- Email: [email protected]

A method based on linguistic aggregation operators for group decision making with linguistic preference relations q Zeshui Xu College of Economics and Management, Southeast University, Nanjing, Jiangsu 210096, China Received 14 July 2003; accepted 14 October 2003

Abstract In this paper, we deﬁne some operational laws of linguistic variables and develop some new aggregation operators such as linguistic geometric averaging (LGA) operator, linguistic weighted geometric averaging (LWGA) operator, linguistic ordered weighted geometric averaging (LOWGA) operator and linguistic hybrid geometric averaging (LHGA) operator, etc., which can be utilized to aggregate preference information taking the form of linguistic variables, and then study some desirable properties of the operators. Based on the LGA and the LHGA operators, we propose a practical method for group decision making with linguistic preference relations. The method is straightforward and has no loss of information. Finally, an illustrative numerical example is also given. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Group decision making; Linguistic preference relations; Aggregation operators; Operational laws

q The work was supported by the National Natural Science Foundation of China (NSFC) under Project 79970093. E-mail address: [email protected] (Z. Xu).

0020-0255/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2003.10.006

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1. Introduction Decision making problems generally consist of ﬁnding the most desirable alternative(s) from a given alternative set. The increasing complexity of the socioeconomic environment makes it less and less possible for single decision maker (DM) to consider all relevant aspects of a problem [1], as a result, many decision making processes, in the real world, take place in group settings. In the process of decision making, DMs generally need to compare a set of decision alternatives with respect to a single criterion, and construct preference relations. In general, the preference relations take the form of multiplicative preference relations ([2– 8], etc.) or fuzzy preference relations ([9–16], etc.), whose elements estimate the dominance of one alternative over another and take the form of exact numerical values. However, the DMs may have vague knowledge about the preference degree of one alternative over another, and cannot estimate their preferences with exact numerical values. Furthermore, it is too complex or too ill-deﬁned to be amenable for description in conventional quantitative expressions. It is more suitable to provide their preferences by means of linguistic variables rather than numerical ones [17–24]. After constructing preference relations, the DMs then need to aggregate the preference information contained in the preference relations, and rank the given alternatives. One of the most common methods for aggregating decision information is the ordered weighted averaging (OWA) operator presented by Yager [25]. Later, some new families of OWA operators were introduced [26]. In the short time since their ﬁrst appearance, the OWA operators have been used in an astonishingly wide range of applications [24,27]. Most of these operators, however, can only be used in situations where the input arguments are the exact values, and few of them can be used to aggregate the linguistic preference information. Recently, Herrera and Martınez [28] developed a linguistic representation model for representing the linguistic information with a pair of values called 2-tuple, composed by a linguistic term and a number. Together with the model, they also presented a computational technique to deal with the 2-tuples without loss of information. Motivated by this idea, in this paper we develop a practical method for group decision making with linguistic preference relations. To do so, this paper is set out as follows. Section 2 deﬁnes some operational laws of linguistic variables and develops some new aggregation operators such as linguistic geometric averaging (LGA) operator, linguistic weighted geometric averaging (LWGA) operator, linguistic ordered weighted geometric averaging (LOWGA) operator and linguistic hybrid geometric averaging (LHGA) operator, etc., and then studies some desirable properties of the operators. Section 3 develops a practical method based on the LGA and the LHGA operators for group decision making with linguistic preference relations, which is straightforward and has no loss of information. Section 4 gives an illustrative example to verify the developed approach and to demonstrate its feasibility and practicality. Section 5 concludes the paper.

