Group decision making based on multiple types of linguistic preference relations

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Information Sciences 178 (2008) 452–467 www.elsevier.com/locate/ins

Group decision making based on multiple types of linguistic preference relations q Zeshui Xu

*

Department of Management Science and Engineering, School of Economics and Management, Tsinghua University, Beijing 100084, China Received 27 October 2006; received in revised form 4 May 2007; accepted 8 May 2007

Abstract In this paper, we investigate group decision making problems with multiple types of linguistic preference relations. The paper has two parts with similar structures. In the first part, we transform the uncertain additive linguistic preference relations into the expected additive linguistic preference relations, and present a procedure for group decision making based on multiple types of additive linguistic preference relations. By using the deviation measures between additive linguistic preference relations, we give some straightforward formulas to determine the weights of decision makers, and propose a method to reach consensus among the individual preferences and the group’s opinion. In the second part, we extend the above results to group decision making based on multiple types of multiplicative linguistic preference relations, and finally, a practical example is given to illustrate the application of the results. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Group decision making; Linguistic preference relation; Aggregation operator; Expectation function; Linguistic label

1. Introduction Linguistic preference relations are usually used by decision makers (DMs) to express their linguistic preference information based on pairwise comparisons of alternatives. Group decision making based on linguistic preference relations is a hot research topic, which has received a great deal of attention from researchers recently [11,13,18–20,24,45,49,50]. Herrera et al. [18] developed a consensus model for group decision making under linguistic assessments, which is based on the use of fuzzy majority of consensus represented by means of a linguistic quantifier [34,61]. Herrera et al. [19] presented several models of group decision making based on the rational properties of the linguistic ordered weighted averaging (LOWA) operator. Herrera et al. [20] presented a consensus model for the consensus reaching process based on a linguistic framework in group decision making. Herrera and Herrera-Viedma [13] defined various linguistic choice sets of alternatives, and depending on the q

The work was supported by the National Natural Science Foundation of China (No. 70571087 and No. 70321001), China Postdoctoral Science Foundation (No. 20060390051), and the National Science Fund for Distinguished Young Scholars of China (No. 70625005). * Tel.: +86 1 62795845. E-mail address: [email protected] 0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.05.018

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required precision, gave two application mechanisms of linguistic choice functions to obtain the solution set of alternatives. Herrera and Herrera-Viedma [12] analyzed the steps to follow in the linguistic decision analysis of a group decision making problem. Xu [45] introduced some aggregation operators, such as the linguistic geometric averaging (LGA) operator and linguistic hybrid geometric averaging (LHGA) operator, etc., and developed a method based on these operators for group decision making based on linguistic preference relations. Xu [49] investigated the deviation measures of linguistic preference relations and established a theoretic basis for the application of linguistic preference relations in group decision making. All of these attempts focus on a single type of representation format of linguistic preference relations. Herrera-Viedma et al. [24] presented a model of consensus support system to assist DMs in all phases of the consensus reaching process of group decision making problems with multigranular linguistic preference relations. The developed consensus support system model is based on (1) a multigranular linguistic methodology, (2) two consensus criteria, consensus degrees and proximity measures, and (3) a guidance advice system. The multigranular linguistic methodology permits the unification of different linguistic domains to facilitate the calculus of consensus degrees and proximity measures on the basis of the DMs’ opinions. The consensus degrees assess the agreement amongst all the DMs’ opinions, while the proximity measures are used to find out how far the individual opinions are from the group’s opinion. In [50], we proposed some aggregation operators including the uncertain linguistic geometric mean (ULGM) operator, uncertain linguistic weighted geometric mean (ULWGM) operator, uncertain linguistic ordered weighted geometric (ULOWG) operator, and induced uncertain linguistic ordered weighted geometric (IULOWG) operator. Moreover, based on the ULOWG and IULOWG operators and the formula for comparing uncertain multiplicative linguistic variables, we developed an approach to group decision making based on uncertain multiplicative linguistic preference relations. However, each DM is characterized by his/her own personal experience, learning, situation, state of mind, and so forth, the DMs’ preferences may differ substantially, that is, the DMs generally use different representation formats to express their preferences for each pair of alternatives in a group decision making problem. Herrera et al. [5–7,14–17] studied systematically the group decision making problems in which the decision information about alternatives provided by the DMs takes the form of different numerical preference structures (preference orderings [8,37], utility functions [29,39], fuzzy preference relations [32,39], and multiplicative preference relations [38]). But in many real life situations, such as negotiation processes, project investment, and supply chain management, etc. [48], the decision information provided by the DMs may be presented by means of different linguistic preference representation structures, such as traditional linguistic preference relations [12,13,18–20,24,45,46,49] and uncertain linguistic preference relations [51], etc., because that (1) a decision may be made under time pressure and lack of data, (2) the DMs have limited attention and information processing capabilities, and (3) in group settings, all participants do not have equal expertise about problem domain [25,26,33,36,44]. So far, no method has been proposed to deal with this issue, and thus it is an interesting and promising research field, which is worth paying attention to. In this paper, we will develop an approach to group decision making problems with different representation formats of linguistic preference relations (traditional additive linguistic preference relations, traditional multiplicative linguistic preference relations, uncertain additive linguistic preference relations, and uncertain multiplicative linguistic preference relations). The paper has two independent parts with similar structures. In Section 2, we introduce the additive linguistic labels and their operational laws. In Section 3, we define the concepts of various types of additive linguistic preference relations. Section 4 presents a procedure for group decision making based on multiple types of additive linguistic preference relations. Section 5 introduces the multiplicative linguistic labels and their operational laws. Section 6 defines the concepts of various types of multiplicative linguistic preference relations. Section 7 presents a procedure for group decision making based on multiple types of multiplicative linguistic preference relations. Section 8 gives a practical example to illustrate the developed procedures, and finally, Section 9 concludes the paper. 2. Additive linguistic labels In many actual decision making processes, a DM usually provides his/her preference information by using linguistic labels [2–4,9,10,22,23,27,30,35,40–43,47,51–60]. For example, when evaluating the comfort or design of a car, labels like ‘‘good’’, ‘‘fair’’, ‘‘poor’’ can be used; when evaluating a car’s speed linguistic labels like

