Power-Geometric Operators and Their Use in ... - Semantic Scholar
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
Power-Geometric Operators and Their Use in Group Decision Making Zeshui Xu, Senior Member, IEEE, and Ronald R. Yager, Fellow, IEEE
Abstract—The power-average (PA) operator and the powerordered-weighted-average (POWA) operator are the two nonlinear weighted-average aggregation tools whose weighting vectors depend on the input arguments. In this paper, we develop a powergeometric (PG) operator and its weighted form, which are on the basis of the PA operator and the geometric mean, and develop a power-ordered-geometric (POG) operator and a power-orderedweighted-geometric (POWG) operator, which are on the basis of the POWA operator and the geometric mean, and study some of their properties. We also discuss the relationship between the PA and PG operators and the relationship between the POWA and POWG operators. Then, we extend the PG and POWG operators to uncertain environments, i.e., develop an uncertain PG (UPG) operator and its weighted form, and an uncertain power-orderedweighted-geometric (UPOWG) operator to aggregate the input arguments taking the form of interval of numerical values. Furthermore, we utilize the weighted PG and POWG operators, respectively, to develop an approach to group decision making based on multiplicative preference relations and utilize the weighted UPG and UPOWG operators, respectively, to develop an approach to group decision making based on uncertain multiplicative preference relations. Finally, we apply both the developed approaches to broadband Internet-service selection. Index Terms—Group decision making, multiplicative preference relation, power-average (PA) operator, power-geometric (PG) operator, uncertain multiplicative preference relation, uncertain PG (UPG) operator.
I. INTRODUCTION NFORMATION aggregation is a process that fuses data from various resources by using a proper aggregation technique. The investigation on information aggregation has been receiving extensive attention from researchers and practitioners over the past decades, and a variety of operators have been developed to aggregate data information under various environments [1]–[5], such as the weighted-average operator [6], weighted-geometric-mean operator [7], harmonic-mean operator [8], ordered-weighted-average (OWA) operator [9], ordered-weighted-geometric operator [10], [11], weighted OWA operator [12], induced OWA operator [13], induced orderedweighted-geometric operator [4], uncertain OWA operator [14],
I
Manuscript received February 21, 2009; revised June 19, 2009, and September 14, 2009; accepted October 29, 2009. First published November 17, 2009; current version published February 5, 2010. This work was supported by the National Science Fund for Distinguished Young Scholars of China under Grant 70625005. Z. S. Xu is with the Antai School of Economic and Management, Shanghai Jiaotong University, Shanghai 200052, China, and also with the Institute of Sciences, People’s Liberation Army University of Science and Technology, Nanjing 210096, China (e-mail: [email protected]). R. R. Yager is with the Machine Intelligence Institute, Iona College, New Rochelle, NY 10801 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2009.2036907
hybrid aggregation operator [15], fuzzy-weighted-average operator [16], generalized OWA aggregation operator [17], linguistic aggregation operators [18]–[22], etc. However, yet most of the existing aggregation operators do not take into account the information about the relationship between the values being fused. Yager [23] introduced a tool to provide more versatility in the information-aggregation process, i.e., developed a poweraverage (PA) operator and a power OWA (POWA) operator, whose weighting vectors depend upon the input arguments and allow values being aggregated to support and reinforce each other. In real-life situations, the arguments are usually given by different individuals (decision makers or experts) to express their preferences over a set of alternatives, i.e., X = {x1 , x2 , . . . , xn }. The preferences mainly take the forms of utility values [10], [24]–[26], fuzzy-preference relations [10], [27], [28], and multiplicative preference relations [29]–[31], which can be described as follows, respectively: 1) Utility values ui ∈ [0, 1] (i = 1, 2, . . . , m), where ui represents the utility evaluation given by the decision maker to the alternative xi ; 2) fuzzypreference relation P = (pij )m ×m ⊂ X × X, which satisfies pij ≥ 0, pij + pj i = 1, pii = 0.5, i, j = 1, 2, . . . , m, where pij indicates the preference degree of the alternative xi over xj ; and 3) multiplicative preference relation A = (aij )m ×m ⊂ X × X, which satisfies aij > 0, aij aj i = 1, aii = 1, i, j = 1, 2, . . . , m, where aij is interpreted as the ratio of the preference intensity of the alternative xi to that of xj . If the preference values (arguments) are expressed in utility values or contained in a collection of fuzzy-preference relations provided by different individuals, then we can use the PA and POWA operators to aggregate all these individual preferences into group one. However, considering that both the PA and POWA operators are a weighted-average aggregation tool, these two operators are unsuitable to deal with the arguments taking the forms of multiplicative preference relations, for example, let aij = 5 and bij = 1/5 be two elements taken, respectively, from two multiplicative preference relations provided by two decision makers, and let the weights wk (k = 1 and 2) associated with aij and bij be the same (i.e., w1 = w2 = 0.5), where aij = 3 indicates that the alternative xi is strongly preferred to the alternative xj , and bij = 1/3 indicates that the alternative xj is strongly preferred to the alternative xi . Then, the result aggregated by the PA and POWA operators is 0.5 × 5 + 0.5 × (1/5) = 2.6, which means that xi is approximately strongly preferred to the alternative xj . Obviously, this result does not conform to reality (in this case, the weighted-geometric aggregation tool may get the reasonable result, i.e., 50.5 × (1/5)0.5 = 1, which implies an indifference between xi and xj ). To solve this issue, being
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XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
motivated by the PA and POWA operators, in this paper, we will develop some new geometric aggregation operators, including the power-geometric (PG) operator, weighted PG operator, and power-ordered-weighted-geometric (POWG) operator, and apply them to group decision making based on multiplicative preference relations. In order to do this, we organize the rest of the paper in seven sections. In Section II, we first review the concepts of the PA and POWA operators. Then, we develop a PG operator and its weighted form, which are on the basis of the PA operator and the geometric mean, and a POWG operator, which is on the basis of the POWA operator and the geometric mean, and study some of their properties, such as commutativity, idempotency, boundedness, etc. We also discuss the relationship between the PA and PG operators and the relationship between the POWA and POWG operators. Section III utilizes the weighted PG and POWG operators, respectively, to develop an approach to group decision making based on multiplicative preference relations. In Section IV, we develop an uncertain PG (UPG) operator and its weighted form and an uncertain POWG (UPOWG) operator to aggregate the input arguments, which are expressed in interval numbers. The properties of these operators are also studied. In Section V, we utilize the weighted UPG and UPOWG operators, respectively, to develop an approach to group decision making based on uncertain multiplicative preference relations. A practical example is given in Section VI, and finally, some concluding remarks are provided in Section VII. II. POWER-GEOMETRIC OPERATORS Yager [23] introduced a nonlinear weighted-average aggregation tool, which is called PA operator, and which can be defined as follows: n (1 + T (ai ))ai (1) PA(a1 , a2 , . . . , an ) = i=1 n i=1 (1 + T (ai )) where
95
ωi (i = 1, 2, . . . , n) are a collection of weights such that i i−1 ωi = g −g , i = 1, 2, . . . , n n n
(4)
where g : [0, 1] → [0, 1] is a basic unit-interval monotonic (BUM) function having the following properties. 1) g(0) = 0. 2) g(1) = 1. 3) g(x) ≥ g(y), if x > y. Based on the OWA and PA operators, Yager [23] defined a POWA operator as follows: POWA(a1 , a2 , . . . , an ) =
n
ui aindex(i)
(5)
i=1
where
ui = g
TV =
n
Ri TV
−g
Vindex(i) ,
Ri−1 TV
,
Ri =
i
Vindex(j )
j =1
Vindex(i) = 1 + T (aindex(i) )
(6)
i=1
and T (aindex(i) ) denotes the support of the ith largest argument by all the other arguments, i.e., T (aindex(i) ) =
n
Sup aindex(i) , aindex(j )
(7)
j =1 j = i
where Sup aindex(i) , aindex(j ) indicates the support of jth largest argument for the ith largest argument. Especially, if g(x) = x, then the POWA operator reduces to the PA operator. Based on the PA operator and the geometric mean, in the following, we define a PG operator: n 1 + T ( a i ) n (1+ T (a i )) PG(a1 , a2 , . . . , an ) = ai i = 1
(8)
i=1
T (ai ) =
n
Sup(ai , aj )
(2)
j =1 j = i
and Sup(a, b) is the support for a from b, which satisfies the following three properties. 1) Sup(a, b) ∈ [0, 1]. 2) Sup(a, b) = Sup(b, a). 3) Sup(a, b) ≥ Sup(x, y), if |a − b| < |x − y|. Obviously, the support (i.e., Sup) measure is essentially a similarity index. The more the similarity, the closer the two values are, and the more they support each other. Yager [9] developed an OWA operator, which was defined as. OWA(a1 , a2 , . . . , an ) =
n
ωi aindex(i)
(3)
i=1
where index is an indexing function such that index(i) is the index of the ith largest of the arguments aj (j = 1, 2, . . . , n), and thus, aindex(i) is the ith largest argument of aj (j = 1, 2, . . . , n).
where aj (j = 1, 2, . . . , n) are a collection of arguments, and T (ai ) satisfies the condition (2). Clearly, the PG operator is a nonlinear weighted-geometric aggregation operator, and the weight (1 + T (ai ))/ ni=1 (1 + T (ai )) of the argument ai depends on all the input arguments aj (j = 1, 2, . . . , n) and allows the argument values to support each other in the geometric aggregation process. Lemma 1 [32], [33]: Letting xj > 0, λj > 0, j = 1, 2, . . . , n, and nj=1 λj = 1, then n j =1
(xj )λj ≤
n
λj xj
(9)
j =1
with equality if and only if x1 = x2 = · · · = xn . By Lemma 1, we have the following theorem. Theorem 1: Assuming that aj (j = 1, 2, . . . , n) are a collection of arguments, we then have PG(a1 , a2 , . . . , an ) ≤ PA(a1 , a2 , . . . , an ). Now, we discuss some properties of the PG operator.
(10)
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Theorem 2: Letting Sup(ai , aj ) = k, for all i = j, then n 1/n PG(a1 , a2 , . . . , an ) = ai (11)
Thus, by (16), it follows that PGw (a1 , a2 , . . . , an ) =
i=1
which indicates that when all the supports are the same, the PG operator is simply the geometric mean. Especially, if Sup(ai , aj ) = 0, for all i = j, i.e., all the supports are zero, then there is no support in the geometric aggregation process, and in this case, by (2), we have T (ai ) = 0, i = 1, 2, . . . , n, then 1 1 + T (ai ) = , n i=1 (1 + T (ai ))
n
i = 1, 2, . . . , n
(13)
2) Idempotency: If aj = a, for all j, then PG(a1 , a2 , . . . , an ) = a.
(14)
3) Boundedness: min ai ≤ PG(a1 , a2 , . . . , an ) ≤ max ai . i
i
(15)
In (8), all the objects that are being aggregated are of equal importance. In many situations, the weights of the objects should be taken into account, for example, in group decision making, the decision makers usually have different importance and, thus, need to be assigned different weights. Suppose that each object that is being aggregated has a weight indicating its importance; then, we define the weighted form of (8) as follows: nw i ( 1 + T ( a i ) ) n w i (1+ T (a i )) ai i = 1 PGw (a1 , a2 , . . . , an ) =
n
(16)
where n
T (a˙ i ) =
PGw (a˙ 1 , a˙ 2 , . . . , a˙ n ) =
wj Sup(ai , aj ) = k
n
wj ,
n
j =1 w j j = i
wj Sup(a˙ i , a˙ j )
wi = 1.
(18)
i = 1, 2, . . . , n.
j =1 j = i
(19)
(20)
(21)
w i (1+T ( a˙ i ))
n
a˙ i
i= 1
w i (1+T ( a˙ i ))
.
i=1
(22) Since (T (a˙ 1 ), T (a˙ 2 ), . . . , T (a˙ n )) may not be the permutation of (T (a1 ), T (a2 ), . . . , T (an )), then PGw (a1 , a2 , . . . , an ) = PGw (a˙ 1 , a˙ 2 , . . . , a˙ n ) generally does not hold. Based on the POWA operator and the geometric mean, in the following, we define a power ordered weighted geometric (i.e., POWG) operator as follows: POWG(a1 , a2 , . . . , an ) =
n
i auindex(i)
(23)
i=1
which satisfies the conditions (6) and (7), and aindex(i) is the ith largest argument of aj (j = 1, 2, . . . , n). Especially, if g(x) = x, then the POWG operator reduces to the PG operator. In fact, by (6), we have POWG(a1 , a2 , . . . , an ) =
n
i auindex(i)
i=1 n
(g (R i /T V )−g (R i −1 /T V )) aindex(i)
n
=
n i=1
(V i n d e x ( i ) /T V ) aindex(i)
=
n
1+ T (a i )/
ai
(R /T V −R i −1 /T V )
i aindex(i)
n i= 1
(1+ T (a i ))
i=1
= PG(a1 , a2 , . . . , an ).
Obviously, the weighted PG operator [see (16)] has the properties, as described in Theorem 1, as well as 2) and 3) of Theorem 3. However, Theorem 2, and 1) of Theorem 3 do not hold for the weighted PG operator. In fact, if we let Sup(ai , aj ) = k, for all i = j, then by (2), we have
j =1 j = i
n
n
i=1
i = 1, 2, . . . , n,
1+k
and thus
(17)
i=1
n
i= 1
wi
w i (1+T (a i ))
j =1 j = i
= n
n
i=1
wj Sup(ai , aj )
with the condition
T (ai ) =
j =1 w j j = i
i= 1
and hence, (20) is not equivalent to PGw (a1 , a2 , . . . , an ) = n 1/n . Moreover, if (a˙ 1 , a˙ 2 , . . . , a˙ n ) is any permutation of i=1 ai (a1 , a2 , . . . , an ), then
=
j =1 j = i
wi ∈ [0, 1],
n
n
i=1
i=1
T (ai ) =
ai
1+k
(12)
and thus, by (8) and (12), it is clear that the PG operator reduces to the geometric mean. Theorem 3: Let aj (j = 1, 2, . . . , n) be a collection of arguments, then we have the following properties. 1) Commutativity: If (a˙ 1 , a˙ 2 , . . . , a˙ n ) is any permutation of (a1 , a2 , . . . , an ), then PG(a1 , a2 , . . . , an ) = PG(a˙ 1 , a˙ 2 , . . . , a˙ n ).
=
w i (1+T (a i )) ai
i=1
wi
n
(24)
By Lemma 1, we have the following theorem. Theorem 4: Assuming that aj (j = 1, 2, . . . , n) are a collection of arguments, then we have POWG(a1 , a2 , . . . , an ) ≤ POWA(a1 , a2 , . . . , an ).
(25)
From Theorem 2 and (24), we have the following corollary. Corollary 1: Letting Sup(ai , aj ) = k, for all i = j, and g(x) = x, we then have n 1/n ai (26) POWG(a1 , a2 , . . . , an ) = i=1
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
TABLE I MEAN CI OF RANDOMLY GENERATED MATRICES
which indicates that when all the supports are the same, the POWG operator is simply the geometric mean. Similar to Theorem 3, we have the following theorem. Theorem 5: Letting aj (j = 1, 2, . . . , n) be a collection of arguments, then we have the following properties. 1) Commutativity: If (a˙ 1 , a˙ 2 , . . . , a˙ n ) is any permutation of (a1 , a2 , . . . , an ), then POWG(a1 , a2 , . . . , an ) = POWG(a˙ 1 , a˙ 2 , . . . , a˙ n ). (27) 2) Idempotency: If aj = a, for all j, then POWG(a1 , a2 , . . . , an ) = a.