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2. Some new aggregation operators n

Deﬁnition 1 [26,29]. Let WGA : Rþ ! Rþ , if WGAx ða1 ; a2 ; . . . ; an Þ ¼

n Y

x

aj j

ð1Þ

j¼1 T

where x ¼ P ðx1 ; x2 ; . . . ; xn Þ is the exponential weighting vector of the aj , and xj 2 ½0; 1, nj¼1 xj ¼ 1, then WGA is called the weighted geometric averaging (WGA) operator. Deﬁnition 2 [26,30,31]. An ordered weighted geometric averaging (OWGA) n operator of dimension n is a mapping OWGA : Rþ ! Rþ which has associated with P it an exponential weighting vector w ¼ ðw1 ; w2 ; . . . ; wn ÞT , with wj 2 ½0; 1 n and j¼1 wj ¼ 1, such that f ða1 ; a2 . . . ; an Þ ¼

n Y

w

bj j

ð2Þ

j¼1

where bj is the jth largest of the ai . The WGA and the OWGA operators have only been used in situations in which the input arguments are the exact values. However, judgements of people depend on personal psychological aspects such as experience, learning, situation, state of mind, and so forth. It is more suitable to provide their preferences by means of linguistic variables rather than numerical ones (for example when evaluating the comfort or design of a car, terms like good, fair, poor can be used). In the following, based on these two operators, we shall present some new aggregation operators, which can be used to accommodate the situations where the input arguments are linguistic variables. Let S ¼ fsi g (i ¼ 1; . . . ; t) be a ﬁnite and totally ordered discrete term set. Any label, si , represents a possible value for a linguistic variable, and it must have the following characteristics [32]: (1) (2) (3) (4)

The set is ordered: si P sj if i P j; There is the negation operator: negðsi Þ ¼ sj such that j ¼ t þ 1 i; Max operator: maxðsi ; sj Þ ¼ si if si P sj ; Min operator: minðsi ; sj Þ ¼ si if si 6 sj .

For example, S can be deﬁned so as its elements are uniformly distributed on a scale on which a total order is deﬁned:

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S ¼ fs1 ¼ extremely poor; s2 ¼ very poor; s3 ¼ poor; s4 ¼ slightly poor; s5 ¼ fair; s6 ¼ slightly good; s7 ¼ good; s8 ¼ very good; s9 ¼ extremely goodg To preserve all the given information, we extend the discrete term set S to a continuous linguistic term set S ¼ fsa j s1 < sa 6 st ; a 2 ½1; tg, where, if sa 2 S, then we call sa the original linguistic term, otherwise, we call sa the virtual linguistic term. Consider any two linguistic terms sa ; sb 2 S, and l; l1 ; l2 2 ½0; 1, we deﬁne some operational laws as follows: (1) (2) (3) (4)

l

ðsa Þ ¼ sal ; ðsa Þl1 ðsa Þl2 ¼ ðsa Þl1 þl2 ; ðsa sb Þl ¼ ðsa Þl ðsb Þl ; sa sb ¼ sb sa ¼ sab . n

Deﬁnition 3. Let LWGA : S ! S, if LWGAx ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsa1 Þx1 ðsa2 Þx2 ðsan Þxn ¼ ðsax1 Þ ðsax2 Þ ðsaxn n Þ ¼ sa 1

2

ð3Þ

Qn x T where a ¼ j¼1 aj j , x ¼ ðx 1 ; x2 ; . . . ; xn Þ is the exponential weighting vector P n of the saj , and xj 2 ½0; 1, j¼1 xj ¼ 1, saj 2 S, then LWGA is called the linguistic weighted geometric averaging (LWGA) operator. Especially, if x ¼ ð1=n; 1=n; . . . ; 1=nÞT , then LWGA is called the linguistic geometric averaging (LGA) operator. T

Example 1. Assume x ¼ ð0:3; 0:1; 0:4; 0:2Þ , then LWGAx ðs4 ; s7 ; s3 ; s1 Þ ¼ ðs4 Þ