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‘‘fast’’, ‘‘very fast’’, ‘‘slow’’ can be used [3,27]. For computational convenience, let S ¼ fsa ja ¼ t; . . . ; tg be a finite and totally ordered discrete label set, where t is a positive integer, sa represents a possible value for a linguistic variable, and requires that (1) sa < sb iff a < b, and (2) there is the negation operator: negðsa Þ ¼ sa , especially, negðs0 Þ ¼ s0 . We call this linguistic label set S an additive linguistic label set. Obviously, the mid label s0 represents an assessment of ‘‘indifference’’, and with the rest of the labels being placed symmetrically around it. For example, S can be defined as [48,49]: S ¼ fs4 ¼ extremely poor; s3 ¼ very poor; s2 ¼ poor; s1 ¼ slightly poor; s0 ¼ fair; s1 ¼ slightly good; s2 ¼ good; s3 ¼ very good; s4 ¼ extremely goodg In the process of information aggregation, some results may not exactly match any linguistic labels in S. To preserve all the given information, Herrera and Martı´nez [21–23] developed a fuzzy linguistic representation model based on the symbolic translation. It represents the linguistic information by means of 2-tuples, which are composed by a linguistic label and the symbolic translation represented by a numeric value assessed in ½0:5; 0:5Þ. Xu [48,49] extended the discrete label set S to a continuous label set  S ¼ fsa ja 2 ½q; qg, where qðq > tÞ is a sufficiently large positive integer. If sa 2 S, then sa is termed an original additive linguistic label, otherwise, sa is termed a virtual additive linguistic label. For any two labels sa ; sb 2  S ; Xu [49] defined their operational laws as follows: (1) sa  sb ¼ sb  sa ¼ saþb ; (2) ksa ¼ ska , k 2 ½1; 1; (3) kðsa  sb Þ ¼ ksa  ksb , k 2 ½1; 1. Definition 1. Let sa ; sb 2  S , then dðsa ; sb Þ ¼ ja  bj

ð1Þ

is called a deviation between sa and sb. It is clear that the deviation measure (1) has the following properties: Theorem 1. Let sa ; sb ; sc 2  S , then (1) dðsa ; sb Þ P 0, especially, dðsa ; sb Þ ¼ 0 iff sa ¼ sb ; (2) dðsa ; sb Þ ¼ dðsb ; sa Þ; (3) dðsa ; sb Þ 6 dðsa ; sc Þ þ dðsc ; sb Þ.

Definition 2. A linguistic weighted arithmetic averaging (LWAA) operator of dimension n is a mapping  Þn !  LWAA : ðS S , which is given by LWAAw ðsa1 ; sa2 ; . . . ; san Þ ¼ w1 sa1  w2 sa2      wn san T

ð2Þ Pn

S , with wj P 0 and j¼1 wj ¼ 1. Especially, if where w ¼ ðw1 ; w2 ; . . . ; wn Þ is the weighting vector of saj 2  w ¼ ð1=n; 1=n; . . . ; 1=nÞT , then the LWAA operator is reduced to a linguistic arithmetic averaging (LAA) operator 1 LAAðsa1 ; sa2 ; . . . ; san Þ ¼ ðsa1  sa2      san Þ ð3Þ n T If wi ¼ 1; wk ¼ 0 for all k 6¼ i, then LWAAw ðsa1 ; sa2 ; . . . ; san Þ ¼ sai . Especially, if w ¼ ð1; 0; . . . ; 0Þ , then T LWAAw ðsa1 ; sa2 ; . . . ; san Þ ¼ sa1 ; if w ¼ ð0; 0; . . . ; 1Þ , then LWAAw ðsa1 ; sa2 ; . . . ; san Þ ¼ san . The fundamental aspect of the LWAA operator is that it computes the aggregated linguistic labels taking into account the importance of the sources of information. Definition 3. Let ~s ¼ ½sa ; sb , where sa ; sb 2  S , sa and sb are, respectively, the lower and upper limits of ~s, then we call ~s an uncertain additive linguistic variable [47], and denote e S as the set of all the uncertain additive linguistic variables.

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3. Multiple types of additive linguistic preference relations In the process of group decision making, the DMs generally use different representation formats to express their preferences for each pair of alternatives. Herrera et al. [5–7,14–17] studied systematically the group decision making problems, in which the decision information about alternatives provided by the DMs takes the form of different numerical preference structures, such as preference orderings, utility functions, fuzzy preference relations, and multiplicative preference relations, etc. They also developed some useful techniques to solve these problems. But in many real life situations, such as negotiation processes, project investment, and supply chain management, etc. [48], the decision information provided by the DMs may be presented by means of different linguistic preference representation structures, such as traditional linguistic preference relations [12,13,18–20,24,45,46,49] and uncertain linguistic preference relations [50], etc. It seems that no approach has been developed to dealing with this issue by now. In what follows, we will focus our attention on it. Consider a decision making problem where X ¼ fx1 ; x2 ; . . . ; xn g is the set of alternatives, a DM compares each pair of alternatives in X by means of the linguistic labels in the set S ¼ fsa ja ¼ t; . . . ; tg, and provides his/her linguistic preference information for each pair of alternatives by using one of the representation formats: traditional additive linguistic preference relations and uncertain additive linguistic preference relations, which are defined as follows. Definition 4. An additive linguistic preference relation A ¼ ðaij Þnn on a set of alternatives X is characterized by a function S lA : X  X ! 

ð4Þ

where aij ¼ lA ðxi ; xj Þ, which estimates the preference degree of the alternative xi over xj, and satisfies [49]: aij  aji ¼ s0 ; aii ¼ s0 ;

for all i; j ¼ 1; 2; . . . ; n

ð5Þ

For convenience, here we call this additive linguistic preference relation a traditional additive linguistic preference relation [12,13,49]. Let A1 ; A2 ; . . . ; Am be the traditional additive linguistic preference relations provided by m DMs, where ðkÞ Ak ¼ ðaij Þnn ðk ¼ 1; 2; . . . ; m; i; j ¼ 1; 2; . . . ; nÞ, then the aggregated additive linguistic preference relation A_ ¼ ða_ ij Þnn is also a traditional additive linguistic preference relation [49], where ð1Þ

ð2Þ

ðmÞ

ð1Þ

ð2Þ

ðmÞ

a_ ij ¼ LWAAw ðaij ; aij ; . . . ; aij Þ ¼ w1 aij  w2 aij      wm aij

ð6Þ

and a_ ij  a_ ji ¼ s0 ; a_ ii ¼ s0 ;

for all i; j ¼ 1; 2; . . . ; n

ð7Þ

T

where w ¼ ðw1 ; w2 ; . . . ; wm Þ is the weight vector of A1 ; A2 ; . . . ; Am , with wk P 0 and

Pm

k¼1 wk

¼ 1.

e ¼ ð~aij Þ on a set of alternatives X is charDefinition 5. An uncertain additive linguistic preference relation A nn acterized by a function le : X  X ! e S A

ð8Þ

where ~ aij ¼ leðxi ; xj Þ, which estimates the preference degree of the alternative xi over xj, and satisfies A ðlÞ

ðuÞ

ðlÞ

ðuÞ

ðlÞ

ðuÞ

ðuÞ

ðlÞ

ðlÞ

ðuÞ

~ S ; aij  aji ¼ s0 ; aij  aji ¼ s0 ; aii ¼ aii ¼ s0 ; aij ¼ ½aij ; aij ; aij ; aij 2  for all i; j ¼ 1; 2; . . . ; n

ð9Þ

As is well known, expected value is a central principle in the probability theory, which is used for average estimation of some variable, and the average value of any variable in a long run gets close to its expected value. Expected value has been used in a very wide range of practical applications [28]. In the following we define the expected additive linguistic preference relation of an uncertain additive linguistic preference relation.