(28)
3) Boundedness: min ai ≤ POWG(a1 , a2 , . . . , an ) ≤ max ai . i
i
(29)
From the theoretical analysis described earlier, both the weighted PG and POWG operators can take the given arguments and their relationships into account; the difference between these two operators is that the weighted PG operator emphasizes the importance of each argument, i.e., the closer an argument is to mid one(s), the more weight it is given, while the POWG operator weights the importance of the ordered position of each argument, i.e., the closer the ordered position of the argument is to mid one(s), the more weight it is given. In real-life situations, the arguments sometimes take the form of a collection of multiplicative preference relations provided by different individuals [29]–[31]. The commonly used geometric aggregation techniques, such as the weighted-geometricmean operator [7] and ordered-weighted-geometric-mean operator [10], [11], cannot capture the sophisticated nuances the decision makers want to reflect in the aggregated value, i.e., these aggregation techniques cannot take into account the relationships of the arguments provided by different individuals. However, the weighted PG and POWG operators, which are developed in this paper, can not only embody the relationships among the input arguments by allowing values being aggregated to support and reinforce each other but also measure the similarity degrees of the arguments and reduce the influence of those unduly high or unduly low arguments on the decision result by using the support measure to assign them lower weights. The PA and POWA operators, which have been developed by Yager [23], also have some desirable characteristics described earlier; they are very suitable for aggregation of individual fuzzy-preference relations into the collective one in the process of group decision making but unsuitable to deal with multiplicative preference relations (just as interpreted in Section I). The weighted PG and POWG operators are the useful complements and extensions of the PA and POWA operators and are very suitable for aggregation of individual multiplicative preference relations into the collective one by assigning lower weights to those unduly high or unduly low elements of individual multiplicative preference relations, which can make the group opinion more unanimous. In the next section, we will apply the weighted PG and POWG operators to group decision making based on multiplicative preference relations.
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III. APPROACH TO GROUP DECISION MAKING BASED ON MULTIPLICATIVE PREFERENCE RELATIONS Let us consider a group-decision-making problem; let X = {x1 , x2 , . . . , xn } be a finite set of alternatives, and let E = {e1 , e2 , . . . , em } be a set of decision makers, whose weight vector isw = (w1 , w2 , . . . , wm )T , with wk ≥ 0, k = 1, 2, . . . , m, and m k =1 wk = 1. The decision maker ek compares each pair of (k ) alternatives (xi , xj ) and provides his/her preference value aij over them using a ratio scale, and in particular, as Saaty [29] (k ) showed, the 1–9 scale. The preference value aij is interpreted as the ratio of the preference intensity of the alternative xi to that (k ) of xj , and all the preference values aij (i, j = 1, 2, . . . , n) are contained in a multiplicative preference relation Ak on the set (k ) X, which is defined as a reciprocal matrix Ak = (aij )n ×n ⊂ X × X under the following condition: (k )
aij > 0,
(k )
(k )
aij · aj i = 1,
(k )
aii = 1 for all i, j = 1, 2, . . . , n.
(30)
Saaty [29] presented a consistency index (CI(Ak )), which is given by CI(Ak ) =
λm ax (Ak ) − n n−1
(31)
and a consistency ratio (CR(Ak )), which is given by CR(Ak ) =
CI(Ak ) RI
(32)
to measure the consistency of the multiplicative preference relation Ak , where λm ax (Ak ) is the maximal eigenvalue of Ak , RI is the mean CI of randomly generated multiplicative preference relations, which is given in Table I, and suggested that if CR(Ak ) < 0.1, then the multiplicative preference relation Ak is of satisfactory consistency; otherwise, Ak is of unsatisfactory consistency. In the case where CR(Ak ) ≥ 0.1, we need to return the multiplicative preference relation Ak to the decision maker ek to reconsider the construction of a new multiplicative preference relation according to his/her new judgments and to follow this procedure until the multiplicative preference relation with satisfactory CR(Ak ) is obtained. Then, we utilize the weighted PG operator to develop an approach to group decision making based on multiplicative preference relations, which involves the following steps.
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Approach I: Step 1: Calculate the supports
Approach II: Step 1: Calculate
(k )
a − a(l) (k ) (l) ij ij , Sup aij , aij = 1 − m (k ) (l)
l=1 aij − aij
l = 1, 2, . . . , m
l= k
l= k
(33) which satisfy the support conditions 1)–3) in Section II. m (k ) (l) Especially, if l=1 |aij − aij | = 0, then we stipulate l= k
(k )
(l)
Sup(aij , aij ) = 1. Step 2: Utilize the weights wk (k = 1, 2, . . . , m) of the decision makers ek (k = 1, 2, . . . , m) to calculate the weighted (k ) (k ) support T (aij ) of the preference value aij by the other pref(l)
erence values aij (l = 1, 2, . . . , m, and l = k) T
(k ) (aij )
=
m
(k ) (l) wl Sup aij , aij
(34)
l=1 l= k
(k ) vij
= 1, 2, . . . , m)
(k ) wk 1 + T aij = m , (k ) k =1 wk 1 + T aij
k = 1, 2, . . . , m
,
= 1. It is necessary to point out that the support measure is a similarity measure, which can be used to measure the degree that a preference value provided by a decision maker is close to another one provided by other decision maker in a group-decision-making problem. Thus, in (38), index(k ) index(l) , aij ) denotes the similarity degree between Sup(aij preference value
and the lth largest
index(l) aij . index(k )
Step 2: Calculate the support T (aij
) of the kth largest (l)
m index(k ) index(k ) index(l) = T aij Sup aij , aij
(39)
l=1 l= k (k )
and by (6), calculate the weight uij associated with the kth index(k )
largest preference value aij , where (k −1) (k ) k Rij Rij (k ) (k ) index(l) −g , Rij = Vij uij = g T Vij T Vij l=1
T Vij =
m
index(l)
Vij
,
index(l)
Vij
index(l) = 1 + T aij
(40)
l=1
1 , aij
for all i < j.
(36)
Step 4: Utilize the well-known logarithmic least-square method (LLSM) [34] given by
i=1
l= k index(k )
| = 0, then we stipulate Sup(aij
index(l) aij )
(35)
k =1
n
index(l)
− aij
preference value by the other preference values aij (l = 1, 2, . . . , m, and l = k)
(k ) m (1) (2) (k ) v i j (m ) aij = PGw aij , aij , . . . , aij = aij
vi =
index(k )
|aij
index(k ) aij
(k ) (k ) where vij ≥ 0, k = 1, 2, . . . , m, and m k =1 vij = 1. Step 3: Utilize the weighted PG operator (16) to aggregate all the individual multiplicative preference relations Ak = (k ) (aij )n ×n (k = 1, 2, . . . , m) into the collective multiplicative preference relation A = (aij )n ×n , where
aj i =
ij
(38) which indicates the support of lth largest preference value, index(l) i.e., aij , for the kth largest preference value, i.e., m index(k ) (s) aij , of aij (s = 1, 2, . . . , m). Especially, if l=1
index(k )
(k )
the preference values
ij
the kth largest preference value aij
and calculate the weights vij (k = 1, 2, . . . , m) associated with (k ) aij (k
index(k ) index(l)
a index(k ) index(l) − aij ij = 1 − m index(k ) , aij Sup aij index(l)
−a l=1 a
n j =1
1/n aij
n j =1
1/n ,
i = 1, 2, . . . , n
(37)
aij
to derive the priority vector v = (v1 , v2 , . . . , vn )T of A = (aij )n ×n , where vi > 0, i = 1, 2, . . . , n, and ni=1 vi = 1. Step 5: Rank all the alternatives xi (i = 1, 2, . . . , n) in accordance with the priority weights vi (i = 1, 2, . . . , n). The more the weight vi , the better the alternative xi will be. In the case where the information about the weights of decision makers is unknown, then we utilize the POWG operator to develop an approach to group decision making based on multiplicative preference relations, which can be described as follows.
(k ) (k ) where uij ≥ 0, k = 1, 2, . . . , m, and m k =1 uij = 1, g is the BUM function, as described in Section II. Step 3: Utilize the POWG operator (23) to aggregate all the (k ) individual multiplicative preference relations Ak = (aij )n ×n (k = 1, 2, . . . , m) into the collective multiplicative preference relation A = (aij )n ×n , where aij =
(1) (2) (m ) POWG(aij , aij , . . . , aij )
n index(k ) u (i jk ) aij = i=1
aj i =
1 , aij
for all i < j.
(41)
Step 4: For this step, see Approach I. Step 5: For this step, see Approach I. In the previous two approaches, Approach I considers the situations where the weights of decision makers can be predefined and utilizes the weighted PG operator to aggregate all the individual multiplicative preference relations with satisfactory consistency into the collective multiplicative preference relation
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
and then uses the well-known LLSM method to derive its priority vector, and using this, we can rank and select the given alternatives. While Approach II considers the situations where the information about the weights of decision makers is unknown and utilizes the POWG operator to aggregate all the individual multiplicative preference relations with satisfactory consistency into the collective multiplicative preference relation, it then also uses the LLSM method to find the final decision result. Both the approaches are very suitable for group decision making based on multiplicative preference relations. Usually, in the process of group decision making, some individuals may provide unduly high or unduly low preferences to their preferred or repugnant objects. The approaches can reduce the influence of these unduly high or unduly low arguments on the decision result by using the support measure to assign them lower weights, thereby making the decision more reasonable and reliable. IV. UNCERTAIN POWER-GEOMETRIC OPERATORS In the previous sections, the considered arguments are exact numerical values; in this section, we consider the situations where the input arguments cannot be expressed in exact numerical values, but value ranges (i.e., interval numbers) can be obtained. We first review some operational laws, which are related to interval numbers [35], shortly. Let a ˜ = [aL , aU ] and ˜b = [bL , bU ] be two interval numbers, where aU ≥ aL > 0, and bU ≥ bL > 0; then, we have the following expressions. 1) a ˜ + ˜b = [aL , aU ] + [bL , bU ] = [aL + bL , aU + bU ]. 2) a ˜˜b = [aL , aU ] · [bL , bU ] = [aL bL , aU bU ]. 3) λ˜ a = λ [aL , aU ] = [λ aL , λ aU ], where λ > 0. λ 4) a ˜ = [aL , aU ]λ = [(aL )λ , ( aU )λ ], where λ > 0. In order to rank interval numbers, we now introduce a possibility degree formula [36] for the comparison between the interval numbers a ˜ = [aL , aU ] and ˜b = [bL , bU ] aU − bL p(˜ a ≥ ˜b) = min max U , 0 ,1 (42) a − aL + bU − bL where p(˜ a ≥ ˜b) is called the possibility degree of a ˜ ≥ ˜b, which satisfies 0 ≤ p(˜ a ≥ ˜b) ≤ 1,
p(˜ a ≥ ˜b) + p(˜b ≥ a ˜) = 1 p(˜ a≥a ˜) = 0.5.
(43)
Let a ˜j = [aLj , aUj ] (j = 1, 2, . . . , n) be a collection of interval numbers; then, based on the previous operational laws of interval numbers, we extend the PG operator to uncertain environments and define an UPG operator as follows:
n n 1+T (˜ ai ) (1+T (˜ a i )) i= 1 ˜2 , . . . , a ˜n ) = a ˜i (44) UPG(˜ a1 , a i=1
where T (˜ ai ) =
n j =1 j = i
Sup(˜ ai , a ˜j )
(45)
99
and Sup(˜ a, ˜b) is the support for a ˜ from ˜b, which satisfies the following three properties. 1) Sup(˜ a, ˜b) ∈ [0, 1]. 2) Sup(˜ a, ˜b) = Sup(˜b, a ˜). 3) Sup(˜ a, ˜b) ≥ Sup(˜ x, y˜), if d(˜ a, ˜b) < d(˜ x, y˜), where d is a distance measure. Similar to the PG operator, the UPG operator has the following properties. ˜j ) = k, for all i = j, then Theorem 6: Letting Sup(˜ ai , a 1/n n ˜2 , . . . , a ˜n ) = a ˜i (46) UPG(˜ a1 , a i=1
which indicates that when all the supports are the same, the UPG operator is simply the uncertain geometric mean. Theorem 7: Let a ˜j (j = 1, 2, . . . , n) be a collection of interval numbers, then we have the following properties. ˜2 , . . . , a ˜n ) is any permutation of 1) Commutativity: If (˜ a1 , a ˜2 , . . . , a ˜n ), then (˜ a1 , a UPG(˜ a1 , a ˜2 , . . . , a ˜n ) = UPG(˜ a1 , a ˜2 , . . . , a ˜n ).
(47)
2) Idempotency: If a ˜j = a ˜, for all j, then UPG(˜ a1 , a ˜2 , . . . , a ˜n ) = a ˜.
(48)
3) Boundedness: min a ˜i ≤ UPG(˜ a1 , a ˜2 , . . . , a ˜n ) ≤ max a ˜i . i
i
(49)
If the weights of the objects are taken into account, then we define the weighted form of (44) as follows:
n n w i (1+T (˜ a i )) w i (1+T (˜ a i )) i= 1 UPGw (˜ a1 , a ˜2 , . . . , a ˜n ) = a ˜i i=1
(50) where T (˜ ai ) =
n
wj Sup(˜ ai , a ˜j )
(51)
j =1 j = i
with the condition (18). Obviously, the weighted UPG operator (50) has the properties of 2) and 3) in Theorem 7; however, Theorem 6 and 1) of Theorem 7 do not hold for the weighted UPG operator. Based on the POWG operator (23) and the possibility degree formula (42), here, we define a UPOWG operator as follows: ˜2 , . . . , a ˜n ) = UPOWG(˜ a1 , a
n
i a ˜uindex(i)
(52)
i=1
˜j (j = where a ˜index(i) is the ith largest interval numbers of a 1, 2, . . . , n), and i Ri Ri−1 Vindex(j ) ui = g −g , Ri = TV TV j =1 TV =
n i=1
Vindex(i) ,
Vindex(j ) = 1 + T (˜ aindex(i) )
(53)
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
and T (˜ aindex(i) ) denotes the support of the ith largest interval number by all the other interval numbers, i.e., T (˜ aindex(i) ) =
n
Sup(˜ aindex(i) , a ˜index(j ) )
(54)
j =1
where Sup(˜ aindex(i) , a ˜index(j ) ) indicates the support of jth largest interval number for the ith largest interval number (here, we can use the possibility degree formula (42) to rank interval numbers). Especially, if g(x) = x, then the UPOWG operator reduces to the UPG operator. From Theorem 6, we have the following corollary. ˜j ) = k, for all i = j, and Corollary 2: Letting Sup(˜ ai , a g(x) = x, then n 1/n ˜2 , . . . , a ˜n ) = a ˜i (55) UPOWG(˜ a1 , a i=1
which indicates that when all the supports are the same, the UPOWG operator is simply the uncertain geometric mean. Similar to Theorem 7, we have the following theorem. Theorem 8: Letting a ˜j (j = 1, 2, . . . , n) be a collection of interval numbers, then we have the following properties. ˜2 , . . . , a ˜n ) is any permutation of 1) Commutativity: If (˜ a1 , a ˜2 , . . . , a ˜n ), then (˜ a1 , a UPOWG(˜ a1 , a ˜2 , . . . , a ˜n ) = UPOWG(˜ a1 , a ˜2 , . . . , a ˜n ). (56) ˜, for all j, then 2) Idempotency: If a ˜j = a UPOWG(˜ a1 , a ˜2 , . . . , a ˜n ) = a ˜.
(57)
3) Boundedness: min a ˜i ≤ UPOWG(˜ a1 , a ˜2 , . . . , a ˜n ) ≤ max a ˜i . i
i
(58)
As mentioned in Section III, in the next section, we will apply the weighted UPG and UPOWG operators to group decision making based on uncertain multiplicative preference relations. V. APPROACH TO GROUP DECISION MAKING BASED ON UNCERTAIN MULTIPLICATIVE PREFERENCE RELATIONS Let us consider a group-decision-making problem under uncertainty. Suppose that the decision maker ek compares each pair of alternatives (xi , xj ), provides his/her preference value range (k ) L (k ) U (k ) a ˜ij = [aij , aij ], and constructs an uncertain multiplica(k ) tive preference relation A˜k = (˜ aij )n ×n under the following condition: U (k )
≥ aij
L (k )
= aii
aij
aii
L (k )
> 0,
aij
L (k )
U (k )
U (k )
= 1,
for all i, j = 1, 2, . . . , n.
