Fuzzy group decision-making with generalized probabilistic OWA ...

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Journal of Intelligent & Fuzzy Systems 27 (2014) 783–792 DOI:10.3233/IFS-131036 IOS Press

Fuzzy group decision-making with generalized probabilistic OWA operators José M. Merigóa,b,∗ , Montserrat Casanovasb and Yejun Xuc a Manchester

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Business School, University of Manchester, Manchester, UK of Business Administration, University of Barcelona, Barcelona, Spain c Research Institute of Management Science, Business School, Hohai University, Nanjing, P.R. China

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b Deparment

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Abstract. This paper presents a new fuzzy group decision-making approach by using probabilities and ordered weighted averages (OWA). Thus, we are able to assess the problem considering probabilistic information and the attitudinal character of the decision maker. For doing so, the fuzzy generalized probabilistic OWA (FGPOWA) operator is presented. It is an aggregation operator that unifies the probability and the OWA operator considering the degree of importance that each concept has in the aggregation. It also uses fuzzy numbers that deal with uncertain environments where the available information is imprecise. Moreover, it includes a wide range of particular cases including the fuzzy generalized probabilistic aggregation, the fuzzy POWA operator and the fuzzy GOWA operator. A further generalization to this approach is presented by using quasi-arithmetic means forming the fuzzy quasi-arithmetic POWA (Quasi-FPOWA) operator. An application in fuzzy group decision making is developed by using the multi-person – FGPOWA (MP-FGPOWA) operator.

1. Introduction

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Keywords: Decision making, fuzzy numbers, OWA operator, probabilities, fuzzy group decision-making

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Decision making is a very common process in our life. People are always making decisions [9]. When studying the decision making process we can use a wide range of mathematical and statistical techniques. A very useful tool for doing so is the aggregation operator. In the literature, there are a lot of aggregation operators [1, 2, 11, 32, 39, 40]. For example, we can mention the probabilistic aggregation based on the use of probabilities in the analysis. Another useful aggregation operator is the ordered weighted averaging (OWA) operator [34]. It provides a parameterized family of aggregation operators between the minimum and the maximum. Usually, when using the OWA operator we assume that the information is clearly known. However, in real ∗ Corresponding author. Jos´ e M. Merigó, Manchester Business School, University of Manchester, Booth Street West, M15 6PB Manchester, UK. Tel.: +44 (0)1613063495; E-mail: jose. [email protected].

world situations this may not be the case because the information may be vague or imprecise. Therefore, it is necessary to use another approach such as the use of the fuzzy numbers (FNs). By using FNs we can consider the minimum and the maximum results and the possibility that the internal values of the fuzzy interval will occur. If we extend the OWA operator to a fuzzy environment, we obtain the fuzzy OWA (FOWA) operator. Since its introduction, it has been studied by a lot of authors. For example, Chen and Chen [6] developed an extension by using induced aggregation operators. Liu [13] studied a process by using interval-valued trapezoidal fuzzy numbers. Merigó and Casanovas [18, 19], Merigó and Gil-Lafuente [24] and Wang and Luo [26] developed several generalizations by using generalized and quasi-arithmetic means. Wei [27] and Xu [31] suggested the use of harmonic means in the aggregation. Zeng and Su [52–57] introduced several distance measures by using fuzzy numbers, linguistic variables and intuitionistic fuzzy sets. Wei et al. [28] and Zhao et al.

1064-1246/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

J.M. Merigó et al. / Fuzzy group decision-making with generalized

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of strategies. For doing so, it is used the multi-person FGPOWA (MP-FGPOWA) operator that is able to deal with the opinion of several persons (or experts) in the analysis. It includes a wide range of particular cases including the multi-person FGOWA operator and the multi-person FPOWA operator. This paper is organized as follows. In Section 2 we briefly review some basic preliminaries. Section 3 introduces the FGPOWA operator and Section 4 the Quasi-FPOWA operator. Section 5 presents an application in multi-person decision making and Section 6 an illustrative example. In Section 7 we summarize the main conclusions of the paper.