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e ¼ ð~ Definition 6. Let A aij Þnn be an uncertain additive linguistic preference relation, then we define its expected e ¼ ðEð~aij ÞÞ , where additive linguistic preference relation as Eð AÞ nn ðlÞ

ðuÞ

Eð~ aij Þ ¼ ð1  hÞaij  haij ;

Eð~ aji Þ ¼ Eð~ aij Þ;

ð10Þ

for all i < j

and h is an index that reflects the DM’s risk-bearing attitude. If h > 0:5, then the DM is a risk lover; if h ¼ 0:5, then the attitude of the DM is neutral to the risk; if h < 0:5, then the DM is a risk avertor. In general, h can be given by the DM directly. By the operational laws of additive linguistic labels and Definition 6, we have ðlÞ

ðuÞ

ðlÞ

ðuÞ

Eð~ aij Þ  Eð~ aji Þ ¼ Eð~ aij Þ  ðEð~ aij ÞÞ ¼ ðð1  hÞaij  haij Þ  ððh  1Þaij  ðhaij ÞÞ ðlÞ

ðuÞ

¼ ð1  h þ h  1Þaij  ðh  hÞaij ¼ s0  s0 ¼ s0 ; ðlÞ

ðuÞ

Eð~ aii Þ ¼ ð1  hÞaii  haii ¼ ð1  hÞs0  hs0 ¼ ð1  h þ hÞs0 ¼ s0 ;

for all i; j ¼ 1; 2; . . . ; n

ð11Þ

for all i ¼ 1; 2; . . . ; n

ð12Þ

4. A procedure for group decision making based on multiple types of additive linguistic preference relations The increasing complexity of the socio-economic environment makes it less and less possible for a single DM to consider all relevant aspects of a problem [25]. As a result, many decision making processes, in the real world, take place in group settings. Furthermore, since judgments of people depend on personal psychological aspects, such as experience, learning, situation, state of mind, and so forth, the DMs generally use different representation formats to express their linguistic preferences for each pair of alternatives in a group decision making problem. Below we present a procedure for group decision making based on various types of additive linguistic preference relations. (Procedure I) Step 1. For a group decision making problem with additive linguistic preference relations, there exist a finite T set of alternatives X ¼ fx1 ; x2 ; . . . ; xn g and a finite set of DMs D ¼ fd 1 ; d 2P ; . . . ; d m g. Let w ¼ ðw1 ; w2 ; . . . ; wm Þ m be the weight vector of the DMs d k ðk ¼ 1; 2; . . . ; mÞ, where wk P 0 and k¼1 wk ¼ 1. Bodily [1] developed a straightforward method for deriving the DMs’ weights as a result of designation of voting weights by each DM to a delegation subcommittee made up of other DMs of the group. Here, we utilize Bodily’ method to determine the original weights of the DMs d k ðk ¼ 1; 2; . . . ; mÞ. Based on the additive linguistic label set S ¼ fsa ja ¼ t; . . . ; tg, each DM d k 2 D provides his/her preference information for each pair of alternatives, ðkÞ and constructs an additive linguistic preference relation Ak ¼ ðaij Þnn by using one of the following representation formats: traditional additive linguistic preference relations and uncertain additive linguistic preference relations. Step 2. Utilize (10) to transform all the uncertain additive linguistic preference relations into the expected additive linguistic preference relations. We denote the expected additive linguistic preference relations of all the addiðkÞ ðkÞ tive linguistic preference relations Ak ¼ ðaij Þnn ðk ¼ 1; 2; . . . ; mÞ by EðAk Þ ¼ ðEðaij ÞÞnn ðk ¼ 1; 2; . . . ; mÞ (here, for convenience, we also denote the traditional additive linguistic preference relations by the expected additive linguistic preference relations, for example, if Ai is a traditional additive linguistic preference relation, then its expected additive linguistic preference relation is also Ai, i.e., EðAi Þ ¼ Ai Þ. Step 3. Utilize the LWAA operator ð1Þ

ð2Þ

ðmÞ

a_ ij ¼ LWAAw ðEðaij Þ; Eðaij Þ; . . . ; Eðaij ÞÞ;

for all i; j ¼ 1; 2; . . . ; n

ð13Þ ðkÞ

to aggregate all the expected additive linguistic preference relations EðAk Þ ¼ ðEðaij ÞÞnn ðk ¼ 1; 2; . . . ; mÞ into a collective additive linguistic preference relation A_ ¼ ða_ ij Þnn . To measure the deviation degree between each expected additive linguistic preference relation ðkÞ EðAk Þ ¼ ðEðaij ÞÞnn and the collective additive linguistic preference relation A_ ¼ ða_ ij Þnn , we define n X ðkÞ _ ¼ 1 dðEðAk Þ; AÞ dðEðaij Þ; a_ ij Þ ð14Þ n2 i;j¼1 ðkÞ _ where dðEðaðkÞ _ ij Þ is the deviation between Eðaij Þ and a_ ij , which is as the deviation between EðAk Þ and A, ij Þ; a calculated by using (1).

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Similar to Theorem 1, we have _ P 0, especially, dðEðAk Þ; AÞ _ ¼ 0 iff EðAk Þ ¼ A; _ (1) dðEðAk Þ; AÞ _ _ (2) dðEðAk Þ; AÞ ¼ dðA; EðAk ÞÞ. _ generally reflects the degree of departure of the individual additive linguistic prefThe deviation dðEðAk Þ; AÞ _ erences from the group’s opinion. In the following, we define a function of the deviation dðEðAk Þ; AÞ: _ y ¼ f ðdðEðAk Þ; AÞÞ

ð15Þ

and then, based on (15), we give a formula for determining the DMs’ weights: _ f ðdðEðAk Þ; AÞÞ ; wk ¼ Pm _ k¼1 f ðdðEðAk Þ; AÞÞ

k ¼ 1; 2; . . . ; m

ð16Þ

Now we discuss the following cases: Case 1. If the majority opinion should be emphasized, then the low weights should be assigned to the DM dk _ In this case, y ¼ f ðdðEðAk Þ; AÞÞ _ is a decreasing function of the deviation with high deviation dðEðAk Þ; AÞ. _ dðEðAk Þ; AÞ. For example, if _ _ ¼ edðEðAk Þ;AÞ f ðdðEðAk Þ; AÞÞ

ð17Þ

then _

edðEðAk Þ;AÞ wk ¼ Pm dðEðA Þ;AÞ _ ; k k¼1 e

k ¼ 1; 2; . . . ; m

ð18Þ

Case 2. If the minority opinion should be emphasized, then the high weights should be assigned to the DM dk _ In this case, y ¼ f ðdðEðAk Þ; AÞÞ _ is an increasing function of the deviation with high deviation dðEðAk Þ; AÞ. _ For example, if dðEðAk Þ; AÞ. _