0:3

0:1

ðs7 Þ

ðs3 Þ

0:4

ðs1 Þ

0:2

¼ ðs40:3 Þ ðs70:1 Þ ðs30:4 Þ ðs10:2 Þ ¼ s2:86 Theorem 1 (Bounded) Minðsai Þ 6 LWGAx ðsa1 ; sa2 ; . . . ; san Þ 6 Maxðsai Þ i

i

Proof. Let Maxðsai Þ ¼ sb and Minðsai Þ ¼ sa , then i

i

LWGAx ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsa1 Þx1 ðsa2 Þx2 ðsan Þxn 6 ðsb Þ

x1

x

ðsb Þ 2 ðsb Þ Pn x ¼ ðsb Þ j¼1 j ¼ sb

xn

Z. Xu / Information Sciences 166 (2004) 19–30

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LWGAx ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsa1 Þx1 ðsa2 Þx2 ðsan Þxn P ðsa Þ

x1

ðsa Þ

x2

ðsa Þ

xn

¼ ðsa Þ

Pn j¼1

xj

¼ sa

hence Minðsai Þ 6 LWGAx ðsa1 ; sa2 ; . . . ; san Þ 6 Maxðsai Þ i

i

Deﬁnition 4. A LOWGA operator of dimension n is a mapping n LOWGA : S ! S, which has associated with it Pnan exponential weighting T vector w ¼ ðw1 ; w2 ; . . . ; wn Þ , with wj 2 ½0; 1 and j¼1 wj ¼ 1, such that LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þw1 ðsb2 Þw2 ðsbn Þwn ¼ ðsbw1 Þ ðsbw2 Þ ðsbwn n Þ ¼ sb where b ¼

Qn

wj j¼1 bj ,

1

2

ð4Þ

sbj is the jth largest of the sai . T

Example 2. Assume w ¼ ð0:3; 0:1; 0:4; 0:2Þ , then LOWGAw ðs4 ; s7 ; s3 ; s1 Þ ¼ ðs7 Þ

0:3

ðs4 Þ

0:1

ðs3 Þ

0:4

0:2

ðs1 Þ

¼ ðs70:3 Þ ðs40:1 Þ ðs30:4 Þ ðs10:2 Þ ¼ s3:20 In the following, let us look at some desirable properties associated with the LOWGA operator.

Theorem 2 (Commutativity)

LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ LOWGAw s0a1 ; s0a2 ; . . . ; s0an

where ðs0a1 ; s0a2 ; . . . ; s0an Þ is any permutation of ðsa1 ; sa2 ; . . . ; san Þ. Proof. Let w

w

w

LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þ 1 ðsb2 Þ 2 ðsbn Þ n w w w LOWGAw s0a1 ; s0a2 ; . . . ; s0an ¼ ðs0b1 Þ 1 ðs0b2 Þ 2 ðs0bn Þ n Since ðs0a1 ; s0a2 ; . . . ; s0an Þ is a permutation of ðsa1 ; sa2 ; . . . ; san Þ, we have sbj ¼ s0bj (j ¼ 1; 2; . . . ; n), then LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ LOWGAw ðs0a1 ; s0a2 ; . . . ; s0an Þ

Theorem 3 (Idempotency). If saj ¼ sa , for all j, then LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ sa

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Z. Xu / Information Sciences 166 (2004) 19–30

Proof. Since saj ¼ sa , for all j, it follows that w

w

LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þ 1 ðsb2 Þ 2 ðsbn Þ Pn w ¼ ðsa Þ j¼1 j ¼ sa

wn

Theorem 4 (Monotonicity). If saj 6 ^saj , for all j, then LOWGAw ðsa1 ; sa2 ; . . . ; san Þ 6 LOWGAw ð^sa1 ; ^sa2 ; . . . ; ^san Þ Proof. Let LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þ

w1

ðsb2 Þ

w2

ðsbn Þ

wn

LOWGAw ð^sa1 ; ^sa2 ; . . . ; ^san Þ ¼ ð^sb1 Þ

w1

ð^sb2 Þ

w2

ð^sbn Þ

wn

Since saj 6 ^saj , for all j, it follows that sbj 6 ^sbj , then LOWGAw ðsa1 ; sa2 ; . . . ; san Þ 6 LOWGAw ð^sa1 ; ^sa2 ; . . . ; ^san Þ

