A Procedure for Decision Making Based on Incomplete ... - Springer Link

Fuzzy Optimization and Decision Making, 4, 175–189, 2005. 2005 Springer Science+Business Media, Inc. Printed in The Netherlands.

A Procedure for Decision Making Based on Incomplete Fuzzy Preference Relation1 ZESHUI XU [email protected] College of Economics and Management, Southeast University, Nanjing, Jiangsu, 210096, China Abstract. In this paper, we investigate the decision making problem based on fuzzy preference relation with incomplete information. We first introduce incomplete fuzzy preference relation and present some of its desirable properties. We then develop a system of equations. Based on this system of equations, we propose a procedure for decision making based on incomplete fuzzy preference relation, and finally, a numerical example is presented to illustrate the proposed procedure. Keywords: decision making, fuzzy preference relation, incomplete fuzzy preference relation, priority, transitivity

1. Introduction Fuzzy preference relations have received a great deal of attention from researchers (Orlovsky (1978), Nurmi (1981), Tanino (1984), Kacprzyk (1986), Roubens (1989), Kitainik (1993), Xu (1999), (2002), (2003), (2004a), (2004b), (2005), Chiclana et al (2001), Lipovetsky and Michael Conklin (2002), Xu and Da (2002), (2003), Herrera-Viedma et al (2004)). These researchers focused on the studies of the fuzzy preference relations with complete information (in this paper we call them complete fuzzy preference relations). A complete fuzzy preference relation of order n necessitates the completion of all n(n ) 1)/2 judgements in its entire top triangular portion. Sometimes, however, a decision maker (DM) may develop a fuzzy preference relation with incomplete information (Xu (2004c)) because of time pressure, lack of knowledge, and the DM’s limited expertise related with problem domain. Therefore, this is an important and promising research field. The aim of this paper is to develop a procedure for decision making based on incomplete fuzzy preference relation. To do so, the rest of this paper is organized as follows. Section 2 reviews complete fuzzy preference relation and some of their properties. Section 3 introduces incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation, and presents some properties of incomplete fuzzy preference relation. Section 4 develops a system of equations and a procedure for decision making based on incomplete fuzzy preference relation, and finally, an illustrative example is presented in Section 5. Section 6 includes the concluding remarks.

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2. Complete Fuzzy Preference Relation In the process of decision making, a DM generally needs to compare a set of decision alternatives with respect to a criterion, and constructs a preference relation. Complete fuzzy preference relation is a common preference relation, which can be described as follows: Let X ¼ fx1 ; x2 ; . . . ; xn g be a finite set of alternatives. A complete fuzzy preference relation (Orlovsky (1978)) P on X is a fuzzy set on the product set X · X that is characterized by a membership function lP: X · X fi [0, 1]. The DM’s preferences on X are given by a complete fuzzy preference relation P=(pij)n · n, where pij 2[0, 1], pij+pji=1, pii=0.5, for all i, j, and pij denotes the preference degree or intensity of the alternative xi over xj. In particular, pij=0 indicates that xj is absolutely preferred to xi; pij < 0.5 indicates that xj is preferred to xi; pij=0.5 indicates indifference between xi and xj; pij > 0.5 indicates that xi is preferred to xj; pij=1 indicates that xi is absolutely preferred to xj. Tanino (1984) introduced the concept of multiplicative transitivity: pji pkj pki ¼ ð1Þ pij pjk pik where pij > 0 for all i, j, and interpreted pij/pji as a ratio of the preference intensity for xi to that of xj, that is, xi is pij/pji times as good as xj. Herrera-Viedma et al. (2004), Xu (2003), and Xu and Da (2003) rewrote Eq. (1) as pik pkj pji ¼ pij pjk pki ;

ð2Þ

for all i; j; k

which can be extended to accommodate the case where pij can be equal to 0. Xu (2003), and Xu and Da (2003) pointed out that if a complete fuzzy preference relation P satisfies the multiplicative transitivity Eq. (2), then P is called a multiplicative consistent complete fuzzy preference relation. Saaty (1980) introduced the concept of multiplicative preference relation A on X, which is represented by a matrix A=(aij)n · n X · X, where aij > 0, aij aji=1, aii=1, for all i, j, and aij indicates a ratio of preference intensity for xi to that of xj, i.e., it is interpreted as aij times as good as xj, and is measured using a ratio scale, in particular, 1–9 scale, aij=1 indicates indifference between xi and xj, aij=9 indicates that xi is absolutely preferred to xj, and aij 2 f2; 3; . . . ; 8g indicates intermediate evaluations. Furthermore, A is called a consistent multiplicative preference relation, if aij ¼ aik akj ;

