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The maximizing deviation method for group multiple attribute decision making under linguistic environment
Fuzzy Sets and Systems 158 (2007) 1608 – 1617 www.elsevier.com/locate/fss
The maximizing deviation method for group multiple attribute decision making under linguistic environment Zhibin Wu∗ , Yihua Chen College of Mathematics and Physics, Chonqing University, Chongqing 400044, China Received 21 July 2006; received in revised form 4 November 2006; accepted 26 January 2007 Available online 17 February 2007
Abstract The aim of this paper is to put forward a method for multi-attribute decision making problems with linguistic information, in which the preference values take the form of linguistic variables. An aggregating operator named linguistic weighted arithmetic averaging (LWAA) operator is introduced to aggregate the given decision information to get the overall preference value of each alternative. Some properties of the LWAA operator are also investigated. Based on the idea that the attribute with a larger deviation value among alternatives should be evaluated a larger weight, a method to determine the optimal weighting vector of LWAA operator is developed under the assumption that attribute weights are completely unknown. The based approach is extended to the situation where partially weight information can be obtained by solving a constrained non-linear optimization problem. Then a procedure to group multiple attribute decision making is provided under linguistic environment. Finally, an example of risk investment problem is given to verify the proposed approach; a comparative study to fuzzy ordered weighted averaging (F-OWA) operator methods is also demonstrated. © 2007 Elsevier B.V. All rights reserved. Keywords: Group decision making; Linguistic weighted arithmetic averaging (LWAA) operator; Maximizing deviation method; Linguistic variables
1. Introduction Multiple attribute decision making (MADM) addresses the problem of choosing an optimum choice that has the highest degree of satisfaction from a set of alternatives that are characterized in terms of their attributes. In order to make a decision or choose a best alternative, a decision maker (DM) is often asked to provide his/her preferences either on alternatives or on the relative weights of attributes or on both of them. Usually, the information is expressed by means of numerical values (like exact value, interval values, fuzzy numbers [20,21]). However, there are some situations in which the information may not be quantified because of its nature, and, thus, it may be stated only in linguistic terms (e.g., when evaluating the “comfort” or “design” of a car, terms like “bad,” “poor,” “tolerable,” “average,” or “good” can be used [13]). In other cases, precise quantitative information may not be stated because either it is not available or the cost of its computation is too high, and then an “approximate value” may be tolerated (e.g., when evaluating the speed of a car, linguistic terms like “fast,” “very fast,” or “slow” are used instead of numerical values). So, the use of linguistic labels makes expert judgment more reliable and informative for decision making. ∗ Corresponding author. Tel./fax: +86 23 65111075.
E-mail address: [email protected] (Z. Wu). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.01.013
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Group decision-making problems usually follow a common resolution scheme composed by two phases: aggregation phase and exploitation phase [8,16]. In the literature, many aggregation operators have been developed to aggregation information [29], such as ordered weighted averaging (OWA) operator [30], induced ordered weighted averaging (IOWA) operator [32], ordered weighted geometric (OWG) operator [4,28], induced ordered weighted geometric (IOWG) operator [5], generalized IOWA (GIOWA) operator and generalized IOWGA (GIOWGA) operator [29], continuous ordered weighted averaging (C-OWA) operator [33], etc. One important issue in the OWA-like operators is to determining their associated weights. Xu [25] briefly review the existing methods for determining the weights associated OWA operator, such as the linguistic quantifier’s approach [30], the analytic approach [3]. A number of studies recently focused on group decision making under linguistic environment [1,2,6–15,17,18,21–24,26,27,29,31] and thus a lot of linguistic aggregation operators were proposed. They are linguistic weighted geometric averaging (LWGA) operator and linguistic ordered weighted geometric averaging (LOWGA) operator [22], uncertain linguistic OWA (ULOWA) operator and uncertain linguistic hybrid aggregation (ULHA) operator [23], linguistic aggregation of majority additive (LAWA) operator [15], uncertain linguistic geometric mean (ULGM) operator, uncertain linguistic weighted geometric mean (ULWGM) operator, uncertain linguistic OWG (ULOWG) operator, induced ULOWG (IULOWG) operator [26], etc. The above operators have been found applications in group decision making and management science. These linguistic operators are the extension or generalization of the conventional operators. Herrera and Herrera-Viedma [8] analyzed the steps to follow in the linguistic decision analysis of a group decision-making problem with linguistic preference relations, i.e., choice of the linguistic term set, choice of the aggregation operator for linguistic information, choice of the best alternatives. Herrera-Viedma et al. [11] presented a model of consensus support system to assist the experts in all phases of the consensus reaching process of group decision-making problems with multi-granular linguistic preference relations. Ben-Arieh and Chen [1] also proposed a method to increase the consensus level by updating the importance of the group members. Tang and Zheng [17] developed a new linguistic modeling technique based on the semantic similarity relation among linguistic labels. Huynh and Nakamori [12] proposed a satisfactoryoriented approach in which a random preference is defined for each alternative in the aggregation phase instead of using an aggregation operator to obtain a collective preference value. Most papers in the literature put their emphasis on the extension of the aggregation operators or the consensus level of the group DM. However, when using these operators, the associated weighting vector is more or less determined subjectively and the decision information itself is not taken into consideration sufficiently. In this paper, we focus our attention on developing a method objectively to determine the attribute weights under the conditions that the attribute weights are completely unknown and the input arguments of the decision matrix take the form of linguistic variables. We will follow the steps analyzed in Ref. [8]. To do so, the rest of this paper is organized as follows. In Section 2, we introduce some basic notations and operational laws of linguistic variables. In Section 3, we propose LWAA operator as the aggregation operator and present a method to derive the weight vector of attributes in group decision making. In Section 4, we give a decision procedure for group decision making with linguistic variables. Section 5 gives an illustrative example to verify the developed approach and makes a comparative study to the F-OWA operator method. Section 6 concludes the paper. 2. Preliminaries: linguistic variables Suppose that S = {s | = −t, . . . , −1, 0, 1, . . . , t} is a finite and totally ordered discrete term set whose cardinality value is an odd one, such as 7 and 9, where s represents a possible value for a linguistic variable. The cardinality of S must be small enough so as not to impose useless precision to the experts, and it must be rich enough to allow a discrimination of the performances of each criterion in limited grades. For example, a set of nine terms S could be [21,22] S = {s−4 = extremely poor, s−3 = very poor, s−2 = poor, s−1 = slightly poor, s0 = fair, s1 = slightly good, s2 = good, s3 = very good, s4 = extremely good}. In these cases, it usually requires that si and sj satisfy the following additional characteristics [8,9]: (1) the set is ordered: si sj , if i j ; (2) there is a negation operator: neg(si ) = s−i , especially, neg(s0 ) = s0 ;
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(3) max operator: max(si , sj ) = si , if i j ; (4) min operator: min(si , sj ) = si , if i j . In the process of information aggregation, however, some results may not exactly match any linguistic labels in S. To preserve all the information, extend the discrete linguistic label set S to a continuous linguistic label set S¯ = {s |s−q s sq }, where s meets all the characteristics above and q(q > t)is a sufficiently large positive integer. If s ∈ S, s is called the original term, otherwise, s is called the virtual term. In general, the original term is used to ¯ and evaluate alternatives, while the virtual term is used in operation. Consider any two linguistic terms s , s ∈ S, , 1 , 2 ∈ [0, 1], their operational laws are given by s ⊕ s = s+ ,
s ⊕ s = s ⊕ s ,
(1 + 2 )s = 1 s ⊕ 2 s ,
s = s ,
(s ⊕ s ) = s ⊕ s .
