变权综合理论与多目标决策
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摘要
常权综合仅仅考虑到各类指标在决策中的相对重要性,而忽略了对状态均衡程度的偏好,这种“以不变”(常权)应万变(指标值)的做法具有一定的片面性,为了解决此类问题,我国的汪培庄教授最早提出了变权的思想,并给出一个变权经验公式,李洪兴教授在系统研究因素空间理论时由经验公式出发对变权综合进行了深入的分析,给出了变权、状态变权向量和均衡函数的公理化定义,随后刘文奇教授根据变权综合与效用分析这两大决策理论之间的联系提出了折衷型变权模式,揭示了均衡函数为综合效用的推广形式和单因素效用的类型与均衡函数类型之间的关系,通过折衷型效用函数获得折衷变权,从而扩展了变权方法。李德清教授提出了多层次变权分析、等效关系及变权综合时状态变权向量的确定等一系列理论。
本文在上述成果的基础上完成了以下几个方面的工作：首先介绍了因素空间理论的一些基本概念,接着是文章的主体部分,讨论了均衡函数的代数性质,指出在一定前提下,均衡函数集的加法、乘法运算构成半群,然后按等效关系进行等价类划分,验证了取自不同等价类中的均衡函数可根据代数运算得到与原有均衡函数均不等效的均衡函数,并给出了计算机可以实现的具体算法,同时,为了解决在决策中可以根据人的均衡要求有效地选择变权模型,而引入带参数的均衡函数。因为均衡函数的梯度向量是状态变权向量,所以用类似的方法讨论了状态变权向量的代数性质,得出状态变权向量集对于Hadamard运算也构成半群,按等效性分类,取自不同等价类中的状态变权向量,经过代数运算,可以获得与原状态变权向量均不等效的状态变权向量,同时,也给出了计算机可以实现的具体算法,并利用几何平均值和信息熵构造了两类新的状态变权向量。由于变权综合与多目标决策相结合,可以为许多实际问题提供更加合理的解决方法,因此在文章中给出了基于变权理论的可行方案集优劣顺序的排列方法,接下来用实例验证了该方法的科学性和有效性。最后,文章提出了对未来变权理论的研究设想。
Professor WANG Pei-zhuang proposed most early variable weights thought, and gives one empirical formula of variable weights in order to solve the kind of problem that constant weight synthesis merely considers each kind of indicator in the decision-making relation importance, but has neglected to the condition balanced degree by personal preference, which "remain unchanged" (constant weight) should be changing (indicator value) practice has a certain one-sidedness. Professor LI Hong-xing gives axiomatic definition of variable weights, state variable weights vector and balance function through systematic research factor space theory and thorough analysis variable weights synthesis. Later Professor LIU Wen-qi proposed compromise of the variable weights pattern according to these two big decision theories relation between the utility analysis and variable weights synthesis, concluded balance function is extended form of synthesis utility and between the single factor utility type and balance function relation, obtained through the compromised utility function compromised variable weights, thus expended method of variable weights. Professor LI De-qing proposed a series of theories, for example hierarchy variable weight, the equivalent relations, state variable weights vector s determination when variable weights synthesis and so on.
This article has completed the following several aspect work in the above achievement s foundation. First this article introduces some basic concepts of factor space theory. The main conclusion of the article is that discussed the algebraic properties of balance function, points out under certain premise, balance function collection can constitute semi-group by addition and multiple operation, then carries on the equal kind of division according to the equivalent relation, and verifies that between the new balance function which picks out different equal kinds by addition, multiply, power operation and original balance function are not the equivalent, and has given the concrete algorithm which the computer can do, at the same time, introduces the balance function with parameter in order to solve that variable weight s model can be effective choose, according to balance request of decision-making. Because the gradient vector of balance function is state variable weights vector, this article discusses the algebraic properties of state variable weights vector s with the similar method, obtains the state variable weights vector s collection can also constitute the semi-group by Hadamard multiple operator, then carries on the equal kind of division according to the equivalent relation, and verifies the between the new state variable weights vectors which picks out different equal kinds by Hadamard multiple operation and original state variable weight vectors are not equivalent, and has given the concrete algorithm which the computer can do and give two kind of new state variable weight vectors, based on geometry mean value and the information entropy. Variable weights synthesis and multi-objective decision-makings unify that may provide the more reasonable solution for many actual problems, therefore this article has given arrangement method of the feasible plan collection, then has confirmed this method scientific nature and the validity with the example. Finally, the article proposes the research tentative plan of variable weights theory.
