Fuzzy Generalized Ordered Weighted Averaging Distance Operator

402

International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012

Fuzzy Generalized Ordered Weighted Averaging Distance Operator and Its Application to Decision Making Shouzhen Zeng, Weihua Su, and Anbo Le Abstract1 We develop the fuzzy generalized ordered weighted averaging distance (FGOWAD) operator. It is a new aggregation operator that uses the main characteristics of the generalized OWA (GOWA), the ordered weighted averaging distance (OWAD) and uncertain information represented as fuzzy numbers. This operator includes a wide range of distance measures and aggregation operators such as the fuzzy maximum distance, the fuzzy minimum distance, the fuzzy normalized Hamming distance (FNHD), the fuzzy weighted Hamming distance (FWHD), the fuzzy normalized Euclidean distance (FNED), the fuzzy weighted Euclidean distance (FWED), the fuzzy ordered weighted averaging distance (FOWAD) operator, the fuzzy Euclidean ordered weighted averaging distance (FEOWAD) operator and the fuzzy generalized ordered weighted averaging (FGOWA) operator. We study some of its main properties. Finally, we apply the developed operator to a multi-person decision-making problem regarding the selection of strategies. Keywords: distance measure, GOWA operator, fuzzy number, decision making.

1. Introduction The Hamming distance [1] is an effective technique that has been used in a wide range of applications such as fuzzy set theory, business decisions and multi-criteria decision making [2-12]. Usually, we normalize the Hamming distance by using the arithmetic mean or the weighted average (WA) obtaining the normalized Hamming distance (NHD) and the weighted Hamming disCorresponding Author: Shouzhen Zeng is with the College of Computer and Information, Zhejiang Wanli University, Ningbo, 315100, China. E-mail: [email protected] Weihua Su is with Zhejiang University of Finance and Economics. E-mail: [email protected] Anbo Le is with Zhejiang Wanli University. E-mail: [email protected]. Manuscript received 7 Dec. 2010; revised 21 Oct. 2011; accepted 26 Aug. 2012.

tance (WHD), respectively. However, it is sometimes of interest to consider the possibility of the results from the maximum distance to the minimum distance. In this case, a useful technique is the ordered weighted averaging (OWA) operator [13], providing a parameterized family of aggregation operators that includes the maximum, the minimum and the average criteria as special cases. Recently, motivated by the idea of the OWA operator, Merigó and Gil-Lafuente [6] introduced a new decision-making technique called the ordered weighted averaging distance (OWAD) operator. It is an aggregation operator that uses the OWA operator and Hamming distance in the same formulation. The main advantage of this operator is that we are able to underestimate or overestimate the selection process according to the desired degree of optimism (i.e., the degree of orness). Therefore, we are able to provide decision maker with an approach to the optimal choice according to his or her interests. Another advantage of the OWAD operator is that it provides a parameterized family of distance aggregation operators that ranges from the minimum to the maximum distance. Therefore, they are able to provide a wide range of situations depending on the particular attitude taken by the decision maker in the specific problem considered. Moreover, with the OWAD, it is possible to establish an ideal, though unrealistic, alternative in order to compare it with available options in the decision-making problem. As such, the optimal choice is the alternative closest to the ideal one. Going a step further, Merigó and Gil-Lafuente [14] analyzed the use of the OWAD operator in the selection of human resources in sport management. Note that the use of the OWA operator in other types of distance measures have been studied by several authors [3, 15-18]. A further interesting extension of the OWA operator is the generalized OWA (GOWA) operator [8, 19] that uses generalized means [20] in the OWA operator. The GOWA operator in fact generalizes many situations, including OWA and its particular cases, the ordered weighted geometric (OWG) operator, the ordered weighted harmonic averaging (OWHA) operator, and the generalized mean. It has received increasing attention in recent years [3, 21-29]. Usually, when using the OWAD operator and above distance measures, it is assumed that the available information is clearly known and can be assessed with ex-

© 2012 TFSA

Shouzhen Zeng et al.: Fuzzy Generalized Ordered Weighted Averaging Distance Operator and Its Application to Decision Making