From the above descriptions, we can see that the operation on two linguistic terms can be converted to the operation on lower indices of the corresponding terms. 3. The maximizing deviation method For simplicity, we let M = {1, 2, . . . , m} and N = {1, 2, . . . , n}. Suppose the alternatives are known. Let X = {X1 , X2 , . . . , Xn } denote a discrete set of n(n2) potential alternatives. Attributes are predefined too, let C = {C1 , C2 , . . . , Cm } denote a set of m(m 2) criterions or attributes. Theattribute weights are completely unknown. Let w = (w1 , w2 , . . . , wm )T be the weight vector of attributes, such that m j =1 wj = 1,wj 0, j ∈ M and wj denotes the weight of attribute Cj . Let D = {d1 , d2 , . . . , dt } be the set of DMs, and = (1 , 2 , . . . , t ) be the weight vector (k) of DMs, with k ∈ [0, 1], k = 1, 2, . . . , t, tk=1 k = 1. Suppose that A(k) = [aij ]n×m is the decision matrix given (k)
by the DM dk ∈ D, where aij is a linguistic variable for alternative Xi with respect to the attribute Cj . Definition 1. Let {sa1 , sa2 , . . . , san } be a collection of linguistic arguments, a linguistic weighted arithmetic averaging (LWAA) operator is a mapping LWAA:S n → S, and defined as LWAAw (sa1 , sa2 , . . . , san ) = w1 sa1 ⊕ w2 sa2 ⊕ · · · ⊕ wn san = s , (1) n where = j =1 wj I (aj ), w = (w1 , w2 , . . . , wn )T is the associated weighting vector such that wj ∈ [0, 1], j ∈ N , n j =1 wj = 1, and I (aj ) is the subscript of aj . Example. Assume w = (0.3, 0.2, 0.4, 0.1)T , then LWAAw (s4 , s2 , s−1 , s−3 ) = 0.3 × s4 + 0.2 × s2 + 0.4 × s−1 + 0.1 × s−3 = s0.3×4 + s0.2×2 + s0.4×(−1) + s0.1×(−3) = s0.3×4+0.2×2+0.4×(−1)+0.1×(−3) = s0.9 . It is easy to show that LWAA operator is a special case of the linguistic hybrid aggregation (LHA) operator [21] and has some good properties such as bounded, idempotency, monotonicity. Proposition 1 (Bounded). minj ∈N (saj ) LWAAw (sa1 , sa2 , . . . , san ) maxj ∈N (saj ). Proposition 2 (Idempotency). If saj = sa , for all j, j ∈ N , then LWAAw (sa1 , sa2 , . . . , san ) = sa . Proposition 3 (Monotonicity). If saj sbj , for all j, j ∈ N , then s = LWAAw (sa1 , sa2 , . . . , san ) s = LWAAw (sb1 , sb2 , . . . , sbn ).
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Proposition 4 (LAA operator). If w = (1/n, 1/n, . . . , 1/n)T , then LWAA operator is reduced to LAA operator, i.e., 1 I (aj ). n n
LWAAw (sa1 , sa2 , . . . , san ) = s
where =
j =1
To compare with other techniques of aggregation of linguistic information in the example section, the fuzzy ordered weighted averaging (F-OWA) operator is defined in the following. The F-OWA operator is a special case of generalized induced OWA (GIOWA) operator [21]. It transforms the linguistic scales into triangle fuzzy numbers. The set of corresponding transformation is given by F = {extremely poor = [0, 0.1, 0.2], very poor = [0.1, 0.2, 0.3], poor = [0.2, 0.3, 0.4], slightly poor = [0.3, 0.4, 0.5], s0 = [0.4, 0.5, 0.6], slightly good = [0.5, 0.6, 0.7], good = [0.6, 0.7, 0.8], very good = [0.7, 0.8, 0.9], extremely good = [0.8, 0.9, 1]}. Definition 2. Let {sa1 , sa2 , . . . , san } be a collection of linguistic arguments, a F-OWA operator is defined as F -OWAw (sa1 , sa2 , . . . , san ) =
n
(2)
w j fj ,
j =1
where w = (w1 , w2 , . . . , wn )T is the associated weighting vector such that wj ∈ [0, 1], j ∈ N , is the jth largest triangle fuzzy number in the corresponding transformed set.
n
j =1 wj
= 1, and fj
3.1. Computing the optimal weights: the maximizing deviation method Once each expert has expressed linguistic judgments on each alternative with respect to each criterion or attribute, the first phase is aimed at synthesizing the performance judgments of the alternatives for each expert. In this paper, we will use the LWAA operator as the main aggregation operators. By Definition 1, the overall preference value of alternative Xi for DM k can be expressed as (k)
Zi (w) =
m j =1
(k)
(3)
wj I (aij ).
The maximizing deviation method is proposed by Wang [19] to deal with MADM problems with numerical information. For a MADM problem, we need to compare the collective preference values to rank the alternatives, the larger the ranking value Zi (w) is, the better the corresponding alternative Xi is. If the performance values of each alternative have little differences under an attribute, it shows that such an attribute plays a small important role in the priority procedure. Contrariwise, if some attribute makes the performance values among all the alternatives have obvious differences, such an attribute plays an important role in choosing the best alternative. So to the view of sorting the alternatives, if one attribute has similar attribute values across alternatives, it should be assigned a small weight; otherwise, the attribute which makes larger deviations should be evaluated a bigger weight, in spite of the degree of its own importance. Especially, if all available alternatives score about equally with respect to a given attribute, then such an attribute will be judged unimportant by most DMs. In other word, such an attribute should be assigned a very small weight. Wang [19] suggests that zero should be assigned to the attribute of this kind. The deviation method is selected here to compute the differences of the performance values of each alternative. For the DM k and the attribute Cj , the deviation of alternative Xi to all the other alternatives can be defined as follows: (k)
Hij (w) =
n (k) (k) p I (aij ) − I (alj ) wj , l=1
i ∈ N, j ∈ M,
(4)
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where p(p 1) is an integer. For simplicity, let p = 2, and (4) can be rewritten as (k)
Hij (w) =
n
(k)
(k)
(I (aij ) − I (alj ))2 wj ,
i ∈ N, j ∈ M.
(5)
l=1
Let (k)
Hj (w) =
n
(k)
Hij (w) =
i=1
n n
(k)
(k)
(I (aij ) − I (alj ))2 wj ,
j ∈ M.
