Fuzzy Sets and Systems 157 (2006) 1393 – 1402 www.elsevier.com/locate/fss
The continuous ordered weighted geometric operator and its application to decision making Ronald R. Yagera,∗ , Zeshui Xub a Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA b Department of Management Science and Engineering, School of Economics and Management, Tsinghua University, Beijing 100084, China
Received 6 June 2005; received in revised form 29 November 2005; accepted 3 December 2005 Available online 6 January 2006
Abstract The aim of this paper is to develop a continuous ordered weighted geometric (C-OWG) operator, which is based on the continuous ordered weighted averaging (C-OWA) operator recently introduced by the author and the geometric mean. We study some desirable properties of the C-OWG operator, and present its application to decision making with interval multiplicative preference relation, and finally, an illustrative example is pointed out. © 2006 Elsevier B.V. All rights reserved. Keywords: Decision making; C-OWA operator; C-OWG operator; Interval multiplicative preference relation
1. Introduction The ordered weighted averaging (OWA) operator, developed by Yager [24], is a useful tool in aggregating a finite collection of arguments. Central to this operator is the reordering of arguments, based on their values. Since its appearance in 1988, the OWA operator has been used in a wide range of applications, such as engineering, neural networks, data mining, decision making, image process, expert systems, etc. [4,16,18,26]. Recently, Yager [25] developed a continuous ordered weighted averaging (C-OWA, for short) operator, which is an extension of the OWA operator to the case in which the given argument is a continuous valued interval rather than a finite set of arguments. The C-OWA weights are determined by a basic unit-interval monotonic (BUM) function Q, and the aggregated value derived by the C-OWA operator is associated with the attitudinal character of Q. The C-OWA operator is very suitable for ag = ( gregating decision information taking the form of interval fuzzy preference relation [1,17,18]: P pij )n×n where − + − + + − ij = [pij j i = [pj−i , pj+i ], pij p , pij ], p + pj+i = pij + pj−i = 1, pij pij 0, pii+ = pii− = 0.5 for all i, j = 1, 2, . . . , n, or for aggregating interval-valued intuitionistic fuzzy sets [3,13], etc. However, it is somewhat unsuitable for dealing with multiplicative decision-making problems under uncertainty. For example, it is unsuitable for aggregating deci = ( sion information taking the form of interval multiplicative preference relation: A aij )n×n where aij = [aij− , aij+ ], aj i = [aj−i , aj+i ], aij− · aj+i = aij+ · aj−i = 1, aij+ aij− 0, aii+ = aii− = 1 for all i, j = 1, 2, . . . , n. ∗ Corresponding author. Tel.: +1 212 249 2047.
E-mail addresses:
[email protected] (R.R. Yager),
[email protected] (Z. Xu). 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.12.001
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Interval multiplicative preference relation is very useful in expressing decision maker’s preferences on alternatives under uncertainty [2,15,23,19,28]. Some authors have paid attention to this issue. Saaty and Vargas [15] proposed a method for estimating the probability that an alternative exchanges rank with other alternatives and utilized the principal eigenvector method to calculate the priority vectors of m random samples of multiplicative preference relations, and then combined the priority of each alternative with the probability that it does not change rank, to obtain the final ranking of alternatives. Zahir [28] utilized a recursive algorithm and Saaty’s principal eigenvector method to obtain the priority vector of interval multiplicative preference relation. Xu and Zhai [23] proposed an approach to deriving the fuzzy priority vector of interval multiplicative preference relation by applying the fuzzy extension principle of Zadeh [27] to the altered gradient eigenvector method [8]. Arbel and Vargas [2] developed two approaches for priority vector of interval multiplicative preference relation, one based on a simulation approach and the other based on mathematical programming. The existing research has mainly focused on uncertainty or the priority vectors of interval multiplicative preference relations