act numbers. However, this may not be the real-life situation found in the decision-making problems because the available information is often vague or imprecise, or it is not possible to analyze the situation with exact numbers. Then, it is necessary to use another approach that is able to assess the uncertainty such as the use of fuzzy numbers. With the use of fuzzy numbers, we are able to analyze the best and worst possible scenarios and the possibility that the internal values of the fuzzy interval will occur. As a powerful tool to express data information under different uncertain environments, the fuzzy numbers have been studied by different authors [9-12, 25, 26, 28, 30-45]. The aim of this paper is to present a new decision-making model by extending the OWAD operator to accommodate uncertain situations where the available information is given in the form of fuzzy numbers. In order to do so, we develop the fuzzy generalized ordered weighted averaging distance (FGOWAD) operator, which is an extension of the OWAD operator with the GOWA operator and uncertain information represented in the form of fuzzy numbers. It is a fuzzy aggregation operator that uses the main characteristics of the GOWA and the OWAD operator. Thus, it uses generalized means, distance measure and uncertain information represented in the form of fuzzy numbers. The main advantage of the FGOWAD operator is that it is able to assess the uncertain information in a more complete way because it represents the best and worst scenario and the possibility that the internal values will occur. Another advantage of this operator is that it includes a wide range of distance measures and fuzzy aggregation operators such as the fuzzy maximum distance, the fuzzy minimum distance, the fuzzy normalized Hamming distance (FNHD), the fuzzy weighted Hamming distance (FWHD), the fuzzy normalized Euclidean distance (FNED), the fuzzy weighted Euclidean distance (FWED), the fuzzy ordered weighted averaging distance (FOWAD) operator, the fuzzy Euclidean ordered weighted averaging distance (FEOWAD) operator, the fuzzy ordered weighted geometric averaging distance (FOWGAD), the fuzzy ordered weighted harmonic averaging distance (FOWHAD) operator and the fuzzy generalized ordered weighted averaging (FGOWA) operator [26]. We study some of its main properties and distinguish between different particular cases such as the olympic-FGOWAD, the S-FGOWAD and centered-FGOWAD. We also present an application of the new approach in a multi-person decision-making problem concerning the selection of strategies. The main advantage of the FGOWA operator in this type of problems is that it gives a more complete view of the decision problem because it considers a wide range of distance aggregation operators according to the interests of the decision maker. More-

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over, by using several experts in the analysis, we obtain information that it is more robust because the opinion of several experts is always better than the opinion of one. For doing so, we introduce a new aggregation operator called the multi-person-FGOWAD (MP-GOWAD) operator. This paper is organized as follows. In Section 2, we briefly review basic concepts that are used throughout the paper. In Section 3, we present the FGOWAD operator. Section 4 analyzes different families of FGOWAD operators. In Section 5, we present a method for multi-person decision-making with the FGOWAD operator in investment decisions and Section 6 develops a numerical example of the new approach. Finally, Section 7 summarizes the main conclusions of the paper.

2. Preliminaries In this Section we briefly review some basic concepts about the Hamming distance, the OWA, the OWAD and the GOWA operator. A. The Hamming distance The Hamming distance is a useful technique for calculating the differences between two parameters, such as problems with two elements or two sets. To define the Hamming distance, we first define a distance measure. A distance measure must basically accomplish the following properties: (1) Non-negativity: d ( A1 , A2 ) ≥ 0 ;

(2) Commutativity: d ( A1 , A2 ) = d ( A2 , A1 ) ; (3) Reflexivity: d ( A1 , A1 ) = 0 ; (4) Triangle inequality: d ( A1, A2 ) + d ( A2 , A3 ) ≤ d ( A1, A3 ) . For

two

sets

A = {a1 , a2 ,..., an }

and

B=

{b1 , b2 ,..., bn } , we can define the Hamming distance as follows: Definition 1: A normalized Hamming distance (NHD) of dimension n is a mapping NHD: R n → R , which has the following form 1 n NHD( A, B) = ∑ ai − bi (1) n i =1 where ai and bi is the i th arguments of the sets A

and B , respectively. Sometimes, when normalizing the Hamming distance, we prefer to give different weights to each individual distance. Then, the distance is known as the weighted Hamming distance. It can be defined as follows: Definition 2: A weighted Hamming distance (WHD) of dimension n is a mapping WHD: R n → R that has an

404


associated weighting W with w j ∈ [0,1] and

n

∑w j =1

j

=1

such that: n

WHD( A, B) = ∑ wi ai − bi 

(2)

i =1

where ai and bi is the i th arguments of the sets A and B , respectively. B. The OWA operator The OWA operator introduced by Yager [13] provides a parameterized family of aggregation operators that include the maximum, the minimum and the average criteria as special cases. Since its introduction, the OWA operator has been studied in a wide range of studies such as [3-6, 8-12, 19-29, 46-56]. This operator can be defined as follows: Definition 3: An OWA operator of dimension n is a mapping OWA: R n → R that has an associated weighting W with w j ∈ [0,1] and

n

∑w j =1

j

= 1 such that:

(3)

j =1

where b j is the j th largest of the ai . C. The GOWA operator The generalized OWA (GOWA) operator was introduced in [8, 19]. With this generalization, it is possible to include the special cases found in the OWA operator such as the maximum and the minimum, and the special cases found in the generalized mean such as the geometric and the harmonic mean. It can be defined as follows. Definition 4: A GOWA operator of dimension n is a mapping GOWA: R n → R that has an associated weighting W with w j ∈ [0,1] and

n

∑w j =1

j

weighting W with w j ∈ [0,1] and

n

∑w

OWAD ( a1 , b1 , a2 , b2 ,..., an , bn

j =1

j

= 1 such that: n

) = ∑w d j =1

j

j



(5)

where d j is the j th largest of the ai − bi . When using the OWAD operator, it is assumed that the available information includes exact numbers or crisp numbers. However, this may not be the real situation found in the decision-making problem. Sometimes it is better to use another approach such as the use of fuzzy numbers. In the following, we shall extend the OWAD operator to fuzzy environment.