· aj i
= 1,
L (k )
aj i
U (k )
· aij
=1 (59)
In what follows, we discuss two different cases. Case 1: If the weight vector w = (w1 , w2 , . . . , wm )T of the decision makers ek (k = 1, 2, . . . , m) is predefined, then we utilize the weighted UPG operator to develop an approach to group decision making based on uncertain multiplicative preference relations, which involves the following steps.
Approach III: Step 1: Calculate the supports (k ) (l) (k ) (l) d a ˜ij , a ˜ij ˜ij = 1 − m Sup a ˜ij , a (k ) (l) , ˜ij , a ˜ij l=1 d a
l = 1, 2, . . . , m
l= k
(60) which satisfy the support conditions 1)–3) given in Section IV; here, without loss of generality, we let
U (l) (k ) (l) 1 L (l) L (k ) U (k ) ˜ij = aij − aij + aij − aij . (61) d a ˜ij , a 2 m (k ) (l) Especially, if aij , a ˜ij ) = 0, then we stipulate l=1 d(˜ l= k
(k )
(l)
˜ij ) = 1. Sup(˜ aij , a Step 2: Utilize the weights wk (k = 1, 2, . . . , m) of the decision makers ek (k = 1, 2, . . . , m) to calculate the weighted sup(k ) (k ) port T (˜ aij ) of the uncertain preference value a ˜ij by the other (l)
uncertain preference values a ˜ij (l = 1, 2, . . . , m, and l = k) aij ) = T (˜ (k )
m
(k ) (l) wl Sup a ˜ij , a ˜ij
(62)
l=1 l= k (k )
and calculate the weights v˙ ij
(k = 1, 2, . . . , m), which are (k )
associated with the uncertain preference values a ˜ij (k = 1, 2, . . . , m) (k ) ˜ij wk 1 + T a (k ) v˙ ij = m , k = 1, 2, . . . , m (63) ˜(k ) ij k =1 wk 1 + T a (k ) (k ) where v˙ ij ≥ 0, k = 1, 2, . . . , m, and m k =1 v˙ ij = 1. Step 3: Utilize the weighted UPG operator (50) to aggregate all the individual uncertain multiplicative preference relations (k ) A˜k = (˜ aij )n ×n (k = 1, 2, . . . , m) into the collective uncertain multiplicative preference relation A˜ = (˜ aij )n ×n , where a ˜ij =
(1) (2) (m ) ˜ij , a UPGw a ˜ij , . . . , a ˜ij
(k ) m (k ) v˙ i j = a ˜ij
k =1
a ˜j i = aLji , aUji ,
aLji =
1 , aUij
aUji =
1 , aLij
for all i < j. (64)
Step 4: Utilize the uncertain LLSM (ULLSM) 1/n n ˜ij j =1 a v˜i = 1/n , i = 1, 2, . . . , n n n ˜ij i=1 j =1 a
(65)
to derive the uncertain priority vector v˜ = (˜ v1 , v˜2 , . . . , v˜n )T of ˜ A = (˜ aij )n ×n . Step 5: Compare each pair of the uncertain priority weights v˜i (i = 1, 2, . . . , n) by using the possibility degree formula (42) and construct a possibility degree matrix P = (pij )n ×n , where vi ≥ v˜j ), i, j = 1, 2, . . . , n, which satisfy pij ≥ 0, pij = p(˜
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
pij + pj i = 1, pii = 0.5, i, j = 1, 2, . . . , n. Summing all the elements in each line of the matrix P , we get pi =
n
pij ,
i = 1, 2, . . . , n.
index(k ) index(l) index(k ) index(l) d a ˜ij ,a ˜ij = 1 − m ,a ˜ij Sup a ˜ij index(k ) index(l) ˜ij ,a ˜ij l=1 d a l= k
(67) which indicates the support of lth largest uncertain preference index(l) value, i.e., a ˜ij , for the kth largest uncertain preference (s)
value, i.e., a ˜ij , of a ˜ij (s = 1, 2, . . . , m) (here, we can use Step 5 of Approach III to rank uncertain preference values). index(k ) index(l) aij ,a ˜ij ) = 0, then we stipEspecially, if ml=1 d(˜ ulate
l= k index(k ) index(l) ,a ˜ij ) Sup(˜ aij
= 1. index(k )
Step 2: Calculate the support T (˜ aij index(k ) uncertain preference value a ˜ij (l) erence values a ˜ij (l = 1, 2, . . . , m,
) of the kth largest
by the other uncertain prefand l = k)
m index(k ) index(k ) index(l) T a ˜ij = Sup a ˜ij ,a ˜ij
(68)
l=1 l= k (k )
and by (53), calculate the weight u˙ ij associated with the kth index(k )
largest uncertain preference value, i.e., a ˜ij (k ) u˙ ij
=g
T Vij =
m
(k ) R˙ ij T Vij
−g
index(l)
Vij
,
(k −1) R˙ ij T Vij index(l)
Vij
, where
,
(k ) R˙ ij =
k
index(l) =1+T a ˜ij (k )
index(k ) u˙ i j
(k )
a ˜ij
,
a ˜j i = aLji , aUji ,
aLji =
i=1
aUji =
1 , aLij
for all i < j.
1 aUij (70)
Step 4: For this step, see Approach III. Step 5: For this step, see Approach III. Step 6: For this step, see Approach III. It is clear that Approaches III and IV have characteristics similar to Approaches I and II, respectively. VI. EXAMPLE ILLUSTRATION Four university students share a house, where they intend to have broadband Internet connection installed (adapted from [37]). There are four options available to choose from, which are provided by three Internet-service providers. 1) x1 : 1 Mbps broadband; 2) x2 : 2 Mbps broadband; 3) x3 : 3 Mbps broadband; 4) x4 : 8 Mbps broadband. Since the Internet service and its monthly bill will be shared among the four students ek (k = 1, 2, 3, 4) (whose weight vector w = (0.3, 0.3, 0.2, 0.2)T ), they decide to perform a group decision analysis. Suppose that the students reveal their preference relations for the options independently and anonymously, and construct the following multiplicative preference relations, respectively:
1 4 A1 = 3 4
1 1/2 A3 = 2 1/3
1/4 1 2 3
1/3 1/2 1 2
1/4 1/3 , 1/2 1
2 1 4 3
1/2 1/4 1 2
3 1/3 , 1/2 1
1 3 A2 = 4 2
1/3 1 3 3
1/4 1/3 1 1/2
1/2 1/3 2 1
1 2 A4 = 3 1/2
1/2 1 5 3
1/3 1/5 1 2
2 1/3 . 1/2 1
index(l)
Vij
l=1
m
n
By (32), we calculate the CRs of Ak (k = 1, 2, 3, 4) as follows: CR(A1 ) = 0.0575,
CR(A2 ) = 0.0821
CR(A3 ) = 0.2398,
CR(A4 ) = 0.4899.
(69)
l=1 (k )
(1) (2) (m ) ˜ij , a ˜ij , . . . , a ˜ij a ˜ij = aLij , aUij = UPOWG a =
Then, we can rank the uncertain priority weights v˜i (i = 1, 2, . . . , n) in descending order in accordance with pi (i = 1, 2, . . . , n). Step 6: Rank all the alternatives xi (i = 1, 2, . . . , n) in accordance with the descending order of the uncertain priority weights v˜i (i = 1, 2, . . . , n). Case 2: If the information about the weights of the decision makers ek (k = 1, 2, . . . , m) is unknown, then we utilize the UPOWG operator to develop an approach to group decision making based on uncertain multiplicative preference relations, which can be described as follows. Approach IV: Step 1: Calculate
index(l)
multiplicative preference relation A˜ = (˜ aij )n ×n , where
(66)
j =1
101
where u˙ ij ≥ 0, k = 1, 2, . . . , m, k =1 u˙ ij = 1, and g is the BUM function, as described in Section II. Step 3: Utilize the UPOWG operator (52) to aggregate all the individual uncertain multiplicative preference relations (k ) A˜k = (˜ aij )n ×n (k = 1, 2, . . . , m) into the collective uncertain
Then, CR(A1 ) < 0.1, CR(A2 ) < 0.1, CR(A3 ) > 0.1, and CR(A4 ) > 0.1, i.e., the multiplicative preference relations A1 and A2 are of satisfactory consistency, while the multiplicative preference relations A3 and A4 are of unsatisfactory consistency. In this case, we need to return A3 and A4 to the students e3 and e4 for revaluation, and they construct two new multiplicative preference relations (for convenience, we also denote
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
them by A3 and A4 , respectively) 1 2 1/2 2 1 1 1/2 1 1/2 1 1/4 1/3 1 1 1/3 1/3 A3 = , A4 = 2 4 1 1 2 3 1 1/2 1/2 3 1 1 1 3 2 1
34 31 32 34 = 1, S32 = 0.500, S32 = S32 = 0.750 S31
and then, by (32), we have CR(A3 ) = 0.0773, and CR(A4 ) = 0.0906, which indicate that the reconstructed multiplicative preference relations A3 and A4 are also of satisfactory consistency. Since the weights of the students are predefined, we then utilize Approach I to find the decision result. (k ) (l) We first utilize (33) to calculate the supports Sup(aij , aij )
41 42 43 41 42 S11 = S11 = S11 = 1, S12 = 0.690, S12 = 0.724
(k )
(l)
(i, j, k, l = 1, 2, 3, 4) (for simplicity, we denote Sup(aij , aij ) kl by Sij )
31 32 34 31 32 34 31 S33 = S33 = S33 = S34 = 1, S34 = 0, S34 = 1, S41 = 0.364 32 34 31 32 34 31 S41 = 0.727, S41 = 0.909, S42 = S42 = S42 = S43 =1 32 34 31 32 34 S43 = 0, S43 = S44 = S44 = S44 =1
43 41 42 43 S12 = 0.586, S13 = 0.600, S13 = 0.400, S13 =1 41 42 43 S14 = 0.667, S14 = 0.778, S14 = 0.556 41 42 43 S21 = 0.455, S21 = 0.636, S21 = 0.909 41 42 43 41 42 43 S22 = S22 = S22 = 1, S23 = 0.333, S23 = 1, S23 = 0.667 41 42 43 41 42 43 S24 = S24 = S24 = 1, S31 = 0.667, S31 = 0.333, S31 =1 41 42 43 S32 = 0.500, S32 = 1, S32 = 0.500 41 42 43 41 42 43 S33 = S33 = S33 = S34 = 1, S34 = 0, S34 =1
12 13 14 12 13 S11 = S11 = S11 = 1, S12 = 0.968, S12 = 0.323 14 12 13 14 S12 = 0.710, S13 = 0.800, S13 = 0.600, S13 = 0.600 12 13 14 12 S14 = 0.909, S14 = 0.364, S14 = 0.727, S21 = 0.867
41 42 S41 = 0.333, S41 = 0.778 43 41 42 43 41 42 S41 = 0.889, S42 = S42 = S42 = S43 = 1, S43 =0 43 41 42 43 S43 = 1, S44 = S44 = S44 = 1.
13 14 12 13 14 S21 = 0.533, S21 = 0.600, S22 = S22 = S22 =1 12 13 14 S23 = 0.714, S23 = 0.571, S23 = 0.714 12 13 14 12 13 S24 = S24 = S24 = 1S31 = 0.667, S31 = 0.667 14 12 13 14 S31 = 0.667, S32 = 0.750, S32 = 0.500, S32 = 0.750
Then, we utilize the weight vector w = (0.3, 0.3, 0.2, 0.2)T of the students ek (k = 1, 2, 3, 4) and (34) to calculate the (k ) weighted supports T (aij ) (i, j, k = 1, 2, 3, 4) of the prefer(k )
ence values aij (i, j, k = 1, 2, 3, 4), which are contained in the matrices Tk = (T (aij ))4×4 (k = 1, 2, 3, 4), respectively (k )
12 13 14 12 13 14 S33 = S33 = S33 = 1, S34 = 0, S34 = S34 =1
12 13 14 S41 = 0.692, S41 = 0.462, S41 = 0.538 12 13 14 12 S42 = S42 = S42 = 1, S43 =0
T1
13 14 12 13 14 S43 = S43 = S44 = S44 = S44 =1 21 23 24 21 23 S11 = S11 = S11 = 1, S12 = 0.966, S12 = 0.310 24 S12 21 S14
=
21 0.724, S13
=
23 0.889, S14
=
23 1.000, S13
=
24 0.333, S14
=
24 0.500, S13
= 0.500
=
21 0.778, S21
= 0.818
T2
23 24 21 23 24 S21 = 0.545, S21 = 0.636, S22 = S22 = S22 =1 21 S23
=
23 0.333, S23
=
24 0.667, S23
=
21 S24
=
23 S24
=
24 S24
=1
T3
21 23 24 21 S31 = 0.800, S31 = 0.600, S31 = 0.600, S32 = 0.500 23 24 21 23 24 S32 = 0.500, S32 = S33 = S33 = S33 =1
T4
21 23 24 21 S34 = S34 = S34 = 0.667, S41 = 0.556 23 24 21 23 24 S41 = 0.667, S41 = 0.778, S42 = S42 = S42 =1 21 23 24 21 23 24 S43 = S43 = S43 = 0.667, S44 = S44 = S44 =1 31 S11
=
32 S11
=
34 S11
=
31 1, S12
=
32 0.604, S12
= 0.623
34 31 32 34 S12 = 0.774, S13 = 0.600, S13 = 0.400, S13 =1 31 32 34 31 S14 = 0.588, S14 = 0.647, S14 = 0.765, S21 = 0.462 32 34 31 32 34 S21 = 0.615, S21 = 0.923, S22 = S22 = S22 =1 31 32 34 S23 = 0.400, S23 = 0.800, S23 = 0.800 31 32 34 31 32 S24 = S24 = S24 = 1, S31 = 0.667, S31 = 0.333
0.700 0.487 = 0.467 0.408 0.700 0.482 = 0.480 0.456 0.800 0.508 = 0.500 0.509 0.800 0.509 = 0.500 0.511
0.497 0.700 0.475 0.700
0.480 0.471 0.700 0.400
0.497 0.700 0.450 0.700
0.500 0.433 0.700 0.467
0.523 0.800 0.525 0.800
0.500 0.520 0.800 0.500
0.541 0.800 0.550 0.800
0.500 0.533 0.800 0.500
0.491 0.700 0.400 0.700 0.489 0.700 0.467 0.700 0.524 0.800 0.500 0.800 0.545 0.800 0.500 0.800 (k )
(i, j, k =
(k ) aij
(i, j, k =
and then utilize (35) to calculate the weights vij 1, 2, 3, 4) associated with the preference values
(k )
1, 2, 3, 4), which are contained in the matrices Vk = (vij )4×4 (k = 1, 2, 3, 4), respectively 0.293 0.297 0.297 0.297 0.299 0.293 0.298 0.293 V1 = 0.297 0.297 0.293 0.288 0.289
0.293
0.288
0.293
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
0.298 V2 = 0.299
0.301
0.293 0.292
0.290 0.293
0.293 0.301
0.299
0.293
0.301
0.293
0.207
0.202 V3 = 0.202
0.202
0.201
0.202
0.207 0.204
0.205 0.207
0.207 0.206
0.206
0.207
0.206
0.207
0.207 0.202 V4 = 0.202
0.204 0.207 0.208
0.201 0.207 0.207
0.205 0.207 0.206
0.207
0.207
0.206
0.207
0.296
0.297
0.293
based on which, we utilize the weighted PG operator [see (36)] to aggregate all the individual multiplicative preference relations (k ) Ak = (aij )4×4 (k = 1, 2, 3, 4) into the collective multiplicative preference relation 1 0.5499 0.3598 0.6207 1.8185 1 0.3546 0.3333 A= . 2.7793 2.8201 1 0.7584 1.6111
3.000
1.3184
1
After this, we utilize the LLSM (37) to derive the priority vector of A
If we use Saaty’s approach [8], [29] to solve the previous problem, then we first utilize the weight vector w = (0.3, 0.3, 0.2, 0.2)T and the weighted geometric mean operator [7] (71)
k =1
to aggregate all the individual multiplicative preference relations (k ) Ak = (aij )4×4 (k = 1, 2, 3, 4) into the collective multiplicative preference relation 1 0.5451 0.3596 0.6156 1.8345 1 0.3554 0.3333 A˙ = 2.7809 2.8137 1 0.7579 3.0000
1.3194
1
and then, by the well-known eigenvector method [29], we get the priority vector of A˙ v˙ = (0.1365, 0.1560, 0.3454, 0.3620)T from which the ranking of the options can be derived as follows: x4 x3 x2 x1 .
