Fuzzy Multi-Person Decision Making with Fuzzy Probabilistic ...

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Fuzzy Multi-Person Decision Making with Fuzzy Probabilistic Aggregation Operators José M. Merigó Abstract1 We present a fuzzy multi-person decision making model with fuzzy probabilistic information. For doing so, we present the fuzzy probabilistic ordered weighted averaging (FPOWA) operator. It is an aggregation operator that unifies the fuzzy probabilistic aggregation and the fuzzy OWA (FOWA) operator in the same formulation considering the degree of importance that each concept has in the analysis. We study its applicability and we see 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 approach. We focus on a multi-person decision making problem that unifies risk and uncertain environments in the same formulation. We implement this approach in a political management problem regarding the selection of fiscal policies. Keywords: Fuzzy multi-person decision making, probabilities, OWA operator, fuzzy numbers.

1. Introduction In the literature, we find different methods for dealing with imprecise information. A very useful technique is the fuzzy number (FN) [1-2]. It permits us to represent the information considering the minimum and the maximum and the possibility that the internal values will occur. Since its introduction, it has been used in a wide range of applications [3-4]. One field of application has been the use of fuzzy techniques in all areas of decision theory [5-11]. For dealing with decision making, we can use a lot of methods. A very useful one for fuzzy decision making under uncertainty is the ordered weighted averaging (OWA) operator [12]. It is an aggregation operator that provides a parameterized family of aggregation operators between the minimum and the maximum. Since its introduction, it has been studied by a lot of authors [13-15]. For example, Merigó and Casanovas [16] Corresponding Author: José M. Merigó is with the Department of Business Administration, University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain. Email: [email protected] Manuscript received 02 Jan. 2011; revised 05 Aug. 2011; accepted 22 Aug. 2011.

and Merigó et al. [17] applied the OWA operator in Dempster-Shafer belief structure. Several approaches have considered the use of distance measures such as the OWA distance (OWAD) [18], the induced OWAD (IOWAD) [19], the linguistic OWAD (LOWAD) [20] and the induced heavy OWAD (IHOWAD) [21]. Other approaches have considered the use of interval numbers [22-23]. Zhou and Chen [24] developed a generalization by using logarithmic aggregations. Another interesting extension was developed by Reformat and Golmohammadi [25] that used the OWA operator with semantic similarities. When dealing with fuzzy information, the OWA becomes the fuzzy OWA (FOWA) operator. Its main advantage is that it can assess the OWA operator with uncertain information represented in the form of FNs. The FOWA operator has been studied by a lot of authors [26-29]. For example, Chen and Chen [30] introduced the fuzzy induced OWA (FIOWA) operator. Merigó and Casanovas [6,27] presented several generalizations by using generalized and quasi-arithmetic means. Merigó [26] presented a model with immediate probabilities. Wei [28] and Wei et al. [29] developed several extensions with intuitionistic fuzzy sets. Xu [9] and Zhao et al. [11] also presented several extensions with fuzzy information. Another interesting extension is the probabilistic OWA (POWA) operator [14]. It unifies the probabilistic aggregation and the OWA operator in the same formulation considering the degree of importance that each concept has in the analysis. Thus, we can use the attitudinal character of the decision maker and the probabilistic information of the specific problem considered. The aim of this paper is to present the fuzzy POWA (FPOWA) operator. It is an aggregation operator that unifies the fuzzy probabilistic aggregation and the FOWA operator in the same formulation taking into account the degree of relevance of each concept in the analysis. It provides a parameterized family of aggregation operators between the fuzzy minimum and the fuzzy maximum. It includes a wide range of particular cases such as the fuzzy probabilistic maximum, the fuzzy probabilistic minimum, the fuzzy average, the fuzzy probabilistic aggregation and the FOWA operator. We study the applicability of the FPOWA operator and we see that it is very broad because we can revise and

© 2011 TFSA

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extend all the previous studies that use the probability or the OWA operator with this new approach. We focus on a fuzzy multi-person decision making under risk and uncertainty. That is, we suggest a new model that unifies fuzzy decision making models under risk and under uncertainty in the same formulation and considering the degree of importance that each concept has in the formulation. By using this approach we find a new aggregation operator called the multi-person – FPOWA (MP-FPOWA) operator. Its main advantage is that it can deal with the opinion of several people in the analysis in order to better assess the uncertainty. We focus on a decision making problem in political management regarding the selection of fiscal policies. This paper is organized as follows. In Section 2 we briefly describe some basic concepts concerning FNs, the FOWA operator and the POWA operator. In Section 3 we present the FPOWA operator and some of its main families. Section 4 analyzes the applicability of the FPOWA operator focusing on an application in multi-person decision making. Section 5 summarizes the main results of the paper.

