-4-
数，[]0,1ρ∈，一般取0.5ρ=。
步骤4 计算每个备选方案对直觉模糊正、负理想方案的关联度。 1n i
ij j j w ξξ++==∑，1, 2, , i m =L . (9 1 n i
ij j j w ξξ??==∑，1, 2, , i m =L . (10 式中, i ξ+和i ξ?
分别表示备选方案对直觉模糊正、负理想方案的关联度。 步骤5 计算备选方案对直觉模糊正理想方案相对关联度。
为了确定最优方案，希望其对直觉模糊正理想方案的关联度最大，而对直觉模糊负理想
方案的关联度最小。为此，将每个方案i A 对直觉模糊正理想方案的关联度i ξ+
和对直觉模糊负理想方案的关联度i ξ?
综合，得到方案对直觉模糊正理想方案相对关联度，即 i i i i ξξξξ ++ ?=
+，1, 2, , i m =L . (11 步骤6 方案排序。
依据相对关联度i ξ对所有方案进行排序，即i ξ越大，相应的方案越优。 4. 实例分析
现用文献[6]的算例数据来说明本文提出的方法。假定有3个备选方案123, , A A A ,3个评价属性123, , G G G 。为简单起见，备选方案的评价属性都是效益型属性，每个备选方案对属性的满足程度用直觉模糊集表示，如表1所示。
表1 备选方案直觉模糊集决策表 G 1 2 3 (0.2 A 1 (0.5,0.3,0.2
(0.2,0.6,0.2(0.3,0.5,0.2A 2(0.3,0.3,0.4(0.9,0.0,0.1A 3(0.5,0.2,0.3 (0.8,0.2,0.0
为了确定最优方案，可采用前面给出的直觉模糊集多属性决策灰色关联分析法进行分析。 步骤1 由表1的数据建立直觉模糊决策矩阵
(
((((((((0.5,0.3,0.20.2,0.6,0.20.3,0.5,0.20.4,0.4,0.20.3,0.3,0.40.9,0.0,0.10.4,0.2,0.40.5,0.2,0.30.8,0.2,0.0R ??
??=??????
% 步骤2 根据式(3-6确定直觉模糊正、理想方案。
((((0.5,0.2,0.3, 0.5,0.2,0.3, 0.9,0.0,0.1A += ((((0.4,0.4,0.2, 0.2,0.6,0.2, 0.3,0.5,0.2A ?=
步骤3根据式(7-8计算每个备选方案对直觉模糊正、负理想方案的关联系数。 -5-
( 33
0.73570.43570.33330.61650.61651.00000.73571.00000.6165ij ξ+ ×????=?????? , (33
0.73571.00001.00001.00000.51270.33330.58190.43570.3897ij ξ?×?? ??=??????
步骤 4 根据式(9-10计算每个备选方案对直觉模糊正、负理想方案的关联度。 1230.565,, 0.693, 0.791ξξξ+++=== 1230.868,, 0.720, 0.500ξξξ???===
步骤 5 根据式(11计算每个备选方案对直觉模糊正理想方案的相对关联度。 1230.394, 0.490, 0.613ξξξ===
步骤6 利用(1, 2,3i i ξ=对决策方案进行排序。
方案排序结果为：321A A A f f 。可见备选方案3A 为最优方案。从而和文献[6]的排序结果一致。
5. 结论
直觉模糊集作为模糊集的扩展及一般化形式，运用此理论，使模糊多属性决策问题的研究更加合理、科学。为了解决直觉模糊集多属性决策问题，本文把传统的灰色关联分析法与直觉模糊集结合起来研究模糊多属性决策问题，在模糊环境下进一步扩展了灰色关联分析法的应用范围，最后通过实例分析说明了该方法的有效性。
http://www.paper.edu.cn 参考文献 [1] Zadeh L A. Fuzzy sets [J]. Information and Control, 1965, 8: 338- 356. [2] Atanassov K. Intuitionistic fuzzy sets[J]. Fuzzy Sets and Systems, 1986, 20: 87-96. [3] Atanassov K. More on intuitionistic fuzzy sets[J]. Fuzzy Sets and Systems, 1989, 33: 37- 46. [4] Gau W L, Buehrer D J. Vague sets[J]. IEEE Transactions on Systems,M an and Cybernetics, 1993, 23 (2 : 610-614. [5] Bustine H, Burillo P. Vague sets are intuitionistic fuzzy sets [J]. Fuzzy Sets and System s, 1996, 79: 403-405. [6] 谭春桥,张强. 模糊多属性决策的直觉模糊集方法[J]. 模糊系统与数学,2006,20(5:71-76. [7] Deng J L. Introduction to grey system[J].The Journal of Grey System (UK, 1989,1(1:1-24. [8] Zhang J J, Wu D S, Olson D L. The method of grey related analysis to multiple attribute decision making problems with interval numbers [J]. Mathematical and Computer Modeling, 2005, 42: 991-998. [9] 张吉军.区间数多指标决策问题的灰色关联分析法[J].系统工程与电子技术,2005,27(6: 1030-1033. [10]卫贵武.权重信息不完全的区间数多属性决策GRA方法[J].系统工程与电子技术,2006, 28(12: 1834-1836. Method of grey relational analysis for multiple attribute decision making
based on intuitionistic fuzzy sets Wei Guiwu1,2 1. School of Economics and
Management, Southwest Jiaotong University,Chengdu, China (610031 2. Department of Mathematics, North Sichuan Medical College, Nanchong, China (637007 Abstract The aim of this paper is to extend the grey relational analysis(GRA method to deal with multiple attribute decision making problems in fuzzy environments. This paper presents a novel GRA method for multiple attribute decision making (MADM based on
intuitionistic fuzzy sets theory. The rating of each alternative is expressed in intuitionistic fuzzy value. Then, the distance between two intuitionistic fuzzy sets is introduced. According to the traditional ideas of grey relational analysis, the optimal alternative(s is determined by calculating the relative relational degree of every alternative and positive ideal solution which based on the concept that the optimal alternative should have the largest degree of grey relation from positive ideal solution and the smallest degree of grey relation from the negative ideal solution simultaneously. Finally, an example is shown to high light the procedure of the proposed method at the end of this paper. Keywords: fuzzy multiple attribute decision making; grey relational analysis (GRA; intuitionistic fuzzy sets; intuitionistic fuzzy distances 作者简介：卫贵武（1973-） ，男，四川梓潼人，博士研究生，讲师，从事决策分析，公司理 财及金融工程等研究。 -6-