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 Journal of Zhejiang University SCIENCE A 2005 Vol.6 No.1 P.27-31 http://doi.org/10.1631/jzus.2005.A0027

The construction and combined operation for fuzzy consistent matrixes*

 Author(s):  Min Yao1, Bin Shen1, Jian-hua Luo2 Affiliation(s):  1. School of Computer Science, Zhejiang University, Hangzhou 310027, China; more Corresponding email(s):   myao@zju.edu.cn Key Words:  Fuzzy consistent matrix, Fuzzy consistent relation, Compound operation Share this article to： More <<< Previous Article|Next Article >>>

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.

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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

Fuzziness is one of the general characteristics of human thinking and objective things. On the one hand, there are a lot of decision-making problems difficult to quantify in human activity. On the other hand, in human thought activity, people exchange information in fuzzy language, then carry out inference and integrate judgement, finally implement decision-making. Combining fuzzy intuition with strict and logistic reasoning is one of the characteristics of human thinking activity. Therefore, it is reasonable and necessary to introduce fuzzy techniques in decision-making. Fuzzy techniques are used for fuzzy decision-making (Wang and Li, ; Smolikova and Wachowiak, ; Niskanen, ).

Fan et al.() proposed a new approach to solve the multiple attribute decision making (MADM) problem, where the decision maker gives his or her preference on alternatives in a fuzzy relation. To reflect the decision maker’s preference information, an optimization model is constructed to assess the attribute weights and then to select the most optimal alternatives. Herrera et al.() devised a decision process for resolving selection problem in conditions of uncertainty, supplying a linguistic decision model for evaluating the satisfaction of the objectives by the potential solutions. Williams and Steele () introduced the notion of a fuzzy index to model sets of linguistic terms for which there is no formal measurement scale. Zhou et al.() proposed a fuzzy set approach that integrates objective and subjective information for evaluating grades of journals, which provided a comprehensive method for dealing with incomplete and imprecise information to support the whole evaluation process.

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, ). The fuzzy consistent relation on finite discourse may be expressed by fuzzy consistent matrix. Fuzzy consistent relation has many special characteristics, especially center-division transitivity, that makes fuzzy consistent relation conform to the psychological characteristics of human decision-making. Therefore, fuzzy consistent relation may be used as theoretical foundation for solving certain decision-making problems, and thus find broad applications in soft science such as fuzzy similar selection, fuzzy judgement, analytical hierarchical process, weighting analysis, etc., so fuzzy consistent matrix attracts the interest of many scholars, and is cited by them (Xu, ; Zhang, ; Li, ).

## .  FUZZY CONSISTENT MATRIX AND ITS PROPERTIES

Definition 1 Assume there is a discourse U={ui|iI}, I={1, 2, …} is an index set, a fuzzy subset in U×U $$R:{\text{ }}U \times U \to [0,1]$$

or $$R = \sum\limits_{(i,j)} {\frac{{{\mu _R}({u_i},{u_j})}}{{({u_i},{u_j})}}}$$

is called fuzzy relation in U.

Definition 2 Let R be a fuzzy relation in U; if for any uiU, ujU, for ∀kI, if there is $${\mu _R}\left( {{u_i},{u_j}} \right) = {\mu _R}\left( {{u_i},{u_k}} \right) - {\mu _R}\left( {{u_j},{u_k}} \right) + 0.{\text{5}}$$

then R is called fuzzy consistent relation.

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

Definition 3 Let U={u1, u2, …, um}, then the fuzzy consistent relation R may be denoted by a fuzzy consistent matrix, i.e.