Theorem 5 (Bounded) Minðsai Þ 6 LOWGAw ðsa1 ; sa2 ; . . . ; san Þ 6 Maxðsai Þ i

i

Proof. Let Maxðsai Þ ¼ sb and Minðsai Þ ¼ sa , then i

i

w

w

LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þ 1 ðsb2 Þ 2 ðsbn Þ w

w

6 ðsb Þ 1 ðsb Þ 2 ðsb Þ

wn

wn

Pn w ¼ ðsb Þ j¼1 j ¼ sb

LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þw1 ðsb2 Þw2 ðsbn Þwn w

w

P ðsa Þ 1 ðsa Þ 2 ðsa Þ

wn

¼ ðsa Þ

Pn j¼1

wj

¼ sa

hence Minðsai Þ 6 LOWGAw ðsa1 ; sa2 ; . . . ; san Þ 6 Maxðsai Þ i

i

Theorem 6 (Linguistic geometric average). If the exponential weighting vector w ¼ ð1=n; 1=n; . . . ; 1=nÞT , then the LOWGA operator is reduced to the LGA operator, i.e., LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ sa Q 1=n n where a ¼ . j¼1 aj Proof. Since the exponential weighting vector w ¼ ð1=n; 1=n; . . . ; 1=nÞT , it follows that

Z. Xu / Information Sciences 166 (2004) 19–30

25

LOWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þw1 ðsb2 Þw2 ðsbn Þwn ¼ ðsb1 sb2 sbn Þ ¼ ðsa1 sa2 san Þ

1=n

1=n

¼ sa

The LWGA operator weights the linguistic argument, while the LOWGA operator weights the ordered position of the linguistic argument instead of weighting the argument itself, weights represent diﬀerent aspects in both the LWGA and the LOWGA operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose a LHGA operator. n

Deﬁnition 5. A LHGA operator is a mapping LHGA : S ! S, which has T associated with Pn it an exponential weighting vector w ¼ ðw1 ; w2 ; . . . ; wn Þ , with wj 2 ½0; 1, j¼1 wj ¼ 1, such that LHGAx;w ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þ

w1

ðsb2 Þ

w2

wn

ðsbn Þ

ð5Þ nx

where sbj is the jth largest of the linguistic weighted argument sai (sai ¼ ðsai Þ i , T i ¼ 1; 2; . . . ; n), x ¼ P ðx1 ; x2 ; . . . ; xn Þ is the exponential weighting vector of the n sai , with xj 2 ½0; 1, j¼1 xj ¼ 1, and n is the balancing coeﬃcient. T

T

Example 3. Assume x ¼ ð0:3; 0:1; 0:4; 0:2Þ , w ¼ ð0:3; 0:1; 0:4; 0:2Þ , and sa 1 ¼ s4 ;

sa2 ¼ s7 ;

sa3 ¼ s3 ;

sa4 ¼ s1

By Deﬁnition 5, we have sa1 ¼ ðs4 Þ40:3 ¼ s5:28 ;

sa2 ¼ ðs7 Þ40:1 ¼ s2:18 ;

sa3 ¼ ðs3 Þ40:4 ¼ s5:80 ;

sa4 ¼ ðs1 Þ40:2 ¼ s1 thus sb1 ¼ s5:80 ;

sb2 ¼ s5:28 ;

sb3 ¼ s2:18 ;

sb4 ¼ s1

therefore, LHGAx;w ðs5 ; s6 ; s4 ; s3 Þ ¼ ðs5:80 Þ

0:3

ðs5:28 Þ

0:1

ðs2:18 Þ

0:4

ðs1 Þ

0:2

¼ s2:73

Theorem 7. The LWGA operator is a special case of the LHGA operator.