2. Preliminaries

In this Section we briefly review some basic preliminaries regarding the fuzzy numbers (FNs), the POWA operator, the FOWA operator and the FGOWA operator. 2.1. Fuzzy numbers

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[58] developed several extensions by using intuitionistic fuzzy sets. Xu and Wang [30] and Xu and Xia [33] suggested the use of induced generalized aggregation operators with intuitionistic fuzzy sets. And Zhou et al. [50] introduced the use of intuitionistic fuzzy sets and power aggregation operators in the same formulation. Another interesting extension of the OWA operator is the generalized OWA (GOWA) operator [37]. It uses generalized means in the analysis. By using fuzzy information, we form the fuzzy generalized OWA (FGOWA) operator [19]. It includes the FOWA operator as a special case and a lot of other cases such as the quadratic FOWA (FOWQA) operator and the fuzzy generalized average (FGA). The FGOWA operator can be further extended by using quasi-arithmetic means obtaining the fuzzy quasi-arithmetic OWA (Quasi-FOWA) operator. A further interesting case is the probabilistic OWA (POWA) operator. It unifies the probability and the OWA operator in the same formulation considering the degree of importance that each concept has in the aggregation. Note that the use probabilities and OWA operators in the same formulation has already been considered by other authors. Especially, it is worth noting the concept of immediate probabilities [8, 14, 36, 38]. However, none of the previous models considered the degree of importance of both concepts in the analysis. The aim of this paper is to introduce a more general framework that it is able to deal with uncertain environments that can be assessed with fuzzy information. We introduce the fuzzy generalized probabilistic OWA (FGPOWA) operator. It unifies the FOWA and the fuzzy probabilistic aggregation (FPA) in the same formulation considering the degree of importance that each concept has in the analysis. It also includes a wide range of particular cases including the FPOWA operator, the FOWA, the FPA, the quadratic FPOWA (FPOWQA) operator, the fuzzy generalized probabilistic aggregation (FGPA) and the FGOWA operator. We further extend this approach by using quasiarithmetic means obtaining the fuzzy quasi-arithmetic POWA (Quasi-FPOWA) operator. It generalizes a wide range of aggregation operators including the FGPOWA operator and many others. The applicability of this new approach is also studied and it is seen that it is very broad because all the previous studies that use the probability or the OWA operator can be revised and extended with this new model. The reason is that it includes them as special cases. Thus, we can always reduce the model to the special case we want to consider. We focus on an application in a group decision making problem regarding the selection

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The FNs [3, 42–48] have been studied and applied by a lot of authors [7, 12]. Its main advantage is that it can represent in a more complete way the information because it can consider the maximum and the minimum and the possibility that the internal values may occur. Definition 1. Let R be (–∞, ∞), the set of all real numbers. A FN is a fuzzy subset [41] of R with membership function m: R −→ [0, 1] satisfying the following conditions: • Normality: There exists at least one number a0 ∈ R such that m(a0 ) = 1. • Convexity: m(t) is nondecreasing on (–∞, a0 ] and nonincreasing on [a0 , ∞). The FN can be considered as a generalization of the interval number. In the literature, we find a wide range of FNs such as triangular FNs (TFN), trapezoidal FNs (TpFN), interval-valued FNs, intuitionistic FNs, generalized FNs, type 2 FNs and more complex structures. For example, a trapezoidal FN (TpFN) A of a universe of discourse R can be characterized by a trapezoidal membership function A = (a, a¯ ) such that a(α) = a1 + α(a2 − a1 ), a¯ (α) = a4 − α(a4 − a3 ).

(1)


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1. A + B = (a1 + b1 , a2 + b2 , a3 + b3 ), 2. A − B = (a1 − b3 , a2 − b2 , a3 − b1 ), 3. A × k = (k × a1 , k × a2 , k × a3 ); for k > 0.

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Among the wide range of methods existing in the literature for ranking FNs, we recommend the use of the methods commented by Merigó [15] such as the use of the value found in the highest membership level (␣ = 1) and if it is an interval, the average of the interval. Note that other operations and ranking methods could be studied [7, 12, 24] but in this paper we use these ones. 2.2. The probabilistic OWA operator

The POWA operator is an aggregation operator that provides a parameterized family of aggregation operators between the minimum and maximum that unifies probabilities and OWAs in the same formulation considering the degree of importance that each concept has in the aggregation [15, 16]. It is defined as follows. Definition 2. A POWA operator of dimension n is a mapping POWA: Rn −→ R that has an associated weighting vector W of dimension n such that wj ∈ [0, 1] and nj=1 wj = 1, according to the following formula:

n

pˆ j bj

(3)

j=1

where bj is the jth largest of the ai , each argument ai has an associated probability pi with ni=1 pi = 1 and pi ∈ [0, 1], pˆ j = βwj + (1 − β)pj with ␤ ∈ [0, 1] and pj is the probability pi ordered according to the jth largest of the ai .