ð3Þ

for all i; j; k

Let x ¼ ðx1 ; x2 ; . . . ; xn ÞT be the priority vector of the multiplicative preference n P relation A=(aij)n · n, where xi > 0, i ¼ 1; 2; . . . ; n; xi ¼ 1, xi denotes the priority i¼1

of xi. If A=(aij)n · n is a consistent multiplicative preference relation, then such a preference relation is given by Saaty (1980):

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A PROCEDURE FOR DECISION MAKING

aij ¼

xi ; xj

ð4Þ

for all i; j

In this case, the priority vector of A can be expressed exactly as follows: !T n n n .X .X .X x¼ 1 ai1 ; 1 ai2 ; . . . ; 1 ain i¼1

i¼1

ð5Þ

i¼1

However, judgements of people depend on personal psychological aspects, such as experience, learning, situation, state of mind, and so forth, hence, the consistency condition is rarely satisfied. As a result, in the general cases, Eq. (4) does not hold. Saaty (1980) introduced the well-known eigenvector method to determine the priority vector x of general multiplicative preference relation A: Ax ¼ kmax x

ð6Þ

where kmax is the largest eigenvalue of A. Similarly, let w ¼ ðw1 ; w2 ; . . . ; wn ÞT be the priority vector of the fuzzy preference n P wi ¼ 1. If P=(pij)n · n is a relation P=(pij)n · n, where wi > 0, i ¼ 1; 2; . . . ; n; i¼1

multiplicative consistent complete fuzzy preference relation, then such a preference relation is given by wi pij ¼ ; for all i; j ð7Þ wi þ wj In this case, P can be expressed as 2 w1 w1 w1 3 w1 þw1 w1 þw2 . . . w1 þwn w2 w2 7 6 w2 6 w2 þw1 w2 þw2 . . . w2 þwn 7 P¼6 7 ... ... ... 5 4 ... wn wn wn wn þw1 wn þw2 . . . wn þwn

ð8Þ

From Eq. (8), it follows that the following system of equations can be given wi wi wi ðwi þ w1 Þ þ ðwi þ w2 Þ þ þ ðwi þ wn Þ ¼ nwi ; i ¼ 1; 2;... ;n wi þ w1 wi þ w2 wi þ wn ð9Þ Using the general complete fuzzy preference relation P=(pij)n · n instead of the multiplicative consistent complete fuzzy preference relation in Eq. (9), and using kmax instead of n, it follows that (Lipovetsky and Michael Conklin (2002)) pi1 ðwi þ w1 Þ þ pi2 ðwi þ w2 Þ þ þ pin ðwi þ wn Þ ¼ kmax wi ; where wi > 0, i ¼ 1; 2; . . . ; n;

n P

wi ¼ 1. By solving Eq. (10) together with

i¼1

we can get the value of kmax and the priority vector w ¼ sponding to kmax.

ð10Þ

for all i n P

i¼1 ðw1 ; w2 ; . . . ; wn ÞT

wi ¼ 1, corre-

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The priority vector derived from Eq. (10) is not prone to influence of possible errors among the elements of a complete fuzzy preference relation (Lipovetsky and Michael Conklin (2002)). Apart from the multiplicative transitivity, some other suggested properties can be given for complete fuzzy preference relation as follows: (1) Triangle condition (Luce and Suppes (1965), Herrera-Viedma et al. (2004)): pij þ pjk pik ;

ð11Þ

for all i; j; k

(2) Weak transitivity (Tanino (1984), Herrera-Viedma et al. (2004)): pij 0:5;

pjk 0:5 ) pik 0:5;

for all i; j; k

ð12Þ

(3) Max–min transitivity (Zimmermann (1991), Herrera-Viedma et al. (2004)): pik minfpij ; pjk g;

ð13Þ

for all i; j; k

(4) Max–max transitivity (Dubois and Prade (1980), Zimmermann (1991), HerreraViedma et al. (2004)): pik maxfpij ; pjk g;

ð14Þ

for all i; j; k

(5) Restricted max–min transitivity (Tanino (1984), Herrera-Viedma et al. (2004)): pij 0:5;

pjk 0:5 ) pik minfpij ; pjk g;

for all i; j; k

ð15Þ

(6) Restricted max–max transitivity (Tanino (1984), Herrera-Viedma et al. (2004)): pij 0:5;

pjk 0:5 ) pik maxfpij ; pjk g;

for all i; j; k

ð16Þ

(7) Additive transitivity (Luce and Suppes (1965),Tanino (1984), Herrera-Viedma et al. (2004)): ðpij 0:5Þ þ ðpjk 0:5Þ ¼ ðpik 0:5Þ;

for all i; j; k

ð17Þ

In the following, we shall focus on the decision making problem based on fuzzy preference relation with incomplete information.