2. Preliminaries In this Section we briefly review the fuzzy numbers, the FOWA operator and the POWA operator. A. Fuzzy Numbers The FN was first introduced by [1-2]. Since then, it has been studied and applied by a lot of authors [3-4,14]. 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. Let R be (−∞, ∞), the set of all real numbers. A FN is a fuzzy subset [31] 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 (simple and interval-valued), generalized FNs (simple, interval-valued, intuitionistic and interval-valued intuitionistic), 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 ( a 2 − a1 ), (1) a (α ) = a 4 − α ( a 4 − a3 ). where α ∈ [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: ⎧ 1 if t = [ a2 , a3 ], ⎪ t − a1 if t ∈ [ a1, a2 ], ⎪a −a ⎪ 2 1 (2) m(t ) = ⎨ a − t ⎪ 4 if t ∈ [ a3 , a 4 ], ⎪ a 4 − a3 ⎪⎩ 0 otherwise, 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 ã. 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). Then: 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. Among the wide range of methods existing in the literature for ranking FNs, we recommend the methods commented by Merigó [14] such as the use of the value found in the highest membership level (α = 1). Note that other operations and ranking methods could be studied [3-4] but in this paper we use these ones. B. The Fuzzy OWA Operator The FOWA operator is an extension of the OWA operator that uses uncertain information in the arguments represented in the form of FNs. The reason for using this aggregation operator is that sometimes the available information cannot be assessed with exact numbers and it is necessary to use other techniques such as 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. Definition 1: Let Ψ be the set of FNs. A FOWA operator of dimension n is a mapping FOWA: Ψn → Ψ that has an associated weighting vector W of dimension n with wj ∈ [0, 1] and ∑nj =1 w j = 1 , such that: n

FOWA (ã1, ã2, …, ãn) = ∑ w j b j j =1

(3)

José M. Merigó: Fuzzy Multi-Person Decision Making with Fuzzy Probabilistic Aggregation Operators

where bj is the jth largest of the ãi, and the ãi are FNs. Note that sometimes, it is not clear how to reorder the arguments. Thus, it is necessary to establish a criterion for comparing FNs. For simplicity, we recommend to select the FN with the highest value in its highest membership level, usually, when α = 1. Note that if the membership level α = 1 is an interval, then, we will calculate the average of the interval. 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. C. The Probabilistic OWA Operator The POWA operator is an aggregation operator that unifies the probability and the OWA operator in the same formulation considering the degree of importance that each concept has in the analysis and providing a parameterized family of aggregation operators between the minimum and the maximum [14]. 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 =1w j = 1 , according to the following formula: n

POWA (a1, a2, …, an) = ∑ pˆ j b j j =1

(4)

where bj is the jth largest of the ai, each argument ai has an associated probability pi with ∑in=1 pi = 1 and pi ∈ [0, 1],

pˆ j = β w j + (1 − β ) p j with β ∈ [0, 1] and pj is the

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we can use subjective and objective probabilities. However, in this paper we will focus on the use of objective probabilities. It is worth noting that some previous models already considered the possibility of using the FOWA operator and the fuzzy probabilistic aggregation (fuzzy expected value) in the same formulation. The main model is the concept of fuzzy immediate probability [26,32-35]. This method is very practical for aggregating the information. However, its main disadvantage is that it can not take into account the degree of importance of each concept in the aggregation process. Note that other approaches that could be considered are those used for the weighted average because sometimes it can be seen as a probability. Thus, we can mention the hybrid averaging (HA) [36] and the weighed OWA (WOWA) operator [37]. However, in these methods we find the same disadvantage than the fuzzy immediate probability. In the following, we are going to analyze the FPOWA operator. It can be defined as follows. Definition 3: Let Ψ be the set of FNs. A FPOWA operator of dimension n is a mapping FPOWA: Ψn → Ψ that has associated a weighting vector W of dimension n such ~ ∈ [0, 1] and ∑n w ~ that w , according to the j j =1 j = 1 following formula: n

FPOWA (ã1, ã2, …, ãn) = ∑ pˆ j b j j =1

(5)

where bj is the jth largest of the ãi, the ãi are FNs and each one has an associated probability ~ pi with ~ n ~ ~ + (1 − β~) ~ pi ∈ [0, 1], pˆ j = β w pj ∑i =1 pi = 1 and ~ j ~ p is the probability ~ p orwith β ∈ [0, 1] and ~

probability pi ordered according to bj, that is, according to j i the jth largest of the ai. dered according to bj, that is, according to the jth largest Note that if β = 0, we get the probabilistic aggregation, of the ãi. and if β = 1, we get the OWA operator. The FPOWA operator can also be formulated separating the part that strictly affects the OWA operator and the part that affects the probabilistic aggregation. This rep3. The Fuzzy Probabilistic OWA Operator resentation is useful to see both models in the same forIn this section we present the FPOWA operator and mulation but it does not seem to be as a unique equation that unifies both models. study some of its main families. Definition 4: Let Ψ be the set of FNs. A FPOWA operaA. Main Concepts n The fuzzy probabilistic ordered weighted averaging tor is a mapping FPOWA: Ψ → Ψ of dimension n, if it n ~ (FPOWA) operator is an aggregation operator that pro- has associated a weighting vector W, with ∑ j =1 w j = 1 vides a parameterized family of aggregation operators and w ~ ∈ [0, 1] and a probabilistic vector P, with j between the fuzzy maximum and the fuzzy minimum. Its n pi = 1 and ~ pi ∈ [0, 1], such that: main advantage is that it can unify the fuzzy probabilis- ∑i =1 ~ tic aggregation and the FOWA operator in the same for~ n ~ ~ n~~ mulation and considering the degree of importance that FPOWA (ã1, …, ãn) = β ∑ w j b j + (1 − β ) ∑ pi ai (6) j =1 i =1 each concept has in the aggregation. Thus, we can use probabilistic information and the attitudinal character of where bj is the jth largest of the arguments ãi, the ãi are the decision maker in the same formulation. Note that FNs and β~ ∈ [0, 1].