本文在上述成果的基础上完成了以下几个方面的工作：首先介绍了因素空间理论的一些基本概念,接着是文章的主体部分,讨论了均衡函数的代数性质,指出在一定前提下,均衡函数集的加法、乘法运算构成半群,然后按等效关系进行等价类划分,验证了取自不同等价类中的均衡函数可根据代数运算得到与原有均衡函数均不等效的均衡函数,并给出了计算机可以实现的具体算法,同时,为了解决在决策中可以根据人的均衡要求有效地选择变权模型,而引入带参数的均衡函数。因为均衡函数的梯度向量是状态变权向量,所以用类似的方法讨论了状态变权向量的代数性质,得出状态变权向量集对于Hadamard运算也构成半群,按等效性分类,取自不同等价类中的状态变权向量,经过代数运算,可以获得与原状态变权向量均不等效的状态变权向量,同时,也给出了计算机可以实现的具体算法,并利用几何平均值和信息熵构造了两类新的状态变权向量。由于变权综合与多目标决策相结合,可以为许多实际问题提供更加合理的解决方法,因此在文章中给出了基于变权理论的可行方案集优劣顺序的排列方法,接下来用实例验证了该方法的科学性和有效性。最后,文章提出了对未来变权理论的研究设想。
Professor WANG Pei-zhuang proposed most early variable weights thought, and gives one empirical formula of variable weights in order to solve the kind of problem that constant weight synthesis merely considers each kind of indicator in the decision-making relation importance, but has neglected to the condition balanced degree by personal preference, which "remain unchanged" (constant weight) should be changing (indicator value) practice has a certain one-sidedness. Professor LI Hong-xing gives axiomatic definition of variable weights, state variable weights vector and balance function through systematic research factor space theory and thorough analysis variable weights synthesis. Later Professor LIU Wen-qi proposed compromise of the variable weights pattern according to these two big decision theories relation between the utility analysis and variable weights synthesis, concluded balance function is extended form of synthesis utility and between the single factor utility type and balance function relation, obtained through the compromised utility function compromised variable weights, thus expended method of variable weights. Professor LI De-qing proposed a series of theories, for example hierarchy variable weight, the equivalent relations, state variable weights vector s determination when variable weights synthesis and so on.
This article has completed the following several aspect work in the above achievement s foundation. First this article introduces some basic concepts of factor space theory. The main conclusion of the article is that discussed the algebraic properties of balance function, points out under certain premise, balance function collection can constitute semi-group by addition and multiple operation, then carries on the equal kind of division according to the equivalent relation, and verifies that between the new balance function which picks out different equal kinds by addition, multiply, power operation and original balance function are not the equivalent, and has given the concrete algorithm which the computer can do, at the same time, introduces the balance function with parameter in order to solve that variable weight s model can be effective choose, according to balance request of decision-making. Because the gradient vector of balance function is state variable weights vector, this article discusses the algebraic properties of state variable weights vector s with the similar method, obtains the state variable weights vector s collection can also constitute the semi-group by Hadamard multiple operator, then carries on the equal kind of division according to the equivalent relation, and verifies the between the new state variable weights vectors which picks out different equal kinds by Hadamard multiple operation and original state variable weight vectors are not equivalent, and has given the concrete algorithm which the computer can do and give two kind of new state variable weight vectors, based on geometry mean value and the information entropy. Variable weights synthesis and multi-objective decision-makings unify that may provide the more reasonable solution for many actual problems, therefore this article has given arrangement method of the feasible plan collection, then has confirmed this method scientific nature and the validity with the example. Finally, the article proposes the research tentative plan of variable weights theory.
引文
[1]李洪兴.因素空间理论与知识表示的数学框架(Ⅷ).模糊系统与数学,1995,9(3)：1-9.
[2]汪培庄.模糊集与随机落影.北京：北京师范大学出版社,1985.