(6)
i=1 l=1
(k)
Then Hj (w) represent the deviation value of all alternatives to other alternatives for the attribute Cj and the DM k. Based on the aforementioned analysis, we have to choose the weight vector w to maximize all deviation values for all the attributes and for all the DMs. To do so, we can construct a non-linear programming model as follows: ⎧ m n n t ⎪ (k) (k) ⎪ ⎪ (I (aij ) − I (alj ))2 wj k ⎨ max H (w) = k=1 j =1 i=1 l=1 (M1) m ⎪ ⎪ s.t. w 0, j ∈ M, wj2 = 1. ⎪ j ⎩ j =1
To solve the above model, let L(w, ) =
t
k
m n n j =1 i=1 l=1
k=1
⎞ ⎛ m 1 (k) (k) (I (aij ) − I (alj ))2 wj + ⎝ wj2 − 1⎠ 2
(7)
j =1
denote the Lagrange function of the constrained optimization problem (M1), where is a real number. Then the partial derivatives of L are computed as n t n jL (k) (k) = k (I (aij ) − I (alj ))2 + wj = 0, jwj k=1 i=1 l=1 ⎞ ⎛ m jL 1 ⎝ 2 wj − 1⎠ = 0. = j 2
j ∈ M,
(8)
(9)
j =1
From (8), it follows that (k) (k) − tk=1 k ni=1 nl=1 (I (aij ) − I (alj ))2 wj = , Put (10) into (9), we have
t 2 m n n (k) (k) 2 = − k (I (aij ) − I (alj )) . j =1
k=1
j ∈ M.
(10)
(11)
i=1 l=1
Let Yj =
t k=1
k
n n
(k)
(k)
(I (aij ) − I (alj ))2 ,
j ∈ M.
(12)
i=1 l=1
Then (10) and (11) give Yj wj = m
2 j =1 Yj
,
j ∈ M.
Thus Eq. (13) is the extreme point of model (M1).
(13)
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From (13), it can be verified easily that wj , j ∈ M are positive such that they do satisfy the constrained conditions in model (M1) and the solution is unique. By normalizing wj to let the sum of wj , j ∈ M be a unit, we have wj wj∗ = m
j =1 wj
Yj = m
j =1 Yj
,
j ∈ M.
(14)
As a mater of fact, Yj represents the deviation of all alternatives to other alternatives for the attribute Cj and for all the DMs. Because the larger Yj is, the more important the attribute Rj is, Eq. (14) is obtained directly by using each Yj divide the sum of Yj , which is considered as an alternative approach and an explanation to the maximizing method. The theoretic foundation of this method is based on information theory, that is, the attribute providing more information should be evaluated a bigger weight. Now, the overall weighted assessment value of each alternative can be computed using the following expression and the decision can be made: Zi (w) =
t
k
k=1
m j =1
wj∗ I (aij ). (k)
(15)
3.2. Further analysis We have discussed the situation where the attribute weights are completely unknown in group decision making in the first part. If t = 1, the group decision-making problem is reduced to a single decision-making problem. Following in a similar manner, we can obtain the optimal weight vector as follows: ⎤ ⎡ n n 2 −1 n m n ⎥ ⎢ (k) (k) (k) (k) wj∗ = (I (aij ) − I (alj ))2 ∗ ⎣ (I (aij ) − I (alj ))2 ⎦ , j ∈ M. j =1
i=1 l=1
i=1 l=1
(16) There are actual situations that the information about the weighting vector is not completely unknown but partially known. For these cases, let be the set of weight information known, consequently, the following constrained non-linear programming model is constructed. ⎧ m n n t ⎪ (k) (k) ⎪ ⎪ max H (w) = k (I (aij ) − I (alj ))2 wj ⎨ k=1 j =1 i=1 l=1 (M2) m ⎪ ⎪ wj2 = 1. ⎪ ⎩ s.t. w ∈ , wj 0, j ∈ M, j =1
The solution to the above maximization problem could be found by MatLab software with optimization toolbox or Lindo/Lingo software package. 4. An approach to group decision making with linguistic information Group decision-making problems follow a common resolution scheme [16] composed by the following two phases: (1) Aggregation phase: It combines individual preferences to obtain a collective preference value for each alternative. (2) Exploitation phase: It orders collective preference values to obtain the best alternative(s). The notations in this section are assumed have the same meaning to the previous section. In the following, we develop a practical approach based on LWAA operator to group decision making with linguistic information. (k) Step1: To a group MADM problems, construct the decision matrix A(k) = [aij ]n×m , where k represents the kth DM (k)
and all the arguments aij are linguistic variables. Step2: If the weight vector of attributes is completely unknown, utilize Eq. (14) to obtain the optimal weights.
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Step3: Utilize the LWAA operator to aggregate the decision information. In group decision making, the collective overall preference values Zi (i ∈ N ) are computed by Eq. (15). Step4: Ranking all the alternatives Xi (i ∈ N ) and select the best one(s) in accordance with the value of Zi (i ∈ N ). Step5: End. 5. An illustrative example: a risk investment problem Let us suppose a risk investment company, which wants to invest a sum of money in the best option (adapted from [21, p. 165]). This MADM problem involves the evaluation of four possible alternatives denoted as X1 , X2 , X3 , X4 . The investment company must make a decision according the following seven attributes: C1 —the ability of sale, C2 —the ability of management, C3 —the ability of production, C4 —the ability of technology, C5 —the ability of financing, C6 —the ability to resist venture, C7 —the consistency of corporation strategy. Three DMs (without loss of generality, take weight vector = ( 13 , 13 , 13 ) are asked to evaluate each alternative using the linguistic term set in terms of their performance S = {s−4 = extremely poor, s−3 = very poor, s−2 = poor, s−1 = slightly poor, s0 = fair, s1 = slightly good, s2 = good, s3 = very good, s4 = extremely good}. (k)
The decision matrices A(k) = (aij )n×m (k = 1, 2, 3) are listed in Tables 1–3. Assume the weight vector of the attribute is completely unknown, by applying Eq. (14), we get the optimal weight vector w = (0.1154, 0.0216, 0.2452, 0.0481, 0.1875, 0.1178, 0.2644)T . Table 1 Decision matrices A(1)
X1 X2 X3 X4
C1
C2
C3
C4
C5
C6
C7
s1 s3 s2 s2
s3 s2 s2 s2
s3 s0 s3 s−1
s0 s2 s1 s1
s1 s3 s4 s3
s2 s1 s3 s1
s2 s−1 s2 s1
C1
C2
C3
C4
C5
C6
C7
s1 s0 s3 s0
s2 s1 s1 s1
s3 s0 s2 s0
s0 s1 s2 s1
s2 s2 s4 s0
s2 s2 s4 s1
s4 s1 s1 s−1
C1
C2
C3
C4
C5
C6
C7
s0 s2 s2 s0
s2 s1 s1 s1
s2 s1 s2 s−1
s1 s2 s2 s1
s3 s0 s2 s0
s2 s2 s3 s0
s3 s−1 s2 s1
Table 2 Decision matrices A(2)
X1 X2 X3 X4
Table 3 Decision matrices A(3)
X1 X2 X3 X4
Z. Wu, Y. Chen / Fuzzy Sets and Systems 158 (2007) 1608 – 1617
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Utilize Eq. (15) to aggregate individual preference and thus get the collective overall preference values Zi (w) (i = 1, 2, 3, 4) of alternative Xi Z1 (w) = 2.2011,
Z2 (w) = 0.8037,
Z3 (w) = 2.4087,
Z4 (w) = 0.3446.