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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. Thus, it is very common ~ ≠ 1 and P = ∑n ~ that W = ∑nj =1 w j i =1 pi ≠ 1 . A very practical approach for assessing these problems is by using: ~ n ~ n ~ b + (1 − β ) ∑ ~ f (ã1, ã2, …, ãn) = β ∑ w pi a~i (7) j j W j =1

P

subjective probabilistic bounds. Note that in this paper we focus on an objective perspective. This property can be proved with the following theorem. Theorem 1 (Semi-boundary conditions): Assume f is the FPOWA operator, then: n ~ ~ β × Min{ãi} + (1 − β ) × ∑ ~ pi a~i ≤ f (ã1, ã2, …, ãn) ≤ i =1

~

~

n

β × Max{ãi} + (1 − β ) × ∑ ~ pi a~i

(9) i =1 Different types of FNs could be used in the aggregation including triangular FNs (TFNs) and trapezoidal Proof: Let max{ãi} = c, and min{ãi} = d, then FNs (TpFNs). Note that sometimes, the use of the FNs ~ n ~ ~ ~ n ~ b j + (1 − β ) ∑ ~ pi ai ≤ f (ã1, ã2, …, ãn) = β ∑ w j implies difficulties in the ranking of the arguments bei =1 j =1 cause it is not clear which FN is higher. For simplicity, ~ n ~ ~ n~~ ~ n ~ ~ n~~ we recommend the following criteria. β ∑ w j c + (1 − β ) ∑ pi ai = β c ∑ w j + (1 − β ) ∑ pi ai Step 1: Analyze if there is an order between the FNs. i =1 i =1 j =1 j =1 That is, if all the values of the interval A = (a1, a2, a3) are (10) higher than the values in the interval C = (c1, c2, c3) such n n ~ ~ b + (1 − β~) ∑ ~ pi a~i ≥ f (ã1, ã2, …, ãn) = β ∑ w that a1 > c3. j j = 1 i = 1 j Step 2: If not, we select the FN with highest result n n when α = 1. ~ ~ ~~ ~ n ~ ~ n~~ ~ Step 3: If there is a tie in α = 1, we calculate the aver- β ∑ w j d + (1 − β ) ∑ pi ai = β d ∑ w j + (1 − β ) ∑ pi ai i =1 i =1 j =1 j =1 age of the bounds of the FN. If it is an interval, we cal(11) culate the average of the interval. In the case of tie, we n ~ will select the interval with the lowest increment (a3 − Since ∑ j =1 w j = 1, we get a1). ~ ~ n ~ From a generalized perspective of the reordering step, p i ai f (ã1, ã2, …, ãn) ≤ β c + (1 − β ) ∑ ~ (12) i =1 we can distinguish between the descending FPOWA (DFPOWA) and the ascending FPOWA (AFPOWA) op~ ~ n ~ pi ai f (ã1, ã2, …, ãn) ≥ β d + (1 − β ) ∑ ~ (13) erator by using pˆ j = pˆ * n − j +1 , where pˆ j is the jth i =1 i =1

weight of the DFPOWA and pˆ * n − j +1 the jth weight of

Therefore, n ~ ~ β × Min{ãi} + (1 − β ) × ∑ ~ pi a~i ≤ f (ã1, ã2, …, ãn) ≤

the AFPOWA operator. Note that this reordering process is particularly interesting when analyzing the OWA part i =1 of the aggregation. n ~ ~ ~ T β × Max{ ã } + ■ − × β pi a~i ( 1 ) ∑ i If B is a vector of the ordered arguments bj and P is i =1 the transpose of the weighting vector, then, the FPOWA Following the ideas of [38-39], we can develop a genoperator can be expressed as: erating function for the arguments of the FPOWA operaFPOWA (ã1, ã2, …, ãn) = PT B (8) tor that represents the internal formation of this informaThe FPOWA is monotonic, bounded and idempotent. tion, such as the use of a multi-person process where It is monotonic because if ãi ≥ ũi, for all ãi, then, FPOWA each argument is constituted by the opinion of m persons. (ã1, ã2, …, ãn) ≥ FPOWA (ũ1, ũ2, …, ũn). It is idempotent Moreover, we will also use a weighting function f for the because if ãi = ã, for all ãi, then, FPOWA (ã1, ã2, …, ãn) weighting vector. We call this formulation the mixture = ã. It is bounded because the FPOWA aggregation is FPOWA (MFPOWA) operator and it is defined as foldelimitated by the minimum and the maximum: Min{ãi} lows. Definition 5: A MFPOWA operator of dimension n is a ≤ FPOWA (ã1, ã2, …, ãn) ≤ Max{ãi}. n Additionally, the FPOWA operator also presents more mapping MFPOWA: Ψ → Ψ that has associated a vecm specific bounds by using the probability. Thus, we can tor of weighting functions f, s: Ψ → Ψ, such that: n consider semi-boundary conditions where we use the ∑ f j ( s y (b j ))s y (b j ) maximum and the minimum with the probability. That is, f (sy(ã1),…, sy(ãn)) = j =1 (14) we consider the bounds subject to the available probn ∑ f j ( s y (b j )) abilistic information. Therefore, we form probabilistic j =1 bounds that can be objective probabilistic bounds or