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2.3. The fuzzy OWA operator The FOWA operator is an extension of the OWA operator that uses uncertain information represented in the form of FNs. The FOWA operator provides a parameterized family of aggregation operators that include the fuzzy maximum, the fuzzy minimum and the fuzzy average criteria, among others. It is defined as follows.

Definition 3. Let be the set of FNs. A FOWA operator of dimension n is a mapping FOWA: n −→ that has an associated weighting vector W of dimension n with wj Iˆ [0, 1] and nj=1 wj = 1, such that:

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where a1 , a2 , a3 , a4 ∈ R and a1 ≤ a2 ≤ a3 ≤ a4 . Note that in this paper, especially when developing the illustrative example, we denote the TpFN as (a1 , a2 , a3 , a4 ). Furthermore, we will denote all the FNs in a general way as a˜ . Thus, by providing this abbreviation we will be able to represent all the FNs in the same formulation. In the following, we are going to review the FNs arithmetic operations as follows. Let A and B be two TFNs, where A = (a1 , a2 , a3 ) and B = (b1 , b2 , b3 ), such that:

POWA (a1 , . . . an ) =

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where ␣ ∈ [0, 1] and parameterized by (a1 , a2 , a3 , a4 ) where a1 ≤ a2 ≤ a3 ≤ a4 , are real values. Note that if a1 = a2 = a3 = a4 , then, the FN is a crisp value and if a2 = a3 , the FN is represented by a triangular FN (TFN). Note that the TFN can be parameterized by (a1 , a2 , a4 ). The TpFN can also be represented in the following way: ⎧ if t = [a2 , a3 ], ⎪ ⎪ ⎪ 1 ⎪ ⎪ if t ∈ [a1 , a2 ], ⎪ ⎪ 1 ⎨ at−a 2 −a1 m(t) = (2) a4 −t ⎪ ⎪ a4 −a3 ⎪ if t ∈ [a3 , a4 ], ⎪ ⎪ ⎪ ⎪ ⎩ 0 otherwise,

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FOWA (ã1, a˜ 2 , . . . , a˜ n ) =

n

wj bj ,

(4)

j=1

where bj is the jth largest of the a˜ i , and the a˜ i are FNs. The FOWA operator accomplishes similar properties than the OWA operator such as the distinction between the descending FOWA (DFOWA) operator and the ascending FOWA (AFOWA) operator and so on. 2.4. The fuzzy generalized OWA operator The fuzzy generalized OWA (FGOWA) operator [19] is an extension of the GOWA operator that uses uncertain information in the aggregation represented in the form of FNs. 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 FNs in order to consider the different uncertain results that could happen in the future. It can be defined as follows.

Definition 4. Let be the set of FNs. An FGOWA operator of dimension n is a mapping FGOWA: n −→ that has an associated weighting vector W of dimension n such that the sum of the weights is 1 and wj Î [0,1], then:


⎛ FGOWA (ã1 , a˜ 2 , . . . , a˜ n ) = ⎝

n

⎞1/λ wj bjλ ⎠

,

j=1

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(5) where bj is the jth largest of the a˜ i , the arguments a˜ i are FNs and λ is a parameter such that λ ∈ (−∞, ∞). Note that if λ = 1, we get the FOWA operator and if λ = 2, the quadratic FOWA (FOWQA) operator. The FGOWA operator accomplishes similar properties than the GOWA operator [37].