Then we rank Zi (w) (i = 1, 2, 3, 4) in descending order Z3 (w) Z1 (w) Z2 (w) Z4 (w). From the above results, the maximum value is Z3 (w). Hence X3 is the best option. Represent the results by the original linguistic labels, Z1 = s2.2011 ,
Z2 = s0.8037 ,
Z3 = s2.4087 ,
Z4 = s0.3446 .
We know the performances of X1 and X3 are between “good” and “very good”, but X3 is better. In the following, the F-OWA operator is used to compute the overall preference values. From Definition 2, let (k) w = (0.2, 0.1, 0.15, 0.2, 0.1, 0.15, 0.1)T [21], then the individual overall preference values zi of alternative Xi (i = 1, 2, 3, 4) for the DM k (k = 1, 2, 3) are (1)
z1 = 0.2 × [0.7, 0.8, 0.9] + 0.1 × [0.7, 0.8, 0.9] + 0.15 × [0.6, 0.7, 0.8] + 0.2 × [0.6, 0.7, 0.8] + 0.1 × [0.5, 0.6, 0.7] + 0.15 × [0.5, 0.6, 0.7] + 0.1 ×[0.4, 0.5, 0.6] = [0.585, 0.685, 0.785]. Similarly, we have (1)
z2 = [0.56, 0.66, 0.76],
(1)
z3 = [0.655, 0.755, 0.855],
(1)
(2)
z4 = [0.545, 0.645, 0.745], (2)
z4 = [0.435, 0.535, 0.635],
(3)
z3 = [0.61, 0.71, 0.81],
z3 = [0.65, 0.75, 0.85], z2 = [0.51, 0.61, 0.71],
(2)
z1 = [0.615, 0.715, 0.815], (2)
(3)
z2 = [0.505, 0.605, 0.705], (3)
z1 = [0.595, 0.695, 0.795], (3)
z4 = [0.435, 0.535, 0.635].
By using the F-OWA operator again (here, take = ( 13 , 31 , 13 )T as the DM’s weight vector), thus we get the collective overall preference value zi of alternative Xi z1 = [0.5983, 0.6983, 0.7983],
z2 = [0.5250, 0.5350, 0.6350],
z3 = [0.6383, 0.7383, 0.8383],
z4 = [0.4717, 0.5717, 0.6717].
Rank zi (i = 1, 2, 3, 4) in descending order z3 z1 z2 z4 . Thus the best alternative is X3 . Although the order of the selection is the same, there are differences in the original linguistic domains. For our methods, the grade of alternative X1 and X3 both are better than “good”, while only the alternative X3 is upper the grade “good” in F-OWA operator method. The LWAA operator puts emphasize on the importance of the linguistic arguments and the attribute themselves, while F-OWA operator considers the positions of the arguments. So the method proposed in this paper utilizes the decision information more sufficiently by deciding the attribute weights objectively. The F-OWA method does not take the importance of attribute into consideration and determine the associated weighting vector more subjectively. 6. Concluding remarks In group decision making, the DMs may have vague knowledge about the preference degree of one alternative over another and cannot estimate their preferences with exact numerical values. It is more suitable to provide their preferences
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by means of linguistic variables rather than numerical ones. In this article, based on the idea that the attribute with a larger deviation value among alternatives should be evaluated a larger weight, we have developed a method named the maximizing deviation method to determine the optimal relative weights of attributes under linguistic environment. The prominent characteristic of the developed is that it can relieve the influence of subjectivity of the DMs and utilize the decision information sufficiently. Then we have proposed a general approach on the basis of LWAA operator to group multi-attribute decision-making problems with linguistic information, in which the preference values take the form of linguistic variables. We have also applied the approach to a group decision-making problem of choosing the best option for a risk investment company. The proposed approach is also compared to the F-OWA method to show its advantages and effectiveness. Our method is straightforward and has no loss of information. The given approach can be extended to other domain like uncertain linguistic environment and be applied to many other practical fields. Dealing with MADM problem defined in multi-granular linguistic contexts is the future work. Acknowledgments We are very grateful to the editor and the anonymous referees for their valuable comments and suggestions, which have been very helpful in improving the paper. We also would like to thank Chen Chong for his help to improve the language quality. References [1] D. Ben-Arieh, Z.F. Chen, Linguistic-labels aggregation and consensus measure for autocratic decision making using group recommendations, IEEE Trans. Systems Man Cybernet. Part A 36 (2006) 558–568. [2] G. Bordogna, M. Fedrizzi, G. Pasi, A linguistic modeling of consensus in group decision making based on OWA operators, IEEE Trans. Systems Man Cybernet. Part A 27 (1997) 126–132. [3] S.A. Byeong, On the properties of OWA operator weights functions with constant level of orness, IEEE Trans. Fuzzy Systems 14 (2006) 511– 515. [4] 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. [5] F. Chiclana, E. Herrera-Viedma, F. Herrera, S. Alonso, Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations, Internat. J. Intell. Systems 19 (2004) 891–913. [6] O. Cordon, F. Herrera, I. Zwir, Linguistic modeling by hierarchical systems of linguistic rules, IEEE Trans. Fuzzy Systems 10 (2002) 2–20. [7] F. Herrera, E. Herrera-Viedma, Aggregation operators for linguistic weighted information, IEEE Trans. Systems Man Cybernet. Part A 27 (1997) 646–656. [8] F. Herrera, E. Herrera-Viedma, Linguistic decision analysis: steps for solving decision problems under linguistic information, Fuzzy Sets and Systems 115 (2000) 67–82. [9] F. Herrera, E. Herrera-Viedma, L. Martinez, A fusion approach for managing multi-granularity linguistic term sets in decision making, Fuzzy Sets and Systems 114 (2000) 43–58. [10] F. Herrera, E. Herrera-Viedma, J.L. Verdegay, A sequential selection process in group decision making with linguistic assessment, Inform. Sci. 85 (1995) 223–239. [11] E. Herrera-Viedma, L. Martinez, F. Mata, F. Chiclana,A consensus support system model for group decision-making problems with multigranular linguistic preference relations, IEEE Trans. Fuzzy Systems 13 (2005) 644–658. [12] V.A. Huynh, Y. Nakamori, A satisfactory-oriented approach to multi-expert decision-making with linguistic assessments, IEEE Trans. Systems Man Cybernet. Part B 35 (2005) 184–196. [13] E. Levrat, A. Voisin, S. Bombardier, J. Bremont, Subjective evaluation of car seat comfort with fuzzy techniques, Internat. J. Intell. Systems 12 (1997) 891–913. [14] M. Marimin, M. Uano, I. Hatono, et al., Hierarchical semi-numeric method for pairwise fuzzy group decision making, IEEE Trans. Systems Man Cybernet. Part B 32 (2002) 691–700. [15] J.I. Peláez, J.M. Doˇna, LAMA: a linguistic aggregation of majority additive operator, Internat. J. Intell. Systems 18 (2003) 809–820. [16] M. Roubens, Fuzzy sets and decision analysis, Fuzzy Sets and Systems 90 (1997) 199–206. [17] Y.C. Tang, J.C. Zheng, Linguistic modeling based on semantic similarity relation among linguistic labels, Fuzzy Sets and Systems 157 (2006) 1662–1673. [18] J.H. Wang, J.Y. Hao, A new version of 2-tuple fuzzy linguistic representation model for computing with words, IEEE Trans. Fuzzy Systems 14 (2006) 435–445. [19] Y.M. Wang, Using the method of maximizing deviations to make decision for multi-indices, System Eng. Electron. 7 (1998) 24–26 31. [20] Y.M. Wang, C. Parkan, A general multiple attribute decision-making approach for integrating subjective preferences and objective information, Fuzzy Sets and Systems 157 (2006) 1333–1345. [21] Z.S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Tsinghua University Press, Beijing, 2004. [22] Z.S. Xu, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Inform. Sci. 166 (2004) 19–30.