3. Incomplete Fuzzy Preference Relation From Section 2, we know that a complete fuzzy preference relation of order n necessitates the completion of all n(n ) 1)/2 judgements in its entire top triangular portion. Because of time pressure, lack of knowledge, and the decision maker’s limited expertise related with problem domain, sometimes, however, it is difficult to obtain such a fuzzy preference relation, especially for the fuzzy preference relation

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with high order, the DM may develop an incomplete fuzzy preference relation in which some of the elements can not be provided (Xu (2004c)). In the following we shall pay attention to this issue. Definition 2.1 (Xu (2004c)) Let B=(bij)n · n be a fuzzy preference relation, then B is called an incomplete fuzzy preference relation, if some of its elements can not be given by the DM, which we denote by the unknown number ‘‘x’’, and the others can be provided by the DM, which satisfy bij 2 ½0; 1, bij+bji=1, bii=0.5. Definition 2.2 (Xu (2004c)) Let B=(bij)n · n be an incomplete fuzzy preference relation, then B is called a multiplicative consistent incomplete fuzzy preference relation, if all the known elements satisfy the multiplicative transitivity: bik bkj bji ¼ bij bjk bki ;

ð18Þ

for all i; j; k

For convenience, we let X be the set of all the known elements in the incomplete fuzzy preference relation B. Similar to complete fuzzy preference relation, some other properties of incomplete fuzzy preference relation can be suggested as follows: (1) Triangle condition: bij þ bjk bik ;

for all bij ; bjk ; bik 2 X

ð19Þ

(2) Weak transitivity: bij 0:5;

bjk 0:5 ) bik 0:5;


ð20Þ

(3) Max–min transitivity: bik minfbij ; bjk g;


ð21Þ


ð22Þ

(4) Max–max transitivity: bik maxfbij ; bjk g;

(5) Restricted max–min transitivity: bij 0:5;

bjk 0:5 ) bik minfbij ; bjk g;


ð23Þ


ð24Þ

(6) Restricted max–max transitivity: bij 0:5;

bjk 0:5 ) bik maxfbij ; bjk g;

(7) Additive transitivity: ðbij 0:5Þ þ ðbjk 0:5Þ ¼ ðbik 0:5Þ;


ð25Þ

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Given an incomplete fuzzy preference relation, how to obtain the priority vector of it? To solve this issue, we first consider a complete fuzzy preference relation as follows: 2 3 5 0:5 58 7 6 7 P ¼ 4 38 0:5 35 5 2 2 0:5 7 5 According to Eq. (2), we know that P is a multiplicative consistent complete fuzzy preference relation. By solving Eq. (10) together with w1+w2+w3=1, we get the value of kmax and the priority vector w corresponding to kmax as follows: kmax ¼ 3;

w ¼ ð0:5; 0:3; 0:2ÞT

thus P can be rewritten as 2 w1 w1 P¼

w1 þw1 6 w2 4 w2 þw1 w3 w3 þw1

w1 w1 þw3 w2 w2 þw3 w3 w3 þw3

w1 þw2 w2 w2 þw2 w3 w3 þw2

3 7 5¼

2

0:5 0:5þ0:5 6 0:3 4 0:3þ0:5 0:2 0:2þ0:5

0:5 0:5þ0:3 0:3 0:3þ0:3 0:2 0:2þ0:3

0:5 3 0:5þ0:2 0:3 7 0:3þ0:2 5 0:2 0:2þ0:2

In the case when the elements p23 and p32 are unknown, P is an incomplete fuzzy preference relation, which can be expressed as 2 3 5 0:5 58 7 6 7 P ¼ 4 38 0:5 x 5 2 7

x

0:5

To get the priority vector of P, it is natural to substitute the elements p23 and p32 with w2 /(w2+w3) and w3 /(w3+w2), respectively, i.e., 2 5 5 3 0:5 8 7 6 3 7 2 0:5 w2wþw P¼4 8 3 5 2 7

w3 w3 þw2

0:5

Following the priority method of complete fuzzy preference relation, i.e., by utilizing Eq. (10) and w1+w2+w3=1, we establish the following system of equations: 5 5 0:5ðw1 þ w1 Þ þ ðw1 þ w2 Þ þ ðw1 þ w3 Þ ¼ kmax w1 8 7 3 w2 ðw2 þ w1 Þ þ 0:5ðw2 þ w2 Þ þ ðw2 þ w3 Þ ¼ kmax w2 8 w2 þ w3