the following criteria. First, we analyze if there is an order between the FNs. That is, if the lowest value of an interval is higher than the highest value of another interval: A = (a1 , a2 , a3 ) is strictly higher than C = (c1 , c2 , c3 ) if a1 > c3 . Note that in this case we can guarantee 100% that A > C. If not, we select the FN with the highest value in its highest membership level, usually, when ␣ = 1. Note that if the membership level ␣ = 1 is an interval, then, we calculate the average of the interval. Note also that it is possible to distinguish between the descending FGPOWA (DFGPOWA) and the ascending FGPOWA (AFGPOWA) operator by using wj = w*n − j +1 , where wj is the jth weight of the DFGPOWA and w*n − j +1 the jth weight of the AFGPOWA operator. Note that this reordering is in the OWA aggregation. However, it is possible to consider a more general reordering process by using pˆ j = pˆ n−j+1 . In this case, we consider descending and ascending orders in the FOWA and in the fuzzy probabilistic aggregation (FPA) at the same time. Another interesting transformation is found [35] by using wi *=(1 + wi ) / (m – 1). Furthermore, we can also analyze situations with buoyancy measures [35]. In this case, we assume that wi ≥ wj , for i < j. Note that it is also possible to consider a stronger case known as buoyancy measure extensive where wi > wj , for i < j. And we can also consider the contrary case, that is, wi ≤ wi , for i < j, and the extensive measure wi < wj , for i < j. The FGPOWA operator is monotonic, bounded and idempotent. Additionally, if the weights of the probabilities and the OWAs are also fuzzy, then, we have to establish a method for dealing with these fuzzy weights. Note that it is very common that W = n in these situations n ˜ w = / 1 and P = ˜i = / 1. Thus, a very usej=1 j i=1 p ful method for dealing with these situations is by using:

3. The fuzzy generalized probabilistic OWA operator 3.1. Main concepts

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The fuzzy generalized probabilistic OWA (FGPOWA) operator is an aggregation operator that provides a unified formulation between the probability and the OWA operator in an uncertain environment that can be assessed with FNs. It also uses generalized means providing a general framework that includes a wide range of particular cases. Its main advantage is that it can deal with subjective or objective information and with the attitudinal character of the decision maker in the same framework. Thus, it is possible to evaluate the probabilities according to our degree of optimism or pessimism being able to under or overestimate them. It can be defined as follows.

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Definition 5. Let be the set of FNs. A FGPOWA operator of dimension n is a mapping FGPOWA: n −→ that has an associated weighting vector W of ˜ j ∈ [0, 1] and nj=1 w ˜ j = 1, and a dimension n with w probabilistic vector P of dimension n with p˜ i ∈ [0, 1] and ni=1 p˜ i = 1, such that: FGPOWA (ã1 , a˜ 2 , . . . a˜ n ) = ⎞1/λ ⎛ 1/δ n n λ⎠ δ ˜ ˜β ⎝ ˜ j bj w + (1 − β) p˜ i a˜ i , (6) j=1

i=1

where bj is the jth largest a˜ i , the a˜ i are FNs, β˜ ∈ [0, 1], and λ is a parameter such that λ ∈ (−∞, ∞). Note that different types of FNs could be used in the aggregation including TFNs, TpFNs, IVFNs, type-2 FNs and more complex structures. Often, when dealing with FNs in the FGPOWA operator, it is not clear which FN is higher. Therefore, we need to establish a criteria for ranking the FNs. For simplicity, we recommend

FGPOWA (ã1, a˜ 2, . . . , a˜ n ) = ⎞1/λ ⎛ 1/δ n n ˜ β˜ ⎝ (1 − β) λ⎠ δ ˜ j bj + w p˜ i a˜ i . (7) W P j=1

i=1

A further interesting issue is the measures for characterizing the weighting vector. Following a similar methodology as it has been developed for the OWA operator [19, 35] we could formulate the orness measure (attitudinal character) and the entropy of dispersion. The orness measure of the FGPOWA operator is formulated as follows:


⎛ ⎞

λ 1/λ n n−j ⎠ ˆ = β˜ ⎝ ˜j α(P) + w n−1

• If β = 1, we get the FGOWA operator. • If β = 0, we get the fuzzy generalized probabilistic aggregation (FGPA).

j=1

p˜ j

n−j n−1

δ

⎞1/δ ⎠

, (8)

where p˜ j represents the probabilistic weights reordered according to the values of the arguments bj . Note that if β˜ = 1, we get the orness measure of the FGOWA operator and if β˜ = 0, the orness measure of the fuzzy generalized probabilistic aggregation (FGPA). However, it is worth noting that the orness measure seen from the perspective of the attitudinal character, can also be formulated assuming that the probabilistic aggregation is neutral. Thus, we can establish a fixed value for this part of the aggregation, for example, assuming it is 0.5. ⎛ ⎞