3. The FGOWAD operator

n

OWA(a1 , a2 ,..., an ) = ∑ w j b j 

tension of the traditional normalized Hamming distance by using OWA operators. For two sets A = {a1 , a2 ,..., an } and B = {b1 , b2 ,..., bn } , the OWAD operator can be defined as follows: Definition 5: An OWAD operator of dimension n is a mapping OWAD: R n × R n → R that has an associated

= 1 such that: 1λ

⎛ n ⎞ (4) GOWA(a1 , a2 ,..., an ) = ⎜ ∑ w j b λj ⎟  ⎝ j =1 ⎠ where b j is the j th largest of the ai , λ is a pa-

rameter such that λ ∈ ( −∞, ∞ ) . Generalizing the reordering step, it is possible to distinguish between descending (DGOWA) and ascending (AGOWA) orders. As demonstrated in previous studies, the GOWA operator is commutative, monotonic, bounded, and idempotent. It can also be demonstrated that it yields the maximum, the minimum, and the generalized mean as special cases. D. The OWAD operator The OWAD (or Hamming OWAD) operator is an ex-

In this section, we will review basic concepts about the fuzzy numbers and develop the FGOWAD operator. E. Fuzzy Numbers In the literature, we find a wide range of fuzzy numbers [30-37]. For practical reasons we use, however, the notation introduced by Van Laarhoven and Pedrycz [37]. According to this notation, a triangular fuzzy number (TFN) aˆ may be expressed as following: Definition 6: Let aˆ = [a L , a M , aU ] , where a L ≤ a M ≤ aU , then aˆ is called a triangular fuzzy number (TFN), where a L and aU stand for the lower and upper values of aˆ , and a M stands for the modal value. Especially, if a L = a M = aU , then aˆ is reduced to a real number. Definition 7: Let aˆ = [aL , aM , aU ] and bˆ = [bL , bM , bU ] be two triangular fuzzy numbers (TFNs), then 1 d aˆ , bˆ = a L − b L + a M − b M + aU − bU (6) 3

( ) (

)

is called the distance between aˆ and bˆ . The above defined distance between TFNs accomplishes the following properties: Theorem 1: For any three TFNs aˆ = [a L , a M , aU ] , bˆ = [b L , b M , bU ] and cˆ = [c L , c M , cU ] , then:

( ) d ( aˆ , bˆ ) = d ( bˆ, aˆ ) ;

(1) Non-negativity: d aˆ, bˆ ≥ 0 ; (2) Commutativity:

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(3) Reflexivity: d ( aˆ , aˆ ) = 0 ;

Definition 10: A fuzzy generalized ordered weighted averaging distance (FGOWAD) operator of dimension n is a mapping FGOWAD: Ψ n × Ψ n → R that has an as-

( ) ( )

(4) Triangle inequality: d ( aˆ, cˆ) ≤ d aˆ,bˆ + d bˆ, cˆ . The proofs of these properties are straightforward and thus omitted. Let Ψ be the set of all TFNs, Aˆ = (aˆ1 , aˆ2 ,..., aˆn ) and Bˆ = (bˆ1 , bˆ2 ,..., bˆn ) be two collections of TFNs, we can define a fuzzy normalized Hamming distance (FNHD) as following: Definition 8: A fuzzy normalized Hamming distance of dimension n is a mapping FNHD: Ψ n × Ψ n → R , such that 1 n FNHD aˆ1 , bˆ1 ,..., aˆn , bˆn = ∑ aî − bî (7) n i =1 where aˆ and bˆ are the i th arguments of the sets Aˆ

(

i

)

i

and Bˆ , respectively. aî − bî

is the distance between

the aî and bî , defined by the (6). Sometimes, the weight of each individual distance should be taken into account. In this case, we shall define the fuzzy weighted Hamming distance (FWHD) as follows. Definition 9: A fuzzy weighted Hamming distance of dimension n is a mapping FWHD: Ψ n × Ψ n → R that has an associated weighting W with w j ∈ [0,1] and n

∑w j =1

j

= 1 such that:

(

FWHD aˆ1 , bˆ1 ,..., aˆn , bˆn

) = ∑w aˆ − bˆ n

i =1

i

i

i

(8)

F. The FGOWAD operator The fuzzy generalized OWAD (FGOWAD) operator is an extension of the OWAD operator that uses generalized means and fuzzy numbers in the aggregation. The main difference between the FGOWAD and OWAD operators is that the FGOWAD operator addresses uncertain information represented using fuzzy numbers, while the OWAD operator uses exact numbers. The reason for using this operator is that sometimes, the uncertain factors that affect our decisions are not clearly known and in order to assess the problem we need to use fuzzy numbers in order to consider the different uncertain results that could happen in the future. Moreover, by using the generalized means, we obtain a generalization that includes a wide range of fuzzy aggregation operators, such as the fuzzy maximum distance, the fuzzy minimum distance, the FNHD, FWHD, FNED, FWED, FOWAD, FEOWAD, FGOWA operators and so on. It can be defined as follows.