in the analytic hierarchy process, and the developed approaches are of high computational complexity. In this paper, we shall present a straightforward and practical approach to decision making with interval multiplicative preference relation. To do that, the remainder of this paper is organized as follows. In Section 2, we review some basic operators. Section 3 introduces a continuous ordered weighted geometric (C-OWG) operator, which is based on the C-OWA operator and the geometric mean. In Section 4, we develop a C-OWG operator based approach to decision making with interval multiplicative preference relation. In Section 5, a decision-making problem of determining which candidate should be the best qualified for the post of professor in Operations Research at a university is used to illustrate the proposed approach, and finally, Section 6 concludes this paper. 2. Preliminaries In [24], Yager introduced a non-linear operator called the OWA operator for aggregating a finite collection of arguments, whose fundamental aspect is the re-ordering step. The OWA operator was defined as follows: n An OWA operator of dimension n is a mapping n f : R → R which has an associated weighting vector w = T (w1 , w2 , . . . , wn ) such that wj ∈ [0, 1] and j =1 wj = 1 where f (1 , 2 , n ) =
n
w j j
(1)
j =1
with j being the j th largest of the i . Later, some authors [5–7,9,20,21] investigated the OWG operators, which are based on the OWA operators and on the geometric mean. We define the OWG operator in the following: +n → R + which has associated with it an exponential weighting An OWG operator of dimension n is a mapping g : R vector w = (w1 , w2 , . . . , wn )T with wj ∈ [0, 1] and nj=1 wj = 1, such that g(1 , 2 , . . . , n ) =
n
w
j j ,
(2)
j =1
where j is the j th largest of the i . In the latest decade, the OWA and OWG operators and their families have been used in an astonishingly wide range of applications [4,16,18,26]. Recently, Yager [25] further developed a continuous ordered weighted averaging (C-OWA) operator that extends the OWA operator to the case where the arguments to be aggregated are all the values in a closed interval [a, b], which was defined as follows: 1 dQ(y) fQ ([a, b]) = (b − y(b − a)) dy, (3) 0 dy where Q is a basic unit-interval monotonic (BUM) function Q : [0, 1] → [0, 1] and is monotonic with the properties: (1) Q(0) = 0; (2) Q(1) = 1; and (3) Q(x)Q(y) if x > y.
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The C-OWA operator has some properties, for example, this operator is monotonic and bounded. The C-OWA operator is a tool useful in many applications. For example, it can be used for decision making under ignorance in the case in which the possible payoffs associated with an alternative lie in a range rather than being a finite set of values [25], it can also be very suitable for aggregating decision information taking the form of interval fuzzy preference relation [1,17,18]. However, the C-OWA operator is sometimes unsuitable for dealing with multiplicative decision-making problems under uncertainty. Therefore, it is necessary to pay attention to this issue.
3. C-OWG operator In the following, we shall develop a new aggregation operator called continuous ordered weighted geometric (C-OWG) operator, which is based on the C-OWA operator and the geometric mean. Definition 1. A continuous ordered weighted geometric (C-OWG) operator is a mapping g : + → R + which has associated with it a BUM function: Q : [0, 1] → [0, 1] having the properties: (1) Q(0) = 0; (2) Q(1) = 1; and (3) Q(x)Q(y) if x > y, such that gQ ([a, b]) = b ·
a 1 (dQ(y)/dy)y dy 0
b
,
(4)
where + is the set of closed intervals, in which the lower limits of all closed intervals are positive, R + is the set of positive real numbers, and [a, b] is an closed intervals in + . The C-OWG operator has some desirable properties similar to the C-OWA operator. Theorem 1. If a1 a2 and b1 b2 , then gQ ([a1 , b1 ])gQ ([a2 , b2 ]) for all Q. Theorem 2. If Q1 and Q2 are such that Q1 (y)Q2 (y) for all y ∈ [0, 1], then we define this as Q1 Q2 , in this case, gQ1 ([a, b])gQ2 ([a, b]) holds. Theorem 3. a gQ ([a, b]) b for all Q. Let us look at gQ ([a, b]) for some special cases of Q: (1) If we consider Q(y) = y r , then dQ(y)/dy = ry r−1 . Thus, we have gQ ([a, b]) = a r/(r+1) · b1/(r+1) .