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[23] Z.S. Xu, Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment, Inform. Sci. 168 (2004) 171–184. [24] Z.S. Xu, Deviation measures of linguistic preference relations in group decision making, Omega 33 (2005) 249–254. [25] Z.S. Xu, An overview of methods for determining OWA weights, Internat. J. Intell. Systems 20 (2005) 843–865. [26] Z.S. Xu, An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations, Deci. Support Systems 41 (2006) 488–499. [27] Z.S. Xu, Induced uncertain linguistic OWA operators applied to group decision making, Inform. Fusion 7 (2006) 231–238. [28] Z.S. Xu, Q.L. Da, The ordered weighted geometric averaging operators, Internat. J. Intell. Systems 17 (2002) 709–716. [29] Z.S. Xu, Q.L. Da, An overview of operators for aggregating information, Internat. J. Intell. Systems 18 (2003) 953–969. [30] R.R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Trans. Systems Man Cybernet. 18 (1988) 183–190. [31] R.R. Yager, Non-numerical multi-criteria multi-person decision making, Group Deci. Negot. 2 (1993) 81–93. [32] R.R. Yager, D.P. Filev, Induced ordered weighted averaging operators, IEEE Trans. Systems Man Cybernet. Part B 29 (1999) 141–150. [33] R.R. Yager, Z.S. Xu, The continuous ordered weighted geometric operator and its application to decision making, Fuzzy Sets and Systems 157 (2006) 1393–1402.
The maximizing deviation method for group multiple attribute decision making under linguistic environment Zhibin Wu∗ , Yihua Chen College of Mathematics and Physics, Chonqing University, Chongqing 400044, China Received 21 July 2006; received in revised form 4 November 2006; accepted 26 January 2007 Available online 17 February 2007
Abstract The aim of this paper is to put forward a method for multi-attribute decision making problems with linguistic information, in which the preference values take the form of linguistic variables. An aggregating operator named linguistic weighted arithmetic averaging (LWAA) operator is introduced to aggregate the given decision information to get the overall preference value of each alternative. Some properties of the LWAA operator are also investigated. Based on the idea that the attribute with a larger deviation value among alternatives should be evaluated a larger weight, a method to determine the optimal weighting vector of LWAA operator is developed under the assumption that attribute weights are completely unknown. The based approach is extended to the situation where partially weight information can be obtained by solving a constrained non-linear optimization problem. Then a procedure to group multiple attribute decision making is provided under linguistic environment. Finally, an example of risk investment problem is given to verify the proposed approach; a comparative study to fuzzy ordered weighted averaging (F-OWA) operator methods is also demonstrated. © 2007 Elsevier B.V. All rights reserved. Keywords: Group decision making; Linguistic weighted arithmetic averaging (LWAA) operator; Maximizing deviation method; Linguistic variables
1. Introduction Multiple attribute decision making (MADM) addresses the problem of choosing an optimum choice that has the highest degree of satisfaction from a set of alternatives that are characterized in terms of their attributes. In order to make a decision or choose a best alternative, a decision maker (DM) is often asked to provide his/her preferences either on alternatives or on the relative weights of attributes or on both of them. Usually, the information is expressed by means of numerical values (like exact value, interval values, fuzzy numbers [20,21]). However, there are some situations in which the information may not be quantified because of its nature, and, thus, it may be stated only in linguistic terms (e.g., when evaluating the “comfort” or “design” of a car, terms like “bad,” “poor,” “tolerable,” “average,” or “good” can be used [13]). In other cases, precise quantitative information may not be stated because either it is not available or the cost of its computation is too high, and then an “approximate value” may be tolerated (e.g., when evaluating the speed of a car, linguistic terms like “fast,” “very fast,” or “slow” are used instead of numerical values). So, the use of linguistic labels makes expert judgment more reliable and informative for decision making. ∗ Corresponding author. Tel./fax: +86 23 65111075.