n

∑w

sociated weighting W with w j ∈ [0,1] and

j =1

j

=1

such that:

(

1λ

)

⎛ n λ⎞ = ⎜ ∑wd (9) i j ⎟ ⎝ i=1 ⎠ is j th largest d (aî , bî ) , λ is a parameter

FGOWAD aˆ1 , bˆ1 ,..., aˆn , bˆn where d j

such that λ ∈ ( −∞, +∞ ) . Example 1: Let Aˆ = ( (3, 4,5),(6,7,9),(4,6,7),(2, 4,5) )

and Bˆ = ( (4,6,8),(3, 4,6),(2,5,7),(3, 4,6) ) be two set of TFNs, then by (6), 1 d aˆ1 , bˆ1 = ( 3 − 4 + 4 − 6 + 5 − 8 ) = 2 3 Similarly, we have d aˆ2 , bˆ2 = 3, d aˆ3 , bˆ3 = 1, d aˆ4 , bˆ4 = 0.67

(

(

)

)

(

)

(

)

Assume the following weighting vector W = ( 03,0.2,0.4,0.1) and without loss of generality, let λ = 2 , then we can calculate the distance between Aˆ and Bˆ by using the FGOWAD operator: FGOWAD Aˆ , Bˆ = (0.3 × 32 + 0.2 × 22 +

(

)

+0.4 × 1 + 0.1 × 0.67 2 )1 2 = 2.18 From a generalized perspective of the reordering step, it is possible distinguish to distinguish between descending and ascending orders by using w j = w *n − j +1 , 2

where w j is the

j th weight of the descending

FGOWAD (DFGOWAD) operator and w *n − j +1 is the

j th weight of the ascending FGOWAD (AFGOWAD) operator. Note that if the weighting vector is not normalized, n

i.e., W = ∑ w j ≠ 1 , then the FGOWAD operator can be j =1

expressed as:

(

)

1λ

⎛1 n ⎞ FGOWAD aˆ1, bˆ1 ,..., aˆn , bˆn = ⎜ ∑wj d λj ⎟ (10) ⎝ W j =1 ⎠ Note that this situation is very common when the weights are also uncertain and represented in the form of fuzzy numbers. Note also that more complex types of fuzzy numbers could be considered but in this paper we focus on the use of TFNs. The FGOWAD operator is commutative, monotonic, bounded, idempotent, nonnegative and reflexive but it does not satisfy the triangle inequality in general. These

406


properties can be proved with the following theorems: Theorem 2 (Commutativity - OWA aggregation): Assume f is the FGOWAD operator, then

( )= f ( ) (11) where ( cˆ , dˆ ,..., cˆ , dˆ ) is any permutation of the arguments ( aˆ , bˆ ,..., aˆ , bˆ ) . f

aˆ1 , bˆ1 ,..., aˆn , bˆn 1

1

n

1

cˆ1 , dˆ1 ,..., cˆn , dˆn

n

1

n

n

Note that the commutativity of the FGOWAD can also be studied from the context of a distance measure, which can be proved with the following theorem: Theorem 3 (Commutativity - distance measure): Assume f is the FGOWAD operator, then f

( aˆ , bˆ 1

,..., aˆn , bˆn

1

) = f ( bˆ , aˆ 1

)

,..., bˆn , aˆn

1

(12)

Theorem 4 (Monotonic): Assume f is the FGOWAD operator, if d (aî , bî ) ≥ d (cî , dî ) for all i , then f

( aˆ , bˆ 1

1

) ≥ f ( cˆ , dˆ

,..., aˆn , bˆn

1

1

,..., cˆn , dˆn

)

(13)

Theorem 5 (Boundary): Assume f is the FGOWAD operator, then min d (aî , bî ) ≤ FGOWAD Aˆ, Bˆ ≤ max d (aî , bî ) (14)

{

}

{

( )

}

Theorem 6 (Idempotency): Assume f is the FGOWAD operator, if d (aˆ , bˆ ) = d for all i , then i

i

f

( aˆ , bˆ 1

1

,..., aˆn , bˆn

)=d

(15) is the

Theorem 7 (Non-negativity): Assume f FGOWAD operator, then f aˆ1 , bˆ1 ,..., aˆn , bˆn ≥ 0

(

)

(16)

Theorem 8 (Reflexivity): Assume f is the FGOWAD operator, then f ( aˆ1 , aˆ1 ,..., aˆn , aˆn ) = 0 (17) The proofs of these properties are straightforward and thus omitted. Note that the FGOWAD operator does not always satisfy the triangle inequality because we may find some special situations where FGOWAD Aˆ , Bˆ + FGOWAD Bˆ , Cˆ (18) ≤ FGOWAD Aˆ , Cˆ