[3]李洪兴.因素空间理论与知识表示的数学框架(Ⅰ).北京师范大学学报(自然科学版),1996,32(4)：452-459.
[4]李洪兴.因素空间理论与知识表示的数学框架(Ⅱ).北京师范大学学报(自然科学版),1997,33(2)：151-157.
[5]李洪兴.因素空间理论与知识表示的数学框架(Ⅲ).系统工程学报,1996,11(4)：7-16.
[6]李洪兴.因素空间理论与知识表示的数学框架(Ⅳ)[J].系统工程学报,1997,12(4)：30-38.
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[8]李洪兴.因素空间理论与知识表示的数学框架(Ⅵ).系统工程学报,1999,14(1)：1-8.
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[13]刘文奇.变权综合中的惩罚—激励效用.系统工程理论与实践.1998,18(4)：41-47.
[14]刘文奇.变权综合中的激励策略及其解法.系统工程理论与实践.1998,18(12)：40-43.
[15]刘文奇.变权的一般原理与多目标决策.系统工程理论与实践.1999,19(6)：1-10.
[16]刘文奇.Fuzzy集的表现理论及其应用.云南：云南科技出版社.1999.
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[19]李德清,冯艳彬等.两类均衡函数的结构分析与一类状态变权向量的构造.北京师范大学学报,2003,39(5)：595-600.
[20]蔡前凤,李洪兴.均衡度与变权.系统工程理论与实践,2001(10)：83-87.
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[22]姚炳学,李洪兴.局部变权的公理体系.系统工程理论与实践,2000,20(1)：106-109.
[23]李德清,戴敦敬.一种基于变权的人才选优决策模型.大学数学,2005,21(2)：128-131.
[24]李德清,崔红梅,李洪兴.基于层次变权的多因素决策.系统工程学报,2005,20,(3)：245-249.
[25]鲁风菊,谷云东.关于均衡度与变权的注记.北京师范大学学报,2002,38,(6)：739-743.
[26]崔红梅,谷云东,孙明魁,李洪兴.关于状态变权向量公理化体系的注记,北京师范大学学报,2004,40,(1)：l-7.
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[28]李德清,赵彩霞.状态变权向量的等效分析以及一类均衡函数的构造.模糊系统与数学,2004,18(2)：8]3-88.
[29]李德清,赵彩霞,谷云东.等效均衡函数的性质及均衡函数的构造.模糊系统与数学,2005,19(1)：87-92.
[30]蔡前凤,刘伟.带参数的变权综合模型.广东工业大学学报,2002,19(1)：110-114.
[31]李德清,谷云东等.变权综合与可加型标准综合函数.模糊系统与数学,2005,19(3)：101-104.
[32]张剑湖,何湘藩.变权决策方法及应用.云南大学学报,1999,21(6)：443-445.
[33]于莹莹,赵一娜.个人投资的多目标决策分析.投资分析,2006(471)：150.
[34]罗君君.基于熵权的公路建设项目排列多目标决策分析.华中科技大学学报,2006,23(3)：62-64.
[35]王兆玲,李德胜等.房地产投资的多目标决策分析与模型.聊城大学学报,2006,19(3)：27-29.
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[39]Li Hongxing. Multifactorial fuzzy sets and multifactorial degree of nearness. Fuzzy Sets and Systems,19(1986):291-297.
[40]Li Hongxing. Multifactorial functions in fuzzy sets theory.Fuzzy Sets and Systems,1990,33(1):69-84.
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[2]汪培庄.模糊集与随机落影.北京：北京师范大学出版社,1985.
[3]李洪兴.因素空间理论与知识表示的数学框架(Ⅰ).北京师范大学学报(自然科学版),1996,32(4)：452-459.
[4]李洪兴.因素空间理论与知识表示的数学框架(Ⅱ).北京师范大学学报(自然科学版),1997,33(2)：151-157.
[5]李洪兴.因素空间理论与知识表示的数学框架(Ⅲ).系统工程学报,1996,11(4)：7-16.
[6]李洪兴.因素空间理论与知识表示的数学框架(Ⅳ)[J].系统工程学报,1997,12(4)：30-38.