λ 1/λ n n − j ˜ × 0.5, ⎠ + (1 − β) ˆ = β˜ ⎝ ˜j α(P) w n−1

• The FPOWA operator: When λ = 1. • The geometric FPOWA (FGPOWA) operator: When λ −→ 0. • The harmonic FPOWA (FHPOWA) operator: When λ = −1. • The quadratic FPOWA (FQPOWA) operator: When λ = 2. • The fuzzy maximum: When λ −→ ∞. • The fuzzy minimum: When λ −→ –∞.

Moreover, it is possible to consider other families by using different values in the parameter λ and δ. For example, we can form the following aggregation operators: • If λ = 2 and δ = 1, we get the fuzzy probabilistic ordered weighted quadratic average (FPOWQA). That is:

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j=1

The more β approaches to 1, the more importance we give to the FGOWA operator, and vice versa. If we analyze different values of the parameter λ and δ, we obtain another group of particular cases.

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˜ ⎝ +(1 − β)

n

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j=1

⎛

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(9) In this case, if β˜ = 0, the attitudinal character is neutral, that is, 0.5. Note that we could also use other values instead of 0.5 depending on the assumptions we make regarding neutrality since a lot of decision making theories may establish at a different level, especially when dealing with utility theory. The entropy of dispersion [34] measures the amount of information being used in the aggregation. For the FGPOWA operator, it is defined as follows. ⎛ ⎞ n n ˜ ˆ = − ⎝β˜ ˜ j ) + (1 − β) ˜ j ln(w H(P) p˜ i ln(p˜ i )⎠ w j=1

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FGPOWA (ã1, a˜ 2, . . . , a˜ n ) ⎞1/2 ⎛ n n ˜ ˜ j bj2 ⎠ + (1 − β) = β˜ ⎝ p˜ i a˜ i . w j=1

(11)

i=1

• If λ = 1 and δ = 2, then, we obtain the fuzzy probabilistic quadratic ordered weighted averaging (FPQOWA) operator:

i=1

(10) Note that p˜ i is the ith weight of the FPA aggregation. As we can see, if β˜ = 1, we get the entropy of dispersion of the FGOWA operator and if β˜ = 0, we extend the classical Shannon entropy [25] by using fuzzy generalized probabilities. 3.2. Families of FGPOWA operators Another interesting issue is the analysis of different families of FGPOWA operators by analyzing particular cases in the coefficient ␤, in the parameter λ and δ and in the weighting vector W. If we analyze the coefficient ␤, we get the following:

FGPOWA (ã1, a˜ 2 . . . , a˜ n ) = β˜

n

˜ ˜ j bj + (1 − β) w

j=1

n

1/2 p˜ i a˜ i2

.(12)

i=1

• If λ = 1 and δ = 3, we obtain the fuzzy probabilistic cubic ordered weighted averaging (FPCOWA) operator: FGPOWA (ã1, a˜ 2, . . . , a˜ n ) = β˜

n j=1

˜ ˜ j bj + (1 − β) w

n i=1

1/3 p˜ i a˜ i3

.(13)


• The maximum fuzzy probabilistic aggregation (Max-FPA) (w1 = 1 and wj = 0, for all j = / 1). • The minimum fuzzy probabilistic aggregation (Min-FPA) (wn = 1 and wj = 0, for all j = / n). • The fuzzy generalized mean (FGM) (wj = 1/n, and pi = 1/n, for all a˜ i ). • The arithmetic FGOWA (pi = 1/n, for all a˜ i ). • The arithmetic FGPA (wj = 1/n, for all a˜ i ). Note that other families of FGPOWA operators could be studied following [15, 16, 35, 40].

4. The Quasi-FPOWA operator The FGPOWA operator can be further generalized by using quasi-arithmetic means [1, 10, 20–22]. Thus, we obtain the Quasi-FPOWA operator that can be defined as follows.