(

)

(

)

(

)

Next, we present a numerical example which shows that the FGOWAD operator does not always satisfy the triangle inequality. Example 2: Let Aˆ = ( (2, 4,5),(4,5,10),(6,8,10) ) , Bˆ = ( (4,5,10),(2, 4,6),(6,8,10) ) and Cˆ = ( (6,8,10), (2, 4,6),

(4,5,11) ) . We assume the following weighting vector

W = ( 0,0,1)

and λ = 1 , then

( )

( )

FGOWAD Aˆ , Bˆ = 0 ,

( )

FGOWAD Bˆ , Cˆ = 0 , FGOWAD Aˆ , Cˆ = 2 . As we can see, 0 + 0 < 2 ; and therefore, we have proved that the FGOWAD operator does not always satisfy the triangle inequality. An interesting issue to consider is the measures for characterizing the weighting vector W such as the attitudinal character, the entropy of dispersion, the balance operator and the divergence of W . The attitudinal character can be defined as follows: 1λ

λ ⎛ n ⎛n− j⎞ ⎞ α (W ) = ⎜ ∑ w j ⎜ (19) ⎜ j =1 ⎝ n − 1 ⎟⎠ ⎟⎟ ⎝ ⎠ Note that it is possible to develop different types of measures of entropy but the most common one is based on the Shannon entropy and for the FGOWAD operator is defined as follows:

H (W ) = −∑ w j ln ( w j ) n

(20)

j =1

The balance operator measures the balance of the weights against the orness versus andness. It can be defined as follows: n ⎛ n +1− 2 j ⎞ Bal (W ) = ∑ w j ⎜ (21) ⎟ ⎝ n −1 ⎠ j =1 And the divergence of W : n ⎛n− j ⎞ Div (W ) = ∑ w j ⎜ − α (W ) ⎟ 1 n − ⎝ ⎠ j =1

2

(22)

4. Families of FGOWAD operator In this section, we consider different types of FGOWAD operators. We distinguish between two main classes. The first class focuses on the weighting vector W , and the second class on the parameter λ . The main advantage of using these particular cases is that we can select for each problem the particular case that we believe is closest to our interests. G. Analyzing the Weighting Vector W By choosing a different manifestation of the weighting vector in the FGOWAD operator, we are able to obtain different types of aggregation operators. For example, we can obtain fuzzy maximum distance, the fuzzy minimum distance, the fuzzy normalized generalized distance (FNGD) and the fuzzy weighted generalized distance (FWGD). Remark 1: Some of the most basic families applied to the FGOWAD operator are obtained as follows. • The fuzzy maximum distance is found if w1 = 1


and w j = 0 , for all j ≠ 1 .

407

decaying when i < j ≤ ( n + 1) 2 then wi < w j and

• The fuzzy minimum distance if wn = 1 and w j = 0 , when i > j ≥ (n + 1) 2 then wi < w j . It is inclusive if for all j ≠ n w j > 0 . Note that it is possible to consider a softening • More generally, if wk = 1 and w j = 0 for all j ≠ k , of the second condition by using w ≤ w instead of i j we get the step-FGOWAD operator. • The FNGD is formed when w j = 1 n for all j .

wi < w j . We shall refer to this as softly decaying centered-FGOWAD operator. Another particular situation of

• The FWGD is obtained when the ordered position of the centered-FGOWAD operator appears if we remove the i is the same as the ordered position of the j . the third condition. We shall refer to it as a non-inclusive • Note that the FGOWA [26] operator is also included centered-FGOWAD operator.

as a particular case of FGOWAD operator. This situation appears when one of the sets of the FGOWAD operator is empty. Remark 2: Another particular case is the Olympic-FGOWAD. This operator is found when w1 = wn = 0 and for all others w j* = 1 (n − 2) . Note that if n = 3 or n = 4 , the olympic-FGOWAD is transformed in the median- FGOWAD. Remark 3: Note that it is possible to present a general form of the olympic-FGOWAD operator, considering that w j = 0 for j = 1, 2,..., k , n, n − 1,..., n − k + 1 , and for all others, w j* = 1 (n − 2k ) where k < n 2 . Note

Remark 7: Using a similar methodology, we could develop numerous other families of FGOWAD operators. For more information, refer to [15-17, 21, 26, 46, 50]. H. Analyzing the Weighting Vector λ If we analyze different values of the parameter λ , we obtain another group of particular cases, such as the FOWAD, the FEOWAD, the FOWGAD and the FOWHA operator. Remark 8: The FOWAD operator is obtained when the parameter λ = 1 . It can be constructed as a special case of the FGOWAD operator, but it is also possible to construct it by mixing the OWA operator with the fuzzy Hamming distance.