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Z. Xu / Information Sciences 166 (2004) 19–30 T

Proof. Let w ¼ ð1=n; 1=n; . . . ; 1=nÞ , then LHGAx;w ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsb1 Þw1 ðsb2 Þw2 ðsbn Þwn ¼ ðsb1 sb2 sbn Þ where a ¼

Qn

xj j¼1 aj .

¼ ðsa1 Þ

x1

ðsa2 Þ

x2

1=n xn

ðsan Þ

¼ sa

h

Theorem 8. The LOWGA operator is a special case of the LHGA operator. T

Proof. Let x ¼ ð1=n; 1=n; . . . ; 1=nÞ , then sai ¼ sai ; i ¼ 1; 2; . . . ; n which completes the proof of Theorem 8. h From Theorems 7 and 8, we know that the LHA operator generalizes both the LWAA and the EOWA operators, and reﬂects the importance degrees of both the given argument and its ordered position. 3. A method for group decision making with linguistic preference relations Based on the LGA and the LHGA operators, we develop a practical method for group decision making with linguistic preference relations as follows: Step 1. For a group decision making problem with linguistic preference relations, let X ¼ fx1 ; x2 ; . . . ; xn g be the set of alternatives and D ¼ T fd1 ; d2 ; . . . ; dm g be the set of DMs, Pm and let k ¼ ðk1 ; k2 ; . . . ; km Þ be the weight vector of DMs, where kk P 0, k¼1 kk ¼ 1. The DM dk 2 D compares these alternatives with respect to a single criterion by the linguistic terms in the set S ¼ fsi g (i ¼ 1; . . . ; t), and constructs the linguistic preference relation ðkÞ Rk ¼ ðrij Þnn , where the diagonal elements in Rk are expressed as ‘‘–’’, which ðkÞ

ðkÞ

mean ‘‘undeﬁned’’, and rij rji ¼ st , i; j ¼ 1; 2; . . . ; n; i 6¼ j. Step 2. Utilize the LGA operator 1=ðn1Þ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ zi ¼ LGA ri1 ; ri2 ; . . . ; rin ¼ ri1 ri2 rin ; i ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ; m ðkÞ

to aggregate the preference information rij (i 6¼ j) in the ith line of Rk , and ðkÞ then get the preference degree zi of the ith alternative over all the other alternatives (corresponding to dk 2 D). Step 3. Utilize the LHGA operator ð1Þ ð2Þ ðmÞ zi ¼ LHGAk;w zi ; zi ; . . . ; zi

Z. Xu / Information Sciences 166 (2004) 19–30

27

ðkÞ

to aggregate zi (k ¼ 1; 2; . . . ; m) corresponding to the alternative xi , and then get the preference degree zi of the ith alternative over all the other alternatives, T where k ¼ ðk1 ; k2 ; . . . ; km Þ is the weight vector of DMs, where kk P 0, Pm T is the exponential weighting vector of the k¼1 kk ¼ 1; w ¼ ðw1 ; w2 ; . . . ; wm Þ P LHGA operator, with wk 2 ½0; 1, mk¼1 wk ¼ 1. Step 4. Rank all the alternatives and select the optimal one(s) in accordance with the values of zi (i ¼ 1; 2; . . . ; n). Step 5. End.

4. Illustrative example In this section, a group decision making problem involves the evaluation of ﬁve schools xi (i ¼ 1; 2; 3; 4; 5) of a university. One main criterion used is research. There are three DMs dk (k ¼ 1; 2; 3), whose weight vector k ¼ ð0:3; 0:4; 0:3ÞT . The DMs compare these ﬁve schools with respect to the criterion research by using the linguistic terms in the set S ¼ fs1 ¼ extremely poor; s2 ¼ very poor; s3 ¼ poor; s4 ¼ slightly poor; s5 ¼ fair; s6 ¼ slightly good; s7 ¼ good; s8 ¼ very good; s9 ¼ extremely goodg, and construct, respectively, the linguistic preference relations Rk (k ¼ 1; 2; 3) as listed in Tables 1–3. To get the best school(s), the following steps are involved: Step 1. Utilize the LGA operator to aggregate the preference information in ðkÞ the ith line of the Rk (i ¼ 1; 2; 3), and then get the preference degree zi of the ith school over all the other schools: ð1Þ

z2 ¼ s5:57 ;