José M. Merigó: Fuzzy Multi-Person Decision Making with Fuzzy Probabilistic Aggregation Operators

where sy(bj) is the jth largest of the sy(ãi), ãi is the argument variable, and y indicates that each argument is formed using a different function. 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 [12,40-41] we could formulate the orness measure (attitudinal character), the entropy of dispersion, the divergence of W and the balance operator. The orness measure of the FPOWA operator is formulated as follows: ⎛n− j⎞ ~ n ~ ⎛n− j⎞ ~ n α ( Pˆ ) = β ∑ w pj⎜ ⎟ (15) ⎟ + (1 − β ) ∑ ~ j⎜ ⎝ n −1⎠ ⎝ n −1⎠ j =1 j =1 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 FOWA operator ~ and if β = 0, the orness measure of the fuzzy probabilistic aggregation. The entropy of dispersion [12] measures the amount of information being used in the aggregation. For the FPOWA operator, it is defined as follows. n ⎛~ n ⎞ ~ ln( w ~ ) + (1 − β~) ∑ ~ H ( Pˆ ) = −⎜ β ∑ w pi ln( ~ pi ) ⎟ (16) j j ⎜ j =1 ⎟ i =1 ⎝ ⎠ ~ Note that pi is the ith weight of the FPA aggregation. ~ As we can see, if β = 1, we get the entropy of dispersion ~ of the FOWA operator and if β = 0, we extend the classical Shannon entropy by using fuzzy probabilities [42]. The divergence of W [41] measures the divergence of the weights against the degree of or-ness. It is defined as follows. 2 ~⎛ n ~ ⎛ n − j ~ ⎞ ⎞⎟ Div( Pˆ ) = β ⎜ ∑ w − α ( W ) + ⎜ ⎟ ⎜ j =1 j ⎝ n − 1 ⎟ ⎠ ⎝ ⎠ 2 ⎛ ⎛n− j ~ n ~⎞ ⎞ + (1 − β )⎜ ∑ ~ − α ( P )⎟ ⎟ pj⎜ ⎜ j =1 ⎝ n − 1 ⎠ ⎟⎠ ⎝

(17)

The balance operator [40] measures the balance of the weights against the orness or the andness. For the FPOWA operator, it is formulated as follows. ~⎛ n ~ ⎛ n + 1 − 2 j ⎞ ⎞⎟ Bal ( Pˆ ) = β ⎜ ∑ w + ⎜ j =1 j ⎜⎝ n − 1 ⎟⎠ ⎟ ⎝ ⎠ ⎛ n + 1 − 2 j ⎞ ⎞⎟ ~⎛ n (18) pi ⎜ + (1 − β )⎜ ∑ ~ ⎜ j =1 ⎝ n − 1 ⎟⎠ ⎟ ⎝ ⎠ ~ If β = 1, we get the balance operator by using FNs and ~ if β = 0, we obtain the balance operator of the fuzzy probabilistic aggregation. As we can see, Bal (Pˆ ) ∈ [−1, 1].

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B. Families of FPOWA Operators Different types of FPOWA operators can be studied by using different types of weighting vector Pˆ (W or P). Each case is a particular attitude of the decision maker that is useful in some specific situations. For example, we can form the fuzzy probabilistic maximum, the fuzzy probabilistic minimum, the fuzzy arithmetic probabilistic aggregation (FAPA) and the fuzzy arithmetic OWA (FA-OWA) operator. Remark 1: The fuzzy probabilistic maximum is found when w1 = 1 and wj = 0 for all j ≠ 1. The fuzzy probabilistic minimum is formed when wn = 1 and wj = 0 for all j ≠ n. More generally, the step-FPOWA is formed when wk = 1 and wj = 0 for all j ≠ k. Remark 2: The fuzzy maximum is formed when pˆ1 = 1 and pˆ j = 0 for all j ≠ 1. The fuzzy minimum is obtained when pˆ n = 1 and pˆ j = 0 for all j ≠ n. Note that several interpretations are possible for these situations. One interpretation is that β = 0 because otherwise we should accept that one probability is 1 and this implies certainty. Another interpretation is to assume that we have some probabilistic information but we do not strongly believe on it. Therefore, we can accept situations with one probability equal 1 but still be under uncertainty. Remark 3: The FAPA operator is obtained when wj = 1/n for all j. That is: 1~ ~ n ~ pi ai FA-PA (ã1, ã2, …, ãn) = β a~i + (1 − β ) ∑ ~ (19) n i =1 Remark 4: The FA-OWA operator is formed when pi = 1/n for all i. In this case, we get: ~ 1~ ~ n ~ FA-OWA (ã1, ã2, …, ãn) = β ∑ w j b j + (1 − β ) ai (20) n j =1 Note that if w1 = 1 and wj = 0 for all j ≠ 1, the FA-OWA operator becomes the FA-Max that it is also known in the literature as the or-like S-FOWA operator and if wn = 1 and wj = 0 for all j ≠ n, it becomes the FA-Min that it is known as the and-like S-FOWA operator [14,43]. Remark 5: Other particular types of FPOWA operators could be formed following the recent literature for obtaining OWA weights [14,43-44]. For example, we can consider the following ones: • The fuzzy arithmetic mean: If wj = 1/n for all j and pi = 1/n for all i. Note that it could also be found by using pˆ j = 1/ n for all j.

