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YAO Min, SHEN Bin, LUO Jian-hua. The construction and combined operation for fuzzy consistent matrixes[J]. Journal of Zhejiang University Science A, 2005, 6(1): 27-31.

@article{title="The construction and combined operation for fuzzy consistent matrixes",

author="YAO Min, SHEN Bin, LUO Jian-hua",

journal="Journal of Zhejiang University Science A",

volume="6",

number="1",

pages="27-31",

year="2005",

publisher="Zhejiang University Press & Springer",

doi="10.1631/jzus.2005.A0027"

}

%0 Journal Article

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%P 27-31

%@ 1673-565X

%D 2005

%I Zhejiang University Press & Springer

%DOI 10.1631/jzus.2005.A0027

TY - JOUR

T1 - The construction and combined operation for fuzzy consistent matrixes

A1 - YAO Min

A1 - SHEN Bin

A1 - LUO Jian-hua

J0 - Journal of Zhejiang University Science A

VL - 6

IS - 1

SP - 27

EP - 31

%@ 1673-565X

Y1 - 2005

PB - Zhejiang University Press & Springer

ER -

DOI - 10.1631/jzus.2005.A0027

**Abstract: **Fuzziness is one of the general characteristics of human thinking and objective things. Introducing fuzzy techniques into decision-making yields very good results. fuzzy consistent matrix has many excellent characteristics, especially center-division transitivity conforming to the reality of the human thinking process in decision-making. This paper presents a new approach for creating fuzzy consistent matrix from mutual supplementary matrix in fuzzy decision-making. At the same time, based on the distance between individual fuzzy consistent matrix and average fuzzy consistent matrix, a kind of combined operation for several fuzzy consistent matrixes is presented which reflects most opinions of experienced experts. Finally, a practical example shows its flexibility and practicability further.

**
**

. INTRODUCTION

Fan et al.(

In the opinion of set theory, decision-making is very closely associated with relation. The decision-making process deals with relations among affair discourse, countermeasure discourse and benefit discourse. We present a new kind of fuzzy relation for the first time, i.e. fuzzy consistent relation (Yao and Huang,

. FUZZY CONSISTENT MATRIX AND ITS PROPERTIES

or

is called fuzzy relation in

then

It is necessary to note that the fuzzy consistent relation conforms consistently to the thinking process of human decision-making.

here

Apparently, the elements in fuzzy consistent matrix

Besides the general properties of fuzzy relation, fuzzy consistent relation has many special characteristics, especially center-division transitivity as described in the following theorem.

(1)

(2) The sum of elements in column

(3)

(4) The sub-matrix obtained by deleting any row and corresponding column from

(5)

(a) When

(b) When

It is easy to prove the above theorem by means of Definition 2 and Definition 3. Readers may also find the proof of Theorem 1 in Yao and Huang (

(a) Let

(b) Let

less important than

. CREATION OF FCR

then matrix

Generally speaking, in the process of fuzzy deci-

sion-making, the estimation matrix constructed by decision-maker is commonly a fuzzy mutual supplementary matrix

The remaining work is how to rebuild fuzzy consistent matrix from mutual supplementary matrix Yao and Huang (

and performing the following mathematical transform

Then the so-produced matrix

(2)

The significance of Theorem 3 is that because the creation of fuzzy mutual supplementary matrix is simpler, the estimation matrixes built by decision-maker in the actual decision-making process are commonly fuzzy mutual supplementary matrix. At this time, fuzzy consistent matrix can be rebuilt from fuzzy mutual supplementary matrix by means of Theorem 3.

It is necessary to note that in order to adapt daedal decision-making situations, you can present more reasonable and more effective rebuilding algorithms according to the actual problems to be solved.

. COMBINED OPERATION

then fuzzy relation

Then

The proof of Theorem 4 is easy, here is omitted. It is necessary to note that the significance of combined operation of fuzzy consistent relation is that it may synthesize many decision-making results (i.e. many fuzzy consistent relations) effectively, and thus forms the whole fuzzy consistent relation.

Well then, how to determine weight coefficient in combined operation If the degrees of significance for

In fact, the weight coefficient

then fuzzy matrix

be the distance between fuzzy consistent matrix

is the coherence degree between fuzzy consistent matrix

Having the coherence degree of fuzzy consistent matrixes, the formula for computing weight coefficients may be given as follows.

here

Let

By Definition 5, the combined fuzzy consistent matrix is

=

Obviously, the order of four schemes from good to bad is

. CONCLUSION

1. Because the creation of fuzzy mutual supplementary matrix is simpler, fuzzy consistent matrix can be rebuilt from fuzzy mutual supplementary matrix by means of certain algorithm.

2. The significance of combined operation of fuzzy consistent relation is that it may synthesize many decision-making results effectively, and thus forms the whole fuzzy consistent relation. The formula for computing weight coefficients can reflect most opinions and embody the status of experienced experts.

* Project (No. 20040335129) supported by the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP), China

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