181

2 w3 ðw3 þ w1 Þ þ ðw3 þ w2 Þ þ 0:5ðw3 þ w3 Þ ¼ kmax w3 7 w3 þ w2 w1 þ w2 þ w3 ¼ 1 This system of equations can be transformed into the following eigenproblem: P0 w ¼ kmax w together with w1+w2+w3=1, where 2 131 5 5 3 6 P0 ¼ 4

56 3 8 2 7

8 19 8

0

7

7 05

16 7

By solving this eigenproblem, we get the value of kmax and the priority vector w corresponding to kmax as follows: kmax ¼ 3;

w ¼ ð0:5; 0:3; 0:2ÞT

and thus, these results are the same as those of the complete fuzzy preference relation P. The unknown elements p23 and p32 can be obtain by p23 ¼

w2 0:3 3 ¼ ¼ w2 þ w3 0:3 þ 0:2 5

p32 ¼

w3 0:2 2 ¼ ¼ w3 þ w2 0:2 þ 0:3 5

In the following, we develop another approach to determining the unknown elements p23 and p32. In fact, Eq. (18) can be used to determine the value of the unknown element p23 by using the known elements. Since p32=1 ) p23, then by Eq. (18), we have p2k pk3 ð1 p23 Þ ¼ p23 p3k pk2 ;

for all k 2 K23

where K23={k | p2k, p3k 2 X}, i.e., p23 ¼

3 5 p21 p13 3 ¼ 3 85 72 5 ¼ p21 p13 þ p31 p12 8 7 þ 7 8 5

p32 ¼ 1 p23 ¼

2 5

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then by using Eq. (10) together with w1+w2+w3=1, we can get the priority vector of P. However, if the incomplete fuzzy preference relation P is not multiplicative consistent and the order n of P is greater than 3, then it may produce different values for a unknown element pij by using Eq. (18), in this case, we can replace each unknown element pij in P with 1 X pik pkj ð26Þ pij ¼ DKij k2K pik pkj þ pjk pki ij

where Kij={k | rik, rjk 2 X } and DKij is the number of the elements in Kij. From the above analysis, it can be known that the first idea utilizes directly a system of equations to obtain the priority vector of an incomplete fuzzy preference relation, while the second one utilizes first the known elements to determine the unknown elements, and then obtains the priority vector of the incomplete fuzzy preference relation through a system of equations. Since the main aim of this paper is to obtain the priority vector of an incomplete fuzzy preference relation rather than to determine the unknown elements, then the first idea is simpler and more practical than the second one. In the following section, we shall consider the priority problem of an incomplete fuzzy preference relation based on the first idea. 4. A Procedure for Decision Making Based on Incomplete Fuzzy Preference Relation Let v ¼ ðv1 ; v2 ; . . . ; vn ÞT be the priority vector of the incomplete fuzzy preference n P vi ¼ 1. relation B=(bij)n · n, where vi > 0, i ¼ 1; 2; . . . ; n; i¼1

In order to estimate the priority vector v ¼ ðv1 ; v2 ; . . . ; vn ÞT of the incomplete fuzzy preference relation B=(bij)n · n, we first construct an auxiliary preference relation B ¼ ðbij Þnn , which is based on the incomplete fuzzy preference relation B=(bij)n · n, where vi ; bij ¼ x bij ¼ vi þvj ð27Þ bij ; bij 6¼ x then we replace Eq. (10) with the following system of equations: bi1 ðvi þ v1 Þ þ bi2 ðvi þ v2 Þ þ þ bin ðvi þ vn Þ ¼ kmax vi ; where vi > 0, i ¼ 1; 2; . . . ; n;

n P

i ¼ 1; 2; . . . ; n

ð28Þ

vi ¼ 1.