E-mail address: [email protected] (Z. Wu). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.01.013
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Group decision-making problems usually follow a common resolution scheme composed by two phases: aggregation phase and exploitation phase [8,16]. In the literature, many aggregation operators have been developed to aggregation information [29], such as ordered weighted averaging (OWA) operator [30], induced ordered weighted averaging (IOWA) operator [32], ordered weighted geometric (OWG) operator [4,28], induced ordered weighted geometric (IOWG) operator [5], generalized IOWA (GIOWA) operator and generalized IOWGA (GIOWGA) operator [29], continuous ordered weighted averaging (C-OWA) operator [33], etc. One important issue in the OWA-like operators is to determining their associated weights. Xu [25] briefly review the existing methods for determining the weights associated OWA operator, such as the linguistic quantifier’s approach [30], the analytic approach [3]. A number of studies recently focused on group decision making under linguistic environment [1,2,6–15,17,18,21–24,26,27,29,31] and thus a lot of linguistic aggregation operators were proposed. They are linguistic weighted geometric averaging (LWGA) operator and linguistic ordered weighted geometric averaging (LOWGA) operator [22], uncertain linguistic OWA (ULOWA) operator and uncertain linguistic hybrid aggregation (ULHA) operator [23], linguistic aggregation of majority additive (LAWA) operator [15], uncertain linguistic geometric mean (ULGM) operator, uncertain linguistic weighted geometric mean (ULWGM) operator, uncertain linguistic OWG (ULOWG) operator, induced ULOWG (IULOWG) operator [26], etc. The above operators have been found applications in group decision making and management science. These linguistic operators are the extension or generalization of the conventional operators. Herrera and Herrera-Viedma [8] analyzed the steps to follow in the linguistic decision analysis of a group decision-making problem with linguistic preference relations, i.e., choice of the linguistic term set, choice of the aggregation operator for linguistic information, choice of the best alternatives. Herrera-Viedma et al. [11] presented a model of consensus support system to assist the experts in all phases of the consensus reaching process of group decision-making problems with multi-granular linguistic preference relations. Ben-Arieh and Chen [1] also proposed a method to increase the consensus level by updating the importance of the group members. Tang and Zheng [17] developed a new linguistic modeling technique based on the semantic similarity relation among linguistic labels. Huynh and Nakamori [12] proposed a satisfactoryoriented approach in which a random preference is defined for each alternative in the aggregation phase instead of using an aggregation operator to obtain a collective preference value. Most papers in the literature put their emphasis on the extension of the aggregation operators or the consensus level of the group DM. However, when using these operators, the associated weighting vector is more or less determined subjectively and the decision information itself is not taken into consideration sufficiently. In this paper, we focus our attention on developing a method objectively to determine the attribute weights under the conditions that the attribute weights are completely unknown and the input arguments of the decision matrix take the form of linguistic variables. We will follow the steps analyzed in Ref. [8]. To do so, the rest of this paper is organized as follows. In Section 2, we introduce some basic notations and operational laws of linguistic variables. In Section 3, we propose LWAA operator as the aggregation operator and present a method to derive the weight vector of attributes in group decision making. In Section 4, we give a decision procedure for group decision making with linguistic variables. Section 5 gives an illustrative example to verify the developed approach and makes a comparative study to the F-OWA operator method. Section 6 concludes the paper. 2. Preliminaries: linguistic variables Suppose that S = {s | = −t, . . . , −1, 0, 1, . . . , t} is a finite and totally ordered discrete term set whose cardinality value is an odd one, such as 7 and 9, where s represents a possible value for a linguistic variable. The cardinality of S must be small enough so as not to impose useless precision to the experts, and it must be rich enough to allow a discrimination of the performances of each criterion in limited grades. For example, a set of nine terms S could be [21,22] S = {s−4 = extremely poor, s−3 = very poor, s−2 = poor, s−1 = slightly poor, s0 = fair, s1 = slightly good, s2 = good, s3 = very good, s4 = extremely good}. In these cases, it usually requires that si and sj satisfy the following additional characteristics [8,9]: (1) the set is ordered: si sj , if i j ; (2) there is a negation operator: neg(si ) = s−i , especially, neg(s0 ) = s0 ;
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(3) max operator: max(si , sj ) = si , if i j ; (4) min operator: min(si , sj ) = si , if i j . In the process of information aggregation, however, some results may not exactly match any linguistic labels in S. To preserve all the information, extend the discrete linguistic label set S to a continuous linguistic label set S¯ = {s |s−q s sq }, where s meets all the characteristics above and q(q > t)is a sufficiently large positive integer. If s ∈ S, s is called the original term, otherwise, s is called the virtual term. In general, the original term is used to ¯ and evaluate alternatives, while the virtual term is used in operation. Consider any two linguistic terms s , s ∈ S, , 1 , 2 ∈ [0, 1], their operational laws are given by s ⊕ s = s+ ,
s ⊕ s = s ⊕ s ,
(1 + 2 )s = 1 s ⊕ 2 s ,
s = s ,
(s ⊕ s ) = s ⊕ s .
From the above descriptions, we can see that the operation on two linguistic terms can be converted to the operation on lower indices of the corresponding terms. 3. The maximizing deviation method For simplicity, we let M = {1, 2, . . . , m} and N = {1, 2, . . . , n}. Suppose the alternatives are known. Let X = {X1 , X2 , . . . , Xn } denote a discrete set of n(n2) potential alternatives. Attributes are predefined too, let C = {C1 , C2 , . . . , Cm } denote a set of m(m 2) criterions or attributes. Theattribute weights are completely unknown. Let w = (w1 , w2 , . . . , wm )T be the weight vector of attributes, such that m j =1 wj = 1,wj 0, j ∈ M and wj denotes the weight of attribute Cj . Let D = {d1 , d2 , . . . , dt } be the set of DMs, and = (1 , 2 , . . . , t ) be the weight vector (k) of DMs, with k ∈ [0, 1], k = 1, 2, . . . , t, tk=1 k = 1. Suppose that A(k) = [aij ]n×m is the decision matrix given (k)
by the DM dk ∈ D, where aij is a linguistic variable for alternative Xi with respect to the attribute Cj . Definition 1. Let {sa1 , sa2 , . . . , san } be a collection of linguistic arguments, a linguistic weighted arithmetic averaging (LWAA) operator is a mapping LWAA:S n → S, and defined as LWAAw (sa1 , sa2 , . . . , san ) = w1 sa1 ⊕ w2 sa2 ⊕ · · · ⊕ wn san = s , (1) n where = j =1 wj I (aj ), w = (w1 , w2 , . . . , wn )T is the associated weighting vector such that wj ∈ [0, 1], j ∈ N , n j =1 wj = 1, and I (aj ) is the subscript of aj . Example. Assume w = (0.3, 0.2, 0.4, 0.1)T , then LWAAw (s4 , s2 , s−1 , s−3 ) = 0.3 × s4 + 0.2 × s2 + 0.4 × s−1 + 0.1 × s−3 = s0.3×4 + s0.2×2 + s0.4×(−1) + s0.1×(−3) = s0.3×4+0.2×2+0.4×(−1)+0.1×(−3) = s0.9 . It is easy to show that LWAA operator is a special case of the linguistic hybrid aggregation (LHA) operator [21] and has some good properties such as bounded, idempotency, monotonicity. Proposition 1 (Bounded). minj ∈N (saj ) LWAAw (sa1 , sa2 , . . . , san ) maxj ∈N (saj ). Proposition 2 (Idempotency). If saj = sa , for all j, j ∈ N , then LWAAw (sa1 , sa2 , . . . , san ) = sa . Proposition 3 (Monotonicity). If saj sbj , for all j, j ∈ N , then s = LWAAw (sa1 , sa2 , . . . , san ) s = LWAAw (sb1 , sb2 , . . . , sbn ).