(5)
As r → 0, then gQ ([a, b)] = b, in this case, we get the maximum; As r = 21 , then gQ ([a, b)] = a 1/3 · b2/3 ; As r = 1, then gQ ([a, b)] = (ab)1/2 , this is the usual geometric mean; As r → ∞, then gQ ([a, b)] = a, in this case, we get the minimum; As r = s/t, then gQ ([a, b]) = a s/(s+t) · a t/(s+t) . 1 In [25], Yager defined = 0 Q(y) dy as the attitudinal character of Q. As a result, we have Theorem 4. If is the attitudinal character of Q, then gQ ([a, b]) = a 1− · b .
(6)
From Theorem 4, we know that gQ ([a, b]) is always the weighted geometric mean of end points based on the attitudinal character. The C-OWG can be usefully applied to defuzzification, by computing it for each cut and then integrating the results.
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Table 1 1–9 ratio scale Preference intensity
Explanation
1 3 5 7 9 2, 4, 6, 8 Reciprocals
Indifference between two alternatives Moderate preference of one alternative to another Strong preference of one alternative to another Very strong preference of one alternative to another Absolute preference of one alternative to another Intermediate values between the two adjacent judgments If the alternative xi has one of the above numbers assigned to it when compared with the alternative xj , then xj has the reciprocal value when compared with xi .
4. The application of the C-OWG operator to decision making with interval multiplicative preference relation 4.1. Interval multiplicative preference relation Consider a decision-making problem. Let X = {x1 , x2 , . . . , xn } be a finite set of alternatives. In the process of decision making, a decision maker generally needs to compare each pair of alternatives, and constructs a preference relation [10,14,18]. If the decision maker gives his/her preference information on X by means of a multiplicative preference relation [7,14,22] A = (aij )n×n where aij > 0, aij · aj i = 1, aii = 1,
i, j = 1, 2, . . . , n
(7)
and aij indicates a ratio of preference intensity for alternative xi to that of xj , i.e., it is interpreted as xi is aij times as good as xj . Saaty [14] suggested measuring aij using the 1–9 ratio scale as shown in Table 1. However, sometimes, the decision maker may have vague knowledge about the preference degree of one alternative over another, and cannot estimate his/her preference with an exact numerical value, but with an interval number. In this = ( case, the decision maker may construct an interval multiplicative preference relation [2,15,23,19,28]: A aij )n×n − + − + − + + − + − + − where aij = [aij , aij ], aj i = [aj i , aj i ], aij · aj i = aij · aj i = 1, aij aij > 0, aii = aii = 1, i, j = 1, 2, . . . , n, and aij indicates an interval ratio of preference intensity for alternative xi to that of xj . In [15], Saaty and Vargas investigated the effect of uncertainty in judgment on the stability of the rank order of alternatives. They utilized the principal eigenvector method to calculate the priority vectors of m random samples of multiplicative preference relations and constructed confidence intervals for each of the components of the eigenvector to compute the probabilities that an alternative exchanges rank with other alternatives. The derived priority vectors and the probabilities are then used to obtain the final ranking of alternatives. Zahir [28] showed how to incorporate the uncertainty of interval judgments in the analytic hierarchy process and utilized a recursive algorithm and Saaty’s principal eigenvector method to obtain the priority vector of interval multiplicative preference relation. Xu and Zhai [23] used the altered gradient eigenvector method to derive the priority vector of interval multiplicative preference relation. Arbel and Vargas [2] explored two approaches for priority derivation when preferences are expressed as interval judgments, one based on a simulation approach and the other based on mathematical programming. The existing research has mainly focused on uncertainty or the priority vectors of interval multiplicative preference relations in the analytic hierarchy process, and the developed approaches are complex computationally. In the following, we shall first utilize the C-OWG operator to define the expected preference relation of an interval multiplicative preference relation, and then present a straightforward and practical approach to decision making with interval multiplicative preference relation, which derives the rank order of alternatives direct from the expected preference degrees and is of low computational complexity. 4.2. Expected multiplicative preference relation As is well known, expected value is a central principle in the probability theory, which is used for average estimation of some variable, and the average value of any variable in a long run gets close to its expected value. Expected value has been used in a very wide range of practical applications [11]. By using Eq. (4), we define an expected preference relation based on the attitudinal character as follows.