The FGPOWA and the Quasi-FPOWA operators can be applied in a lot of fields including statistics, economics and engineering. The main reason is that they include the classical probabilistic aggregation (expected value) and the OWA operator as particular cases. Therefore, all the previous studies that use the probability or the OWA operator can be revised an extended with this new approach. In this Section, we analyze the use of the FGPOWA operator in fuzzy multi-person problems. Observe that the FGPOWA operator is able to create a unified decision making process between risk and uncertain environments by integrating probabilities and OWA operators in the same formulation. We analyze a fuzzy multi-person decision-making problem in the selection of strategies. We focus on the selection of general strategies in an enterprise. As these decisions are very complex and relevant, it requires the opinion of several experts that assess the problem. Thus, by using a multiperson analysis we can assess the information in the most efficient way. The procedure to select general strategies with the FGPOWA operator in fuzzy multi-person decisionmaking is described as follows. Step 1: Let A=A1 , A2 , . . . , Am be a set of finite alternatives, S=S1 , S2 , . . . , Sn , a set of finite states of nature (or attributes), forming the payoff matrix (ãhi )m ×n . Let E=e1 , e2 , . . . , ep be a finite set of decision-makers. Let U=(u1 , u2 , . . . , up ) bethe weighting vector of the p decision-makers such that k=1 u˜ k = 1 and u˜ k ∈ [0, 1]. Each decision-maker provides his own payoff matrix (ãhi (k ) )m ×n . ˜ + Step 2: Calculate the weighting vector Pˆ = β˜ × W ˜ × P˜ to be used in the FGPOWA aggregation. (1 − β) ˜j =1 Note that W=(w1 , w2 , . . . , wn ) such that nj=1 w ˜ ∈ [0, 1] and P=(p , p , . . . , p ) such that and w 1 2 n n j p ˜ = 1 and p ˜ ∈ [0, 1]. i i=1 i Step 3: Aggregate the information of the decisionmakers E using the weighting vector U. In this example, we use the fuzzy weighted average (FWA). The result is collective payoff matrix (ãhi )m ×n . Thus, a˜ hi = the p k . Note that we can use different types of ˜ k a˜ hi k=1 u aggregation operators instead of the FWA to aggregate this information such as different types of FGPOWA operators. Step 4: Calculate the aggregated results using the FGPOWA operator explained in Eq. (6). Consider different families of FGPOWA operators as described in Section 3.

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Definition 6. Let be the set of FNs. A Quasi-FPOWA operator of dimension n is a mapping QFPOWA: n −→ that has an associated weighting vector W of ˜ j ∈ [0, 1] and nj=1 w ˜ j = 1, and a dimension n with w probabilistic vector P of dimension n with p˜ i ∈ [0, 1] and ni=1 p˜ i = 1, such that:

5. Group decision-making with the FGPOWA

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And if we look to the weighting vector W, we get, for example, the following ones:

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QFPOWA (ã1, a˜ 2, · · · , a˜ n ) ⎛ ⎞ n n −1 −1 ˜ ˜ ⎝ ˜ j g(bj )⎠+(1− β)h = βg w p˜ i h(ãi ) ,

(14)

where bj is the jth largest ai , the a˜ i are FNs, ␤ ∈ [0, 1], and g(b) and h(a) are strictly continuous monotonic functions. Note that if g(b) = bλ and h(ã) = a˜ δ , then, the QuasiFPOWA operator becomes the FGPOWA operator. The Quasi-FPOWA operator includes a wide range of families of aggregation operators in a similar way as the FGPOWA operator. Moreover, it also includes a lot of other families by analysing more complex structures in its continuous monotonic function including the exponential FPOWA operator and the trigonometric FPOWA operators. Note that a lot of other extensions could be developed by using Choquet integrals [21, 23, 24], continuous aggregations [60, 61, 64, 65], logarithmic aggregation operators [59, 63] and power averages [29, 62].


p

MP − FGPOWA((ã11 , . . . , a˜ 1 ), . . . , (ãn1 , . . . , a˜ np )) = =

n

pˆ j bj ,

(15)

j=1

• • • •

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• The multi-person – fuzzy probabilistic aggregation (MP-FPA) operator. • The multi-person – FOWA operator. • The multi-person – fuzzy arithmetic mean (MPFAM) operator. • The multi-person – fuzzy generalized probabilistic aggregation (MP-FGPA). • The multi-person – FGOWA (MP-FGOWA). • The multi-person – fuzzy generalized mean. Note that it is possible to consider other types of aggregation operators to aggregate the experts opinions, although in Definition 6 we assume that the opinions are aggregated by using FWA operators.