[2, 3]
[1, 1]
[2, 3] A˜2 = [2, 4]
[1, 1] [3, 5] [2, 4]
[1, 1]
[2, 3]
[1/3, 1]
[1, 3]
[1/3, 1/2] [1/4, 1/2]
[1/3, 1/2] [1, 1] A˜3 = [2, 3] [3, 4]
x4 x3 x2 x1 .
1.6244
[1, 2]
Using this, we get the ranking of the options as follows:
i, j = 1, 2, 3, 4
In the previous illustrative example, both approaches can simultaneously consider the weights of students and all the preference values provided by these students in the process of aggregating all the individual preference information into the group opinion, and especially, they derive the same ranking of the options. However, Saaty’s approach cannot take into account the relationships of the arguments provided by different individuals, while the approaches developed in this paper can not only embody the relationships among the input arguments by allowing values being aggregated to support and reinforce each other but also measure the similarity degrees of the arguments and reduce the influence of those unduly high or unduly low arguments on the decision result by using the support measure to assign them lower weights, thereby making the group opinion more unanimous. In the case where the preferences constructed by the students ek (k = 1, 2, 3, 4) are uncertain multiplicative preference relations, for example [1, 1] [1/5, 1/3] [1/3, 1] [1/2, 1] [3, 5] [1, 1] [1/4, 1/2] [1/3, 1/2] A˜1 = [1, 3] [2, 4] [1, 1] [1/3, 1]
[1, 3]
v = (0.1338, 0.1539, 0.3530, 0.3592)T .
4 (k ) w k aij , a˙ ij =
103
[1, 1]
[1, 3] A˜4 = [1, 2] [1/2, 2]
[1, 1] [1/3, 1]
[1/5, 1/3] [1/4, 1/2] [1, 1] [1/2, 1] [1, 2]
[1, 1]
[1/3, 1/2]
[1, 3]
[1/2, 1]
[1, 1]
[1/3, 1]
[1/2, 1]
[1/2, 2]
[4, 5] [3, 4]
[1/4, 1/3] [1/5, 1/3] [1, 1] [1, 2]
[3, 5] [1, 1]
[1/5, 1/4] [1/4, 1/3] . [1, 1] [1/2, 1] [1, 2]
[1, 1]
Then, we can utilize Approach III to derive the ranking of the four options, and the following decision steps are needed. (k ) Step 1: Utilize (60)–(63) to calculate the weights v˙ ij (k = 1, 2, 3, 4) associated with the uncertain preference val(k ) ues a ˜ij (k = 1, 2, 3, 4), which are contained in the matrices (k ) V˙ k = (v˙ ij )4×4 (k = 1, 2, 3, 4), respectively 0.293 0.295 0.292 0.297 0.297 0.293 0.297 0.296 V˙ 1 = 0.289 0.295 0.293 0.295 0.294
0.298
0.286
0.293 0.294 V˙ 2 = 0.302
0.296 0.293
0.293 0.291
0.292 0.296
0.293 0.296
0.297
0.293
0.297 0.304 0.294
0.293
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
0.202
0.204 V˙ 3 = 0.206
0.201
0.208
0.207 0.208
0.207 0.207
0.200 0.202
0.203
0.203
0.207
0.207
0.207
0.206 V˙ 4 = 0.204
0.207
0.207
0.204
0.207 0.206
0.206 0.207
0.200 . 0.209
0.205
0.204
0.211
0.207
0.207
Step 2: Utilize the weighted UPG operator (50) to aggregate all the individual uncertain multiplicative preference rela(k ) tions A˜k = (˜ aij )4×4 (k = 1, 2, 3, 4) into the collective uncertain multiplicative preference relation
our operators to aggregate all the individual multiplicative (or uncertain multiplicative) preference relations into the collective multiplicative (or uncertain multiplicative) preference relation under various group-decision-making contexts and then developed some group-decision-making approaches based on multiplicative (or uncertain multiplicative) preference relations. The prominent characteristic of the developed approaches is that they can take all the decision arguments and their relationships into account. Our operators could usefully be applied to many other areas such as data mining, information retrieval, and pattern recognition, which we suggest are the possible paths for future research. ACKNOWLEDGMENT
A˜ =
The authors would like to thank the associate editor and the six anonymous referees for their insightful and constructive [1, 1] [0.411, 0.735] [0.333, 0.707] [0.510, 1.438] comments and suggestions that have led to an improved version [1.361, 2.433] [1, 1] [0.223, 0.354] [0.260, 0.425] . of this paper. [1.414, 3.003] [2.825, 4.484] [1, 1] [0.510, 1.150]
[0.695, 1.961]
[2.353, 3.846]
[0.870, 1.961]
[1, 1]
Step 3: Utilize the ULLSM (65) to derive the uncertain priority vector of A˜ v˜1 = [0.0919, 0.2791],
v˜2 = [0.0948, 0.2149]
v˜3 = [0.2137, 0.5956],
v˜4 = [0.1954, 0.5888].
Step 4: In order to rank v˜i (i = 1, 2, 3, 4), by (42), we construct the possibility degree matrix 0.5 0.5997 0.1149 0.1442 0.4003 0.5 0.0024 0.0380 P = 0.8851 0.9976 0.5 0.5162 0.8558
0.9620
0.4838
0.5
and by (66), we have p1 = 1.3588,
p2 = 0.9407,
p3 = 2.8989,
p4 = 2.8016.
Then, v˜3 > v˜4 > v˜1 > v˜2 . Therefore, we rank the options xi (i = 1, 2, 3, 4) in accordance with the descending order of a ˜i (i = 1, 2, 3, 4) x3 x4 x1 x2 . From the previous numerical results, it can be known that the ranking of the options in the latter case are slightly different from the former case due to the change of the input arguments. VII. CONCLUDING REMARKS Motivated by the idea of the Yager’s PA operator, we have developed some new nonlinear weighted-geometric aggregation operators, including the PG operator, weighted PG operator, POWG operator, UPG operator, weighted UPG operator, and UPOWG operator. We have studied some properties of the developed operators, such as commutativity, idempotency, boundedness, etc. The fundamental aspect of these operators is that the weight of each input argument depends on the other input arguments and allows argument values to support each other in the geometric aggregation process. Moreover, we have applied
REFERENCES [1] R. R. Yager and J. Kacprzyk, The Ordered Weighted Averaging Operator: Theory and Applications. Boston, MA: Kluwer, 1997. [2] T. Calvo, G. Mayor, and R. Mesiar, Aggregation Operators: New Trends and Applications. Heidelberg, Germany: Physica-Verlag, 2002. [3] V. Torra, Information Fusion in Data Mining. New York: SpringerVerlag, 2003. [4] Z. S. Xu and Q. L. Da, “An overview of operators for aggregating information,” Int. J. Intell. Syst., vol. 18, pp. 953–969, 2003. [5] H. Bustince, F. Herrera, and J. Montero, Fuzzy Sets and Their Extensions: Representation, Aggregation and Models. Berlin, Germany: SpringerVerlag, 2007. [6] J. C. Harsanyi, “Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility,” J. Polit. Econ., vol. 63, pp. 309–321, 1955. [7] J. Acz´el and T. L. Saaty, “Procedures for synthesizing ratio judgements,” J. Math. Psychol., vol. 27, pp. 93–102, 1983. [8] P. S. Bullen, D. S. Mitrinovi, and P. M. Vasi, Means and Their Inequalities. Dordrecht, The Netherlands: Reidel, 1988. [9] R. R. Yager, “On ordered weighted averaging aggregation operators in multi-criteria decision making,” IEEE Trans. Syst., Man, Cybern., vol. 18, no. 1, pp. 183–190, Jan./Feb. 1988. [10] F. Chiclana, F. Herrera, and E. Herrera-Viedma, “Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations,” Fuzzy Sets Syst., vol. 122, pp. 277–291, 2001. [11] Z. S. Xu and Q. L. Da, “The ordered weighted geometric averaging operators,” Int. J. Intell. Syst., vol. 17, pp. 709–716, 2002. [12] V. Torra, “The weighted OWA operator,” Int. J. Intell. Syst., vol. 12, pp. 153–166, 1997. [13] R. R. Yager and D. P. Filev, “Induced ordered weighted averaging operators,” IEEE Trans. Syst., Man, Cybern., vol. 29, no. 2, pp. 141–150, Apr. 1999. [14] Z. S. Xu and Q. L. Da, “The uncertain OWA operator,” Int. J. Intell. Syst., vol. 17, pp. 569–575, 2002. [15] Z. S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications. Beijing, China: Tsinghua Univ. Press, 2004. [16] W. M. Dong and F. S. Wong, “Fuzzy weighted averages and implementation of the extension principle,” Fuzzy Sets Syst., vol. 21, pp. 183–199, 1987. [17] R. R. Yager, “Generalized OWA aggregation operator,” Fuzzy Optim. Decision Making, vol. 3, pp. 93–107, 2004. [18] R. R. Yager, “An approach to ordinal decision making,” Int. J. Approx. Reasoning, vol. 12, pp. 237–261, 1995. [19] F. Herrera, E. Herrera-Viedma, and J. L. Verdegay, “A sequential selection process in group decision making with a linguistic assessment approach,” Inf. Sci., vol. 85, pp. 223–239, 1995. [20] F. Herrera and L. Mart´ınez, “A 2-tuple fuzzy linguistic representation model for computing with words,” IEEE Trans. Fuzzy Syst., vol. 8, no. 6, pp. 746–752, Dec. 2000.
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
[21] V. Torra, “Aggregation of linguistic labels when semantics is based on antonyms,” Int. J. Intell. Syst., vol. 16, pp. 513–524, 2001. [22] Z. S. Xu, “A method based on linguistic aggregation operators for group decision making with linguistic preference relations,” Inf. Sci., vol. 166, pp. 19–30, 2004. [23] R. R. Yager, “The power average operator,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 31, no. 6, pp. 724–731, Nov. 2001. [24] R. D. Luce and P. Suppes, “Preference, utility and subject probability,” in Handbook of Mathematical Psychology, vol. III, R. D. Luce, R. R. Bush, and E. Galanter, Eds. New York: Wiley, 1965, pp. 249–410. [25] J. Ma, Z. P. Fan, Y. P. Jiang, and J. Y. Mao, “An optimization approach to multiperson decision making based on different formats of preference information,” IEEE Trans. Syst., Man, Cybern. A, Syst. Humans, vol. 36, no. 5, pp. 876–889, Sep. 2006. [26] Z. S. Xu, “Multiple attribute group decision making with different formats of preference information on attributes,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 37, no. 6, pp. 1500–1511, Dec. 2007. [27] S. A. Orlovsky, “Decision-making with a fuzzy preference relation,” Fuzzy Sets Syst., vol. 13, pp. 155–167, 1978. [28] T. Tanino, “Fuzzy preference ordering in group decision making,” Fuzzy Sets Syst., vol. 12, pp. 117–131, 1984. [29] T. L. Saaty, The Analytic Hierarchy Process. New York: McGraw-Hill, 1980. [30] F. Herrera, E. Herrera-Viedma, and F. Chiclana, “Multiperson decision making based on multiplicative preference relations,” Eur. J. Oper. Res., vol. 129, pp. 372–385, 2001. [31] Z. S. Xu, “On consistency of the weighted geometric mean complex judgment matrix in AHP,” Eur. J. Oper. Res., vol. 126, pp. 683–687, 2000. ¨ [32] O. Holder, “Uber einen Mittelwertsatz,” G¨ottingen Nachrichten, pp. 38– 47, 1889. [33] J. L. Jensen, “Sur les fonctions convexes et les in´egualit´es entre les valeurs moyennes,” Acta Math., vol. 30, pp. 175–193, 1906. [34] G. Crawford and C. Williams, “A note on the analysis of subjective judgment matrices,” J. Math. Psychol., vol. 29, pp. 387–405, 1985. [35] R. N. Xu and X. Y. Zhai, “Extensions of the analytic hierarchy process in fuzzy environment,” Fuzzy Sets Syst., vol. 52, pp. 251–257, 1992. [36] G. Facchinetti, R. G. Ricci, and S. Muzzioli, “Note on ranking fuzzy triangular numbers,” Int. J. Intell. Syst., vol. 13, pp. 613–622, 1998. [37] Y. M. Wang and C. Parkan, “Optimal aggregation of fuzzy preference relations with an application to broadband internet service selection,” Eur. J. Oper. Res., vol. 187, pp. 1476–1486, 2008.
Zeshui Xu (M’08–SM’09) received the Ph.D. degree in management science and engineering from Southeast University, Nanjing, China, in 2003. From April 2003 to May 2005, he was a Postdoctoral Researcher with the School of Economics and Management, Southeast University. From October 2005 to December 2007, he was a Postdoctoral Researcher with the School of Economics and Management, Tsinghua University, Beijing, China. He is an Adjunct Professor with the Antai School of Economic and Management, Shanghai Jiaotong University, Shanghai, China. He is also currently a Professor with the Institute of Sciences, People’s Liberation Army University of Science and Technology, Nanjing. He is a member of the Editorial Boards of Information: An International Journal, the International Journal of Applied Management Science, the International Journal of Data Analysis Techniques and Strategies, and the System Engineering—Theory and Practice and Fuzzy Systems and Mathematics. He has authored the following books: Uncertain Multiple Attribute Decision Making: Methods and Applications (Tsinghua Univ. Press, 2004), Intuitionistic Fuzzy Information: Aggregation Theory and Applications (Science, 2008), and Decision Making with Linguistic Information: Theory and Methods (Science, 2008). He has contributed more than 200 journal articles to professional journals. His current research interests include information fusion, group decision making, computing with words, and aggregation operators.
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Ronald R. Yager (S’66–M’68–SM’93–F’97) received the B.E.E. degree from the City College of New York, New York, and the Ph.D. degree from the Polytechnic University of New York, New York. He was a National Aeronautics and Space Administration (NASA)/Stanford Visiting Fellow and a Research Associate at the University of California, Berkeley. He has been a Lecturer at North Atlantic Treaty Organization (NATO) Advanced Study Institutes. He is currently the Director of the Machine Intelligence Institute, Iona College, New Rochelle, NY, where he is also a Professor of information and decision technologies. He is the Editor-in-Chief of the International Journal of Intelligent Systems. He serves on the Editorial Boards of a number of journals including Neural Networks, Data Mining and Knowledge Discovery, Fuzzy Sets and Systems, Journal of Approximate Reasoning, and International Journal of General Systems. He has been engaged in the area of fuzzy sets and related disciplines of computational intelligence for over 25 years. In addition to his pioneering work in the area of fuzzy logic, he has made fundamental contributions in decision making under uncertainty and the fusion of information. He has authored or coauthored over 500 papers published and 15 books. Prof. Yager was the recipient of the IEEE Computational Intelligence Society Pioneer Award in Fuzzy Systems. He is a Fellow of the New York Academy of Sciences and the Fuzzy Systems Association. He was given an award by the Polish Academy of Sciences for his contributions. He was the Program Director in the Information Sciences program at the National Science Foundation. He is a member of the Editorial Boards of a number of journals, including the IEEE TRANSACTIONS ON FUZZY SYSTEMS and IEEE INTELLIGENT SYSTEMS.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
Power-Geometric Operators and Their Use in Group Decision Making Zeshui Xu, Senior Member, IEEE, and Ronald R. Yager, Fellow, IEEE
Abstract—The power-average (PA) operator and the powerordered-weighted-average (POWA) operator are the two nonlinear weighted-average aggregation tools whose weighting vectors depend on the input arguments. In this paper, we develop a powergeometric (PG) operator and its weighted form, which are on the basis of the PA operator and the geometric mean, and develop a power-ordered-geometric (POG) operator and a power-orderedweighted-geometric (POWG) operator, which are on the basis of the POWA operator and the geometric mean, and study some of their properties. We also discuss the relationship between the PA and PG operators and the relationship between the POWA and POWG operators. Then, we extend the PG and POWG operators to uncertain environments, i.e., develop an uncertain PG (UPG) operator and its weighted form, and an uncertain power-orderedweighted-geometric (UPOWG) operator to aggregate the input arguments taking the form of interval of numerical values. Furthermore, we utilize the weighted PG and POWG operators, respectively, to develop an approach to group decision making based on multiplicative preference relations and utilize the weighted UPG and UPOWG operators, respectively, to develop an approach to group decision making based on uncertain multiplicative preference relations. Finally, we apply both the developed approaches to broadband Internet-service selection. Index Terms—Group decision making, multiplicative preference relation, power-average (PA) operator, power-geometric (PG) operator, uncertain multiplicative preference relation, uncertain PG (UPG) operator.