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= ( Definition 2. Let A aij )n×n be an interval multiplicative preference relation, where aij = [aij− , aij+ ],
aj i = [aj−i , aj+i ],
aij− · aj+i = aij+ · aj−i = 1,
aii+ = aii− = 1,
i, j = 1, 2, . . . , n,
aij+ aij− > 0, (8)
= (gQ ( where gQ ( then we call gQ (A) aij ))n×n the expected multiplicative preference relation corresponding to A, aij ) is the expected value of preference degree aij of the alternative xi to xj , obtained by the C-OWG operator: aij ) = gQ ([aij− , aij+ ]) = aij+ · gQ (
aij−
10 (dQ(y)/dy)y dy ,
aij+
gQ ( aj i ) =
1 for all i j, gQ ( aij )
(9)
where Q is a basic unit-interval monotonic (BUM) function. Obviously, we have gQ ( aij ) > 0,
gQ ( aij ) · gQ ( aj i ) = 1,
In fact, by the result [25]: 1 Q(y) dy = 1 − 0
1 0
= (aij− )
Since 0
1
1
0 (dQ(y)/dy) dy
1
0 (dQ(y)/dy) dy
1
= (aij− )1−
0
0 Q(y) dy 1, 1
gQ ( aii ) = (aii− )1−
0
i, j = 1, 2, . . . , n
dQ(y) y dy dy
(10)
and from Eq. (9), it follows that gQ ( aij ) = (aij− )
gQ ( aii ) = 1,
1
· (aij+ )1− · (aij+ )
· (aij+ )
Q(y) dy
1 0
1
0 (dQ(y)/dy) dy
Q(y) dy
0
Q(y) dy
.
(11)
and aij+ aij− > 0, then gQ ( aij ) > 0, for all i, j . By aii+ = aii− = 1, for all i, j , we have Q(y) dy
· (aii+ )
1 0
Q(y) dy
= 1,
i, j = 1, 2, . . . , n
(12)
and the result gQ ( aij ) · gQ ( aj i ) = 1 follows Eq. (9) directly. From (11), we have gQ ( aij ) = (aij− )
1
0 (dQ(y)/dy) dy
= (aij− ) ·
aij+
1 0
1
· (aij+ )1−
0 (dQ(y)/dy) dy
Q(y) dy
.
aij−
(13)
Then by Eq. (8), Eq. (9) can be rewritten as follows: aij ) = gQ (
gQ ([aij− , aij+ ])
=
aij−
·
aij+ aij−
,
1 gQ ( aj i ) = − aij
aij− aij+
for all i j,
(14)
1 where is the attitudinal character of Q, and = 0 Q(y) dy. Recall that gQ ( aij ) is the expected value of preference degree of the alternative xi to xj , we can get the expected value of preference degree of the alternative xi to all the alternatives by the geometric mean ⎛ ⎞1/n n gQ ( ai ) = ⎝ gQ ( aij )⎠ , i = 1, 2, . . . , n. (15) j =1
Obviously, the greater the value of gQ ( ai ), the better the alternative xi , and thus, we can rank all the alternatives xi (i = 1, 2, . . . , n) in accordance with the values of gQ ( ai ) (i = 1, 2, . . . , n).