i¼1

We can also rewrite Eq. (28) as follows: n X j¼1

bij ðvi þ vj Þ ¼ kmax vi ;

i ¼ 1; 2; . . . ; n

ð29Þ

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By solving Eq. (29) and

n P

vi ¼ 1, we can get the value of kmax and the priority

T i¼1

vector v ¼ ðv1 ; v2 ; . . . ; vn Þ corresponding to kmax. Sometimes, however, if too little information is given, then there must exist more than one solution to Eq. (29). In this case, we shall return the incomplete fuzzy preference relation B=(bij)n · n to the DM and ask him/ her to provide more evaluation information and get an improved fuzzy preference relation based on the incomplete fuzzy preference relation B=(bij)n · n. We repeat this procedure until the unique priority vector can be obtained. Based on the above analysis, in the following we shall develop a procedure for decision making with incomplete fuzzy preference relation. Step 1. For a decision making problem, let X ¼ fx1 ; x2 ; . . . ; xn g be the set of alternatives. The DM compares these alternatives with respect to a single criterion, and constructs an incomplete fuzzy preference relation B=(bij)n · n, where some of its elements cannot be given by the DM, which we denote by the unknown number ‘‘x’’, and the others can be provided by the DM, which satisfy bij 2 ½0; 1, bij+bji=1, bii=0.5. Step 2. Based on the incomplete fuzzy preference relation B=(bij)n · n, we construct an auxiliary preference relation B ¼ ðbij Þnn by using (27). n P Step 3. Solve the system of Eqs. (29) together with vi ¼ 1, if there exists a i¼1

unique priority vector v ¼ ðv1 ; v2 ; . . . ; vn ÞT corresponding to kmax, then go to Step 4; otherwise, return the incomplete fuzzy preference relation B=(bij)n · n to the DM and ask him/ her to provide more evaluation information and get an improved fuzzy preference relation (for convenience, we still denote the improved fuzzy preference relation by B=(bij)n · n). Then go to Step 2. Step 4. Rank all the alternatives and select the most desirable one(s) in accordance with the priority vector v ¼ ðv1 ; v2 ; . . . ; vn ÞT . Step 5. End.

5. Illustrative Example For a decision making problem, there are six decision alternatives xi ði ¼ 1; 2; . . . ; 6Þ. The DM provides his/her preferences over these six decision alternatives, and gives an incomplete fuzzy preference relation as follows: 2

0:5 6 x 6 6 x B¼6 6 0:7 6 4 0:2 0:7

x 0:5 x x x x

x x 0:5 x x x

0:3 x x 0:5 0:6 0:2

0:8 x x 0:4 0:5 0:3

3 0:3 x 7 7 x 7 7 0:8 7 7 0:7 5 0:5

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To get the most desirable alternative(s), the following procedure is involved: Utilize Eq. (27) and the incomplete fuzzy preference relation B=(bij)6 · 6 to construct an auxiliary preference relation B ¼ ðbij Þnn : 2

0:5

6 v2 6 v2 þv1 6 v 6 3 6 v þv B ¼ 6 3 1 6 0:7 6 6 0:2 4 0:7

v1 v1 þv2

v1 v1 þv3 v2 v2 þv3

0:5 v3 v3 þv2 v4 v4 þv2 v5 v5 þv2 v6 v6 þv2

By using Eq. (29) and

0:5 v4 v4 þv3 v5 v5 þv3 v6 v6 þv3 6 P

0:3

0:8

0:3




0:5 0:6

0:4 0:5

0:2

0:3

3

7 7 7 7 7 7 0:8 7 7 0:7 7 5 0:5

vi ¼ 1, we can establish the following system of equations:

i¼1

4:4v1 þ 0:3v4 þ 0:8v5 þ 0:3v6 ¼ kmax v1 6v2 ¼ kmax v2 6v3 ¼ kmax v3 0:7v1 þ 4:9v4 þ 0:4v5 þ 0:8v6 ¼ kmax v4 0:2v1 þ 0:6v4 þ 4:5v5 þ 0:7v6 ¼ kmax v5 0:7v1 þ 0:2v4 þ 0:3v5 þ 4:2v6 ¼ kmax v6 v1 þ v2 þ v3 þ v4 þ v5 þ v6 ¼ 1 Obviously, there exist infinite solutions to the above system. Thus, we return the incomplete fuzzy preference relation B=(bij)n · n to the DM and ask him/ her to provide more evaluation information and get an improved incomplete fuzzy preference relation as follows: 3 2 0:5 0:3 x 0:3 0:8 0:3 6 0:7 0:5 0:7 x 0:6 x 7 7 6 7 6 6 x 0:3 0:5 0:4 x x 7 7 B¼6 6 0:7 x 0:6 0:5 0:4 0:8 7 7 6 7 6 4 0:2 0:4 x 0:6 0:5 0:7 5 0:7