• The median-FPOWA: If n is odd we assign pˆ ( n +1) / 2 = 1 and pˆ j * = 0 for all others. If n is even we assign for example, pˆ n / 2 = pˆ ( n / 2) +1 = 0.5 and pˆ j * = 0 for all others. Note that it is possible to con-

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sider a similar model by only focusing on the weights of the OWA operator. • The olympic-FPOWA: Focussing only on the OWA we could use w1 = wn = 0, and for all others wj* = 1/(n − 2). For the whole FPOWA we have pˆ 1 = pˆ n = 0 , and for all others pˆ j * = 1/( n − 2) .

• The general olympic-FPOWA: If pˆ j = 0 for j = 1, 2, …, k, n, n − 1, …, n − k + 1, and for all others pˆ j * = 1/( n − 2k ) , where k < n/2. • Contrary of the general olympic-FPOWA: When pˆ j = (1/ 2k ) for j = 1, 2, …, k, n, n − 1, …, n − k + 1, and pˆ j = 0 , for all other values, where k < n/2. • Centered-FPOWA: If it is symmetric, strongly decaying and inclusive. It is symmetric if pˆ j = pˆ n − j +1 . It is strongly decaying when i < j ≤ (n + 1)/2 then pˆ i < pˆ j and when i > j ≥ (n + 1)/2 then pˆ i < pˆ j . And it is inclusive if pˆ j > 0 .

Remark 6: Moreover, it is possible to consider more families of FPOWA operators by using them both in the probabilities and in the FOWA operator or separately. Furthermore, we can also consider different types for each case. Remark 7: Additionally, it is also possible to study partial cases by considering one part of the fuzzy interval as a type of aggregation and the other part as another type. For example, we could consider a fuzzy partial maximum by using w1 = (0.8, 1), w2 = (0, 0.1) and wj = 0, for all j ≠ 1 and 2. As we can see, it is not strictly the fuzzy ~ maximum. Thus, α ( P ) < 1.

4. Application in Fuzzy Multi-Person Decision Making In this section, we study the applicability of the FPOWA operator focussing on a fuzzy multi-person decision making problem. First, we present a general overview. Second, we analyse the decision making approach obtaining the MP-FPOWA operator. And third, we develop an illustrative example. A. Introduction The FPOWA operator can be applied in a wide range of disciplines because all the previous studies that use the probability or the OWA operator can be revised and extended with this new approach. For example, we could develop a wide range of applications in statistics, soft computing, economics, engineering and decision theory. Focussing in fuzzy decision theory, first, it is worth noting that there are a lot of methodologies for doing so

including fuzzy multiple criteria decision making, fuzzy group decision making and fuzzy game theory. When making decisions, the information may present different degrees of uncertainty. Note that when using FNs, we assume that the information is imprecise because we cannot assess it with exact numbers. In general terms, we can distinguish between 3 forms of fuzzy decision making environments: • Fuzzy decision making under certainty: We know the information that is going to happen in the future. However, the available information is represented with FNs. • Fuzzy decision making under risk: We know the possible outcomes by using FNs but we do not know which of them is going to happen in the future. However, we can assess the information with fuzzy probabilities. • Fuzzy decision making under uncertainty: The possible outcomes by using FNs are known but we do not know which of them is going to occur in the future and we do not have any probabilistic information. This general framework has been extended by a lot of authors in different ways. For example, considering the new developments presented in this paper, we can analyze the use of fuzzy decision making problems under risk and under uncertainty in the same formulation. As we have mentioned before, fuzzy decision making under risk is usually assessed with fuzzy probabilities and fuzzy decision making under uncertainty with the FOWA operator. Therefore, with the introduction of the FPOWA operator, we can assess these two problems in the same formulation and considering the degree of importance that each concept has in the analysis. Thus, with the FPOWA operator we can formulate a new decision making approach: • Fuzzy decision making under risk and uncertainty A. If β = 1, we get fuzzy decision making under uncertainty. B. If β = 0, we get fuzzy decision making under risk. Additionally, when dealing with the FPOWA operator we can use subjective and objective probabilities in the analysis. Thus, we can formulate the following decision making approaches: • Fuzzy decision making under subjective risk and uncertainty A. If β = 0, we get fuzzy decision making under subjective risk. • Fuzzy decision making under objective risk and uncertainty A. If β = 0, we get fuzzy decision making under objective risk. A further interesting issue to consider is the meaning of the set of arguments aggregated in the fuzzy decision making process because this may lead to different inter-

José M. Merigó: Fuzzy Multi-Person Decision Making with Fuzzy Probabilistic Aggregation Operators

pretations of the information. In decision making problems it is very interesting to consider a set of arguments a that depend on a set of states of nature S and a set of alternatives A. This information can be represented in the following matrix shown in Table 1. Table 1. Matrix with states of nature and alternatives. A1 … Ah … Ak FPOWA