n that if k = 1 , then this general form becomes the usual FGOWAD aˆ1 , bˆ1 ,..., aˆn , bˆn = ∑ w j d j (23) olympic-FGOWAD. If k = ( n − 1) 2 , then it becomes j =1 the median-FGOWAD operator. With the FOWAD operator it is also possible to obtain Remark 4: Additionally, it is also possible to present the another parameterized family of aggregation operators contrary case of the general olympic-FGOWAD operator. such as the fuzzy maximum distance, the fuzzy miniw j = 1 (2k ) for mum distance, the FNHD and the FWHD. The fuzzy In this case, j = 1, 2,..., k , n, n − 1,..., n − k + 1 , and w j = 0 , for all maximum and the fuzzy minimum distances are obtained as it has been explained with the FGOWAD operator. others, where k < n 2 . Note that if k = 1 , then we get The FNHD is obtained when w = 1 n for all j . The j the contrary case of the median-FGOWAD. FWHD is obtained when the ordered position of d j is Remark 5: A further interesting family is the S-FGOWAD operator. It can be subdivided in three the same as the position of d (aî , bî ) . classes: the “orlike”, the “andlike” and the generalized Remark 9: If λ = 2 , the FGOWAD operator becomes S-FGOWAD operators. The generalized S-FGOWAD the FEOWAD operator: operator is obtained when w1 = (1 n)(1 − (α + β )) + α , 12 ⎛ n 2⎞ ˆ ˆ FGOWAD aˆ1 , b1 ,..., aˆn , bn = ⎜ ∑ wi d j ⎟ (24) wn = (1 n)(1 − (α + β )) + β and w j = (1 n)(1 − (α + β )) ⎝ i =1 ⎠ for j ≠ 1, n ; where α , β ∈ [ 0,1] and α + β ≤ 1 . Note With the FEOWAD operator it is also possible to obthat if α = 0 , the generalized S-FGOWAD operator tain another parameterized family of aggregation operabecomes the “andlike” S-FGOWAD, and if β = 0 , it tors that include for example, the fuzzy maximum disbecomes the “orlike” S-FGOWAD. Note also that if tance, the fuzzy minimum distance, the fuzzy normalized Euclidean distance (FNED) and the fuzzy weighted α + β = 1 , we get the fuzzy Hurwicz distance criteria. Euclidean distance (FWED). Remark 6: Another family of aggregation operator that Remark 10: If λ = 0 , the FGOWAD operator becomes could be used is the centered-FGOWAD operator. We the FOWGAD operator: could define a FGOWAD operator as a centered aggren w gation operator if it is symmetric, strongly decaying and FGOWAD aˆ1 , bˆ1 ,..., aˆn , bˆn = ∏ d j j (25) j =1 inclusive. It is symmetric if w j = w j + n −1 . It is strongly

(

(

)

)

(

)

408


Remark 11: If λ = −1 , the FGOWAD operator becomes the FOWHAD operator: n w j (26) FGOWAD aˆ1 , bˆ1 ,..., aˆn , bˆn = ∑ d j =1 j

(

)

Note that we could analyze other families by using different values of the parameter λ . Note also that it is possible to study these families individually in a similar way as it has been developed in Section 3.

5. Multi-person decision-making with the FGOWAD operator The FGOWAD operator is applicable in a wide range of situations such as in decision making, statistics, engineering and economics. In this paper, we will consider a decision making application in the selection of strategies by using a multi-person analysis. The main motivation for using this model is that the representation of the information is very complex and we need to use fuzzy numbers and the opinion of several persons (experts) in order to correctly assess the problem. The process to follow in the selection of strategies with the FGOWAD operator in multi-person decision making can be summarized as follows. Step 1: Let A = { A1 , A2 ,..., An } be a discrete set of alternatives, and C = {C1 , C2 ,..., Cn } be a set of finite characteristics (or attributes), forming the matrix

( xˆhi )m×n .

Step 4: Use the weighted average (WA) to aggregate the information of the decision makers E by using the weighting vector V . The result is the collective payoff

t

∑v k =1

k

with

Each decision maker provides his own payoff matrix xˆ (hik ) .

( )

m× n

Step 2: Fixing the ideal levels of each characteristic to form the ideal investment (see Table 1), where I is the ideal investment expressed by a fuzzy subset, Ci is the i th characteristic to consider and yî is a fuzzy number for the i th characteristic. Each decision maker provides his own ideal investment yˆ i( k ) . Step 3: Compare the ideal investment with the different alternatives considered for each expert (person). In this step, the objective is to express numerically the distance between the ideal investment and the different alternatives considered. Table 1: Ideal strategy.

I

C1 yˆ1

C2 yˆ 2

… …

Cn yˆ n

t

∑v k =1

)

t

, thus xˆhi − yˆ hi = ∑ vk xˆ (hik ) − yˆ (hik ) . m× n k =1

k

= 1 and vk ≥ 0 , and a weighting W of di-

mension n with w j ∈ [0,1] and

(

n

∑w j =1

j

= 1 such that:

MP − FGOWAD ( xˆ11 ,..., xˆ1t ),( yˆ11 ,..., yˆ1t ) ,...

