solve fuzzy shortest path problems, Jornal of Southeast University, 2001,17(1):18-21.
[106]李引珍.模糊最短路的一种算法[J].运筹与管理,2004,13(5):7-11.
[107] Copeland T E, Weston J F. Financial theory and corporate policy [M]. Addison-Wesley Publishing Company, 1992.85-99.
[108] Mas-Colell, Whinston A M, Green J. Microe-conomic theory[M]. New York: Oxford University Press, 1995.167-205.
[109] Ford-Fulkerson. Operations on fuzzy numbers[J]. International Journal for Systems Sciences,1978,9(7):613-626.
[110]胡劲松、吴斐、钟永光.模糊网络最大流算法研究[J].数学的实践与认识,2006,36(8):293-299.
[111] Nishizaki I,Sakawa M.Fuzzy and multiobjective game for conflict resolution [M].Heidelberg,Germany:Physica-verleg,2001.
[112] Bector C R,Chandra S ,Vijay V. Duality in linear programming with fuzzy prrameters and matrix games with fuzzy pay-offs [J].Fuzzy Sets and Systems,2004,146(2):253-269.
[113]胡劲松、金毅、达庆丽.模糊矩阵对策问题及求解方法[J].管理工程学报,1998,12(2):30-34.