I. INTRODUCTION NFORMATION aggregation is a process that fuses data from various resources by using a proper aggregation technique. The investigation on information aggregation has been receiving extensive attention from researchers and practitioners over the past decades, and a variety of operators have been developed to aggregate data information under various environments [1]–[5], such as the weighted-average operator [6], weighted-geometric-mean operator [7], harmonic-mean operator [8], ordered-weighted-average (OWA) operator [9], ordered-weighted-geometric operator [10], [11], weighted OWA operator [12], induced OWA operator [13], induced orderedweighted-geometric operator [4], uncertain OWA operator [14],
I
Manuscript received February 21, 2009; revised June 19, 2009, and September 14, 2009; accepted October 29, 2009. First published November 17, 2009; current version published February 5, 2010. This work was supported by the National Science Fund for Distinguished Young Scholars of China under Grant 70625005. Z. S. Xu is with the Antai School of Economic and Management, Shanghai Jiaotong University, Shanghai 200052, China, and also with the Institute of Sciences, People’s Liberation Army University of Science and Technology, Nanjing 210096, China (e-mail: [email protected]). R. R. Yager is with the Machine Intelligence Institute, Iona College, New Rochelle, NY 10801 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2009.2036907
hybrid aggregation operator [15], fuzzy-weighted-average operator [16], generalized OWA aggregation operator [17], linguistic aggregation operators [18]–[22], etc. However, yet most of the existing aggregation operators do not take into account the information about the relationship between the values being fused. Yager [23] introduced a tool to provide more versatility in the information-aggregation process, i.e., developed a poweraverage (PA) operator and a power OWA (POWA) operator, whose weighting vectors depend upon the input arguments and allow values being aggregated to support and reinforce each other. In real-life situations, the arguments are usually given by different individuals (decision makers or experts) to express their preferences over a set of alternatives, i.e., X = {x1 , x2 , . . . , xn }. The preferences mainly take the forms of utility values [10], [24]–[26], fuzzy-preference relations [10], [27], [28], and multiplicative preference relations [29]–[31], which can be described as follows, respectively: 1) Utility values ui ∈ [0, 1] (i = 1, 2, . . . , m), where ui represents the utility evaluation given by the decision maker to the alternative xi ; 2) fuzzypreference relation P = (pij )m ×m ⊂ X × X, which satisfies pij ≥ 0, pij + pj i = 1, pii = 0.5, i, j = 1, 2, . . . , m, where pij indicates the preference degree of the alternative xi over xj ; and 3) multiplicative preference relation A = (aij )m ×m ⊂ X × X, which satisfies aij > 0, aij aj i = 1, aii = 1, i, j = 1, 2, . . . , m, where aij is interpreted as the ratio of the preference intensity of the alternative xi to that of xj . If the preference values (arguments) are expressed in utility values or contained in a collection of fuzzy-preference relations provided by different individuals, then we can use the PA and POWA operators to aggregate all these individual preferences into group one. However, considering that both the PA and POWA operators are a weighted-average aggregation tool, these two operators are unsuitable to deal with the arguments taking the forms of multiplicative preference relations, for example, let aij = 5 and bij = 1/5 be two elements taken, respectively, from two multiplicative preference relations provided by two decision makers, and let the weights wk (k = 1 and 2) associated with aij and bij be the same (i.e., w1 = w2 = 0.5), where aij = 3 indicates that the alternative xi is strongly preferred to the alternative xj , and bij = 1/3 indicates that the alternative xj is strongly preferred to the alternative xi . Then, the result aggregated by the PA and POWA operators is 0.5 × 5 + 0.5 × (1/5) = 2.6, which means that xi is approximately strongly preferred to the alternative xj . Obviously, this result does not conform to reality (in this case, the weighted-geometric aggregation tool may get the reasonable result, i.e., 50.5 × (1/5)0.5 = 1, which implies an indifference between xi and xj ). To solve this issue, being
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XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
motivated by the PA and POWA operators, in this paper, we will develop some new geometric aggregation operators, including the power-geometric (PG) operator, weighted PG operator, and power-ordered-weighted-geometric (POWG) operator, and apply them to group decision making based on multiplicative preference relations. In order to do this, we organize the rest of the paper in seven sections. In Section II, we first review the concepts of the PA and POWA operators. Then, we develop a PG operator and its weighted form, which are on the basis of the PA operator and the geometric mean, and a POWG operator, which is on the basis of the POWA operator and the geometric mean, and study some of their properties, such as commutativity, idempotency, boundedness, etc. We also discuss the relationship between the PA and PG operators and the relationship between the POWA and POWG operators. Section III utilizes the weighted PG and POWG operators, respectively, to develop an approach to group decision making based on multiplicative preference relations. In Section IV, we develop an uncertain PG (UPG) operator and its weighted form and an uncertain POWG (UPOWG) operator to aggregate the input arguments, which are expressed in interval numbers. The properties of these operators are also studied. In Section V, we utilize the weighted UPG and UPOWG operators, respectively, to develop an approach to group decision making based on uncertain multiplicative preference relations. A practical example is given in Section VI, and finally, some concluding remarks are provided in Section VII. II. POWER-GEOMETRIC OPERATORS Yager [23] introduced a nonlinear weighted-average aggregation tool, which is called PA operator, and which can be defined as follows: n (1 + T (ai ))ai (1) PA(a1 , a2 , . . . , an ) = i=1 n i=1 (1 + T (ai )) where
95
ωi (i = 1, 2, . . . , n) are a collection of weights such that i i−1 ωi = g −g , i = 1, 2, . . . , n n n
(4)
where g : [0, 1] → [0, 1] is a basic unit-interval monotonic (BUM) function having the following properties. 1) g(0) = 0. 2) g(1) = 1. 3) g(x) ≥ g(y), if x > y. Based on the OWA and PA operators, Yager [23] defined a POWA operator as follows: POWA(a1 , a2 , . . . , an ) =
n
ui aindex(i)
(5)
i=1
where
ui = g
TV =
n
Ri TV
−g
Vindex(i) ,
Ri−1 TV
,
Ri =
i
Vindex(j )
j =1
Vindex(i) = 1 + T (aindex(i) )
(6)
i=1
and T (aindex(i) ) denotes the support of the ith largest argument by all the other arguments, i.e., T (aindex(i) ) =
n
Sup aindex(i) , aindex(j )
(7)
j =1 j = i
where Sup aindex(i) , aindex(j ) indicates the support of jth largest argument for the ith largest argument. Especially, if g(x) = x, then the POWA operator reduces to the PA operator. Based on the PA operator and the geometric mean, in the following, we define a PG operator: n 1 + T ( a i ) n (1+ T (a i )) PG(a1 , a2 , . . . , an ) = ai i = 1
(8)
i=1
T (ai ) =
n
Sup(ai , aj )
(2)
j =1 j = i
and Sup(a, b) is the support for a from b, which satisfies the following three properties. 1) Sup(a, b) ∈ [0, 1]. 2) Sup(a, b) = Sup(b, a). 3) Sup(a, b) ≥ Sup(x, y), if |a − b| < |x − y|. Obviously, the support (i.e., Sup) measure is essentially a similarity index. The more the similarity, the closer the two values are, and the more they support each other. Yager [9] developed an OWA operator, which was defined as. OWA(a1 , a2 , . . . , an ) =
n
ωi aindex(i)
(3)
i=1
where index is an indexing function such that index(i) is the index of the ith largest of the arguments aj (j = 1, 2, . . . , n), and thus, aindex(i) is the ith largest argument of aj (j = 1, 2, . . . , n).
where aj (j = 1, 2, . . . , n) are a collection of arguments, and T (ai ) satisfies the condition (2). Clearly, the PG operator is a nonlinear weighted-geometric aggregation operator, and the weight (1 + T (ai ))/ ni=1 (1 + T (ai )) of the argument ai depends on all the input arguments aj (j = 1, 2, . . . , n) and allows the argument values to support each other in the geometric aggregation process. Lemma 1 [32], [33]: Letting xj > 0, λj > 0, j = 1, 2, . . . , n, and nj=1 λj = 1, then n j =1
(xj )λj ≤
n
λj xj
(9)
j =1
with equality if and only if x1 = x2 = · · · = xn . By Lemma 1, we have the following theorem. Theorem 1: Assuming that aj (j = 1, 2, . . . , n) are a collection of arguments, we then have PG(a1 , a2 , . . . , an ) ≤ PA(a1 , a2 , . . . , an ). Now, we discuss some properties of the PG operator.
(10)
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
Theorem 2: Letting Sup(ai , aj ) = k, for all i = j, then n 1/n PG(a1 , a2 , . . . , an ) = ai (11)
Thus, by (16), it follows that PGw (a1 , a2 , . . . , an ) =
i=1
which indicates that when all the supports are the same, the PG operator is simply the geometric mean. Especially, if Sup(ai , aj ) = 0, for all i = j, i.e., all the supports are zero, then there is no support in the geometric aggregation process, and in this case, by (2), we have T (ai ) = 0, i = 1, 2, . . . , n, then 1 1 + T (ai ) = , n i=1 (1 + T (ai ))
n
i = 1, 2, . . . , n
(13)
2) Idempotency: If aj = a, for all j, then PG(a1 , a2 , . . . , an ) = a.
(14)
3) Boundedness: min ai ≤ PG(a1 , a2 , . . . , an ) ≤ max ai . i
i
(15)
In (8), all the objects that are being aggregated are of equal importance. In many situations, the weights of the objects should be taken into account, for example, in group decision making, the decision makers usually have different importance and, thus, need to be assigned different weights. Suppose that each object that is being aggregated has a weight indicating its importance; then, we define the weighted form of (8) as follows: nw i ( 1 + T ( a i ) ) n w i (1+ T (a i )) ai i = 1 PGw (a1 , a2 , . . . , an ) =
n
(16)
where n
T (a˙ i ) =
PGw (a˙ 1 , a˙ 2 , . . . , a˙ n ) =
wj Sup(ai , aj ) = k
n
wj ,
n
j =1 w j j = i
wj Sup(a˙ i , a˙ j )
wi = 1.
(18)
i = 1, 2, . . . , n.
j =1 j = i
(19)
(20)
(21)
w i (1+T ( a˙ i ))
n
a˙ i
i= 1
w i (1+T ( a˙ i ))
.
i=1
(22) Since (T (a˙ 1 ), T (a˙ 2 ), . . . , T (a˙ n )) may not be the permutation of (T (a1 ), T (a2 ), . . . , T (an )), then PGw (a1 , a2 , . . . , an ) = PGw (a˙ 1 , a˙ 2 , . . . , a˙ n ) generally does not hold. Based on the POWA operator and the geometric mean, in the following, we define a power ordered weighted geometric (i.e., POWG) operator as follows: POWG(a1 , a2 , . . . , an ) =
n
i auindex(i)
(23)
i=1
which satisfies the conditions (6) and (7), and aindex(i) is the ith largest argument of aj (j = 1, 2, . . . , n). Especially, if g(x) = x, then the POWG operator reduces to the PG operator. In fact, by (6), we have POWG(a1 , a2 , . . . , an ) =
n
i auindex(i)
i=1 n
(g (R i /T V )−g (R i −1 /T V )) aindex(i)
n
=
n i=1
(V i n d e x ( i ) /T V ) aindex(i)
=
n
1+ T (a i )/
ai
(R /T V −R i −1 /T V )
i aindex(i)
n i= 1
(1+ T (a i ))
i=1
= PG(a1 , a2 , . . . , an ).
Obviously, the weighted PG operator [see (16)] has the properties, as described in Theorem 1, as well as 2) and 3) of Theorem 3. However, Theorem 2, and 1) of Theorem 3 do not hold for the weighted PG operator. In fact, if we let Sup(ai , aj ) = k, for all i = j, then by (2), we have
j =1 j = i
n
n
i=1
i = 1, 2, . . . , n,
1+k
and thus
(17)
i=1
n
i= 1
wi
w i (1+T (a i ))
j =1 j = i
= n
n
i=1
wj Sup(ai , aj )
with the condition
T (ai ) =
j =1 w j j = i
i= 1
and hence, (20) is not equivalent to PGw (a1 , a2 , . . . , an ) = n 1/n . Moreover, if (a˙ 1 , a˙ 2 , . . . , a˙ n ) is any permutation of i=1 ai (a1 , a2 , . . . , an ), then
=
j =1 j = i
wi ∈ [0, 1],
n
n
i=1
i=1
T (ai ) =
ai
1+k
(12)
and thus, by (8) and (12), it is clear that the PG operator reduces to the geometric mean. Theorem 3: Let aj (j = 1, 2, . . . , n) be a collection of arguments, then we have the following properties. 1) Commutativity: If (a˙ 1 , a˙ 2 , . . . , a˙ n ) is any permutation of (a1 , a2 , . . . , an ), then PG(a1 , a2 , . . . , an ) = PG(a˙ 1 , a˙ 2 , . . . , a˙ n ).
=
w i (1+T (a i )) ai
i=1
wi
n
(24)
By Lemma 1, we have the following theorem. Theorem 4: Assuming that aj (j = 1, 2, . . . , n) are a collection of arguments, then we have POWG(a1 , a2 , . . . , an ) ≤ POWA(a1 , a2 , . . . , an ).
(25)
From Theorem 2 and (24), we have the following corollary. Corollary 1: Letting Sup(ai , aj ) = k, for all i = j, and g(x) = x, we then have n 1/n ai (26) POWG(a1 , a2 , . . . , an ) = i=1
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
TABLE I MEAN CI OF RANDOMLY GENERATED MATRICES
which indicates that when all the supports are the same, the POWG operator is simply the geometric mean. Similar to Theorem 3, we have the following theorem. Theorem 5: Letting aj (j = 1, 2, . . . , n) be a collection of arguments, then we have the following properties. 1) Commutativity: If (a˙ 1 , a˙ 2 , . . . , a˙ n ) is any permutation of (a1 , a2 , . . . , an ), then POWG(a1 , a2 , . . . , an ) = POWG(a˙ 1 , a˙ 2 , . . . , a˙ n ). (27) 2) Idempotency: If aj = a, for all j, then POWG(a1 , a2 , . . . , an ) = a.