R=(rij)m×m

here $${r_{ij}} = {\mu _R}\left( {{u_i},{u_j}} \right)$$

Apparently, the elements in fuzzy consistent matrix R satisfy $${r_{ij}} = {r_{ik}} - {r_{jk}} + 0.{\text{5}}\;\forall k$$

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

Theorem 1 Fuzzy consistent matrix R=(rij)m×m has the following properties,

(1) rii=0.5;

(2) The sum of elements in column i and row i equals to m;

(3) RT=RC, RT and RC are transpose and complementary matrix of R, respectively. They are fuzzy consistent matrixes, too;

(4) The sub-matrix obtained by deleting any row and corresponding column from R, is still fuzzy consistent matrix;

(5) R satisfies center-division transitivity,

(a) When λ≥0.5, if rijλ, rjkλ then rikλ;

(b) When λ≤0.5, if rijλ, rjkλ then rikλ.

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 (). It is necessary to note that the center-division transitivity conforms is consistent with the thinking process of human decision making, namely,

(a) Let λ>0.5, if ui is more important than uj (rijλ), uj is more important than uk (rjkλ), then ui is more important than uk (rikλ) certainly;

(b) Let λ<0.5, if ui is less important than uj (rijλ), uj is less important than uk (rjkλ), then ui is

less important than uk (rikλ) certainly.

## .  CREATION OF FCR

Definition 4 Let fuzzy matrix F=(fij)m×m; if $${f_{ij}} + {f_{ji}} = {\text{1}}$$

then matrix F is fuzzy mutual supplementary matrix.

Theorem 2 Fuzzy consistent matrix must be fuzzy mutual supplementary matrix.

Proof Based on Definition 2, rij=rikrjk+0.5, then

rij+rji=(rikrjk+0.5)+(rjkrik+0.5)=0.5+0.5=1

Generally speaking, in the process of fuzzy deci-

sion-making, the estimation matrix constructed by decision-maker is commonly a fuzzy mutual supplementary matrix F=(fij)m×m, not a fuzzy consistent matrix. For example, assume that the object set to be estimated is U={u1, u2, …, um}. The estimation matrix F=(fij)m×m is constructed by binary antitheses, here $${f_{ij}} = \left\{ \begin{matrix} 0 {\text{if }}{u_j}{\text{ outbalances }}{u_i} \\ 0.5 {\text{if }}{u_i}{\text{ is the same as }}{u_j} \\ 1.0 {\text{if }}{u_i}{\text{ outbalances }}{u_j} \\ \end{matrix} \right.$$ then the matrix F is a fuzzy mutual supplementary matrix.

The remaining work is how to rebuild fuzzy consistent matrix from mutual supplementary matrix Yao and Huang () presented an approach for this task. Here, based on the characteristics of fuzzy mutual supplementary matrix and fuzzy consistent matrix, we propose another approach for constructing fuzzy consistent matrix, as described in the following theorem.

Theorem 3 If summing by column in fuzzy mutual supplementary matrix F=(fij)m×m, written as $${r_j} = \sum\limits_{k = 1}^m {{f_{kj}}} \;j = {\text{1,}}...,m$$

and performing the following mathematical transform $${r_{ij}} = 0.5 - \frac{{{r_i} - {r_j}}}{{2m}}$$

Then the so-produced matrix R=(rij)m×m is fuzzy consistent.

Proof (1) rij+rji$$= 0.5 - \frac{{{r_i} - {r_j}}}{{2m}} + 0.5 - \frac{{{r_j} - {r_i}}}{{2m}} = 1$$ Thus R is fuzzy mutual supplementary;

(2) $${r_{ij}} = 0.5 - \frac{{{r_i} - {r_j}}}{{2m}}$$ $$= 0.5 - \frac{{{r_i} - {r_k} - ({r_j} - {r_k})}}{{2m}}$$ $$= 0.5 - \frac{{{r_i} - {r_k}}}{{2m}} - (0.5 - \frac{{{r_j} - {r_k}}}{{2m}}) + 0.5$$ $$= {r_{ik}} - {r_{jk}} + 0.5$$ Thus R is fuzzy consistent.

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

Definition 5 Assume $${R^l} = {(r_{ij}^l)_{m \times m}}, l = 1, \cdots ,n$$ be n fuzzy relations in U, if $${r_{ij}} = \sum\limits_{l = 1}^n {{w_l}} r_{ij}^l$$ $$\sum\limits_{l = 1}^n {{w_l}} = 1$$

then fuzzy relation R=(rij)m×m is called combined relation of Rl (l=1,…,n), written as R=R1R2⊕…⊕Rn.