x

x

0:2

0:3

0:5

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Utilize this improved incomplete preference relation as follows: 2 1 0:5 0:3 v1vþv 0:3 3 6 2 6 0:7 0:5 0:7 v vþv 2 4 6 6 v3 6 v þv 0:3 0:5 0:4 6 3 1 B ¼ 6 4 6 0:7 v vþv 0:6 0:5 4 2 6 6 v5 6 0:2 0:4 0:6 v5 þv3 4 0:7

v6 v6 þv2

By using Eq. (29) and

v6 v6 þv3

6 P

0:2

preference relation to construct an auxiliary 0:8

0:3

0:6


v3 v3 þv5

0:4 0:5 0:3

3

7 7 7 7 7 7 7 0:8 7 7 7 0:7 7 5 0:5


i¼1

3:7v1 þ 0:3v2 þ 0:3v4 þ 0:8v5 þ 0:3v6 ¼ kmax v1 0:7v1 þ 5v2 þ 0:7v3 þ 0:6v5 ¼ kmax v2 0:3v2 þ 4:7v3 þ 0:4v4 ¼ kmax v3 0:7v1 þ 0:6v3 þ 4:5v4 þ 0:4v5 þ 0:8v6 ¼ kmax v4 0:2v1 þ 0:4v2 þ 0:6v4 þ 3:9v5 þ 0:7v6 ¼ kmax v5 0:7v1 þ 0:2v4 þ 0:3v5 þ 4:2v6 ¼ kmax v6 v1 þ v2 þ v3 þ v4 þ v5 þ v6 ¼ 1 This system of equations can be transformed into the following eigenproblem: Av ¼ kmax v together with

6 P

vi ¼ 1, where

i¼1

2

3:7 6 0:7 6 6 0 A¼6 6 0:7 6 4 0:2 0:7

0:3 0 5 0:7 0:3 4:7 0 0:6 0:4 0 0 0

0:3 0 0:4 4:5 0:6 0:2

0:8 0:6 0 0:4 3:9 0:3

3 0:3 0 7 7 0 7 7 0:8 7 7 0:7 5 4:2

By solving this eigenproblem, we get the value of kmax and the priority vector v corresponding to kmax as follows:

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kmax ¼ 6:000045 v ¼ ð0:13101; 0:27563; 0:12752; 0:20773; 0:15778; 0:10033ÞT Rank all the alternatives xi ði ¼ 1; 2; . . . ; 6Þ in accordance with the priority vector v, we have f

f

f

f

f

x2 x4 x5 x1 x3 x6 and thus, the most desirable alternative is x2. By the auxiliary preference relation B and the priority vector v, we can also get a complete fuzzy preference relation as follows: 2

0:5 0:7

6 6 6 6 0:49325 0 B ¼ 6 6 0:7 6 6 4 0:2 0:7

0:3 0:5

0:50675 0:7

0:3 0:57024

0:3

0:5

0:4

0:42976 0:4

0:6 0:55303

0:5 0:6

0:26686

0:44033

0:2

3 0:3 0:73314 7 7 7 0:44697 0:55967 7 7 0:4 0:8 7 7 7 0:5 0:7 5 0:8 0:6

0:3

0:5

If we apply the second idea to solve the problem, then by Eq. (26), the unknown elements can be obtained as follows: 1 b12 b23 b12 b23 b13 ¼ þ ¼ 0:58027 2 b12 b23 þ b32 b21 b12 b23 þ b32 b21 b31 ¼ 1 b13 ¼ 0:41973 1 b21 b14 b23 b34 b25 b54 ¼ 0:60033 b24 ¼ þ þ 3 b21 b14 þ b41 b12 b23 b34 þ b43 b32 b25 b54 þ b45 b52 b42 ¼ 1 b24 ¼ 0:39967 1 b21 b16 b25 b56 b26 ¼ þ ¼ 0:77350 2 b21 b16 þ b61 b12 b25 b56 þ b65 b52 b62 ¼ 1 b26 ¼ 0:22650 1 b32 b25 b34 b45 b35 ¼ þ ¼ 0:49476 2 b32 b25 þ b52 b23 b34 b45 þ b54 b43