(28)
3) Boundedness: min ai ≤ POWG(a1 , a2 , . . . , an ) ≤ max ai . i
i
(29)
From the theoretical analysis described earlier, both the weighted PG and POWG operators can take the given arguments and their relationships into account; the difference between these two operators is that the weighted PG operator emphasizes the importance of each argument, i.e., the closer an argument is to mid one(s), the more weight it is given, while the POWG operator weights the importance of the ordered position of each argument, i.e., the closer the ordered position of the argument is to mid one(s), the more weight it is given. In real-life situations, the arguments sometimes take the form of a collection of multiplicative preference relations provided by different individuals [29]–[31]. The commonly used geometric aggregation techniques, such as the weighted-geometricmean operator [7] and ordered-weighted-geometric-mean operator [10], [11], cannot capture the sophisticated nuances the decision makers want to reflect in the aggregated value, i.e., these aggregation techniques cannot take into account the relationships of the arguments provided by different individuals. However, the weighted PG and POWG operators, which are developed in this paper, can not only embody the relationships among the input arguments by allowing values being aggregated to support and reinforce each other but also measure the similarity degrees of the arguments and reduce the influence of those unduly high or unduly low arguments on the decision result by using the support measure to assign them lower weights. The PA and POWA operators, which have been developed by Yager [23], also have some desirable characteristics described earlier; they are very suitable for aggregation of individual fuzzy-preference relations into the collective one in the process of group decision making but unsuitable to deal with multiplicative preference relations (just as interpreted in Section I). The weighted PG and POWG operators are the useful complements and extensions of the PA and POWA operators and are very suitable for aggregation of individual multiplicative preference relations into the collective one by assigning lower weights to those unduly high or unduly low elements of individual multiplicative preference relations, which can make the group opinion more unanimous. In the next section, we will apply the weighted PG and POWG operators to group decision making based on multiplicative preference relations.
97
III. APPROACH TO GROUP DECISION MAKING BASED ON MULTIPLICATIVE PREFERENCE RELATIONS Let us consider a group-decision-making problem; let X = {x1 , x2 , . . . , xn } be a finite set of alternatives, and let E = {e1 , e2 , . . . , em } be a set of decision makers, whose weight vector isw = (w1 , w2 , . . . , wm )T , with wk ≥ 0, k = 1, 2, . . . , m, and m k =1 wk = 1. The decision maker ek compares each pair of (k ) alternatives (xi , xj ) and provides his/her preference value aij over them using a ratio scale, and in particular, as Saaty [29] (k ) showed, the 1–9 scale. The preference value aij is interpreted as the ratio of the preference intensity of the alternative xi to that (k ) of xj , and all the preference values aij (i, j = 1, 2, . . . , n) are contained in a multiplicative preference relation Ak on the set (k ) X, which is defined as a reciprocal matrix Ak = (aij )n ×n ⊂ X × X under the following condition: (k )
aij > 0,
(k )
(k )
aij · aj i = 1,
(k )
aii = 1 for all i, j = 1, 2, . . . , n.
(30)
Saaty [29] presented a consistency index (CI(Ak )), which is given by CI(Ak ) =
λm ax (Ak ) − n n−1
(31)
and a consistency ratio (CR(Ak )), which is given by CR(Ak ) =
CI(Ak ) RI
(32)
to measure the consistency of the multiplicative preference relation Ak , where λm ax (Ak ) is the maximal eigenvalue of Ak , RI is the mean CI of randomly generated multiplicative preference relations, which is given in Table I, and suggested that if CR(Ak ) < 0.1, then the multiplicative preference relation Ak is of satisfactory consistency; otherwise, Ak is of unsatisfactory consistency. In the case where CR(Ak ) ≥ 0.1, we need to return the multiplicative preference relation Ak to the decision maker ek to reconsider the construction of a new multiplicative preference relation according to his/her new judgments and to follow this procedure until the multiplicative preference relation with satisfactory CR(Ak ) is obtained. Then, we utilize the weighted PG operator to develop an approach to group decision making based on multiplicative preference relations, which involves the following steps.
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
Approach I: Step 1: Calculate the supports
Approach II: Step 1: Calculate
(k )
a − a(l) (k ) (l) ij ij , Sup aij , aij = 1 − m (k ) (l)
l=1 aij − aij
l = 1, 2, . . . , m
l= k
l= k
(33) which satisfy the support conditions 1)–3) in Section II. m (k ) (l) Especially, if l=1 |aij − aij | = 0, then we stipulate l= k
(k )
(l)
Sup(aij , aij ) = 1. Step 2: Utilize the weights wk (k = 1, 2, . . . , m) of the decision makers ek (k = 1, 2, . . . , m) to calculate the weighted (k ) (k ) support T (aij ) of the preference value aij by the other pref(l)
erence values aij (l = 1, 2, . . . , m, and l = k) T
(k ) (aij )
=
m
(k ) (l) wl Sup aij , aij
(34)
l=1 l= k
(k ) vij
= 1, 2, . . . , m)
(k ) wk 1 + T aij = m , (k ) k =1 wk 1 + T aij
k = 1, 2, . . . , m
,
= 1. It is necessary to point out that the support measure is a similarity measure, which can be used to measure the degree that a preference value provided by a decision maker is close to another one provided by other decision maker in a group-decision-making problem. Thus, in (38), index(k ) index(l) , aij ) denotes the similarity degree between Sup(aij preference value
and the lth largest
index(l) aij . index(k )
Step 2: Calculate the support T (aij
) of the kth largest (l)
m index(k ) index(k ) index(l) = T aij Sup aij , aij
(39)
l=1 l= k (k )
and by (6), calculate the weight uij associated with the kth index(k )
largest preference value aij , where (k −1) (k ) k Rij Rij (k ) (k ) index(l) −g , Rij = Vij uij = g T Vij T Vij l=1
T Vij =
m
index(l)
Vij
,
index(l)
Vij
index(l) = 1 + T aij
(40)
l=1
1 , aij
for all i < j.
(36)
Step 4: Utilize the well-known logarithmic least-square method (LLSM) [34] given by
i=1
l= k index(k )
| = 0, then we stipulate Sup(aij
index(l) aij )
(35)
k =1
n
index(l)
− aij
preference value by the other preference values aij (l = 1, 2, . . . , m, and l = k)
(k ) m (1) (2) (k ) v i j (m ) aij = PGw aij , aij , . . . , aij = aij
vi =
index(k )
|aij
index(k ) aij
(k ) (k ) where vij ≥ 0, k = 1, 2, . . . , m, and m k =1 vij = 1. Step 3: Utilize the weighted PG operator (16) to aggregate all the individual multiplicative preference relations Ak = (k ) (aij )n ×n (k = 1, 2, . . . , m) into the collective multiplicative preference relation A = (aij )n ×n , where
aj i =
ij
(38) which indicates the support of lth largest preference value, index(l) i.e., aij , for the kth largest preference value, i.e., m index(k ) (s) aij , of aij (s = 1, 2, . . . , m). Especially, if l=1
index(k )
(k )
the preference values
ij
the kth largest preference value aij
and calculate the weights vij (k = 1, 2, . . . , m) associated with (k ) aij (k
index(k ) index(l)
a index(k ) index(l) − aij ij = 1 − m index(k ) , aij Sup aij index(l)
−a l=1 a
n j =1
1/n aij
n j =1
1/n ,
i = 1, 2, . . . , n
(37)
aij
to derive the priority vector v = (v1 , v2 , . . . , vn )T of A = (aij )n ×n , where vi > 0, i = 1, 2, . . . , n, and ni=1 vi = 1. Step 5: Rank all the alternatives xi (i = 1, 2, . . . , n) in accordance with the priority weights vi (i = 1, 2, . . . , n). The more the weight vi , the better the alternative xi will be. In the case where the information about the weights of decision makers is unknown, then we utilize the POWG operator to develop an approach to group decision making based on multiplicative preference relations, which can be described as follows.
(k ) (k ) where uij ≥ 0, k = 1, 2, . . . , m, and m k =1 uij = 1, g is the BUM function, as described in Section II. Step 3: Utilize the POWG operator (23) to aggregate all the (k ) individual multiplicative preference relations Ak = (aij )n ×n (k = 1, 2, . . . , m) into the collective multiplicative preference relation A = (aij )n ×n , where aij =
(1) (2) (m ) POWG(aij , aij , . . . , aij )
n index(k ) u (i jk ) aij = i=1
aj i =
1 , aij
for all i < j.
(41)
Step 4: For this step, see Approach I. Step 5: For this step, see Approach I. In the previous two approaches, Approach I considers the situations where the weights of decision makers can be predefined and utilizes the weighted PG operator to aggregate all the individual multiplicative preference relations with satisfactory consistency into the collective multiplicative preference relation
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
and then uses the well-known LLSM method to derive its priority vector, and using this, we can rank and select the given alternatives. While Approach II considers the situations where the information about the weights of decision makers is unknown and utilizes the POWG operator to aggregate all the individual multiplicative preference relations with satisfactory consistency into the collective multiplicative preference relation, it then also uses the LLSM method to find the final decision result. Both the approaches are very suitable for group decision making based on multiplicative preference relations. Usually, in the process of group decision making, some individuals may provide unduly high or unduly low preferences to their preferred or repugnant objects. The approaches can reduce the influence of these unduly high or unduly low arguments on the decision result by using the support measure to assign them lower weights, thereby making the decision more reasonable and reliable. IV. UNCERTAIN POWER-GEOMETRIC OPERATORS In the previous sections, the considered arguments are exact numerical values; in this section, we consider the situations where the input arguments cannot be expressed in exact numerical values, but value ranges (i.e., interval numbers) can be obtained. We first review some operational laws, which are related to interval numbers [35], shortly. Let a ˜ = [aL , aU ] and ˜b = [bL , bU ] be two interval numbers, where aU ≥ aL > 0, and bU ≥ bL > 0; then, we have the following expressions. 1) a ˜ + ˜b = [aL , aU ] + [bL , bU ] = [aL + bL , aU + bU ]. 2) a ˜˜b = [aL , aU ] · [bL , bU ] = [aL bL , aU bU ]. 3) λ˜ a = λ [aL , aU ] = [λ aL , λ aU ], where λ > 0. λ 4) a ˜ = [aL , aU ]λ = [(aL )λ , ( aU )λ ], where λ > 0. In order to rank interval numbers, we now introduce a possibility degree formula [36] for the comparison between the interval numbers a ˜ = [aL , aU ] and ˜b = [bL , bU ] aU − bL p(˜ a ≥ ˜b) = min max U , 0 ,1 (42) a − aL + bU − bL where p(˜ a ≥ ˜b) is called the possibility degree of a ˜ ≥ ˜b, which satisfies 0 ≤ p(˜ a ≥ ˜b) ≤ 1,
p(˜ a ≥ ˜b) + p(˜b ≥ a ˜) = 1 p(˜ a≥a ˜) = 0.5.
(43)
Let a ˜j = [aLj , aUj ] (j = 1, 2, . . . , n) be a collection of interval numbers; then, based on the previous operational laws of interval numbers, we extend the PG operator to uncertain environments and define an UPG operator as follows:
n n 1+T (˜ ai ) (1+T (˜ a i )) i= 1 ˜2 , . . . , a ˜n ) = a ˜i (44) UPG(˜ a1 , a i=1
where T (˜ ai ) =
n j =1 j = i
Sup(˜ ai , a ˜j )
(45)
99
and Sup(˜ a, ˜b) is the support for a ˜ from ˜b, which satisfies the following three properties. 1) Sup(˜ a, ˜b) ∈ [0, 1]. 2) Sup(˜ a, ˜b) = Sup(˜b, a ˜). 3) Sup(˜ a, ˜b) ≥ Sup(˜ x, y˜), if d(˜ a, ˜b) < d(˜ x, y˜), where d is a distance measure. Similar to the PG operator, the UPG operator has the following properties. ˜j ) = k, for all i = j, then Theorem 6: Letting Sup(˜ ai , a 1/n n ˜2 , . . . , a ˜n ) = a ˜i (46) UPG(˜ a1 , a i=1
which indicates that when all the supports are the same, the UPG operator is simply the uncertain geometric mean. Theorem 7: Let a ˜j (j = 1, 2, . . . , n) be a collection of interval numbers, then we have the following properties. ˜2 , . . . , a ˜n ) is any permutation of 1) Commutativity: If (˜ a1 , a ˜2 , . . . , a ˜n ), then (˜ a1 , a UPG(˜ a1 , a ˜2 , . . . , a ˜n ) = UPG(˜ a1 , a ˜2 , . . . , a ˜n ).
(47)
2) Idempotency: If a ˜j = a ˜, for all j, then UPG(˜ a1 , a ˜2 , . . . , a ˜n ) = a ˜.
(48)
3) Boundedness: min a ˜i ≤ UPG(˜ a1 , a ˜2 , . . . , a ˜n ) ≤ max a ˜i . i
i
(49)
If the weights of the objects are taken into account, then we define the weighted form of (44) as follows:
n n w i (1+T (˜ a i )) w i (1+T (˜ a i )) i= 1 UPGw (˜ a1 , a ˜2 , . . . , a ˜n ) = a ˜i i=1
(50) where T (˜ ai ) =
n
wj Sup(˜ ai , a ˜j )
(51)
j =1 j = i
with the condition (18). Obviously, the weighted UPG operator (50) has the properties of 2) and 3) in Theorem 7; however, Theorem 6 and 1) of Theorem 7 do not hold for the weighted UPG operator. Based on the POWG operator (23) and the possibility degree formula (42), here, we define a UPOWG operator as follows: ˜2 , . . . , a ˜n ) = UPOWG(˜ a1 , a
n
i a ˜uindex(i)
(52)
i=1
˜j (j = where a ˜index(i) is the ith largest interval numbers of a 1, 2, . . . , n), and i Ri Ri−1 Vindex(j ) ui = g −g , Ri = TV TV j =1 TV =
n i=1
Vindex(i) ,
Vindex(j ) = 1 + T (˜ aindex(i) )
(53)
100
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
and T (˜ aindex(i) ) denotes the support of the ith largest interval number by all the other interval numbers, i.e., T (˜ aindex(i) ) =
n
Sup(˜ aindex(i) , a ˜index(j ) )
(54)
j =1
where Sup(˜ aindex(i) , a ˜index(j ) ) indicates the support of jth largest interval number for the ith largest interval number (here, we can use the possibility degree formula (42) to rank interval numbers). Especially, if g(x) = x, then the UPOWG operator reduces to the UPG operator. From Theorem 6, we have the following corollary. ˜j ) = k, for all i = j, and Corollary 2: Letting Sup(˜ ai , a g(x) = x, then n 1/n ˜2 , . . . , a ˜n ) = a ˜i (55) UPOWG(˜ a1 , a i=1
which indicates that when all the supports are the same, the UPOWG operator is simply the uncertain geometric mean. Similar to Theorem 7, we have the following theorem. Theorem 8: Letting a ˜j (j = 1, 2, . . . , n) be a collection of interval numbers, then we have the following properties. ˜2 , . . . , a ˜n ) is any permutation of 1) Commutativity: If (˜ a1 , a ˜2 , . . . , a ˜n ), then (˜ a1 , a UPOWG(˜ a1 , a ˜2 , . . . , a ˜n ) = UPOWG(˜ a1 , a ˜2 , . . . , a ˜n ). (56) ˜, for all j, then 2) Idempotency: If a ˜j = a UPOWG(˜ a1 , a ˜2 , . . . , a ˜n ) = a ˜.
(57)
3) Boundedness: min a ˜i ≤ UPOWG(˜ a1 , a ˜2 , . . . , a ˜n ) ≤ max a ˜i . i
i
(58)
As mentioned in Section III, in the next section, we will apply the weighted UPG and UPOWG operators to group decision making based on uncertain multiplicative preference relations. V. APPROACH TO GROUP DECISION MAKING BASED ON UNCERTAIN MULTIPLICATIVE PREFERENCE RELATIONS Let us consider a group-decision-making problem under uncertainty. Suppose that the decision maker ek compares each pair of alternatives (xi , xj ), provides his/her preference value range (k ) L (k ) U (k ) a ˜ij = [aij , aij ], and constructs an uncertain multiplica(k ) tive preference relation A˜k = (˜ aij )n ×n under the following condition: U (k )
≥ aij
L (k )
= aii
aij
aii
L (k )
> 0,
aij
L (k )
U (k )
U (k )
= 1,
for all i, j = 1, 2, . . . , n.