Theorem 4 Assume Rl (l=1, …, n) to be fuzzy consistent relations in U, fuzzy relation R is called combined relation of Rl (l=1, …, n), i.e. $$R = {R^{\text{1}}} \oplus {R^{\text{2}}} \oplus \ldots \oplus {R^n}$$

Then R is also fuzzy consistent.

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 n individuals are equal, then wl=1/n is appropriate, the element of the whole fuzzy consistent matrix is $${r_{ij}} = \frac{1}{n}\sum\limits_{l = 1}^n {r_{ij}^l}$$

In fact, the weight coefficient wl (l=1, …, n) in combined operation is not only relevant with personnel weights, but also relevant with the degree of coherence among decision-making results determined by n individuals. The former lies on the decision-making individuals’ social status and experience. For example, if some decision-making individual is company leader or principal engineer, then his or her personnel weight should be larger than that of the others. The latter embodies most decision-making opinions adequately so as to prevent the leading function being determined by individual prejudice, especially that of decision-maker with larger personnel weight. If some individual decision-making result differs much from the results of others, then the weight of his or her decision-making opinion in the whole decision-making should be smaller than that of others. The quantitative analysis for coherence degree in decision-making is as follows.

Definition 6 Assume Rl (l=1, …, n) to be n fuzzy consistent matrixes in U, let $${\bar r_{ij}} = \frac{1}{n}\sum\limits_{l = 1}^n {r_{ij}^l}$$

then fuzzy matrix $$\bar R = {({\bar r_{ij}})_{m \times m}}$$ is called average fuzzy matrix.

Theorem 5 Average fuzzy matrix defined by Definition 6 is fuzzy consistent matrix.

Proof $${\bar r_{ij}} = \frac{1}{n}\sum\limits_{l = 1}^n {r_{ij}^l}$$ $$= \frac{1}{n}\sum\limits_{l = 1}^n {(r_{ik}^l - r_{jk}^l + 0.5)}$$ $$= \frac{1}{n}\sum\limits_{l = 1}^n {r_{ik}^l} - \frac{1}{n}\sum\limits_{l = 1}^n {r_{jk}^l} + \frac{1}{n}\sum\limits_{l = 1}^n {0.5}$$ $$= {\bar r_{ik}} - {\bar r_{jk}} + 0.5$$ Definition 7 Assume Rl (l=1, …, n) to be n fuzzy consistent matrixes in U, $$\bar R$$ average fuzzy matrix, let $${D_l} = \left\| {{R_l} - \bar R} \right\|$$

be the distance between fuzzy consistent matrix Rl (l=1, …, n) and $$\bar R$$ Definition 8 If Dl (l=1, …, n) be the distance between fuzzy consistent matrix Rl (l=1, …, n) and their average matrix, then $${B_l} = 1 - {{{D_l}} \mathord{\left/ {\vphantom {{{D_l}} {\sum\limits_{k = 1}^n {{D_k}} }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{k = 1}^n {{D_k}} }}$$

is the coherence degree between fuzzy consistent matrix Rl and Rk (l=1, …, n, kl).

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

Definition 9 Let Al (l=1, …, n) be the personnel weight in creating n fuzzy consistent matrix Rl (l=1, …, n) in U, then the weight coefficients in the combined operation of fuzzy consistent matrixes Rl is $${w_l} = \alpha {A_l} + \beta {B_l}\;l = {\text{1,}}...{\text{,}}n$$

here α and β are appropriate constants satisfying α+β=1. The above formula can reflect most opinions and embody the status of experienced experts.