Let

= 1 ).

hi

− yˆ hi

Step 5: Calculate the aggregated results by using the FGOWAD operator explained in Eq. (9). Consider different particular manifestations of the FGOWAD operator by using different expressions in the weighting vector, as explained in Section 4. Step 6: Adopt decisions according to the results found in the previous steps. Select the alternative/s that provides the best result/s. Moreover, establish an ordering or a ranking of the alternatives from the most to the least preferred alternative to enable consideration of more than one selection. Note that this aggregation process can be summarized using the following aggregation operator, we call the multi-person-FGOWAD (MP-FGOWAD) operator. Definition 11: Let Ψ be the set of all triangular fuzzy numbers, a MP-FGOWAD operator is an aggregation operator that has a weighting vector V of dimension t

E = {e1 , e2 ,..., et } be the set of decision makers (whose weight vector is V = ( v1 , v2 ,..., vt ) , vk ≥ 0 ,

( xˆ

matrix

where

)

t n

1

t

(27)

1λ

⎛ n ⎞ ..., ( xˆn ,..., xˆ ),( yˆ n ,..., yˆ n ) = ⎜ ∑wi d λj ⎟ ⎝ i =1 ⎠ d j is j th largest xî − yî 1

,

and

t

xî − yî =  ∑ vk xˆ i( k ) − yˆ i( k ) , λ is a parameter such that k =1

λ ∈ ( −∞, +∞ ) . Note that the MP-FGOWAD operator has properties similar to those explained in Section 3, such as the distinction between descending and ascending orders, and so on. The MP-FGOWAD operator includes a wide range of particular cases following the methodology explained in Section 4. Note that it is also possible to consider more complex situations by using different types of aggregation operators in the aggregation of the experts’ opinion because in Definition 11 we assume that their opinions are aggregated by using WA operator. However, it is also possible to use the OWA and the GOWA operators.

6. Illustrative example In the following, we are going to develop an example

409


in which we will see the applicability of the new approach. We will focus on a multi-person decision-making problem about selection of strategies. Note that other business decision making applications could be developed such as financial decision making and human resource selection. Assume a company that operates in Europe and North America is analyzing the general policy for the next year and they consider five possible strategies to follow (adopted from [38]): (1) A1 = expand to the Asian market; (2) A2 = expand to the African market; (3) A3 = expand to the South American market; (4) A4 = expand to all three continents; (5) A5 = do not develop any expansion. In order to evaluate these strategies, the group of experts considers that the key factor is the economic situation of the next year. Then, depending on the situation, the expected benefits for the company will be different. The experts have considered five possible situations for the next year: (1) C1 = negative-growth rate; (2) C2 = growth rate near 0; (3) C3 = low-growth rate; (4) C4 = medium-growth rate; (5) C5 = high-growth rate. The group of experts of the company is constituted by three persons who give their own opinion about the expected results that may occur in the future. The expected results depending on the situation Ci and the alternative Ak are shown in Tables 2-4. Note that the results are TFNs. According to the objectives of the decision-maker, each expert establishes his own ideal investment. The results are shown in Table 5. With this information, we can aggregate the available information in order to make a decision. First, we aggregate the information of the three experts to obtain a unified payoff matrix represented in the form of individual distances between the available and ideal alternatives. We use the WA to obtain this matrix assuming that V = (0.3,0.3,0.4) . The results are shown in Table 6. It is now possible to develop different methods based on the FGOWAD operator in order to make a decision. In this example, we consider the fuzzy maximum distance, the fuzzy minimum distance, the FWHD, the FWED, the FOWAD, the FEOWAD and the FOWGAD. We assume the following weighting vector W = (0.1,0.2,0.2,0.2,0.3) . The results are shown in Table 7. Another interesting issue is to establish an ordering of the alternatives. This becomes useful when we want to consider more than one alternative. The results are shown in Table 8.

Table 2. Fuzzy payoff matrix-expert 1.

A1 A2 A3 A4 A5

C1

C2

C3

C4

C5

[60,70,80]

[30,40,50]

[50,60,70]

[70,80,90]

[30,40,50]

[50,60,70]

[70,80,90]

[20,30,40]

[50,60,70]

[40,50,60]

[10,20,30]

[30,40,50]

[40,50,60]

[60,70,80]

[70,80,90]

[20,30,40]

[40,50,60]

[60,70,80]

[80,90,100]

[70,80,90]

[30,40,50]

[40,50,60]

[70,80,90]

[20,30,40]

[60,70,80]


A1 A2 A3 A4 A5

C1

C2

C3

C4

C5

[50,60,70]

[60,70,80]

[80,90,100]

[20,30,40]

[70,80,90]

[60,70,80]

[20,30,40]

[50,60,70]

[30,40,50]

[40,50,60]

[60,70,80]

[50,60,70]

[20,30,40]

[70,80,90]

[60,70,80]

[70,80,90]

[[10,20,30]

[40,50,60]

[70,80,90]

[20,30,40]

[30,40,50]

[50,60,70]

[40,50,60]