· aj i
= 1,
L (k )
aj i
U (k )
· aij
=1 (59)
In what follows, we discuss two different cases. Case 1: If the weight vector w = (w1 , w2 , . . . , wm )T of the decision makers ek (k = 1, 2, . . . , m) is predefined, then we utilize the weighted UPG operator to develop an approach to group decision making based on uncertain multiplicative preference relations, which involves the following steps.
Approach III: Step 1: Calculate the supports (k ) (l) (k ) (l) d a ˜ij , a ˜ij ˜ij = 1 − m Sup a ˜ij , a (k ) (l) , ˜ij , a ˜ij l=1 d a
l = 1, 2, . . . , m
l= k
(60) which satisfy the support conditions 1)–3) given in Section IV; here, without loss of generality, we let
U (l) (k ) (l) 1 L (l) L (k ) U (k ) ˜ij = aij − aij + aij − aij . (61) d a ˜ij , a 2 m (k ) (l) Especially, if aij , a ˜ij ) = 0, then we stipulate l=1 d(˜ l= k
(k )
(l)
˜ij ) = 1. Sup(˜ aij , a Step 2: Utilize the weights wk (k = 1, 2, . . . , m) of the decision makers ek (k = 1, 2, . . . , m) to calculate the weighted sup(k ) (k ) port T (˜ aij ) of the uncertain preference value a ˜ij by the other (l)
uncertain preference values a ˜ij (l = 1, 2, . . . , m, and l = k) aij ) = T (˜ (k )
m
(k ) (l) wl Sup a ˜ij , a ˜ij
(62)
l=1 l= k (k )
and calculate the weights v˙ ij
(k = 1, 2, . . . , m), which are (k )
associated with the uncertain preference values a ˜ij (k = 1, 2, . . . , m) (k ) ˜ij wk 1 + T a (k ) v˙ ij = m , k = 1, 2, . . . , m (63) ˜(k ) ij k =1 wk 1 + T a (k ) (k ) where v˙ ij ≥ 0, k = 1, 2, . . . , m, and m k =1 v˙ ij = 1. Step 3: Utilize the weighted UPG operator (50) to aggregate all the individual uncertain multiplicative preference relations (k ) A˜k = (˜ aij )n ×n (k = 1, 2, . . . , m) into the collective uncertain multiplicative preference relation A˜ = (˜ aij )n ×n , where a ˜ij =
(1) (2) (m ) ˜ij , a UPGw a ˜ij , . . . , a ˜ij
(k ) m (k ) v˙ i j = a ˜ij
k =1
a ˜j i = aLji , aUji ,
aLji =
1 , aUij
aUji =
1 , aLij
for all i < j. (64)
Step 4: Utilize the uncertain LLSM (ULLSM) 1/n n ˜ij j =1 a v˜i = 1/n , i = 1, 2, . . . , n n n ˜ij i=1 j =1 a
(65)
to derive the uncertain priority vector v˜ = (˜ v1 , v˜2 , . . . , v˜n )T of ˜ A = (˜ aij )n ×n . Step 5: Compare each pair of the uncertain priority weights v˜i (i = 1, 2, . . . , n) by using the possibility degree formula (42) and construct a possibility degree matrix P = (pij )n ×n , where vi ≥ v˜j ), i, j = 1, 2, . . . , n, which satisfy pij ≥ 0, pij = p(˜
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
pij + pj i = 1, pii = 0.5, i, j = 1, 2, . . . , n. Summing all the elements in each line of the matrix P , we get pi =
n
pij ,
i = 1, 2, . . . , n.
index(k ) index(l) index(k ) index(l) d a ˜ij ,a ˜ij = 1 − m ,a ˜ij Sup a ˜ij index(k ) index(l) ˜ij ,a ˜ij l=1 d a l= k
(67) which indicates the support of lth largest uncertain preference index(l) value, i.e., a ˜ij , for the kth largest uncertain preference (s)
value, i.e., a ˜ij , of a ˜ij (s = 1, 2, . . . , m) (here, we can use Step 5 of Approach III to rank uncertain preference values). index(k ) index(l) aij ,a ˜ij ) = 0, then we stipEspecially, if ml=1 d(˜ ulate
l= k index(k ) index(l) ,a ˜ij ) Sup(˜ aij
= 1. index(k )
Step 2: Calculate the support T (˜ aij index(k ) uncertain preference value a ˜ij (l) erence values a ˜ij (l = 1, 2, . . . , m,
) of the kth largest
by the other uncertain prefand l = k)
m index(k ) index(k ) index(l) T a ˜ij = Sup a ˜ij ,a ˜ij
(68)
l=1 l= k (k )
and by (53), calculate the weight u˙ ij associated with the kth index(k )
largest uncertain preference value, i.e., a ˜ij (k ) u˙ ij
=g
T Vij =
m
(k ) R˙ ij T Vij
−g
index(l)
Vij
,
(k −1) R˙ ij T Vij index(l)
Vij
, where
,
(k ) R˙ ij =
k
index(l) =1+T a ˜ij (k )
index(k ) u˙ i j
(k )
a ˜ij
,
a ˜j i = aLji , aUji ,
aLji =
i=1
aUji =
1 , aLij
for all i < j.
1 aUij (70)
Step 4: For this step, see Approach III. Step 5: For this step, see Approach III. Step 6: For this step, see Approach III. It is clear that Approaches III and IV have characteristics similar to Approaches I and II, respectively. VI. EXAMPLE ILLUSTRATION Four university students share a house, where they intend to have broadband Internet connection installed (adapted from [37]). There are four options available to choose from, which are provided by three Internet-service providers. 1) x1 : 1 Mbps broadband; 2) x2 : 2 Mbps broadband; 3) x3 : 3 Mbps broadband; 4) x4 : 8 Mbps broadband. Since the Internet service and its monthly bill will be shared among the four students ek (k = 1, 2, 3, 4) (whose weight vector w = (0.3, 0.3, 0.2, 0.2)T ), they decide to perform a group decision analysis. Suppose that the students reveal their preference relations for the options independently and anonymously, and construct the following multiplicative preference relations, respectively:
1 4 A1 = 3 4
1 1/2 A3 = 2 1/3
1/4 1 2 3
1/3 1/2 1 2
1/4 1/3 , 1/2 1
2 1 4 3
1/2 1/4 1 2
3 1/3 , 1/2 1
1 3 A2 = 4 2
1/3 1 3 3
1/4 1/3 1 1/2
1/2 1/3 2 1
1 2 A4 = 3 1/2
1/2 1 5 3
1/3 1/5 1 2
2 1/3 . 1/2 1
index(l)
Vij
l=1
m
n
By (32), we calculate the CRs of Ak (k = 1, 2, 3, 4) as follows: CR(A1 ) = 0.0575,
CR(A2 ) = 0.0821
CR(A3 ) = 0.2398,
CR(A4 ) = 0.4899.
(69)
l=1 (k )
(1) (2) (m ) ˜ij , a ˜ij , . . . , a ˜ij a ˜ij = aLij , aUij = UPOWG a =
Then, we can rank the uncertain priority weights v˜i (i = 1, 2, . . . , n) in descending order in accordance with pi (i = 1, 2, . . . , n). Step 6: Rank all the alternatives xi (i = 1, 2, . . . , n) in accordance with the descending order of the uncertain priority weights v˜i (i = 1, 2, . . . , n). Case 2: If the information about the weights of the decision makers ek (k = 1, 2, . . . , m) is unknown, then we utilize the UPOWG operator to develop an approach to group decision making based on uncertain multiplicative preference relations, which can be described as follows. Approach IV: Step 1: Calculate
index(l)
multiplicative preference relation A˜ = (˜ aij )n ×n , where
(66)
j =1
101
where u˙ ij ≥ 0, k = 1, 2, . . . , m, k =1 u˙ ij = 1, and g is the BUM function, as described in Section II. Step 3: Utilize the UPOWG operator (52) to aggregate all the individual uncertain multiplicative preference relations (k ) A˜k = (˜ aij )n ×n (k = 1, 2, . . . , m) into the collective uncertain
Then, CR(A1 ) < 0.1, CR(A2 ) < 0.1, CR(A3 ) > 0.1, and CR(A4 ) > 0.1, i.e., the multiplicative preference relations A1 and A2 are of satisfactory consistency, while the multiplicative preference relations A3 and A4 are of unsatisfactory consistency. In this case, we need to return A3 and A4 to the students e3 and e4 for revaluation, and they construct two new multiplicative preference relations (for convenience, we also denote
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them by A3 and A4 , respectively) 1 2 1/2 2 1 1 1/2 1 1/2 1 1/4 1/3 1 1 1/3 1/3 A3 = , A4 = 2 4 1 1 2 3 1 1/2 1/2 3 1 1 1 3 2 1
34 31 32 34 = 1, S32 = 0.500, S32 = S32 = 0.750 S31
and then, by (32), we have CR(A3 ) = 0.0773, and CR(A4 ) = 0.0906, which indicate that the reconstructed multiplicative preference relations A3 and A4 are also of satisfactory consistency. Since the weights of the students are predefined, we then utilize Approach I to find the decision result. (k ) (l) We first utilize (33) to calculate the supports Sup(aij , aij )
41 42 43 41 42 S11 = S11 = S11 = 1, S12 = 0.690, S12 = 0.724
(k )
(l)
(i, j, k, l = 1, 2, 3, 4) (for simplicity, we denote Sup(aij , aij ) kl by Sij )
31 32 34 31 32 34 31 S33 = S33 = S33 = S34 = 1, S34 = 0, S34 = 1, S41 = 0.364 32 34 31 32 34 31 S41 = 0.727, S41 = 0.909, S42 = S42 = S42 = S43 =1 32 34 31 32 34 S43 = 0, S43 = S44 = S44 = S44 =1
43 41 42 43 S12 = 0.586, S13 = 0.600, S13 = 0.400, S13 =1 41 42 43 S14 = 0.667, S14 = 0.778, S14 = 0.556 41 42 43 S21 = 0.455, S21 = 0.636, S21 = 0.909 41 42 43 41 42 43 S22 = S22 = S22 = 1, S23 = 0.333, S23 = 1, S23 = 0.667 41 42 43 41 42 43 S24 = S24 = S24 = 1, S31 = 0.667, S31 = 0.333, S31 =1 41 42 43 S32 = 0.500, S32 = 1, S32 = 0.500 41 42 43 41 42 43 S33 = S33 = S33 = S34 = 1, S34 = 0, S34 =1
12 13 14 12 13 S11 = S11 = S11 = 1, S12 = 0.968, S12 = 0.323 14 12 13 14 S12 = 0.710, S13 = 0.800, S13 = 0.600, S13 = 0.600 12 13 14 12 S14 = 0.909, S14 = 0.364, S14 = 0.727, S21 = 0.867
41 42 S41 = 0.333, S41 = 0.778 43 41 42 43 41 42 S41 = 0.889, S42 = S42 = S42 = S43 = 1, S43 =0 43 41 42 43 S43 = 1, S44 = S44 = S44 = 1.
13 14 12 13 14 S21 = 0.533, S21 = 0.600, S22 = S22 = S22 =1 12 13 14 S23 = 0.714, S23 = 0.571, S23 = 0.714 12 13 14 12 13 S24 = S24 = S24 = 1S31 = 0.667, S31 = 0.667 14 12 13 14 S31 = 0.667, S32 = 0.750, S32 = 0.500, S32 = 0.750
Then, we utilize the weight vector w = (0.3, 0.3, 0.2, 0.2)T of the students ek (k = 1, 2, 3, 4) and (34) to calculate the (k ) weighted supports T (aij ) (i, j, k = 1, 2, 3, 4) of the prefer(k )
ence values aij (i, j, k = 1, 2, 3, 4), which are contained in the matrices Tk = (T (aij ))4×4 (k = 1, 2, 3, 4), respectively (k )
12 13 14 12 13 14 S33 = S33 = S33 = 1, S34 = 0, S34 = S34 =1
12 13 14 S41 = 0.692, S41 = 0.462, S41 = 0.538 12 13 14 12 S42 = S42 = S42 = 1, S43 =0
T1
13 14 12 13 14 S43 = S43 = S44 = S44 = S44 =1 21 23 24 21 23 S11 = S11 = S11 = 1, S12 = 0.966, S12 = 0.310 24 S12 21 S14
=
21 0.724, S13
=
23 0.889, S14
=
23 1.000, S13
=
24 0.333, S14
=
24 0.500, S13
= 0.500
=
21 0.778, S21
= 0.818
T2
23 24 21 23 24 S21 = 0.545, S21 = 0.636, S22 = S22 = S22 =1 21 S23
=
23 0.333, S23
=
24 0.667, S23
=
21 S24
=
23 S24
=
24 S24
=1
T3
21 23 24 21 S31 = 0.800, S31 = 0.600, S31 = 0.600, S32 = 0.500 23 24 21 23 24 S32 = 0.500, S32 = S33 = S33 = S33 =1
T4
21 23 24 21 S34 = S34 = S34 = 0.667, S41 = 0.556 23 24 21 23 24 S41 = 0.667, S41 = 0.778, S42 = S42 = S42 =1 21 23 24 21 23 24 S43 = S43 = S43 = 0.667, S44 = S44 = S44 =1 31 S11
=
32 S11
=
34 S11
=
31 1, S12
=
32 0.604, S12
= 0.623
34 31 32 34 S12 = 0.774, S13 = 0.600, S13 = 0.400, S13 =1 31 32 34 31 S14 = 0.588, S14 = 0.647, S14 = 0.765, S21 = 0.462 32 34 31 32 34 S21 = 0.615, S21 = 0.923, S22 = S22 = S22 =1 31 32 34 S23 = 0.400, S23 = 0.800, S23 = 0.800 31 32 34 31 32 S24 = S24 = S24 = 1, S31 = 0.667, S31 = 0.333
0.700 0.487 = 0.467 0.408 0.700 0.482 = 0.480 0.456 0.800 0.508 = 0.500 0.509 0.800 0.509 = 0.500 0.511
0.497 0.700 0.475 0.700
0.480 0.471 0.700 0.400
0.497 0.700 0.450 0.700
0.500 0.433 0.700 0.467
0.523 0.800 0.525 0.800
0.500 0.520 0.800 0.500
0.541 0.800 0.550 0.800
0.500 0.533 0.800 0.500
0.491 0.700 0.400 0.700 0.489 0.700 0.467 0.700 0.524 0.800 0.500 0.800 0.545 0.800 0.500 0.800 (k )
(i, j, k =
(k ) aij
(i, j, k =
and then utilize (35) to calculate the weights vij 1, 2, 3, 4) associated with the preference values
(k )
1, 2, 3, 4), which are contained in the matrices Vk = (vij )4×4 (k = 1, 2, 3, 4), respectively 0.293 0.297 0.297 0.297 0.299 0.293 0.298 0.293 V1 = 0.297 0.297 0.293 0.288 0.289
0.293
0.288
0.293
XU AND YAGER: POWER-GEOMETRIC OPERATORS AND THEIR USE IN GROUP DECISION MAKING
0.298 V2 = 0.299
0.301
0.293 0.292
0.290 0.293
0.293 0.301
0.299
0.293
0.301
0.293
0.207
0.202 V3 = 0.202
0.202
0.201
0.202
0.207 0.204
0.205 0.207
0.207 0.206
0.206
0.207
0.206
0.207
0.207 0.202 V4 = 0.202
0.204 0.207 0.208
0.201 0.207 0.207
0.205 0.207 0.206
0.207
0.207
0.206
0.207
0.296
0.297
0.293
based on which, we utilize the weighted PG operator [see (36)] to aggregate all the individual multiplicative preference relations (k ) Ak = (aij )4×4 (k = 1, 2, 3, 4) into the collective multiplicative preference relation 1 0.5499 0.3598 0.6207 1.8185 1 0.3546 0.3333 A= . 2.7793 2.8201 1 0.7584 1.6111
3.000
1.3184
1
After this, we utilize the LLSM (37) to derive the priority vector of A
If we use Saaty’s approach [8], [29] to solve the previous problem, then we first utilize the weight vector w = (0.3, 0.3, 0.2, 0.2)T and the weighted geometric mean operator [7] (71)
k =1
to aggregate all the individual multiplicative preference relations (k ) Ak = (aij )4×4 (k = 1, 2, 3, 4) into the collective multiplicative preference relation 1 0.5451 0.3596 0.6156 1.8345 1 0.3554 0.3333 A˙ = 2.7809 2.8137 1 0.7579 3.0000
1.3194
1
and then, by the well-known eigenvector method [29], we get the priority vector of A˙ v˙ = (0.1365, 0.1560, 0.3454, 0.3620)T from which the ranking of the options can be derived as follows: x4 x3 x2 x1 .