187


b53 ¼ 1 b35 ¼ 0:50524 b36 ¼

b34 b46 ¼ 0:72727 b34 b46 þ b64 b43

b63 ¼ 1 b36 ¼ 0:27273 and then, based on the incomplete fuzzy preference relation B, we get a complete fuzzy preference relation as follows: 3 2 0:5 0:3 0:58027 0:3 0:8 0:3 0:5 0:7 0:60033 0:6 0:77350 7 6 0:7 7 6 0:3 0:5 0:4 0:49476 0:72727 7 6 0:41973 0 B ¼6 7 0:39967 0:6 0:5 0:4 0:8 7 6 0:7 4 0:2 0:4 0:50524 0:6 0:5 0:7 5 0:7 0:22650 0:27273 0:2 0:3 0:5 By using Eq. (10) and

6 P


i¼1

3:28027v1 þ 0:3v2 þ 0:58027v3 þ 0:3v4 þ 0:8v5 þ 0:3v6 ¼ kmax v1 0:7v1 þ 4:37383v2 þ 0:7v3 þ 0:60033v4 þ 0:6v5 þ 0:77350v6 ¼ kmax v2 0:41973v1 þ 0:3v2 þ 3:34176v3 þ 0:4v4 þ 0:49476v5 þ 0:72727v6 ¼ kmax v3 0:7v1 þ 0:39967v2 þ 0:6v3 þ 3:89967v4 þ 0:4v5 þ 0:8v6 ¼ kmax v4 0:2v1 þ 0:4v2 þ 0:50524v3 þ 0:6v4 þ 3:40524v5 þ 0:7v6 ¼ kmax v5 0:7v1 þ 0:22650v2 þ 0:27273v3 þ 0:2v4 þ 0:3v5 þ 2:69923v6 ¼ kmax v6 v1 þ v2 þ v3 þ v4 þ v5 þ v6 ¼ 1 This system of equations can be transformed into the following eigenproblem: A0 v ¼ kmax v 6 P together with vi ¼ 1, where 2

i¼1

3:28027 6 0:7 6 6 0:41973 A0 ¼ 6 6 0:7 6 4 0:2 0:7

0:3 4:37383 0:3 0:39967 0:4 0:22650

0:58027 0:3 0:7 0:60033 3:34176 0:4 0:6 3:89967 0:50524 0:6 0:27273 0:2

0:8 0:6 0:49476 0:4 3:40524 0:3

3 0:3 0:77350 7 7 0:72727 7 7 0:8 7 7 0:7 5 2:69923

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XU

By solving this eigenproblem, we get the value of kmax and the priority vector v corresponding to kmax as follows: kmax ¼ 6:000015;

v ¼ ð0:13702; 0:28853; 0:13614; 0:20104; 0:15124; 0:08603ÞT

Rank all the alternatives xi ði ¼ 1; 2; . . . ; 6Þ in accordance with the priority vector v, we have f

f

f

f

f

x2 x4 x5 x1 x3 x6 and thus, the most desirable alternative is x2. From the above results, we know that both the approaches can be used to determine the priority vector of an incomplete fuzzy preference relation and produce the same ranking, and both the complete fuzzy preference relations B0 ðkmax ¼ 6:000045Þ and B0 ðkmax ¼ 6:000015Þ produced by using these two approaches, respectively, are of desirable consistency. However, compared with the second approach, the first approach is simpler and more practical in decision making with incomplete fuzzy preference relation because it utilizes directly a system of equations to obtain the priority vector of an incomplete fuzzy preference relation while the second one utilizes first the known elements to determine the unknown elements and then obtains the priority vector of the incomplete fuzzy preference relation through a system of equations. 6. Concluding Remarks In this paper, we have developed a system of equations to determine the priority vector of an incomplete fuzzy preference relation, and then developed a procedure for decision making based on incomplete fuzzy preference relation. The prominent characteristic of this procedure is that if there exists a unique solution to this system of equations, then the obtained solution shall be used to rank alternatives and then to select the most desirable one; otherwise, it requires the DM to provide more evaluation information such that the unique priority vector can be obtained.

Acknowledgements The author is very grateful to the anonymous referees for their valuable comments and suggestions. Note 1. The work was supported by China Postdoctoral Science Foundation under Project 2003034366.


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