_ ¼ edðEðAk Þ;AÞ f ðdðEðAk Þ; AÞÞ

ð19Þ

then _

edðEðAk Þ;AÞ wk ¼ Pm dðEðA Þ;AÞ _ ; k k¼1 e

k ¼ 1; 2; . . . ; m

ð20Þ

Case 3. If both the majority and minority opinions should be emphasized simultaneously, then we can use the following formula to determine the DMs’ weights: _ _ f1 ðdðEðAk Þ; AÞÞ f2 ðdðEðAk Þ; AÞÞ wk ¼ ð1  aÞ Pm þ a Pm ; _ _ k¼1 f1 ðdðEðAk Þ; AÞÞ k¼1 f2 ðdðEðAk Þ; AÞÞ

k ¼ 1; 2; . . . ; m

ð21Þ

_ and y 2 ¼ f2 ðdðEðAk Þ; AÞÞ _ are, where a 2 ½0; 1 (which can be given by the DMs directly), y 1 ¼ f1 ðdðEðAk Þ; AÞÞ _ respectively, the decreasing and increasing functions of the deviation dðEðAk Þ; AÞ. For example, if _ and f2 ðdðEðAk Þ; AÞÞ _ are, respectively, taken from (17) and (19), then (21) can be rewritten as: f1 ðdðEðAk Þ; AÞÞ _

_

edðEðAk Þ;AÞ edðEðAk Þ;AÞ þ a Pm dðEðA Þ;AÞ wk ¼ ð1  aÞ Pm dðEðA Þ;AÞ _ _ ; k k k¼1 e k¼1 e

k ¼ 1; 2; . . . ; m

ð22Þ

Especially, if a = 0, then (22) is reduced to (18); if a = 1, then (22) is reduced to (20). Therefore, Cases 1 and 2 are the special cases of Case 3. Furthermore, if _ 6s dðEðAk Þ; AÞ

ð23Þ

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then EðAk Þ and A_ are called of acceptable deviation, where s is the threshold of acceptable deviation, which are predefined by all the DMs (who participate in determining the DMs’ original weights) in practical applica_ ¼ 0, then EðAk Þ and A_ are equal. However, if dðEðAk Þ; AÞ _ > s, then we will retions. Especially, if dðEðAk Þ; AÞ _ turn Ak together with EðAk Þ and A to the DM dk for revaluation. We repeat this process (in which the DMs’ original weights are replaced by the new weights derived from (22) for obtaining the new collective additive linguistic preference relation A_ by using (13)) until EðAk Þ and A_ are of acceptable deviation or the process will stop as the repetition times reach the maximum number predefined by the DMs, and then go to the next step. Step 4. Utilize the LAA operator a_ i ¼ LAAða_ i1 ; a_ i2 ; . . . ; a_ in Þ;

for all i ¼ 1; 2; . . . ; n

ð24Þ

_ and then get the averaged preference degree a_ i to aggregate the preference information a_ ij in the ith line of A, of the alternative xi over all the other alternatives. Step 5. Rank all the alternatives xi ði ¼ 1; 2; . . . ; nÞ and then select the best one in accordance with the values of a_ i ði ¼ 1; 2; . . . ; nÞ. Step 6. End. 5. Multiplicative linguistic labels In this section, we define the linguistic label set S (given in Section 2) by another form S 0 ¼ fsa ja ¼ 1=t; . . . ; tg, whose cardinality value is odd [46], t is a positive integer, and the linguistic label sa has the following characteristics: (1) the set is ordered: if a > b, then sa > sb , and (2) there is the reciprocal operator: recðsa Þ ¼ sb such that ab ¼ 1. Especially, recðs1 Þ ¼ s1 . We call this linguistic label set S 0 a multiplicative linguistic label set. The cardinality of S 0 must be small enough so as not to impose useless precision to the DMs and it must be rich enough in order to allow a discrimination of the performances of each alternative in a limited number of grades [3]. The cardinality of the linguistic set S 0 should be properly determined in accordance with the actual situation in practical application. In general, it can be taken as 7 and 9 [3,12,13,18–20,24,45,49]. For example, S 0 can be defined as: S 0 ¼ fs1=5 ¼ extremely low; s1=4 ¼ very low; s1=3 ¼ low; s1=2 ¼ slightly low; s1 ¼ fair; s2 ¼ slightly high; s3 ¼ high; s4 ¼ very high; s5 ¼ extremely highg To preserve all the given information, we extend the discrete linguistic label set S 0 to a continuous linguistic label set  S 0 ¼ fsa ja 2 ½1=q; qg, where qðq > tÞ is a sufficiently large positive integer. If sa 2 S 0 , then we call sa an original multiplicative linguistic label, otherwise, we call sa a virtual multiplicative linguistic label. Consider any two labels sa ; sb 2  S 0 , we define their operational laws as follows [45]: (1) sa  sb ¼ sb  sa ¼ sab ; k (2) ðsa Þ ¼ sak , k 2 ½1; 1; (3) ðsa  sb Þk ¼ ðsa Þk  ðsb Þk , k 2 ½1; 1. Similar to Definition 1, we give the following definition: Definition 7. Let sa, sb 2  S 0 , then we call   a b ; d 0 ðsa ; sb Þ ¼ max b a a deviation between sa and sb. Obviously, by (25), we have Theorem 2. Let sa ; sb ; sc 2  S 0 , then   (1) d 0 sa ; sb P 1, especially, d 0 ðsa ; sb Þ ¼ 1 iff sa ¼ sb ;

ð25Þ

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(2) d 0 ðsa ; sb Þ ¼ d 0 ðsb ; sa Þ; (3) If d 0 ðsa ; sc Þ ¼ 1 and d 0 ðsc ; sb Þ ¼ 1, then d 0 ðsa ; sb Þ ¼ 1. A linguistic weighted geometric averaging (LWGA) operator of dimension n is a mapping  0 Þn !  LWGA : ðS S 0 , which is given by w1

LWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ ðsa1 Þ

 ðsa2 Þ

w2

     ðsan Þ

wn

T

ð26Þ Pn

0

where w ¼ ðw1 ; w2 ; . . . ; wn Þ is the exponential weighting vector of saj 2 S , with wj P 0 and j¼1 wj ¼ 1. Especially, if w ¼ ð1=n; 1=n; . . . ; 1=nÞT , then the LWGA operator is reduced to a linguistic geometric averaging (LGA) operator 1=n