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For convenience, by Eq. (14), we further extend Eq. (15) to the following: ⎞ ⎛ ⎞⎤1/n ⎡⎛ ⎞ ⎛ ⎞⎤/n ⎡⎛ aij+ 1 aj−i ⎠·⎝ ⎠·⎝ ⎠⎦ . gQ ( a i ) = ⎣⎝ aj−i ⎠⎦ · ⎣⎝ − + − a a a j i j i ij j 0, i > 0, if k > i , i.e., k /i > 1, then we have k log > 0. i Since 0 1 and 0 + 1, then from Eq. (25), it follows that log(i /k ) − min 1 − , − . log(k /i )
1399
(26)
(27)
ai )gQ ( ak ), then by Eq. (19), we have Since gQ ( i i k k ,
(28)
i.e., i k
k , i
thus, if k = i , then we have k = 1, i hence, for any − 1 − , Eq. (23) always holds. If k < i , i.e., 0 < k /i < 1, then we have k log < 0. i Since 0 1 and 0 + 1, then from Eq. (25), it follows that log(i /k ) max −, − 1 − . log(k /i )
(29)
(30)
(31)
(32)
( ( ai ) gQ ak ). This completes the proof of Theorem 5. Conversely, we can prove that if Eq. (21) holds, then gQ Theorem 5 provides a necessary and sufficient condition for the invariance of the ranking of every two aggregated values obtained by the C-OWG operator with the attitudinal character , which not only lays a theoretic basis for the application of C-OWG operator to decision making, but also is very useful in helping a decision maker choose the BUM function Q in the process of decision making.
4.3. A C-OWG operator based approach In this subsection, we shall try to present the application of the C-OWG operator to decision making with interval multiplicative preference relation, and give a C-OWG operator based approach whose steps are as follows: Step 1: Consider a decision-making problem. Let X = {x1 , x2 , . . . , xn } be a finite set of alternatives. A decision maker = ( gives his/her preference information on X by means of an interval multiplicative preference relation: A aij )n×n − + − + − + + − + − + − where aij = [aij , aij ], aj i = [aj i , aj i ], aij · aj i = aij · aj i = 1, aij aij > 0, aii = aii = 1, i, j = 1, 2, . . . , n. = (gQ ( Step 2: By the C-OWG operator, we construct the expected multiplicative preference relation gQ (A) aij ))n×n − + − − + − where gQ ( aij ) = aij · (aij /aij ) , gQ ( aj i ) = (aij /aij ) /aij , for all i j , is the attitudinal character of the BUM 1 function Q, and = 0 Q(y) dy (some notable examples of the BUM functions, please see [25]). Step 3: Calculate the expected value gQ ( ai ) of preference degree of the alternative xi to all the alternatives by Eq. (15). Step 4: Rank all the alternatives xi (i = 1, 2, . . . , n) and select the most desirable one(s) in accordance with the values of gQ ( ai ) (i = 1, 2, . . . , n). Step 5: End.
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In the above approach, we first construct the expected interval multiplicative preference relation of the interval multiplicative preference relation by mean of the C-OWG operator, and then utilize the expected interval multiplicative preference relation to calculate the expected preference degree of each alternative over all the alternatives. All these expected preference degrees are used to rank the alternatives. Clearly, the developed approach derives the rank order of alternatives direct from the expected preference degrees and has no need to derive the priority vector of the interval multiplicative preference relation, and thus can reduce the computational complexity of the approach drastically.
5. Numerical example In this section, a decision-making problem of determining which candidate should be the best qualified for the post of professor in Operations Research at a university (adapted from [12]) is used to illustrate the proposed approach. Suppose that at a university the post of professor in Operations Research is vacant. After a first selection, five serious candidates xi (i = 1, 2, 3, 4, 5) remain. A decision maker has been asked to give an advice, as to which applicant is the best quantified for the job. The decision maker provides his/her preferences over these five candidates, and gives an interval multiplicative preference relation as follows: ⎤ [1, 1] [1/4, 1/3] [1/2, 2] [2, 4] [7, 8] ⎢ [3, 4] [1, 1] [2, 3] [1/6, 1/4] [3, 5] ⎥ ⎥ ⎢ = ⎢ [1/2, 2] [1/3, 1/2] [1, 1] [4, 5] [1, 2] ⎥ A ⎥. ⎢ ⎣ [1/4, 1/2] [4, 6] [1/5, 1/4] [1, 1] [1/4, 1/3] ⎦ [1/8, 1/7] [1/5, 1/3] [1/2, 1] [3, 4] [1, 1] ⎡
To get the best candidate(s), the following steps are involved: √ Step 1: Suppose that the BUM function Q(y) = y, then we get the attitudinal character of the BUM function Q: =
1
Q(y) =
0
1
√
y dy =
0
2 . 3
By the C-OWG operator, we construct the expected multiplicative preference relation as follows: ⎡
1 ⎢ 3.3014 ⎢ = ⎢ 0.7937 gQ (A) ⎢ ⎣ 0.3150 0.1307
0.3029 1 0.3816 4.5788 0.2371
1.2599 2.6207 1 0.2154 0.6300
3.1748 0.2184 4.6416 1 3.3014
⎤ 7.6517 4.2172 ⎥ ⎥ 1.5874 ⎥ ⎥. 0.3029 ⎦ 1
ai ) of preference degree of the candidate xi to all the candidates by Step 2: Calculate the expected value gQ ( Eq. (15), we have a1 ) = 1.5611, gQ (
gQ ( a2 ) = 1.5145,
a4 ) = 0.6233, gQ (
gQ ( a5 ) = 0.5779.
