Using fuzzy numbers and OWA operators in the weighted average and its application in decision making

1 USING FUZZY NUMBERS AND OWA OPERATORS IN THE WEIGHTED AVERAGE AND ITS APPLICATION IN DECISION MAKING José M. Merigó Lindahl, [email protected], University of Barcelona Montserrat Casanovas Ramón, mcas[email protected], University of Barcelona RESUMEN Se presenta un nuevo método para tratar situaciones de incertidumbre en los que se utiliza el operador OWAWA (media ponderada – media ponderada ordenada). A este operador se le denomina operador OWAWA borroso (FOWAWA). Su principal ventaja se encuentra en la posibilidad de representar la información incierta del problema mediante el uso de números borrosos los cuales permiten una mejor representación de la información ya que consideran el mínimo y el máximo resultado posible y la posibilidad de ocurrencia de los valores internos. Se estudian diferentes propiedades y casos particulares de este nuevo modelo. También se analiza la aplicabilidad de este operador y se desarrolla un ejemplo numérico sobre toma de decisiones en la selección de políticas fiscales. Palabras clave: Toma de decisiones; Operador OWA; Media ponderada; Números borrosos; Selección de políticas fiscales. ABSTRACT We present a new approach for dealing with an uncertain environment when using the ordered weighted averaging – weighted averaging (OWAWA) operator. We call it the fuzzy OWAWA (FOWAWA) operator. The main advantage of this new aggregation operator is that it is able to represent the uncertain information with fuzzy numbers. Thus, we are able to give more complete information because we can consider the maximum and the minimum of the problem and the internal information between these two results. We study different properties and different particular cases of this approach. We also analyze the applicability of the new model and we develop a numerical example in a decision making problem about selection of fiscal policies. Keywords: Decision making; OWA operator; Weighted average; Fuzzy numbers; Selection of fiscal policies. 2 1. INTRODUCTION The weighted average (WA) is one of the most common aggregation operators found in the literature. It can be used in a wide range of different problems including statistics, economics, engineering, etc. Another interesting aggregation operator is the ordered weighted averaging (OWA) operator (Yager, 1988). The OWA operator provides a parameterized family of aggregation operators that range from the maximum to the minimum. For further reading on the OWA operator and some of its applications, refer to (Beliakov et al., 2007; Calvo et al., 2002; Casanovas and Merigó, 2007; Merigó, 2008; Merigó and Casanovas, 2007; 2008a; 2008b; Merigó and Gil-Lafuente, 2009; Sadiq and Tesfamariam, 2008; Torra, 1997; Torra and Narukawa, 2007; Xu, 2006; 2007; Xu and Da, 2003; Xu and Yager, 2008; Yager, 1993; 1996; 2007; Yager and Filev, 1994; Yager and Kacprzyk, 1997). Usually, when using these approaches it is considered that the available information are exact numbers. However, this may not be the real situation found in the specific problem considered. Sometimes, the available information is vague or imprecise and it is not possible to analyze it 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 (FNs). In order to develop the fuzzy approach, we will follow the ideas of (Chang and Zadeh, 1972; Dubois and Prade, 1980; Kaufmann and Gil-Aluja, 1987; Kaufmann and Gupta, 1985). Note that in the literature, there are a lot of studies dealing with uncertain information represented in the form of FNs in different problems such as (Casanovas and Merigó, 2007; Chen and Chen, 2003; 2008; Kaufmann and Gil-Aluja, 1987; Merigó, 2008; Merigó and Casanovas, 2007; 2008a; Sadiq and Tesfamariam, 2008; Xu, 2007; Xu and Yager, 2008). Recently, some authors have tried to unify the WA and the OWA in the same formulation. It is worth noting the work developed by Torra (1997) with the introduction of the weighted OWA (WOWA) operator and the work of Xu and Da (2003) about the hybrid averaging (HA) operator. Both models arrived to a unification between the OWA and the WA because both concepts were included in the formulation as particular cases. However, as it has been studied in (Merigó, 2008), these models seem to be a partial unification but not a real one because they can unify them but they cannot consider how relevant these concepts are in the specific problem considered. For example, in some problems we may prefer to give more importance to the OWA operator because we believe that it is more relevant and vice versa. This problem is solved with the ordered weighted averaging – weighted averaging (OWAWA) operator (Merigó, 2008). In this paper, we present a new approach to unify the OWA operator with the WA when the available information is uncertain and can be assessed with FNs. We call it the fuzzy ordered weighted averaging – weighted averaging (FOWAWA) operator. The main advantage of this approach is that it unifies the OWA and the WA taking into account the degree of importance of