LGAðsa1 ; sa2 ; . . . ; san Þ ¼ ðsa1  sa2      san Þ

ð27Þ T

If wi ¼ 1; wk ¼ 0 for all k 6¼ i; then LWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ sai . Especially, if w ¼ ð1; 0; . . . ; 0Þ , then T LWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ sa1 ; if w ¼ ð0; 0; . . . ; 1Þ , then LWGAw ðsa1 ; sa2 ; . . . ; san Þ ¼ san . Let ~s ¼ ½sa ; sb , where sa ; sb 2  S 0 , sa and sb are, respectively, the lower and upper limits of ~s, then we call ~s an uncertain multiplicative linguistic variable [50], and denote e S 0 as the set of all the uncertain multiplicative linguistic variables. 6. Multiple types of multiplicative linguistic preference relations Consider a decision making problem, the DM compares each pair of alternatives in X by the linguistic labels in the set S 0 ¼ fsa ja ¼ 1=t; . . . ; tg, and provides his/her linguistic preference information for each pair of alternatives by using one of the representation formats: traditional multiplicative linguistic preference relations and uncertain multiplicative linguistic preference relations, which are presented as follows. A multiplicative linguistic preference relation B ¼ ðbij Þnn on a set of alternatives X is characterized by a function lB : X  X !  S0

ð28Þ

where bij ¼ lB ðxi ; xj Þ, which estimates the preference intensity of the alternative xi over xj, and satisfies [45]: bij  bji ¼ s1 ; bii ¼ s1 ;

for all i; j ¼ 1; 2; . . . ; n

ð29Þ

For convenience, here we call this multiplicative linguistic preference relation a traditional multiplicative linguistic preference relation. Let B1 ; B2 ; . . . ; Bm be the traditional multiplicative linguistic preference relations provided by m DMs, where ðkÞ Bk ¼ ðbij Þnn ðk ¼ 1; 2; . . . ; m; i; j ¼ 1; 2; . . . ; nÞ, then the aggregated multiplicative linguistic preference rela_ tion B ¼ ðb_ ij Þnn is also a traditional multiplicative linguistic preference relation [46], where ð1Þ ð2Þ ðmÞ ð1Þ ð2Þ ðmÞ b_ ij ¼ LWGAw ðbij ; bij ; . . . ; bij Þ ¼ ðbij Þw1  ðbij Þw2      ðbij Þwm

ð30Þ

and b_ ij  b_ ji ¼ s1 ; b_ ii ¼ s1 ; T

for all i; j ¼ 1; 2; . . . ; n

where w ¼ ðw1 ; w2 ; . . . ; wm Þ is the weight vector of B1 ; B2 ; . . . ; Bm , with wk P 0 and

ð31Þ Pm

k¼1 wk

¼ 1.

e ¼ ð~bij Þ Definition 8. An uncertain multiplicative linguistic preference relation B nn on a set of alternatives X is characterized by a function e0 le : X  X ! S ð32Þ B ~ij ¼ l ðxi ; xj Þ, which estimates the preference intensity of the alternative xi over xj, and satisfies [50]: where b eB ðlÞ ðuÞ ðlÞ ðuÞ ðuÞ ðlÞ ðlÞ ðuÞ ~ij ¼ ½bij ; bijðuÞ ; bðlÞ S 0 ; bij  bji ¼ s1 ; bij  bji ¼ s1 ; bii ¼ bii ¼ s1 ; b ij ; bij 2  for all i; j ¼ 1; 2; . . . ; n

ð33Þ

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In the following we further define the expected multiplicative linguistic preference relation of an uncertain multiplicative linguistic preference relation. e ¼ ð~ Definition 9. Let B bij Þnn be an uncertain multiplicative linguistic preference relation, then the expected e is defined as Eð BÞ e ¼ ðEð~bij ÞÞ , where multiplicative linguistic preference relation of B nn ðlÞ ðuÞ bji Þ ¼ ðEð~ bij ÞÞ1 ; Eð~ bij Þ ¼ ðbij Þ1h  ðbij Þh ; Eð~

ð34Þ

for all i < j

and h is an index that reflects the DM’s risk-bearing attitude. By the operational laws of multiplicative linguistic labels and Definition 9, we have ðlÞ 1h ðuÞ h ðlÞ 1h ðuÞ h 1 1 bji Þ ¼ Eð~ bij Þ  ðEð~ bij ÞÞ ¼ ðbij Þ  ðbij Þ  ððbij Þ  ðbij Þ Þ Eð~ bij Þ  Eð~ ðlÞ 1h

¼ ðbij Þ ¼ s1 ; ðlÞ

ðuÞ h

ðlÞ ð1hÞ

 ðbij Þ  ðbij Þ

ðuÞ h

 ðbij Þ

ðlÞ 1hð1hÞ

¼ ðbij Þ

ðuÞ hh

 ðbij Þ

for all i; j ¼ 1; 2; . . . ; ðuÞ

Eð~ bii Þ ¼ ðbii Þ1h  ðbii Þh ¼ ðs1 Þ1h  ðs1 Þh ¼ ðs1 Þ1hþh ¼ s1 ;

¼ s1  s1 ð35Þ

for all i ¼ 1; 2; . . . ; n

ð36Þ

7. A procedure for group decision making based on multiple types of multiplicative linguistic preference relations In this section, we present a procedure for group decision making based on various types of multiplicative linguistic preference relations, which is descried as follows: (Procedure II) Step 1. For a group decision making problem with multiplicative linguistic preference relations, there exist a finite set of alternatives X ¼ fx1 ; x2 ; . . . ; xn g and a finite set of DMs D ¼ fd 1 ; d 2 ; . . P . ; d m g. Let T m w ¼ ðw1 ; w2 ; . . . ; wm Þ be the weight vector of the DMs dk ðk ¼ 1; 2; . . . ; mÞ, where wk P 0 and k¼1 wk ¼ 1 (here, we also utilize Bodily’s method [1] to determine the original weights of the DMs d k ðk ¼ 1; 2; . . . ; mÞÞ. Based on the multiplicative linguistic label set S 0 ¼ fsa ja ¼ 1=t; . . . ; tg, each DM d k 2 D provides his/her linguistic preference information for each pair of alternatives, and constructs a multiplicative linguistic preferðkÞ ence relation Bk ¼ ðbij Þnn by using one of the following representation formats: traditional multiplicative linguistic preference relations and uncertain multiplicative linguistic preference relations. Step 2. Utilize (34) to transform all the uncertain multiplicative linguistic preference relations into the expected multiplicative linguistic preference relations. We denote the expected multiplicative linguistic preferðkÞ ence relations of all the multiplicative linguistic preference relations Bk ¼ ðbij Þnn ðk ¼ 1; 2; . . . ; mÞ by ðkÞ EðBk Þ ¼ ðEðbij ÞÞnn ðk ¼ 1; 2; . . . ; mÞ (here, for convenience, we also denote the traditional multiplicative linguistic preference relations by the expected multiplicative linguistic preference relations). Step 3. Utilize the LWGA operator ð1Þ ð2Þ ðmÞ ð37Þ b_ ij ¼ LWGAw ðEðbij Þ; Eðbij Þ; . . . ; Eðbij ÞÞ; for all i; j ¼ 1; 2; . . . ; n ðkÞ

to aggregate all the expected multiplicative linguistic preference relations EðBk Þ ¼ ðEðbij ÞÞnn ðk ¼ 1; 2; . . . ; mÞ into a collective multiplicative linguistic preference relation B_ ¼ ðb_ ij Þnn . To measure the deviation degree between each expected multiplicative linguistic preference relation ðkÞ EðBk Þ ¼ ðEðbij ÞÞnn and the collective multiplicative linguistic preference relation B_ ¼ ðb_ ij Þnn , we define ! 2 n   1=n Y ðkÞ 0 0 _ ¼ d ðEðBk Þ; BÞ d Eðbij Þ; b_ ij ð38Þ i;j¼1 ðkÞ _ _ _ where d 0 ðEðbðkÞ as the deviation between EðBk Þ and B, ij Þ; bij Þ is the deviation between Eðbij Þ and bij , which is calculated by using (25). Similar to Theorem 2, we have