ð2Þ

z2 ¼ s5:60 ;

ð3Þ

z2 ¼ s4:43 ;

z1 ¼ s3:60 ; z1 ¼ s4:36 ; z1 ¼ s4:28 ;

ð1Þ

z3 ¼ s3:94 ;

ð1Þ

z4 ¼ s5:63 ;

ð2Þ

z3 ¼ s4:56 ;

ð3Þ

z3 ¼ s5:38 ;

ð1Þ

z5 ¼ s4:74

ð2Þ

z4 ¼ s4:90 ;

ð3Þ

z4 ¼ s5:01 ;

ð1Þ

ð2Þ

z5 ¼ s4:95

ð3Þ

z5 ¼ s4:76

ð2Þ ð3Þ

Step 2. Utilize the LHGA operator (whose exponential weighting vector ðkÞ T w ¼ ð0:3; 0:5; 0:2Þ ) to aggregate zi (k ¼ 1; 2; 3) corresponding to the school xi , and then get the preference degree zi of the ith school over all the other schools: Table 1 Linguistic preference relation R1 x1 x2 x3 x4 x5

x1

x2

x3

x4

x5

– s8 s6 s7 s3

s2 – s5 s6 s4

s4 s5 – s8 s6

s3 s4 s2 – s7

s7 s6 s4 s3 –

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Z. Xu / Information Sciences 166 (2004) 19–30

Table 2 Linguistic preference relation R2 x1 x2 x3 x4 x5

x1

x2

x3

x4

x5

– s7 s6 s4 s5

s3 – s3 s6 s5

s4 s7 – s6 s4

s6 s4 s4 – s6

s5 s5 s6 s4 –

Table 3 Linguistic preference relation R3 x1 x2 x3 x4 x5

x1

x2

x3

x4

x5

– s8 s4 s6 s3

s2 – s6 s7 s6

s6 s4 – s5 s3

s4 s3 s5 – s7

s7 s4 s7 s3 –

z1 ¼ s4:12 ;

z2 ¼ s5:26 ;

z3 ¼ s4:72 ;

z4 ¼ s5:15 ;

z5 ¼ s4:75

Step 3. Utilize the values of zi (i ¼ 1; 2; 3; 4; 5) to rank the schools: x2 x4 x5 x3 x1 and thus the best school is x2 . 5. Conclusions In this paper, we have deﬁned some operational laws of linguistic variables and developed some new aggregation operators, which can be utilized to aggregate preference information taking the form of linguistic variables. Based on the LGA and the LHGA operators, we have proposed a practical method for group decision making with linguistic preference relations. Theoretical analyses and the numerical results all show that the method is straightforward and has no loss of information.

Acknowledgements I am grateful to three anonymous referees for their valuable comments and suggestions.