S1 ã11 … ãh1 … ãk1 Y1

… … … … … …

Si ã1i … ãhi … ãki Yi

… … … … … …

Sn ã1n … ãhn … ãkn Yn

FPOWA T1 … Th … Tk

As we can see, we can aggregate the arguments in different ways. In general, we can summarize the problem in three types of fuzzy decision-making methodologies: • Fuzzy decision making “ex-ante”: We select an action and see its potential results (aggregation of a row). • Fuzzy decision making “ex-post”: We assume that a state of nature occurs and see how we can react (aggregation of a column). • Fuzzy decision making “ex-ante” and “ex-post”: We mix both cases in the same fuzzy decision making process. Next, we could consider a wide range of fuzzy decision making (FDM) approaches by mixing these concepts with the previous ones. Thus, we could get the following models shown in Table 2. Note that these approaches could also be studied with a multi-person analysis or more generally with a multi-aggregation process. Thus, we get: • Fuzzy multi-person decision making under certainty. • Fuzzy multi-person decision making under risk. A. Fuzzy multi-person decision making under objective risk. B. Fuzzy multi-person decision making under subjective risk. • Fuzzy multi-person decision making under uncertainty. • Fuzzy multi-person decision making under risk and uncertainty. A. Fuzzy multi-person decision making under objective risk and uncertainty. B. Fuzzy multi-person decision making under subjective risk and uncertainty. Finally, note that there are a lot of other methods and techniques for dealing with fuzzy decision making. Therefore, we can construct a lot of other decision-making approaches as it has been constructed in the previous analysis.

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B. Fuzzy Decision Making Approach In this paper, we consider a fuzzy decision making application in political management concerning national decision making. We study the selection of fiscal policies in a country. For doing so, we use a multi-person analysis as these decisions are usually conditioned by the opinion of a lot of persons such as in the parliament. The process to follow for the selection of fiscal policies with the FPOWA operator in fuzzy multi-person decision making is described as follows. Note that many other decision-making models have been discussed in the literature [45-55]. 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, …, eq} be a finite set of decision makers. Let U = (u~1, u~2 ,..., u~q ) be the weighting vector of the decision makers such that ∑q u~ = 1 and u~ ∈ [0, 1]. k

k =1 k

Each decision maker provides his own payoff matrix (ãhi(k))m×n. Step 2: Calculate the weighting vector ~ ~ Pˆ = β × W + (1 − β ) × P to be used in the FPOWA ag~ ,w ~ ~ gregation. Note that W = ( w 1 2 ,..., wn ) such that ~ = 1 and w ~ ∈ [0, 1] and P = ( ~ p ,~ p ,..., ~ p ) ∑n w j =1

j

j

1

2

n

such that ∑in=1 ~ pi = 1 and ~ pi ∈ [0, 1]. Step 3: Use the WA to aggregate the information of the decision makers E using the weighting vector U. The result is the collective payoff matrix (ãhi)m×n. Thus, ãhi = k ∑q u~ a~ . Note that we can also use other types of k =1 k hi

aggregation operators instead of the WA to aggregate this information. Step 4: Calculate the aggregated results using the FPOWA operator explained in Eq. (5) and consider different families of FPOWA operators as described in Section 3. Step 5: Adopt decisions according to the results found in the previous steps. Select the alternative (s) that provides the best result (s). Moreover, establish a ranking of the alternatives from the most- to the least-preferred alternative, enabling consideration of more than one selection. This process can be summarized by using the following aggregation operator that we call the multi-person – FPOWA (MP-FPOWA) operator. Definition 6: Let Ψ be the set of FNs. A MP-FPOWA operator is a mapping MP-FPOWA: Ψn × Ψp → Ψ that has a weighting vector U of dimension p with ∑qk =1u~q = 1 and u~k ∈ [0, 1] and a weighting vector W ~ = 1 and w ~ ∈ [0, 1], of dimension n with ∑n w j =1

j

j

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such that: n

MP-FPOWA ((ã11, …, ã1p), …, (ãn1, …, ãnp)) = ∑ pˆ j b j j =1

(21) where bj is the jth largest of the ãi, each argument ãi is a FN and has an associated probability ~ pi with ~ ~ ~ n ~ ~ p j with ∑i =1 pi = 1 and pi ∈ [0, 1], pˆ j = β w j + (1 − β ) ~ ~ β ∈ [0, 1] and ~ p is the probability ~ p ordered ac-

that the government selects, are presented in Tables 3, 4 and 5. The experts assume the following weighting vector for the three group of experts: U = (0.3, 0.3, 0.4). They assume the following weighting vector for the FOWA: W = (0.1, 0.1, 0.2, 0.3, 0.3); for the fuzzy probability: P =