_ P 1, especially, d 0 ðEðBk Þ; BÞ _ ¼ 1 iff EðBk Þ ¼ B; _ (1) d 0 ðEðBk Þ; BÞ 0 _ ¼ d 0 ðB; _ EðBk ÞÞ. (2) d ðEðBk Þ; BÞ

Z. Xu / Information Sciences 178 (2008) 452–467

461

_ generally reflects the degree of departure of the individual multiplicative linguisThe deviation d 0 ðEðBk Þ; BÞ _ tic preferences from the group opinion. Below we define a function of the deviation d 0 ðEðBk Þ; BÞ: _ y ¼ gðd 0 ðEðBk Þ; BÞÞ

ð39Þ

and then, based on (39), we give a formula for determining the DMs’ weights: _ gðd 0 ðEðBk Þ; BÞÞ ; wk ¼ P m _ gðd 0 ðEðBk Þ; BÞÞ

k ¼ 1; 2; . . . ; m

ð40Þ

k¼1

In what follows, we discuss three cases: Case 1. If the majority opinion should be emphasized, then the low weights should be assigned to the DM dk _ In this case, y ¼ gðd 0 ðEðBk Þ; BÞÞ _ is a decreasing function of the deviation with high deviation d 0 ðEðBk Þ; BÞ. 0 _ For example, if d ðEðBk Þ; BÞ. 1 _ ¼ ð41Þ gðd 0 ðEðBk Þ; BÞÞ _ d 0 ðEðBk Þ; BÞ then 1 _ d 0 ðEðBk Þ;BÞ ; 1 _ k¼1 d 0 ðEðBk Þ;BÞ

wk ¼ Pm

k ¼ 1; 2; . . . ; m

ð42Þ

Case 2. If the minority opinion should be emphasized, then the high weights should be assigned to the DM dk _ In this case, y ¼ gðd 0 ðEðBk Þ; BÞÞ _ is an increasing function of the deviation with high deviation d 0 ðEðBk Þ; BÞ. 0 _ For example, if d ðEðBk Þ; BÞ. _ ¼ d 0 ðEðBk Þ; BÞ _ gðd 0 ðEðBk Þ; BÞÞ

ð43Þ

_ d 0 ðEðBk Þ; BÞ ; wk ¼ Pm 0 _ k¼1 d ðEðBk Þ; BÞ

ð44Þ

then k ¼ 1; 2; . . . ; m

Case 3. If both the majority and minority opinions should be emphasized simultaneously, then we can use the following formula to determine the DMs’ weights: _ _ g ðd 0 ðEðBk Þ; BÞÞ g ðd 0 ðEðBk Þ; BÞÞ wk ¼ ð1  aÞ Pm 1 þ a Pm 2 ; k ¼ 1; 2; . . . ; m ð45Þ 0 0 _ _ g1 ðd ðEðBk Þ; BÞÞ g2 ðd ðEðBk Þ; BÞÞ k¼1

k¼1

_ and y 2 ¼ g2 ðd 0 ðEðBk Þ; BÞÞ _ are, where a 2 ½0; 1 (which can be given by the DMs directly), y 1 ¼ g1 ðd 0 ðEðBk Þ; BÞÞ 0 _ respectively, the decreasing and increasing functions of the deviation d ðEðBk Þ; BÞ. For example, if _ and g2 ðd 0 ðEðBk Þ; BÞÞ _ are, respectively, taken from (41) and (43), then (45) can be rewritten as: g1 ðd 0 ðEðBk Þ; BÞÞ 1 _ d 0 ðEðBk Þ;BÞ

wk ¼ ð1  aÞ P m

k¼1

1 _ d 0 ðEðBk Þ;BÞ

_ d 0 ðEðBk Þ; BÞ ; þaP m _ d 0 ðEðBk Þ; BÞ

k ¼ 1; 2; . . . ; m

ð46Þ

k¼1

Especially, if a = 0, then (46) is reduced to (42). If a = 1, then (46) is reduced to (44). Therefore, Cases 1 and 2 are the special cases of Case 3. Furthermore, we give the following definition: If _ 6g d 0 ðEðBk Þ; BÞ

ð47Þ

then EðBk Þ and B_ are called of acceptable deviation, where g is the threshold of acceptable deviation, which can be predefined by all the DMs (who participate in determining the DMs’ original weights) in practical appli_ ¼ 1, then EðBk Þ and B_ are equal. However, if d 0 ðEðBk Þ; BÞ _ > g, then we will cations. Especially, if d 0 ðEðBk Þ; BÞ

462

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return Bk together with EðBk Þ and B_ to the DM dk for revaluation. We repeat this process (in which the DMs’ original weights are replaced by the new weights derived from (46) for obtaining the new collective additive linguistic preference relation B_ by using (37)) until EðBk Þ and B_ are of acceptable deviation or the process will stop as the repetition times reach the maximum number predefined by the DMs, and then go to Step 4. Step 4. Utilize the LGA operator b_ i ¼ LGAðb_ i1 ; b_ i2 ; . . . ; b_ in Þ;

