﻿ 一种带有属性偏好的模糊多属性决策方法
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1. 西南交通大学 数学学院, 四川 成都 610031;
2. 齐齐哈尔大学 理学院, 黑龙江 齐齐哈尔 161006;
3. 吉林大学 数学学院, 吉林 长春 130012

A method for fuzzy multi-attribute decision-making with preference to attribute
LYU Zhiying1,2 , HUANG Tianmin1, LIANG Xuezhang3
1. College of Mathematics, Southwest Jiaotong University, Chengdu 610031, China;
2. Department of Mathematics, Qiqihar University, Qiqihar 161006, China;
3. College of Mathematics, Jilin University, Changchun 130012, China
Abstract: In this paper, a method for fuzzy multi-attribute decision-making problem considering the characteristics of investment decisions is investigated. The preference information about attribute weights given by the decision makers is in the form of complementary judgment matrixes and the attribute value in the decision matrix is trapezoidal fuzzy number. The attribute weight and expert vectors are received by exploring the feather information of judgment matrixes given by experts about attribute. A new method for ranking alternatives by using ranking trapezoidal fuzzy number based on the area between circumcenter of centroids of a fuzzy number and origin is proposed. Finally, a project evaluation problem for services showed that the proposed method is an effective tool to solve the fuzzy multi-attribute decision-making problems,featuring by simple operations and easy implementation on a computer.
Key words: multi-attribute decision making     trapezoidal fuzzy number     complementary judgment matrix     similar approach degree     superiority index     investment decision

1 模糊多属性决策方法

1.1 权重的确定

1)若pijk > 0.5，则专家ek认为属性ci优于属性cj，记为cik > cjk

2)若0≤pijk < 0.5，则专家ek认为属性ci劣于属性cj，记为cik < cjk

3)若pijk=0.5，则专家ek认为属性ci与属性cj同样重要，记为cik=cjk

rij=ci优于cj的优先度指数，Ri=ci(iM)在C中的优先度指数。

1)cicj当且仅当对kT，有cikcjk且存在k0T满足cik0>cjk0

2)ci > cj当且仅当对kT都有cik > cjk

1)如果存在ciC，使得cpci成立，则称cpC中的劣属性；

2)如果不存在ciC，使得cpci成立，则称cpC中的非劣属性；

3)如果对C，都有cicp，则称cpC中的优属性；

4)如果对C(ip)，都有ci < cp，则称cpC中的最优属性。

1)如果cpC中的优属性，则有Rp=

2)设cp,cqC，如果cqcp，则有① rqq < rpq；② Rq < Rp

3)令cpC中的一个属性，如果Rp=，则cpC中的一个非劣属性。

Pk=(pijk)m×m(kT)为专家ek给出的关于属性集C的互补判断矩阵，根据定义1可以得出专家ek给出的属性的优劣顺序。由定义2判断Pk是否具有可接受一致性，如果Pk不具有可接受一致性，必须进行相应的调整，否则无法判断属性的优劣。如果Pk具有可接受一致性，当考虑到专家的权重时，可将属性ci优于cj的优先度指数定义为

 图 1 属性权重的确定Fig. 1 The step of determining the attribute weights
1.2 方案的排序或择优

 图 2 梯形模糊数Fig. 2 Trapezoidal fuzzy number

2 应用案例

1)首先由定义1，得

2)根据式(1)~(3)计算出决定属性偏好的专家的权重为

3)由定义3得出属性ci的优先度指数矩阵U i=(rijk)4×4(k=1,2,…,4)分别为

4)由式(4)计算每个属性ciC中的优先度指数分别为 R1=2.4155，R2=2.564，R3=1.3715，R4=1.649进而由式(5)计算属性的权重为ω1=0.302，ω2=0.321，ω3=0.171，ω4=0.206。可见客户的权重最大，其次是财务指标的权重，这与服务性行业的服务性有很大的联系，因此顾客的满意度对投资能否实起到了关键的作用。

5)由式(8)集结权重向量ω和模糊决策矩阵F，得到如图 3所示的关于每个备选方案Ai,i=1,2,…,5的模糊评价值为

 图 3 梯形模糊数Fig. 3 Trapezoidal fuzzy numbers and their circumcenters of centroids

6)模糊综合评估值Fi,(i=1,2,…,5)按照如图 2所示的方法分块后得到的外接圆的圆心坐标分别为

3 结束语

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DOI: 10.3969/j.issn.1673-4785.201312036

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#### 文章信息

LYU Zhiying, HUANG Tianmin, LIANG Xuezhang

A method for fuzzy multi-attribute decision-making with preference to attribute

CAAI Transactions on Intelligent Systems, 2015, 10(02): 227-233.
DOI: 10.3969/j.issn.1673-4785.201312036