~

(0.1, 0.2, 0.2, 0.2, 0.3); and β = 30%. First, we aggregate the information of the three groups into one collective matrix that represents all the experts of the problem. i j The results are shown in Table 6. cording to bj, that is, according to the jth largest of the ãi, Next, we aggregate the information obtaining the exa~i = ∑ qk =1u~k a~ik and a~ik is the argument variable pro- pected results for each state of nature in order to make a decision by using Eq. (5). In Table 7 and 8, we present vided by each person. The MP-FPOWA operator has similar properties than the results obtained using different FPOWA operators. With this information, we can rank the alternatives the FPOWA operator such as the distinction between from the most preferred to the less preferred. The results descending and ascending orders, the aggregation with are shown in Table 9. fuzzy weights, and so on. Note that depending on the particular type of FPOWA The MP-FPOWA operator includes a wide range of particular cases following the methodology explained in operator used, the results may lead to different decisions. Section 3.2. Thus, it includes the multi-person – FPA In this example, it seems clear that A5 is the optimal (MP-FPA) operator, the multi-person – FOWA choice. (MP-FOWA) operator, the multi-person – fuzzy average 5. Conclusions (MP-FA) operator, the multi-person – fuzzy arithmetic-PA (MP-FAPA) operator and the multi-person – fuzzy We have presented the FPOWA operator. It is an agarithmetic-OWA (MP-FAOWA) operator. gregation operator that provides a parameterized family C. Illustrative Example Next, we analyze an illustrative example of the new approach in a fuzzy multi-person decision-making problem regarding political management. We study a country that is planning its fiscal policy for the next year. Step 1: Assume the government of a country has to decide on the type of optimal fiscal policy for the next year. They consider five alternatives: • A1 = Develop a strong expansive fiscal policy. • A2 = Develop an expansive fiscal policy. • A3 = Do not make any change in the fiscal policy. • A4 = Develop a contractive fiscal policy. • A5 = Develop a strong contractive fiscal policy. In order to analyze these fiscal policies, the government has brought together a group of experts. This group considers 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 occur in the future: • S1 = Very bad economic situation. • S2 = Bad economic situation. • S3 = Regular economic situation. • S4 = Good economic situation. • S5 = Very good economic situation. The experts are classified in 3 groups. Each group gives different opinions than the other two. The results depending on the state of nature Si and the alternative Ak

of aggregation operators between the fuzzy minimum and the maximum. Its main advantage is that it unifies the fuzzy probabilities and the FOWA operator in the same formulation considering the degree of importance that each concept has in the aggregation. Moreover, it deals with environments where the information is imprecise and cannot be represented with precise numbers but it is possible to use FNs. Thus, we can consider the minimum and the maximum of the imprecise information and the possibility that the internal values will occur. We have studied several properties and particular cases such as the FOWA operator, the fuzzy probabilistic aggregation, the fuzzy probabilistic maximum and the fuzzy probabilistic minimum. We have also studied the applicability of the FPOWA operator in a fuzzy multi-person decision making. This approach permits to unify fuzzy decision making under risk and under uncertainty in the same formulation and considering the degree of importance of each case in the specific problem considered. We have also seen that this approach permits to represent fuzzy decision making problems “ex-ante” and “ex-post”. Moreover, by using a multi-person analysis, we have obtained the MPFPOWA operator. It is an aggregation operator that it is able to deal with the opinion of several persons in the analysis. Finally, we have also developed an illustrative example in political management regarding the selection of fiscal policies.

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Table 2. Fuzzy decision making approaches. FDM under certainty

FDM under risk

FDM – uncertainty

FDM “ex-ante”

FDM under certainty “ex-ante”

FDM under risk “ex-ante”

FDM – uncertainty “ex-ante”

FDM “ex-post”

FDM under certainty “ex-post”

FDM under risk “ex-post”

FDM – uncertainty “ex-post”

FDM “ex-ante” and “ex-post”

FDM under certainty “ex-ante” and “ex-post”

FDM under risk “ex-ante” and “ex-post”

FDM – uncertainty “ex-ante” and “ex-post”

Table 3. Expert 1. A1 A2 A3 A4 A5

S1 (60,70,80) (40,50,60) (30,40,50) (80,90,100) (20,30,40)

S2 (70,80,90) (30,40,50) (50,60,70) (70,80,90) (60,70,80)

S3 (40,50,60) (60,70,80) (50,60,70) (40,50,60) (70,80,90)

S4 (50,60,70) (70,80,90) (60,70,80) (40,50,60) (50,60,70)

S5 (60,70,80) (60,70,80) (70,80,90) (30,40,50) (60,70,80)

Table 4. Expert 2. A1 A2 A3 A4 A5

S1 (40,50,60) (40,50,60) (60,70,80) (40,50,60) (70,80,90)

S2 (60,70,80) (30,40,50) (50,60,70) (30,40,50) (60,70,80)

S3 (50,60,70) (40,50,60) (40,50,60) (40,50,60) (50,60,70)

S4 (50,60,70) (70,80,90) (80,90,100) (40,50,60) (50,60,70)

S5 (60,70,80) (60,70,80) (60,70,80) (70,80,90) (60,70,80)

Table 5. Expert 3. A1 A2 A3 A4 A5

S1 (30,40,50) (40,50,60) (20,30,40) (10,20,30) (50,60,70)

S2 (40,50,60) (70,80,90) (50,60,70) (20,30,40) (60,70,80)

S3 (30,40,50) (60,70,80) (50,60,70) (40,50,60) (60,70,80)

S4 (60,70,80) (30,40,50) (60,70,80) (40,50,60) (50,60,70)