for all i ¼ 1; 2; . . . ; n ð48Þ _ and then get the mean preference intensity b_ i of to aggregate the preference information b_ ij in the ith line of B, the alternative xi over all the other alternatives. Step 5. Rank all the alternatives xi ði ¼ 1; 2; . . . ; nÞ and select the best one in accordance with the values of b_ i ði ¼ 1; 2; . . . ; nÞ. Step 6. End. We have proposed two procedures (Procedures I and II) to solve the group decision making problems in which the preference information about alternatives provided by the DMs takes the form of different linguistic preference structures, namely, (1) traditional additive linguistic preference relations, (2) uncertain additive linguistic preference relations, traditional multiplicative linguistic preference relations, and (3) uncertain multiplicative linguistic preference relations. In these two procedures, we have introduced the deviation measure between additive linguistic preference relations and the deviation measure between multiplicative linguistic preference relations, respectively. Based on these two deviation measures, we have given some straightforward formulas to determine the DMs’ weights, and presented two methods to reach consensus among the individual preferences and the group’s opinion. From Procedures I and II, we know that both these two procedures can be used to derive the rating of each alternative. Procedure I utilizes the additive fuzzy expectation function to transform each individual uncertain additive linguistic preference relation into the uniformed format: traditional additive linguistic preference relation. Then, Procedure I uses the LWAA operator to fuse the uniformed additive linguistic preference relations into the collective additive linguistic preference relation, and uses the LAA operator to aggregate the collective additive linguistic preference relation so as to select the optimal alternative. Procedure II utilizes the multiplicative fuzzy expectation function to transform each individual uncertain multiplicative linguistic preference relation into the uniformed multiplicative linguistic preference relation. Then, Procedure II uses the LWGA operator to fuse the uniformed multiplicative linguistic preference relations into the collective multiplicative linguistic preference relation, and uses the LGA operator to aggregate the collective linguistic preference relation so as to derive the rating of each alternative. In the actual applications, we find that it can produce similar decision results by using these two procedures, but yet Procedure I is more straightforward and convenient than Procedure II from the point of view of computation. Therefore, Procedure I is more practical in group decision making based on multiple types of linguistic preference relations. 8. Illustrative example In this section, a group decision making problem involves the prioritization of a set of six information technology improvement projects (adapted from [31]) is used to illustrate the developed procedures. The information management steering committee of Midwest American Manufacturing Corp. (MAMC), which comprises (1) d1 – the Chief Executive Officer, (2) d2 – the Chief Information Officer, and (3) d3 – the Chief Operating Officer, must prioritize for development and implementation a set of six information technology improvement projects xj ðj ¼ 1; 2; . . . ; 6Þ, which have been proposed by area managers. The committee is concerned that the projects are prioritized from highest to lowest potential contribution to the firm’s strategic goal of gaining competitive advantage in the industry. In assessing the potential contribution of each project, one main factor considered is productivity. The productivity factor assesses the potential of a proposed project to increase the effectiveness and efficiency of the firm’s manufacturing and service operations. The following is the list of proposed information systems projects: (1) x1 – Quality Management Information, (2) x2 – Inventory Control, (3) x3 – Customer Order Tracking, (4) x4 – Materials Purchasing Management, (5) x5 – Fleet Management, and (6) x6 – Design Change Management.

Z. Xu / Information Sciences 178 (2008) 452–467

463

Suppose that the committee members d k ðk ¼ 1; 2; 3; 4Þ compare each pair of these projects with respect to the factor productivity by using the additive linguistic labels in the set S ¼ fs4 ¼ extremely low; s3 ¼ very low; s2 ¼ low; s1 ¼ slightly low; s0 ¼ fair; s1 ¼ slightly high; s2 ¼ high; s3 ¼ very high; s4 ¼ extremely highg and construct the traditional additive linguistic preference relations and uncertain additive linguistic preference relations as follows: 3 2 s0 s1 s2 s3 s1 s1 6 s1 s0 s1 s2 s0 s1 7 7 6 7 6 6 s2 s1 s0 s1 s2 s2 7 7 6 A1 ¼ 6 s2 s3 7 7 6 s3 s2 s1 s0 7 6 4 s1 s0 s2 s2 s0 s2 5 s1 s0

s1 s2

s2 s3

s3 s1

6 s2 6 6 6 s3 A2 ¼ 6 6s 6 1 6 4 s2

s0

s2

s3

s1

s2

s0

s2

s1

s3

s2

s0

s3

s1

s1

s3

s0

2

2

s2 s2 ½s0 ; s0 

6 ½s2 ; s1  6 6 6 ½s1 ; s2  A3 ¼ 6 6 ½s ; s  6 4 2 6 4 ½s1 ; s1  ½s0 ; s1  2 ½s0 ; s0  6 ½s3 ; s2  6 6 6 ½s0 ; s2  A4 ¼ 6 6 ½s ; s  6 2 1 6 4 ½s2 ; s0  ½s1 ; s2 

s2 s2

s0 3 s2 s2 7 7 7 s0 7 7 s1 7 7 7 s1 5

s0 s1 ½s1 ; s2 

s1 s0 ½s2 ; s1 

½s2 ; s4 

½s1 ; s1 

½s1 ; s0 

½s0 ; s0 

½s1 ; s0 

½s1 ; s3 

½s1 ; s1 

½s0 ; s1 

½s0 ; s0 

½s1 ; s2 

½s2 ; s1 

½s3 ; s1 

½s2 ; s1 

½s0 ; s0 

½s1 ; s2 

½s1 ; s2 

½s2 ; s1 

½s0 ; s0 

½s1 ; s2  7 7 7 ½s1 ; s2  7 7 ½s2 ; s3  7 7 7 ½s2 ; s1  5

½s1 ; s1  ½s2 ; s1  ½s2 ; s3 

3

½s0 ; s2 

½s1 ; s2  ½s0 ; s0  3 ½s0 ; s2  ½s2 ; s1  ½s2 ; s1  ½s2 ; s4  ½s2 ; s0  ½s0 ; s1  7 7 7 ½s0 ; s0  ½s0 ; s1  ½s1 ; s1  ½s2 ; s3  7 7 ½s1 ; s0  ½s0 ; s0  ½s2 ; s3  ½s1 ; s2  7 7 7 ½s1 ; s1  ½s3 ; s2  ½s0 ; s0  ½s0 ; s1  5

½s1 ; s0 

½s3 ; s2  ½s2 ; s1  ½s1 ; s0 

½s0 ; s0  ½s1 ; s2  ½s4 ; s2 

½s2 ; s1  ½s3 ; s2  ½s2 ; s0  ½s1 ; s2 

½s0 ; s0 

To prioritize the projects xj ðj ¼ 1; 2; . . . ; 6Þ, the following steps are involved: Step 1. Suppose that the committee members d k ðk ¼ 3; 4Þ give the indices of their risk-bearing attitudes ðkÞ h3 ¼ 0:3 and h4 ¼ 0:6, respectively. Then we utilize (10) to transform Ak ¼ ðaij Þ66 ðk ¼ 3; 4Þ into ðkÞ the expected additive linguistic preference relation EðAk Þ ¼ ðEðaij ÞÞ66 ðk ¼ 3; 4Þ: 3 2 s0 s1:3 s1:7 s2:6 s0:4 s0:7 7 6 s0 s0:7 s1:6 s0:4 s1:3 7 6 s1:3 7 6 6 s s0:7 s0 s0:1 s1:7 s1:3 7 7 6 1:7 EðA3 Þ ¼ 6 7 6 s2:6 s1:6 s0:1 s s s 0 1:3 2:3 7 7 6 6 s s0:4 s1:7 s1:3 s0 s1:7 7 5 4 0:4 s0:7 s1:3 s1:3 s2:3 s1:7 s0