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Proposition 4 (LAA operator). If w = (1/n, 1/n, . . . , 1/n)T , then LWAA operator is reduced to LAA operator, i.e., 1 I (aj ). n n
LWAAw (sa1 , sa2 , . . . , san ) = s
where =
j =1
To compare with other techniques of aggregation of linguistic information in the example section, the fuzzy ordered weighted averaging (F-OWA) operator is defined in the following. The F-OWA operator is a special case of generalized induced OWA (GIOWA) operator [21]. It transforms the linguistic scales into triangle fuzzy numbers. The set of corresponding transformation is given by F = {extremely poor = [0, 0.1, 0.2], very poor = [0.1, 0.2, 0.3], poor = [0.2, 0.3, 0.4], slightly poor = [0.3, 0.4, 0.5], s0 = [0.4, 0.5, 0.6], slightly good = [0.5, 0.6, 0.7], good = [0.6, 0.7, 0.8], very good = [0.7, 0.8, 0.9], extremely good = [0.8, 0.9, 1]}. Definition 2. Let {sa1 , sa2 , . . . , san } be a collection of linguistic arguments, a F-OWA operator is defined as F -OWAw (sa1 , sa2 , . . . , san ) =
n
(2)
w j fj ,
j =1
where w = (w1 , w2 , . . . , wn )T is the associated weighting vector such that wj ∈ [0, 1], j ∈ N , is the jth largest triangle fuzzy number in the corresponding transformed set.
n
j =1 wj
= 1, and fj
3.1. Computing the optimal weights: the maximizing deviation method Once each expert has expressed linguistic judgments on each alternative with respect to each criterion or attribute, the first phase is aimed at synthesizing the performance judgments of the alternatives for each expert. In this paper, we will use the LWAA operator as the main aggregation operators. By Definition 1, the overall preference value of alternative Xi for DM k can be expressed as (k)
Zi (w) =
m j =1
(k)
(3)
wj I (aij ).
The maximizing deviation method is proposed by Wang [19] to deal with MADM problems with numerical information. For a MADM problem, we need to compare the collective preference values to rank the alternatives, the larger the ranking value Zi (w) is, the better the corresponding alternative Xi is. If the performance values of each alternative have little differences under an attribute, it shows that such an attribute plays a small important role in the priority procedure. Contrariwise, if some attribute makes the performance values among all the alternatives have obvious differences, such an attribute plays an important role in choosing the best alternative. So to the view of sorting the alternatives, if one attribute has similar attribute values across alternatives, it should be assigned a small weight; otherwise, the attribute which makes larger deviations should be evaluated a bigger weight, in spite of the degree of its own importance. Especially, if all available alternatives score about equally with respect to a given attribute, then such an attribute will be judged unimportant by most DMs. In other word, such an attribute should be assigned a very small weight. Wang [19] suggests that zero should be assigned to the attribute of this kind. The deviation method is selected here to compute the differences of the performance values of each alternative. For the DM k and the attribute Cj , the deviation of alternative Xi to all the other alternatives can be defined as follows: (k)
Hij (w) =
n (k) (k) p I (aij ) − I (alj ) wj , l=1
i ∈ N, j ∈ M,
(4)
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where p(p 1) is an integer. For simplicity, let p = 2, and (4) can be rewritten as (k)
Hij (w) =
n
(k)
(k)
(I (aij ) − I (alj ))2 wj ,
i ∈ N, j ∈ M.
(5)
l=1
Let (k)
Hj (w) =
n
(k)
Hij (w) =
i=1
n n
(k)
(k)
(I (aij ) − I (alj ))2 wj ,
j ∈ M.
(6)
i=1 l=1
(k)
Then Hj (w) represent the deviation value of all alternatives to other alternatives for the attribute Cj and the DM k. Based on the aforementioned analysis, we have to choose the weight vector w to maximize all deviation values for all the attributes and for all the DMs. To do so, we can construct a non-linear programming model as follows: ⎧ m n n t ⎪ (k) (k) ⎪ ⎪ (I (aij ) − I (alj ))2 wj k ⎨ max H (w) = k=1 j =1 i=1 l=1 (M1) m ⎪ ⎪ s.t. w 0, j ∈ M, wj2 = 1. ⎪ j ⎩ j =1
To solve the above model, let L(w, ) =
t
k
m n n j =1 i=1 l=1
k=1
⎞ ⎛ m 1 (k) (k) (I (aij ) − I (alj ))2 wj + ⎝ wj2 − 1⎠ 2
(7)
j =1
denote the Lagrange function of the constrained optimization problem (M1), where is a real number. Then the partial derivatives of L are computed as n t n jL (k) (k) = k (I (aij ) − I (alj ))2 + wj = 0, jwj k=1 i=1 l=1 ⎞ ⎛ m jL 1 ⎝ 2 wj − 1⎠ = 0. = j 2
j ∈ M,
(8)
(9)
j =1
From (8), it follows that (k) (k) − tk=1 k ni=1 nl=1 (I (aij ) − I (alj ))2 wj = , Put (10) into (9), we have
t 2 m n n (k) (k) 2 = − k (I (aij ) − I (alj )) . j =1
k=1
j ∈ M.
(10)
(11)
i=1 l=1
Let Yj =
t k=1
k
n n
(k)
(k)
(I (aij ) − I (alj ))2 ,
j ∈ M.
(12)
i=1 l=1
Then (10) and (11) give Yj wj = m
2 j =1 Yj
,
j ∈ M.
Thus Eq. (13) is the extreme point of model (M1).
(13)
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From (13), it can be verified easily that wj , j ∈ M are positive such that they do satisfy the constrained conditions in model (M1) and the solution is unique. By normalizing wj to let the sum of wj , j ∈ M be a unit, we have wj wj∗ = m
j =1 wj
Yj = m
j =1 Yj
,
j ∈ M.
(14)
As a mater of fact, Yj represents the deviation of all alternatives to other alternatives for the attribute Cj and for all the DMs. Because the larger Yj is, the more important the attribute Rj is, Eq. (14) is obtained directly by using each Yj divide the sum of Yj , which is considered as an alternative approach and an explanation to the maximizing method. The theoretic foundation of this method is based on information theory, that is, the attribute providing more information should be evaluated a bigger weight. Now, the overall weighted assessment value of each alternative can be computed using the following expression and the decision can be made: Zi (w) =
t
k
k=1
m j =1
wj∗ I (aij ). (k)
(15)
3.2. Further analysis We have discussed the situation where the attribute weights are completely unknown in group decision making in the first part. If t = 1, the group decision-making problem is reduced to a single decision-making problem. Following in a similar manner, we can obtain the optimal weight vector as follows: ⎤ ⎡ n n 2 −1 n m n ⎥ ⎢ (k) (k) (k) (k) wj∗ = (I (aij ) − I (alj ))2 ∗ ⎣ (I (aij ) − I (alj ))2 ⎦ , j ∈ M. j =1
i=1 l=1
i=1 l=1
(16) There are actual situations that the information about the weighting vector is not completely unknown but partially known. For these cases, let be the set of weight information known, consequently, the following constrained non-linear programming model is constructed. ⎧ m n n t ⎪ (k) (k) ⎪ ⎪ max H (w) = k (I (aij ) − I (alj ))2 wj ⎨ k=1 j =1 i=1 l=1 (M2) m ⎪ ⎪ wj2 = 1. ⎪ ⎩ s.t. w ∈ , wj 0, j ∈ M, j =1
The solution to the above maximization problem could be found by MatLab software with optimization toolbox or Lindo/Lingo software package. 4. An approach to group decision making with linguistic information Group decision-making problems follow a common resolution scheme [16] composed by the following two phases: (1) Aggregation phase: It combines individual preferences to obtain a collective preference value for each alternative. (2) Exploitation phase: It orders collective preference values to obtain the best alternative(s). The notations in this section are assumed have the same meaning to the previous section. In the following, we develop a practical approach based on LWAA operator to group decision making with linguistic information. (k) Step1: To a group MADM problems, construct the decision matrix A(k) = [aij ]n×m , where k represents the kth DM (k)
and all the arguments aij are linguistic variables. Step2: If the weight vector of attributes is completely unknown, utilize Eq. (14) to obtain the optimal weights.