S5 (60,70,80) (60,70,80) (40,50,60) (80,90,100) (60,70,80)

S4 (54,64,74) (54,64,74) (66,76,86) (40,50,60) (50,60,70)

S5 (60,70,80) (60,70,80) (55,65,75) (62,72,82) (60,70,80)

Table 6. Collective results. A1 A2 A3 A4 A5

S1 (42,52,62) (40,50,60) (35,45,55) (40,50,60) (47,57,67)

S2 (55,65,75) (46,56,66) (50,60,70) (38,48,58) (60,70,80)

S3 (39,49,59) (54,64,74) (47,57,67) (40,50,60) (60,70,80)

Table 7. Aggregated results 1. A1 A2 A3 A4 A5

FMin (39,49,59) (40,50,60) (35,45,55) (38,48,58) (47,57,67)

Min-FPA (47.9,57.9,67.9) (48.9,58.9,68.9) (47.3,57.3,67.3) (43.7,53.7,63.7) (53.7,63.7,73.7)

Max-FPA (54.2,64.2,74.2) (54.9,64.9,74.9) (56.6,66.6,76.6) (50.9,60.9,70.9) (57.6,67.6,77.6)

FMax (60,70,80) (60,70,80) (66,76,86) (62,72,82) (60,70,80)

Table 8. Aggregated results 2. A1 A2 A3 A4 A5

FA (50,60,70) (50.8,60.8,70.8) (50.6,60.6,70.6) (44,54,64) (55.4,65.4,75.4)

FPA (51.8,61.8,71.8) (52.8,62.8,72.8) (52.6,62.6,72.6) (46.2,56.2,66.2) (56.7,66.7,76.7)

FOWA (46.6,56.6,66.6) (48,58,68) (46.7,56.7,66.7) (41.6,51.6,61.6) (53.1,63.1,73.1)

FPOWA (50.2,60.2,70.2) (51.3,61.3,71.3) (50.8,60.8,70.8) (44.8,54.8,64.8) (55.6,65.6,75.6)

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Table 9. Ordering of the policies. FMin Min-FPA Max-FPA FMax FA FPA FOWA FPOWA

Ordering A5⎬A2⎬A1⎬A4⎬A3 A5⎬A2⎬A1⎬A3⎬A6 A5⎬A3⎬A1⎬A2⎬A4 A3⎬A4⎬A1=A2=A5 A5⎬A2=A3⎬A1⎬A4 A5⎬A2⎬A3⎬A1⎬A4 A5⎬A2⎬A3⎬A1⎬A4 A5⎬A2⎬A3⎬A1⎬A4

In future research, we expect to develop further improvements by adding new concepts in the analysis such as the use of generalized, induced and unified aggregation operators. We will also consider other applications giving special attention in decision theory and statistics.

Acknowledgment We would like to thank the associate editor and the anonymous referees for valuable comments that have improved the quality of the paper. Support from the project 099311 from the University of Barcelona is gratefully acknowledged.

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[48] J. M. Merigó, “The uncertain probabilistic Making and Uncertainty. weighted average and its application in the theory of expertons,” African Journal of Business Management, vol. 5, no. 15, pp. 6092-6102, 2011. [49] J. M. Merigó, “A unified model between the weighted average and the induced OWA operator,” Expert Systems with Applications, vol. 38, no. 9, pp. 11560-11572, 2011. [50] J. M. Merigó and A. M. Gil-Lafuente, “OWA operators in human resource management,” Economic Computation and Economic Cybernetics Studies and Research, vol. 45, no. 2, pp. 153-168, 2011. [51] J. M. Merigó, A. M. Gil-Lafuente, and J. Gil-Aluja, “Decision making with the induced generalized adequacy coefficient,” Applied and Computational Mathematics, vol. 2, no. 2, pp. 321-339, 2011. [52] J. M. Merigó and G. W. Wei, “Probabilistic aggregation operators and their application in uncertain multi-person decision making,” Technological and Economic Development of Economy, vol. 17, no. 2, pp. 335-351, 2011. [53] G. W. Wei, “Uncertain linguistic hybrid mean operator and its application to group decision making under uncertain linguistic environment,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 17, no. 2, pp. 251-267, 2009. [54] Z. S. Xu, “A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making,” Group Decision and Negotiation, vol. 19, no. 1, pp. 57-76, 2010. [55] X. Zhang and P. Liu, “Method for multiple attribute decision-making under risk with interval numbers,” International Journal of Fuzzy Systems, vol. 12, no. 3, pp. 237-242, 2010. José M. Merigó has a MSc and a PhD degree in Business Administration obtained in 2003 and 2009, respectively, from University of Barcelona, Spain. His PhD obtained the Extraordinary Award from the University of Barcelona. He also holds a Bachelor Degree in Economics and a Master in European Business Administration and Business Law from Lund University, Sweden. He is an Assistant Professor in the Department of Business Administration at the University of Barcelona. He has published more than 150 papers in journals, books and conference proceedings including journals such as Information Sciences, International Journal of Intelligent Systems, Expert Systems with Applications, International Journal of Information Technology and Decision Making, International Journal of Computational Intelligence Systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems and Computers & Industrial Engineering. He is currently interested in Aggregation Operators, Decision