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2

s0

s2:6

s0:8

s1:6

s1:2

s1:4

6s 6 2:6 6 6 s0:8 EðA4 Þ ¼ 6 6s 6 1:6 6 4 s1:2

s0 s1:4

s1:4 s0

s3:2 s0:6

s0:8 s0:2

s0:6 s2:6

s3:2 s0:8

s0:6 s0:2

s0 s2:6

s2:6 s0

s1:6 s0:6

s1:4

s0:6

s2:6

s1:6

s0:6

s0

3 7 7 7 7 7 7 7 7 5

Step 2. Let the original weight vector of the committee members d k ðk ¼ 1; 2; 3; 4Þ be T w ¼ ð0:35; 0:20; 0:15; 0:30Þ , which are determined by using Bodily’ method [1], and assume that these four members predefine the threshold of acceptable deviation as s ¼ 0:6. Then, we utilize the LWAA operator ð1Þ

ð2Þ

ð2Þ

ð4Þ

a_ ij ¼ LWAAw ðEðaij Þ; Eðaij Þ; Eðaij ; Eðaij ÞÞÞ; to aggregate A1 ; A2 2 s0 6s 6 1:73 6 6 s _A ¼ 6 1:80 6s 6 2:32 6 4 s1:05 s1:28

for all i; j ¼ 1; 2; . . . ; 6

ð49Þ

and EðAk Þ ðk ¼ 3; 4Þinto the collective additive linguistic preference relation A_ ¼ ða_ ij Þ66 : 3 s1:73 s1:80 s2:32 s1:05 s1:28 s0 s1:28 s2:5 s0:10 s0:93 7 7 7 s1:28 s0 s0:92 s1:10 s2:08 7 7 s2:5 s0:92 s0 s2:28 s2:08 7 7 7 s010 s1:10 s2:28 s0 s0:58 5 s0:93

s2:08

s2:08

s0:58

s0

_ Step 3. Utilize (1) and (14) to calculate the deviation between EðAk Þ and A: _ ¼ 0:393; dðEðA1 Þ; AÞ _ ¼ 0:517; dðEðA3 Þ; AÞ

_ ¼ 0:761 dðEðA2 Þ; AÞ _ ¼ 0:513 dðEðA4 Þ; AÞ

_ > 0:6, then, here we utilize (22) (without loss of generality, let a ¼ 0:3Þ to recalculate the Since dðEðA2 Þ; AÞ weights of the committee members d k ðk ¼ 1; 2; 3; 4Þ: w ¼ ð0:2659; 0:2321; 0:2508; 0:2512ÞT

ð50Þ

and return A2 together with A_ to the committee member d2 for revaluation. Assume that the committee member d2 provides the revaluated traditional additive linguistic preference relation as follows: 3 2 s0 s2 s2 s2 s1 s1 6s s0 s1 7 7 6 2 s0 s1 s3 7 6 6 s s s s s s 2 1 0 1 1 2 7 7 A02 ¼ 6 6s s3 s2 7 7 6 2 s3 s1 s0 7 6 4 s1 s0 s1 s3 s0 s1 5 s1

s1

s2

s2

s1

s0

Then we utilize (13) and (50) to aggregate A1 ; A02 and EðAk Þ ðk ¼ 3; 4Þ into the collective additive linguistic preference relation A_ 0 ¼ ða_ 0ij Þ66 : 2

s0

3

s1:71

s1:62

s2:32

s0:70

s1:03

s0

s2:45 s0:62

s0:30 s1:14

s0

7 7 7 7 7 s2:24 7 7 7 s1:04 5

6s 6 1:71 6 6 s1:62 A_ 0 ¼ 6 6s 6 2:32 6 4 s0:70

s1:03

s1:03 s0

s2:45 s0:30

s0:62 s1:14

s2:21

s2:21 s0

s1:03

s0:97

s1:98

s2:24

s1:04

s0:97 s1:98

s0

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465

By (1) and (14), we have dðEðA1 Þ; A_ 0 Þ ¼ 0:339;

dðEðA02 Þ; A_ 0 Þ ¼ 0:213

dðEðA3 Þ; A_ 0 Þ ¼ 0:411;

dðEðA4 Þ; A_ 0 Þ ¼ 0:552

_ < 0:6 ðj ¼ 1; 3; 4Þ and dðEðA0 Þ; AÞ _ < 0:6, i.e., all the deviations are acceptable, then go to Since all dðEðAj Þ; AÞ 2 the next step. Step 4. Utilize (24) to aggregate the preference information a_ 0ij in the ith line of A_ 0 , and then get the averaged preference degree a_ 0i of the alternative xi over all the other alternatives a_ 01 ¼ s0:35 ;

a_ 02 ¼ s0:06 ;

a_ 03 ¼ s0:69 ;

a_ 04 ¼ s0:16 ;

a_ 05 ¼ s0:42 ;

a_ 06 ¼ s0:52

Step 5. Rank all the alternatives xj ðj ¼ 1; 2; . . . ; 6Þ in accordance with the values of a_ j ðj ¼ 1; 2; . . . ; 6Þ: x3  x1  x2  x4  x5  x6 and thus, the project x3 has the highest potential contribution to the firm’s strategic goal of gaining competitive advantage in the industry. 9. Conclusions In the process of group decision making, the DMs generally use different representation formats to express their preferences for each pair of alternatives, for example, in many real life situations, such as negotiation processes, project investment, and supply chain management, etc., the decision information provided by the DMs may be presented by means of different linguistic preference representation structures, such as traditional additive linguistic preference relations, traditional multiplicative linguistic preference relations, uncertain additive linguistic preference relations, and uncertain multiplicative linguistic preference relations, etc. In this paper, we have developed two procedures (Procedures I and II) for solving this issue. Procedure I transforms the uncertain additive linguistic preference relations into the expected additive linguistic preference relations, and fuses all the individual additive linguistic preference relations into the collective additive linguistic preference relation by using the LWAA operator. Then Procedure I uses the LAA operator to get the averaged preference degree of one alternative over all the other alternatives. Similarly, Procedure II transforms the uncertain multiplicative linguistic preference relations into the expected multiplicative linguistic preference relations, and fuses all the individual multiplicative linguistic preference relations into the collective multiplicative linguistic preference relation by using the LWGA operator. Then Procedure II uses the LGA operator to get the mean preference intensity of one alternative over all the other alternatives. In these two procedures, we have introduced the deviation measure between additive linguistic preference relations and the deviation measure between multiplicative linguistic preference relations, respectively. Based on these deviation measures, we have given some straightforward formulas to determine the DMs’ weights, and presented two methods to reach consensus among the individual preferences and the group’s opinion. Furthermore, we have analyzed and verified the practicality and effectiveness of Procedure I with an illustrative example (we can analyze Procedure II in a similar way). However, there may still be situations where the DMs provide incomplete linguistic preference relations (that is, some of the preferences are missing). This is a problem left for further research in the near future. Acknowledgements The author is very grateful to the Editor-in-Chief, Professor W. Pedrycz, and the three anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper.

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