A1 A2 A3 A4

= Expand to the Asian market. = Expand to the South American market. = Expand to the African market. = Do not make any expansion.

In order to evaluate these strategies, the company is assessed by three independent experts. These experts consider that the key factor is the economic situation of the world economy for the next period. They consider 5 possible states of nature that could happen in the future:

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where bj is the jth largest of the a˜ i , each argument a˜ i has an associated probability p˜ i with ni=1 p˜ i = 1 and ˜ p˜ j with β˜ ∈ [0, 1] and ˜ j + (1 − β) p˜ i ∈ [0, 1], pˆ j = β˜ w p˜ j is the probability p˜ i ordered according to bj , that is, p according to the jth largest of the a˜ i , a˜ i = k=1 u˜ k a˜ ik , a˜ ik is the argument variable provided by each person (or expert). Note that the MP-FGPOWA operator has similar properties to those explained in Section 3, such as the distinction between descending and ascending orders, and so on. The MP-FGPOWA operator includes a wide range of particular cases following the methodology explained in Section 3. Thus, it includes:

In the following, we present a numerical example of the new approach in a fuzzy multi-person decisionmaking problem regarding the selection of strategies. We focus on an economic problem regarding the general strategy of an enterprise for the next year. Note that there are many other types of decision making problems in the literature [4, 5, 17, 49–51, 66]. Step 1: Assume an enterprise that it is planning its general strategy for the next year and considers 5 possible strategies.

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Definition 7. A MP-FGPOWA operator is a mapping MP-FGPOWA: n ×p −→ that has a weighting p vector U of dimension p with k=1 u˜ k = 1 and u˜ k Î [0, n1] and a weighting vector W of dimension n with ˜ j = 1 and w ˜ j Î [0, 1], such that: j=1 w

6. Illustrative example

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Step 5: Adopt decisions according to the results found in the previous steps. Select the alternative (s) that provides the best result (s) and establish a ranking of the alternatives. The previous fuzzy multi-person decision process can be summarized using the following aggregation operator that we call the multi-person – FGPOWA (MPFGPOWA) operator.

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• • • • •

S1 = Very bad economic situation. S2 = Bad economic situation. S3 = Regular economic situation. S4 = Good economic situation. S5 = Very good economic situation.

Each expert gives different opinions. The results of the available strategies, depending on the state of nature Si and the alternative Ak that the decision-maker chooses, are shown in Tables 1, 2 and 3. Step 2: In this problem, we assume the following weighting vector for the three experts: U=(0.4, 0.3, 0.3). The experts assume the following weighting vector for the FGOWA: W=(0.1, 0.1, 0.2, 0.3, 0.3). The three groups assume the following fuzzy probabilistic information for each state of nature: P=(0.1, 0.2, 0.3, 0.3, 0.1). Note that for simplicity we assume that the weights are not fuzzy so it is easier to deal with them. Note that the FOWA operator has an importance of 30% while the FPA has 70% in this particular example. Step 3: First, we aggregate the information of the three experts into one collective matrix that represents the information of all the experts of the problem. The results are shown in Table 4. Step 4: With this information, we can aggregate the expected results for each state of nature in order to make a decision. For this, we use Eq. (6) to calculate the FGPOWA aggregation. In Table 5, we present

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J.M. Merigó et al. / Fuzzy group decision-making with generalized Table 1 Decision matrix provided by expert 1

A1 A2 A3 A4

Table 6 Ranking of the strategies

S1

S2

S3

S4

S5

(30,40,50) (30,40,50) (40,50,60) (40,50,60)

(50,60,70) (20,30,40) (60,70,80) (30,40,50)

(60,70,80) (60,70,80) (70,80,90) (60,70,80)

(80,90,100) (50,60,70) (80,90,100) (60,70,80)

(90,100,110) (70,80,90) (70,80,90) (70,80,90)