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Step3: Utilize the LWAA operator to aggregate the decision information. In group decision making, the collective overall preference values Zi (i ∈ N ) are computed by Eq. (15). Step4: Ranking all the alternatives Xi (i ∈ N ) and select the best one(s) in accordance with the value of Zi (i ∈ N ). Step5: End. 5. An illustrative example: a risk investment problem Let us suppose a risk investment company, which wants to invest a sum of money in the best option (adapted from [21, p. 165]). This MADM problem involves the evaluation of four possible alternatives denoted as X1 , X2 , X3 , X4 . The investment company must make a decision according the following seven attributes: C1 —the ability of sale, C2 —the ability of management, C3 —the ability of production, C4 —the ability of technology, C5 —the ability of financing, C6 —the ability to resist venture, C7 —the consistency of corporation strategy. Three DMs (without loss of generality, take weight vector = ( 13 , 13 , 13 ) are asked to evaluate each alternative using the linguistic term set in terms of their performance S = {s−4 = extremely poor, s−3 = very poor, s−2 = poor, s−1 = slightly poor, s0 = fair, s1 = slightly good, s2 = good, s3 = very good, s4 = extremely good}. (k)
The decision matrices A(k) = (aij )n×m (k = 1, 2, 3) are listed in Tables 1–3. Assume the weight vector of the attribute is completely unknown, by applying Eq. (14), we get the optimal weight vector w = (0.1154, 0.0216, 0.2452, 0.0481, 0.1875, 0.1178, 0.2644)T . Table 1 Decision matrices A(1)
X1 X2 X3 X4
C1
C2
C3
C4
C5
C6
C7
s1 s3 s2 s2
s3 s2 s2 s2
s3 s0 s3 s−1
s0 s2 s1 s1
s1 s3 s4 s3
s2 s1 s3 s1
s2 s−1 s2 s1
C1
C2
C3
C4
C5
C6
C7
s1 s0 s3 s0
s2 s1 s1 s1
s3 s0 s2 s0
s0 s1 s2 s1
s2 s2 s4 s0
s2 s2 s4 s1
s4 s1 s1 s−1
C1
C2
C3
C4
C5
C6
C7
s0 s2 s2 s0
s2 s1 s1 s1
s2 s1 s2 s−1
s1 s2 s2 s1
s3 s0 s2 s0
s2 s2 s3 s0
s3 s−1 s2 s1
Table 2 Decision matrices A(2)
X1 X2 X3 X4
Table 3 Decision matrices A(3)
X1 X2 X3 X4
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Utilize Eq. (15) to aggregate individual preference and thus get the collective overall preference values Zi (w) (i = 1, 2, 3, 4) of alternative Xi Z1 (w) = 2.2011,
Z2 (w) = 0.8037,
Z3 (w) = 2.4087,
Z4 (w) = 0.3446.
Then we rank Zi (w) (i = 1, 2, 3, 4) in descending order Z3 (w) Z1 (w) Z2 (w) Z4 (w). From the above results, the maximum value is Z3 (w). Hence X3 is the best option. Represent the results by the original linguistic labels, Z1 = s2.2011 ,
Z2 = s0.8037 ,
Z3 = s2.4087 ,
Z4 = s0.3446 .
We know the performances of X1 and X3 are between “good” and “very good”, but X3 is better. In the following, the F-OWA operator is used to compute the overall preference values. From Definition 2, let (k) w = (0.2, 0.1, 0.15, 0.2, 0.1, 0.15, 0.1)T [21], then the individual overall preference values zi of alternative Xi (i = 1, 2, 3, 4) for the DM k (k = 1, 2, 3) are (1)
z1 = 0.2 × [0.7, 0.8, 0.9] + 0.1 × [0.7, 0.8, 0.9] + 0.15 × [0.6, 0.7, 0.8] + 0.2 × [0.6, 0.7, 0.8] + 0.1 × [0.5, 0.6, 0.7] + 0.15 × [0.5, 0.6, 0.7] + 0.1 ×[0.4, 0.5, 0.6] = [0.585, 0.685, 0.785]. Similarly, we have (1)
z2 = [0.56, 0.66, 0.76],
(1)
z3 = [0.655, 0.755, 0.855],
(1)
(2)
z4 = [0.545, 0.645, 0.745], (2)
z4 = [0.435, 0.535, 0.635],
(3)
z3 = [0.61, 0.71, 0.81],
z3 = [0.65, 0.75, 0.85], z2 = [0.51, 0.61, 0.71],
(2)
z1 = [0.615, 0.715, 0.815], (2)
(3)
z2 = [0.505, 0.605, 0.705], (3)
z1 = [0.595, 0.695, 0.795], (3)
z4 = [0.435, 0.535, 0.635].
By using the F-OWA operator again (here, take = ( 13 , 31 , 13 )T as the DM’s weight vector), thus we get the collective overall preference value zi of alternative Xi z1 = [0.5983, 0.6983, 0.7983],
z2 = [0.5250, 0.5350, 0.6350],
z3 = [0.6383, 0.7383, 0.8383],
z4 = [0.4717, 0.5717, 0.6717].
Rank zi (i = 1, 2, 3, 4) in descending order z3 z1 z2 z4 . Thus the best alternative is X3 . Although the order of the selection is the same, there are differences in the original linguistic domains. For our methods, the grade of alternative X1 and X3 both are better than “good”, while only the alternative X3 is upper the grade “good” in F-OWA operator method. The LWAA operator puts emphasize on the importance of the linguistic arguments and the attribute themselves, while F-OWA operator considers the positions of the arguments. So the method proposed in this paper utilizes the decision information more sufficiently by deciding the attribute weights objectively. The F-OWA method does not take the importance of attribute into consideration and determine the associated weighting vector more subjectively. 6. Concluding remarks In group decision making, the DMs may have vague knowledge about the preference degree of one alternative over another and cannot estimate their preferences with exact numerical values. It is more suitable to provide their preferences
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by means of linguistic variables rather than numerical ones. In this article, based on the idea that the attribute with a larger deviation value among alternatives should be evaluated a larger weight, we have developed a method named the maximizing deviation method to determine the optimal relative weights of attributes under linguistic environment. The prominent characteristic of the developed is that it can relieve the influence of subjectivity of the DMs and utilize the decision information sufficiently. Then we have proposed a general approach on the basis of LWAA operator to group multi-attribute decision-making problems with linguistic information, in which the preference values take the form of linguistic variables. We have also applied the approach to a group decision-making problem of choosing the best option for a risk investment company. The proposed approach is also compared to the F-OWA method to show its advantages and effectiveness. 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