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References [1] S.H. Kim, S.H. Choi, J.K. Kim, An interactive procedure for multiple attribute group decision making with incomplete information: range-based approach, European Journal of Operational Research 118 (1999) 139–152. [2] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, NY, 1980. [3] L.G. Vargas, Reciprocal matrices with random coeﬃcients, Mathematical Modelling 3 (1982) 69–81. [4] Z.S. Xu, C.P. Wei, A consistency improving method in the analytic hierarchy process, European Journal of Operational Research 116 (1999) 443–449. [5] Z.S. Xu, On consistency of the weighted geometric mean complex judgement matrix in AHP, European Journal of Operational Research 126 (2000) 683–687. [6] Z.S. Xu, Generalized chi square method for the estimation of weights, Journal of Optimization Theory and Applications 107 (2000) 183–192. [7] F. Herrera, E. Herrera-Viedma, F. Chiclana, Multiperson decision-making based on multiplicative preference relations, European Journal of Operational Research 129 (2001) 372–385. [8] Z.S. Xu, A weighted least squares method for priorities in AHP, Information: An International Journal, in press. [9] T. Tanino, Fuzzy preference orderings in group decision making, Fuzzy Set and Systems 12 (1984) 117–131. [10] S.A. Orlovsky, Decision-making with a fuzzy preference relation, Fuzzy Sets and Systems 1 (1978) 155–167. [11] Z.S. Xu, Study on the relation between two classes of scales in AHP, Systems Engineering–– Theory and Practice 19 (9) (1999) 97–101. [12] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Set and Systems 97 (1998) 33–48. [13] Z.S. Xu, Algorithm for priority of fuzzy complementary judgement matrix, Journal of Systems Engineering 16 (2001) 311–314. [14] Z.S. Xu, Q.L. Da, An approach to improving consistency of fuzzy preference matrix, Fuzzy Optimization and Decision Making 2 (1) (2003) 3–12. [15] Z.S. Xu, Two methods for ranking alternatives in group decision-making with diﬀerent preference information, Information: An International Journal 6 (4) (2003) 389–394. [16] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating multiplicative preference relations in a multipurpose decision making model based on fuzzy preference relations, Fuzzy Sets and Systems 122 (2001) 277–291. [17] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. Part 1, 2 and 3, Information Sciences 8 (1975) 199–249, 301–357; 9 (1976) 43–80. [18] J. Kacprzyk, Group decision making with a fuzzy linguistic majority, Fuzzy Sets and Systems 18 (1986) 105–118. [19] M. Delgado, J.L. Verdegay, M.A. Vila, On aggregation operations of linguistic labels, International Journal of Intelligent Systems 8 (1993) 351–370. [20] F. Herrera, A sequential selection process in group decision making with linguistic assessment, Information Sciences 85 (1995) 223–239. [21] V. Torra, Negation functions based semantics for ordered linguistic labels, International Journal of Intelligent Systems 11 (1996) 975–988. [22] G. Bordogna, M. Fedrizzi, G. Passi, A linguistic modeling of consensus in group decision making based on OWA operator, IEEE Transactions on Systems, Man, and Cybernetics 27 (1997) 126–132. [23] F. Herrera, E. Herrera-Viedma, Choice functions and mechanisms for linguistic preference relations, European Journal of Operational Research 120 (2000) 144–161.

30

Z. Xu / Information Sciences 166 (2004) 19–30

[24] Z.S. Xu, Study on methods for multiple attribute decision making under some situations, Ph.D. Thesis, Southeast University, Nanjing, China, 2002. [25] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decisionmaking, IEEE Transactions on Systems, Man, and Cybernetics 18 (1988) 183–190. [26] Z.S. Xu, Q.L. Da, An overview of operators for aggregating information, International Journal of Intelligent Systems 18 (2003) 953–969. [27] R.R. Yager, J. Kacprzyk, The Ordered Weighted Averaging Operators: Theory and Applications, Kluwer, Norwell, MA, 1997. [28] F. Herrera, L. Martınez, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 8 (2000) 746–752. [29] J. Aczel, T.L. Saaty, Procedures for synthesizing ratio judgements, Journal of Mathematical Psychology 27 (1983) 93–102. [30] F. Herrera, E. Herrera-Viedma, F. Chiclana, Multiperson decision-making based on multiplicative preference relations, European Journal of Operational Research 129 (2001) 372–385. [31] Z.S. Xu, Q.L. Da, The ordered weighted geometric averaging operators, International Journal of Intelligent Systems 17 (2002) 709–716. [32] F. Herrera, E. Herrera-Viedma, J.L. Verdegay, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets and Systems 78 (1996) 73–87.

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