Ordering FAM FPA FOWA

A3 A1 A4 A2 A3 A1 A4 A2 A3 A4 A1 A2

Ordering FPOWA A-FPA

A3 A 1 A 4 A 2 A3 A 1 A 4 A 2

7. Conclusions

S2

S3

S4

S5

(20,30,40) (30,40,50) (40,50,60) (60,70,80)

(40,50,60) (20,30,40) (60,70,80) (60,70,80)

(70,80,90) (60,70,80) (70,80,90) (50,60,70)

(80,90,100) (50,60,70) (70,80,90) (60,70,80)

(90,100,110) (80,90,100) (70,80,90) (70,80,90)

Table 3 Decision matrix provided by expert 3 S1

S2

S3

S4

S5

(40,50,60) (30,40,50) (40,50,60) (50,60,70)

(60,70,80) (20,30,40) (60,70,80) (60,70,80)

(60,70,80) (50,60,70) (70,80,90) (70,80,90)

(70,80,90) (50,60,70) (70,80,90) (60,70,80)

(90,100,110) (80,90,100) (70,80,90) (70,80,90)

Table 4 Collective results S2

S3

S4

(50,60,70) (20,30,40) (60,70,80) (48,58,68)

(63,73,83) (57,67,77) (70,80,90) (60,70,80)

(77,87,97) (50,60,70) (74,84,94) (60,70,80)

S5

TH

S1 (30,40,50) (30,40,50) (40,50,60) (49,59,69)

(90,100,110) (76,86,96) (70,80,90) (70,80,90)

AU

A1 A2 A3 A4

OR

A1 A2 A3 A4

The FGPOWA operator has been presented. It is an aggregation operator that unifies the FGPA and the FGOWA operator in the same formulation considering the degree of importance that each concept has in the aggregation. It also uses generalized means providing a general framework that includes a wide range of particular cases including the FPOWA operator and the FPOWQA operator. By using fuzzy information it is able to deal with uncertain environments where the information is vague or imprecise. We have further generalized this approach by using quasi-arithmetic means forming the Quasi-FPOWA operator. We have also studied the applicability of this new approach in a fuzzy group decision-making problem regarding the selection of general strategies in a company. We have used the MP-FGPOWA operator in order to assess the opinion of several experts in the analysis. We have seen that each particular case of FGPOWA operator may lead to different results and decisions. The main advantage of this approach in decision making is that it integrates the attitudinal character of the decision maker and the probabilistic information in the same formulation and considering the degree of importance that each concept has. Thus, it is possible to under or overestimate the neutral probabilistic information according to our attitude. Thus, the FGPOWA permits to be more risk averse or risk taker than the normal expected value. This is useful because in risky and uncertain environments we do not know what it is going to happen in the future. Therefore, it is important to consider the normal situations that should occur but also any possibility from the minimum to the maximum. Note that the applicability of the FGPOWA operator is very broad because all the previous studies that

CO

A1 A2 A3 A4

S1

PY

Table 2 Decision matrix provided by expert 2

the results obtained using different types of FGPOWA operators. Step 5: If we establish a ranking of the alternatives, we get the results shown in Table 6. Note that the first alternative is the optimal choice. Obviously, the order preference may be different depending on the aggregation operator used. Therefore, the decision about which strategy to select may be also different.

Table 5 Aggregated results by different fuzzy aggregation operators A1 A2 A3 A4

FAM

FPA

FOWA

FPOWA

A-FPA

(62,72,82) (46.6,56.6,66.6) (62.8,72.8,82.8) (57.4,67.4,77.4)

(64,74,84) (46.7,56.7,66.7) (66.2,76.2,86.2) (57.5,67.5,77.5)

(53.3,63.3,73.3) (38.3,48.3,58.3) (58.4,68.4,78.4) (54.1,64.1,74.1)

(60.79,70.79,80.79) (44.18,54.18,64.18) (63.86,73.86,83.86) (56.48,66.48,76.48)

(63.4,73.4,83.4) (46.6,56.6,66.6) (65.1,75.1,85.1) (57.4,67.4,77.4)


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Acknowledgments [19]

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We would like to thank the anonymous reviewers for valuable comments that have improved the quality of the paper. Support from the MAPFRE Foundation and project PIEF-GA-2011-300062 from the European Commission is gratefully acknowledged.

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