gQ ( a3 ) = 1.1742,
ai ) (i = 1, 2, 3, 4, 5), we have Step 3: Rank all the candidates xi (i = 1, 2, 3, 4, 5) in accordance with gQ ( x1 x2 x3 x4 x5 , thus, the best candidate is x1 . Below, we give a sensitivity analysis with respect to the attitudinal character of the BUM function Q:
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= (gQ ( We first calculate the expected multiplicative preference relation gQ (A) aij ))5×5 by using Eq. (14), and get the following: ⎡ ⎤ 1 4 22−1 2+1 7 · ( 87 ) 1 4 · (3) ⎢ ⎥ ⎢ 4 · ( 43 ) 1 2 · ( 23 ) 16 · ( 23 ) 3 · ( 53 ) ⎥ ⎢ ⎥ ⎥ =⎢ gQ (A) 1 4 · ( 45 ) 2 ⎥ . ⎢ 21−2 21 · ( 23 ) ⎢ ⎥ ⎢ 2−−1 6 · ( 2 ) 1 · ( 4 ) 1 4 ⎥ 1 · ( ) ⎣ ⎦ 3 4 5 4 3 1 7
· ( 78 )
1 3
· ( 35 )
2−
4 · ( 43 )
1
By Eq. (15), we get the expected value of preference degree of the candidate xi to all the candidates 1/5 /5 /5 7 256 /5 45 5 a1 ) = · , gQ ( a2 ) = 41/5 · , gQ ( a3 ) = 41/5 · gQ ( 4 21 16 12 1/5 /5 1/5 3 16 4 63 /5 gQ ( a4 ) = · , gQ ( a5 ) = · , 16 45 21 320 then, we have x1 x2 x3 x4
x2 if and only if 0.18325 < 1, x3 if and only if 0 1, x4 if and only if 0 1, x5 if and only if 0.02664 1. From the above analysis, we get the ranking results of the candidates xi (i = 1, 2, 3, 4, 5) as follows: (1) If 0.18325 < 1, then x1 x2 x3 x4 x5 . (2) If = 0.18325, then x1 ∼ x2 x3 x4 x5 . (3) If 0.02664 < < 0.18325, then x2 x1 x3 x4 x5 . (4) If = 0.02664, then x2 x1 x3 x4 ∼ x5 . (5) If 0 < 0.02664, then x2 x1 x3 x5 x4 .
Note. The notation “” in the above example indicates one alternative is preferred to another, and the notation “∼” indicates indifference between two alternatives. 6. Conclusions We have introduced a continuous ordered weighted geometric (C-OWG) operator, and studied some of its desirable properties. We have also given a sensitivity analysis with respect to the attitudinal character, and presented the application of the C-OWG operator to decision making with interval multiplicative preference relation. Then, we have developed a C-OWG operator based approach, which has no need to derive the priority vector of the interval multiplicative preference relation, and thus can reduce the computational complexity drastically. The practicality and effectiveness of the developed approach have been verified by an illustrative example.
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