Example Let five experts evaluate four schemes Sm(m=1, …, 4) of one task, and the five fuzzy mutual supplementary matrixes Fn (n=1, …, 5) are obtained as follows, $${F^{1 - 2}} = \left[ \begin{matrix} 0.5 1 1 1 \\ 0 0.5 1 1 \\ 0 0 0.5 1 \\ 0 0 0 0.5 \\ \end{matrix} \right]{\text{,}}{F^3} = \left[ \begin{matrix} 0.5 0.5 0 0 \\ 0.5 0.5 0 0 \\ 1 1 0.5 0.5 \\ 1 1 0.5 0.5 \\ \end{matrix} \right]$$ $${F^4} = \left[ \begin{matrix} 0.5 0 0 0 \\ 1 0.5 0 0 \\ 1 1 0.5 0 \\ 1 1 1 0.5 \\ \end{matrix} \right]{\text{,}}{F^5} = \left[ \begin{matrix} 0.5 1 1 0 \\ 0 0.5 0 0 \\ 0 1 0.5 0 \\ 1 1 1 0.5 \\ \end{matrix} \right]$$ Then they can be transformed into five fuzzy consistent matrixes Rl (l=1, …, 5) by Theorem 3 $${R^{1 - 2}} = \left[ {\begin{array}{*{20}{c}} {0.5}&{0.375}&{0.25}&{0.125} \\ {0.625}&{0.5}&{0.375}&{0.25} \\ {0.75}&{0.625}&{0.5}&{0.375} \\ {0.875}&{0.75}&{0.625}&{0.5} \end{array}} \right]$$ $${R^3} = \left[ {\begin{array}{*{20}{c}} {0.5}&{0.5}&{0.75}&{0.75} \\ {0.5}&{0.5}&{0.75}&{0.75} \\ {0.25}&{0.25}&{0.5}&{0.5} \\ {0.25}&{0.25}&{0.5}&{0.5} \end{array}} \right]$$ $${R^4} = \left[ {\begin{array}{*{20}{c}} {0.5}&{0.625}&{0.75}&{0.875} \\ {0.375}&{0.5}&{0.625}&{0.75} \\ {0.25}&{0.375}&{0.5}&{0.625} \\ {0.125}&{0.25}&{0.375}&{0.5} \end{array}} \right]$$ $${R^5} = \left[ {\begin{array}{*{20}{c}} {0.5}&{0.25}&{0.375}&{0.625} \\ {0.75}&{0.5}&{0.625}&{0.875} \\ {0.625}&{0.375}&{0.5}&{0.75} \\ {0.375}&{0.125}&{0.25}&{0.5} \end{array}} \right]$$ Their average fuzzy matrix $$\bar R$$ is $$\bar R = \left[ {\begin{array}{*{20}{c}} {0.5}&{0.425}&{0.475}&{0.5} \\ {0.575}&{0.5}&{0.55}&{0.575} \\ {0.525}&{0.45}&{0.5}&{0.525} \\ {0.5}&{0.425}&{0.475}&{0.5} \end{array}} \right]$$ and the coherence degrees between fuzzy consistent matrixes are computed by Eq.(15)

B1=0.776 B2=0.776 B3=0.842

B4=0.776 B5=0.829

Let A1=A2=A3=A4=A5=0.2, β=α=0.5, then the normalized weight coefficients are computed by Eq.(16)

w1=0.195 w2=0.195 w3=0.209

w4=0.195 w5=0.206

By Definition 5, the combined fuzzy consistent matrix is

R=R1R2R3R4R5

=$$\left[ {\begin{array}{*{20}{c}} {0.5}&{0.424}&{0.478}&{0.505} \\ {0.576}&{0.5}&{0.554}&{0.581} \\ {0.522}&{0.446}&{0.5}&{0.527} \\ {0.495}&{0.419}&{0.473}&{0.5} \end{array}} \right]$$ Finally, the preferential values of all schemes may be obtained by means of root-squaring method,

v1=0.4756 v2=0.5518

v3=0.4977 v4=0.4706

Obviously, the order of four schemes from good to bad is S4<S1<S3<S2.

## .  CONCLUSION

From the previous discussion and analysis, we have the following conclusions:

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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