[7]李洪兴.因素空间理论与知识表示的数学框架(V).系统工程学报,1998,13(1)：12-20.
[8]李洪兴.因素空间理论与知识表示的数学框架(Ⅵ).系统工程学报,1999,14(1)：1-8.
[9]李洪兴.因素空间理论与知识表示的数学框架(Ⅶ).模糊系统与数学,1995,9(2)：16-24.
[10]李洪兴.因素空间理论与知识表示的数学框架(Ⅸ).模糊系统与数学,1996,10(2)：12-19.
[11]李洪兴.因素空间理论与知识表示的数学框架(X).模糊系统与数学,1996,10(4)：10-18.
[12]刘文奇.均衡函数及其在变权综合中的应用.系统工程理论与实践.1997,17(4)：58-64.
[13]刘文奇.变权综合中的惩罚—激励效用.系统工程理论与实践.1998,18(4)：41-47.
[14]刘文奇.变权综合中的激励策略及其解法.系统工程理论与实践.1998,18(12)：40-43.
[15]刘文奇.变权的一般原理与多目标决策.系统工程理论与实践.1999,19(6)：1-10.
[16]刘文奇.Fuzzy集的表现理论及其应用.云南：云南科技出版社.1999.
[17]李德清,李洪兴.变权决策中变权效果分析与状态变权向量的确定.控制与决策,2004,19(11)：1241-1245.
[18]李德清,李洪兴.状态变权向量的性质与构造.北京师范大学学报.2002,38,(4)：455-461.
[19]李德清,冯艳彬等.两类均衡函数的结构分析与一类状态变权向量的构造.北京师范大学学报,2003,39(5)：595-600.
[20]蔡前凤,李洪兴.均衡度与变权.系统工程理论与实践,2001(10)：83-87.
[21]朱勇珍,李洪兴.状态变权的公理化体系和均衡函数的构造.系统工程理论与实践.1997,7：116-118.
[22]姚炳学,李洪兴.局部变权的公理体系.系统工程理论与实践,2000,20(1)：106-109.
[23]李德清,戴敦敬.一种基于变权的人才选优决策模型.大学数学,2005,21(2)：128-131.
[24]李德清,崔红梅,李洪兴.基于层次变权的多因素决策.系统工程学报,2005,20,(3)：245-249.
[25]鲁风菊,谷云东.关于均衡度与变权的注记.北京师范大学学报,2002,38,(6)：739-743.
[26]崔红梅,谷云东,孙明魁,李洪兴.关于状态变权向量公理化体系的注记,北京师范大学学报,2004,40,(1)：l-7.
[27]李德清,谷云东,李洪兴.关于状态变权向量公理化定义的若干结果.系统工程理论与实践,2004,24,(5)：97-102.
[28]李德清,赵彩霞.状态变权向量的等效分析以及一类均衡函数的构造.模糊系统与数学,2004,18(2)：8]3-88.
[29]李德清,赵彩霞,谷云东.等效均衡函数的性质及均衡函数的构造.模糊系统与数学,2005,19(1)：87-92.
[30]蔡前凤,刘伟.带参数的变权综合模型.广东工业大学学报,2002,19(1)：110-114.
[31]李德清,谷云东等.变权综合与可加型标准综合函数.模糊系统与数学,2005,19(3)：101-104.
[32]张剑湖,何湘藩.变权决策方法及应用.云南大学学报,1999,21(6)：443-445.
[33]于莹莹,赵一娜.个人投资的多目标决策分析.投资分析,2006(471)：150.
[34]罗君君.基于熵权的公路建设项目排列多目标决策分析.华中科技大学学报,2006,23(3)：62-64.
[35]王兆玲,李德胜等.房地产投资的多目标决策分析与模型.聊城大学学报,2006,19(3)：27-29.
[36]汪培庄,李洪兴.模糊系统理论与模糊计算机.北京：科学出版社,1996.
[37]李洪兴,汪培庄.模糊数学.北京：国防工业出版社,1994.
[38]岳超源.决策理论与方法.北京：科学出版社,2003.
[39]Li Hongxing. Multifactorial fuzzy sets and multifactorial degree of nearness. Fuzzy Sets and Systems,19(1986):291-297.
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