[50,60,70]

[30,40,50]


A1 A2 A3 A4 A5

C1

C2

C3

C4

C5

[20,30,40]

[60,70,80]

[40,50,60]

[70,80,90]

[40,50,60]

[40,50,60]

[10,20,30]

[70,80,90]

[50,60,70]

[50,60,70]

[40,50,60]

[70,80,90]

[80,90,100]

[30,40,50]

[50,60,70]

[70,80,90]

[80,90,100]

[20,30,40]

[50,60,70]

[20,30,40]

[60,70,80]

[50,60,70]

[60,70,80]

[30,40,50]

[30,40,50]

Table 5. Ideal strategy.

e1 e2 e3

C1

C2

C3

C4

C5

[70,80,90]

[80,90,100]

[70,80,90]

[80,90,100]

[80,90,100]

[80,90,100]

[70,80,90]

[80,90,100]

[70,80,90]

[80,90,100]

[70,80,90]

[80,90,100]

[80,90,100]

[70,80,90]

[70,80,90]

Table 6. Collective results in the form of individual distances. C1

C2

C3

C4

C5

A1

32

26

22

18

30

A2

24

50

28

29

36

A3

36

25

27

22

17

A4

19

30

43

8

41

A5

31

30

20

42

37

Table 7. Aggregated Results. Max

Min

FWHD

FWED

FOWAD

FEOWAD

FOWGAD

A1

32

18

25.6

26.11

24.2

24.73

23.66

A2

50

24

33.4

34.63

30.8

31.73

30

A3

36

17

25.4

26.16

23.5

24.16

22.86

A4

43

8

28.2

31.16

24.7

28.15

20.32

A5

42

20

32

32.58

29.8

30.69

28.84

410


Table 8. Ordering of the Strategies. Ordering Max Min FWHD FWED FOWAD FEOWAD FOWGAD

A1 A4 A3 A1 A3 A3 A4

A3 A3 A1 A3 A1 A1 A3

A5 A1 A4 A4 A4 A4 A1

A4 A5 A5 A5 A5 A5 A5

A2 A2 A2 A2 A2 A2 A2

12YD66YB) and Zhejiang Province Natural Science Foundation (No. Y6110777).

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As we can see, depending on the aggregation used, the ordering of the strategies is different. Therefore, depending on the aggregation operator used, the results may lead to different decisions.

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7. Conclusions We have introduced the FGOWAD operator. It is an aggregation operator that uses generalized means, distance measure and uncertain information represented in the form of fuzzy numbers. It is very appropriate for uncertain situations where the decision maker can not assess the information with exact numbers but it is possible to assess it with fuzzy numbers. This generalization includes a wide range of fuzzy distance measures and fuzzy aggregation operators such as the fuzzy maximum distance, the fuzzy minimum distance, the FNHD, the FWHD, the FNED, the FWED, the FOWAD, the FEOWAD operator, the FOWGAD and the FGOWA operator. We have analyzed an application of the new approach in a multi-person decision-making problem regarding the selection of strategies. To do so, we have introduced the MP-FGOWAD operator. We have seen that this approach provides better information for decision-making since it is able to consider a wide range of scenarios depending on the interests of the decision maker. We have also seen that, depending on the particular type of aggregation operator used, the results may lead to different decisions. In future research we expect to develop further extensions by adding new characteristics in the problem such as the use of inducing variables or probabilistic aggregations. We will also consider other decision making applications such as human resource management, investment selection and product management.

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tional Research, vol. 198, no. 1, pp. 259-265, 2009. Shouzhen Zeng was born in 1981. He graduated from the Tianjin University and obtained the Master degree in applied mathematics in 2007. At present, he is studying his in-service doctor of applied statistics in Zhejiang Gongshang University. He has published more than 40 papers in journals, books and conference proceedings including journals such as Statistics Research, Knowledge-based Systems and Group decision and Negotiation. His main research fields are Aggregation Operators, Decision Making, Comprehensive evaluation and Uncertainty. Now he is a full-time Lecture in Zhejiang Wanli University. Weihua Su has a MSc and a PhD degree in statistics from Xiamen University, China. He is a Professor in College of Mathematics and Statistics, Zhejiang University of Finance and Economics. He has published more than 90 papers in journals, books and conference proceedings including journals such as Statistics Research, Economic Research, Knowledge-based Systems and Group decision and Negotiation. He has published 3 books. He has participated in several scientific committees and serves as a reviewer in a wide range of journals including Computers & Industrial Engineering, International Journal of Information Technology and Decision Making, Information Sciences and European Journal of Operational Research. He is currently interested in Aggregation Operators, Decision Making and Comprehensive evaluation. Anbo Le has a MSc and a PhD degree in statistics from Zhejiang University, China. He has published more than 100 papers in journals and conference proceedings including journals such as International Journal of Intelligent Systems, Cybernetics & Systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems and International Journal of Fuzzy Systems. He is currently interested in Financial Management, Decision Making and Uncertainty.