[2, 3]
[1, 1]
[2, 3] A˜2 = [2, 4]
[1, 1] [3, 5] [2, 4]
[1, 1]
[2, 3]
[1/3, 1]
[1, 3]
[1/3, 1/2] [1/4, 1/2]
[1/3, 1/2] [1, 1] A˜3 = [2, 3] [3, 4]
x4 x3 x2 x1 .
1.6244
[1, 2]
Using this, we get the ranking of the options as follows:
i, j = 1, 2, 3, 4
In the previous illustrative example, both approaches can simultaneously consider the weights of students and all the preference values provided by these students in the process of aggregating all the individual preference information into the group opinion, and especially, they derive the same ranking of the options. However, Saaty’s approach cannot take into account the relationships of the arguments provided by different individuals, while the approaches developed in this paper can not only embody the relationships among the input arguments by allowing values being aggregated to support and reinforce each other but also measure the similarity degrees of the arguments and reduce the influence of those unduly high or unduly low arguments on the decision result by using the support measure to assign them lower weights, thereby making the group opinion more unanimous. In the case where the preferences constructed by the students ek (k = 1, 2, 3, 4) are uncertain multiplicative preference relations, for example [1, 1] [1/5, 1/3] [1/3, 1] [1/2, 1] [3, 5] [1, 1] [1/4, 1/2] [1/3, 1/2] A˜1 = [1, 3] [2, 4] [1, 1] [1/3, 1]
[1, 3]
v = (0.1338, 0.1539, 0.3530, 0.3592)T .
4 (k ) w k aij , a˙ ij =
103
[1, 1]
[1, 3] A˜4 = [1, 2] [1/2, 2]
[1, 1] [1/3, 1]
[1/5, 1/3] [1/4, 1/2] [1, 1] [1/2, 1] [1, 2]
[1, 1]
[1/3, 1/2]
[1, 3]
[1/2, 1]
[1, 1]
[1/3, 1]
[1/2, 1]
[1/2, 2]
[4, 5] [3, 4]
[1/4, 1/3] [1/5, 1/3] [1, 1] [1, 2]
[3, 5] [1, 1]
[1/5, 1/4] [1/4, 1/3] . [1, 1] [1/2, 1] [1, 2]
[1, 1]
Then, we can utilize Approach III to derive the ranking of the four options, and the following decision steps are needed. (k ) Step 1: Utilize (60)–(63) to calculate the weights v˙ ij (k = 1, 2, 3, 4) associated with the uncertain preference val(k ) ues a ˜ij (k = 1, 2, 3, 4), which are contained in the matrices (k ) V˙ k = (v˙ ij )4×4 (k = 1, 2, 3, 4), respectively 0.293 0.295 0.292 0.297 0.297 0.293 0.297 0.296 V˙ 1 = 0.289 0.295 0.293 0.295 0.294
0.298
0.286
0.293 0.294 V˙ 2 = 0.302
0.296 0.293
0.293 0.291
0.292 0.296
0.293 0.296
0.297
0.293
0.297 0.304 0.294
0.293
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2010
0.202
0.204 V˙ 3 = 0.206
0.201
0.208
0.207 0.208
0.207 0.207
0.200 0.202
0.203
0.203
0.207
0.207
0.207
0.206 V˙ 4 = 0.204
0.207
0.207
0.204
0.207 0.206
0.206 0.207
0.200 . 0.209
0.205
0.204
0.211
0.207
0.207
Step 2: Utilize the weighted UPG operator (50) to aggregate all the individual uncertain multiplicative preference rela(k ) tions A˜k = (˜ aij )4×4 (k = 1, 2, 3, 4) into the collective uncertain multiplicative preference relation
our operators to aggregate all the individual multiplicative (or uncertain multiplicative) preference relations into the collective multiplicative (or uncertain multiplicative) preference relation under various group-decision-making contexts and then developed some group-decision-making approaches based on multiplicative (or uncertain multiplicative) preference relations. The prominent characteristic of the developed approaches is that they can take all the decision arguments and their relationships into account. Our operators could usefully be applied to many other areas such as data mining, information retrieval, and pattern recognition, which we suggest are the possible paths for future research. ACKNOWLEDGMENT
A˜ =
The authors would like to thank the associate editor and the six anonymous referees for their insightful and constructive [1, 1] [0.411, 0.735] [0.333, 0.707] [0.510, 1.438] comments and suggestions that have led to an improved version [1.361, 2.433] [1, 1] [0.223, 0.354] [0.260, 0.425] . of this paper. [1.414, 3.003] [2.825, 4.484] [1, 1] [0.510, 1.150]
[0.695, 1.961]
[2.353, 3.846]
[0.870, 1.961]
[1, 1]
Step 3: Utilize the ULLSM (65) to derive the uncertain priority vector of A˜ v˜1 = [0.0919, 0.2791],
v˜2 = [0.0948, 0.2149]
v˜3 = [0.2137, 0.5956],
v˜4 = [0.1954, 0.5888].
Step 4: In order to rank v˜i (i = 1, 2, 3, 4), by (42), we construct the possibility degree matrix 0.5 0.5997 0.1149 0.1442 0.4003 0.5 0.0024 0.0380 P = 0.8851 0.9976 0.5 0.5162 0.8558
0.9620
0.4838
0.5
and by (66), we have p1 = 1.3588,
p2 = 0.9407,
p3 = 2.8989,
p4 = 2.8016.
Then, v˜3 > v˜4 > v˜1 > v˜2 . Therefore, we rank the options xi (i = 1, 2, 3, 4) in accordance with the descending order of a ˜i (i = 1, 2, 3, 4) x3 x4 x1 x2 . From the previous numerical results, it can be known that the ranking of the options in the latter case are slightly different from the former case due to the change of the input arguments. VII. CONCLUDING REMARKS Motivated by the idea of the Yager’s PA operator, we have developed some new nonlinear weighted-geometric aggregation operators, including the PG operator, weighted PG operator, POWG operator, UPG operator, weighted UPG operator, and UPOWG operator. We have studied some properties of the developed operators, such as commutativity, idempotency, boundedness, etc. The fundamental aspect of these operators is that the weight of each input argument depends on the other input arguments and allows argument values to support each other in the geometric aggregation process. Moreover, we have applied
REFERENCES [1] R. R. Yager and J. Kacprzyk, The Ordered Weighted Averaging Operator: Theory and Applications. Boston, MA: Kluwer, 1997. [2] T. Calvo, G. Mayor, and R. Mesiar, Aggregation Operators: New Trends and Applications. Heidelberg, Germany: Physica-Verlag, 2002. [3] V. Torra, Information Fusion in Data Mining. New York: SpringerVerlag, 2003. [4] Z. S. Xu and Q. L. Da, “An overview of operators for aggregating information,” Int. J. Intell. Syst., vol. 18, pp. 953–969, 2003. [5] H. Bustince, F. Herrera, and J. Montero, Fuzzy Sets and Their Extensions: Representation, Aggregation and Models. Berlin, Germany: SpringerVerlag, 2007. [6] J. C. Harsanyi, “Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility,” J. Polit. Econ., vol. 63, pp. 309–321, 1955. [7] J. Acz´el and T. L. Saaty, “Procedures for synthesizing ratio judgements,” J. Math. Psychol., vol. 27, pp. 93–102, 1983. [8] P. S. Bullen, D. S. Mitrinovi, and P. M. Vasi, Means and Their Inequalities. Dordrecht, The Netherlands: Reidel, 1988. [9] R. R. Yager, “On ordered weighted averaging aggregation operators in multi-criteria decision making,” IEEE Trans. Syst., Man, Cybern., vol. 18, no. 1, pp. 183–190, Jan./Feb. 1988. [10] F. Chiclana, F. Herrera, and E. Herrera-Viedma, “Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations,” Fuzzy Sets Syst., vol. 122, pp. 277–291, 2001. [11] Z. S. Xu and Q. L. Da, “The ordered weighted geometric averaging operators,” Int. J. Intell. Syst., vol. 17, pp. 709–716, 2002. [12] V. Torra, “The weighted OWA operator,” Int. J. Intell. Syst., vol. 12, pp. 153–166, 1997. [13] R. R. Yager and D. P. Filev, “Induced ordered weighted averaging operators,” IEEE Trans. Syst., Man, Cybern., vol. 29, no. 2, pp. 141–150, Apr. 1999. [14] Z. S. Xu and Q. L. Da, “The uncertain OWA operator,” Int. J. Intell. Syst., vol. 17, pp. 569–575, 2002. [15] Z. S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications. Beijing, China: Tsinghua Univ. Press, 2004. [16] W. M. Dong and F. S. Wong, “Fuzzy weighted averages and implementation of the extension principle,” Fuzzy Sets Syst., vol. 21, pp. 183–199, 1987. [17] R. R. Yager, “Generalized OWA aggregation operator,” Fuzzy Optim. Decision Making, vol. 3, pp. 93–107, 2004. [18] R. R. Yager, “An approach to ordinal decision making,” Int. J. Approx. Reasoning, vol. 12, pp. 237–261, 1995. [19] F. Herrera, E. Herrera-Viedma, and J. L. Verdegay, “A sequential selection process in group decision making with a linguistic assessment approach,” Inf. Sci., vol. 85, pp. 223–239, 1995. [20] F. Herrera and L. Mart´ınez, “A 2-tuple fuzzy linguistic representation model for computing with words,” IEEE Trans. Fuzzy Syst., vol. 8, no. 6, pp. 746–752, Dec. 2000.
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[21] V. Torra, “Aggregation of linguistic labels when semantics is based on antonyms,” Int. J. Intell. Syst., vol. 16, pp. 513–524, 2001. [22] Z. S. Xu, “A method based on linguistic aggregation operators for group decision making with linguistic preference relations,” Inf. Sci., vol. 166, pp. 19–30, 2004. [23] R. R. Yager, “The power average operator,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 31, no. 6, pp. 724–731, Nov. 2001. [24] R. D. Luce and P. Suppes, “Preference, utility and subject probability,” in Handbook of Mathematical Psychology, vol. III, R. D. Luce, R. R. Bush, and E. Galanter, Eds. New York: Wiley, 1965, pp. 249–410. [25] J. Ma, Z. P. Fan, Y. P. Jiang, and J. Y. Mao, “An optimization approach to multiperson decision making based on different formats of preference information,” IEEE Trans. Syst., Man, Cybern. A, Syst. Humans, vol. 36, no. 5, pp. 876–889, Sep. 2006. [26] Z. S. Xu, “Multiple attribute group decision making with different formats of preference information on attributes,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 37, no. 6, pp. 1500–1511, Dec. 2007. [27] S. A. Orlovsky, “Decision-making with a fuzzy preference relation,” Fuzzy Sets Syst., vol. 13, pp. 155–167, 1978. [28] T. Tanino, “Fuzzy preference ordering in group decision making,” Fuzzy Sets Syst., vol. 12, pp. 117–131, 1984. [29] T. L. Saaty, The Analytic Hierarchy Process. New York: McGraw-Hill, 1980. [30] F. Herrera, E. Herrera-Viedma, and F. Chiclana, “Multiperson decision making based on multiplicative preference relations,” Eur. J. Oper. Res., vol. 129, pp. 372–385, 2001. [31] Z. S. Xu, “On consistency of the weighted geometric mean complex judgment matrix in AHP,” Eur. J. Oper. Res., vol. 126, pp. 683–687, 2000. ¨ [32] O. Holder, “Uber einen Mittelwertsatz,” G¨ottingen Nachrichten, pp. 38– 47, 1889. [33] J. L. Jensen, “Sur les fonctions convexes et les in´egualit´es entre les valeurs moyennes,” Acta Math., vol. 30, pp. 175–193, 1906. [34] G. Crawford and C. Williams, “A note on the analysis of subjective judgment matrices,” J. Math. Psychol., vol. 29, pp. 387–405, 1985. [35] R. N. Xu and X. Y. Zhai, “Extensions of the analytic hierarchy process in fuzzy environment,” Fuzzy Sets Syst., vol. 52, pp. 251–257, 1992. [36] G. Facchinetti, R. G. Ricci, and S. Muzzioli, “Note on ranking fuzzy triangular numbers,” Int. J. Intell. Syst., vol. 13, pp. 613–622, 1998. [37] Y. M. Wang and C. Parkan, “Optimal aggregation of fuzzy preference relations with an application to broadband internet service selection,” Eur. J. Oper. Res., vol. 187, pp. 1476–1486, 2008.
Zeshui Xu (M’08–SM’09) received the Ph.D. degree in management science and engineering from Southeast University, Nanjing, China, in 2003. From April 2003 to May 2005, he was a Postdoctoral Researcher with the School of Economics and Management, Southeast University. From October 2005 to December 2007, he was a Postdoctoral Researcher with the School of Economics and Management, Tsinghua University, Beijing, China. He is an Adjunct Professor with the Antai School of Economic and Management, Shanghai Jiaotong University, Shanghai, China. He is also currently a Professor with the Institute of Sciences, People’s Liberation Army University of Science and Technology, Nanjing. He is a member of the Editorial Boards of Information: An International Journal, the International Journal of Applied Management Science, the International Journal of Data Analysis Techniques and Strategies, and the System Engineering—Theory and Practice and Fuzzy Systems and Mathematics. He has authored the following books: Uncertain Multiple Attribute Decision Making: Methods and Applications (Tsinghua Univ. Press, 2004), Intuitionistic Fuzzy Information: Aggregation Theory and Applications (Science, 2008), and Decision Making with Linguistic Information: Theory and Methods (Science, 2008). He has contributed more than 200 journal articles to professional journals. His current research interests include information fusion, group decision making, computing with words, and aggregation operators.
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Ronald R. Yager (S’66–M’68–SM’93–F’97) received the B.E.E. degree from the City College of New York, New York, and the Ph.D. degree from the Polytechnic University of New York, New York. He was a National Aeronautics and Space Administration (NASA)/Stanford Visiting Fellow and a Research Associate at the University of California, Berkeley. He has been a Lecturer at North Atlantic Treaty Organization (NATO) Advanced Study Institutes. He is currently the Director of the Machine Intelligence Institute, Iona College, New Rochelle, NY, where he is also a Professor of information and decision technologies. He is the Editor-in-Chief of the International Journal of Intelligent Systems. He serves on the Editorial Boards of a number of journals including Neural Networks, Data Mining and Knowledge Discovery, Fuzzy Sets and Systems, Journal of Approximate Reasoning, and International Journal of General Systems. He has been engaged in the area of fuzzy sets and related disciplines of computational intelligence for over 25 years. In addition to his pioneering work in the area of fuzzy logic, he has made fundamental contributions in decision making under uncertainty and the fusion of information. He has authored or coauthored over 500 papers published and 15 books. Prof. Yager was the recipient of the IEEE Computational Intelligence Society Pioneer Award in Fuzzy Systems. He is a Fellow of the New York Academy of Sciences and the Fuzzy Systems Association. He was given an award by the Polish Academy of Sciences for his contributions. He was the Program Director in the Information Sciences program at the National Science Foundation. He is a member of the Editorial Boards of a number of journals, including the IEEE TRANSACTIONS ON FUZZY SYSTEMS and IEEE INTELLIGENT SYSTEMS.