each case in the formulation and considering that the information is given with FNs. Thus, we are able to consider situations where we give more or less importance to the FOWA and the FWA depending on our interests and the problem analysed. We study different properties of the FOWAWA operator and different particular cases. Moreover, we are also able to unify the fuzzy arithmetic mean (or simple fuzzy average) with the FOWA operator when the weights of the FWA are equal. We study other families such as the step-FOWAWA, the median-FOWAWA, the olympic-FOWAWA, the S- FOWAWA, the centered-FOWAWA, etc. 3 We also analyze the applicability of the new approach and we see that it is possible to develop an astonishingly wide range of applications. For example, we can apply it in a lot of problems about statistics, economics, engineering, decision theory, etc. In this paper, we focus on a decision making problem about selection of fiscal policies. The main advantage of the FOWAWA in these problems is that it is possible to consider the subjective probability (or the degree of importance) and the attitudinal character of the decision maker at the same time. This paper is organized as follows. In Section 2 we revise the FNs, the FWA and the FOWA operator. In Section 3 we present the new approach. Section 4 analyzes different families of FOWAWA operators. In Section 5 we analyze the applicability of the new approach in a decision making problem. Section 6 presents a numerical example and in Section 7 we summarize the main conclusions of the paper. 2. PRELIMINARIES In this Section, we briefly describe some basic concepts to be used throughout the paper such as the FNs, the OWAWA operator and the FOWA operator. 2.1. FUZZY NUMBERS A FN A is defined as a fuzzy subset of a universe of discourse that is both convex (i.e., μ A(λx1 + (1 − λ)x2 ≥ min( μ A(x1), μ A(x2)); for ∀x1, x2 ∈ R and λ ∈ [0, 1]) and normal (i.e., supx∈R μ A(x) = 1). Note that the FN may be considered as a generalization of the interval number although it is not strictly the same because the interval numbers may have different meanings. In the literature, we find a wide range of FNs (Casanovas and Merigó, 2007; Chen and Chen, 2003; 2008; Kaufmann and Gil-Aluja, 1987; Merigó, 2008; Merigó and Casanovas, 2007; 2008a; Sadiq and Tesfamariam, 2008; Xu, 2007; Xu and Yager, 2008) such as the TFN, the TpFN, the IVFN, the IFN, the GFN, the IVGFN, etc. For example, a TpFN A of a universe of discourse R can be characterized by a trapezoidal membership function ( α −cut representation) ),( aaA = such that ).()( ),()( 344 121 aaaa aaaa −−= −+= αα αα (1) 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 TFN. Note that the TFN can be parameterized by (a1, a2, a4). In the following, we are going to review some basic FN 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) 4 2) A − B = (a1 − b3, a2 − b2, a3 − b1) 3) A × k = (k × a1, k × a2, k × a3); for k > 0. Note that other operations could be studied but in this paper we will focus on these ones. For more complete overviews about FNs, see for example (Chang and Zadeh, 1972; Dubois and Prade, 1980; Kaufmann and Gil-Aluja, 1987; Kaufmann and Gupta, 1985). 2.2. THE OWAWA OPERATOR The ordered weighted averaging – weighted averaging (OWAWA) operator is an aggregation operator that unifies the WA and the OWA operator in the same formulation (Merigó, 2008). It can be defined as follows. Definition 1. An OWAWA operator of dimension n is a mapping OWAWA: Rn → R that has an associated weighting vector W of dimension n such that wj ∈ [0, 1] and ∑= = n jj w 11, according to the following formula: OWAWA (a1, …, an) = ∑ = n jjjbv 1 ˆ (2) where bj is the jth largest of the ai, each argument ai has an associated weight (WA) vi with ∑= n ii v 1 = 1 and vi ∈ [0, 1], jjj vwv )1( ˆ ββ −+= with β ∈ [0, 1] and vj is the weight (WA) vi ordered according to bj, that is, according to the jth largest of the ai. Note that it is possible to distinguish between descending (DOWAWA) and ascending (AOWAWA) orders. The weights of these operators are related by wj = w*n − j+1, where wj is the jth weight of the DOWAWA and w*n − j+1 the jth weight of the AOWAWA operator. By choosing a different manifestation in the weighting vector, we are able to obtain a wide range of particular types of OWAWA operators (Merigó, 2008). Especially, when β = 0, we get the WA, and if β = 1, we get the OWA operator. 2.3. 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. 5 Definition 2. Let Ψ be the set of FNs. A FOWA operator of dimension n is a mapping FOWA: Ψ n → Ψ that has an associated weighting vector W of dimension n such that the sum of the weights is 1 and wj ∈ [0,1], then: FOWA (ã1, ã2…, ãn) = ∑ = n jjjbw 1 (3) 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. Then, it is necessary to establish a criterion for comparing FNs. For simplicity, we recommend the following method. 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. If there is still a tie, then, we recommend to use an average or a weighted average of the FN according to the interests of the decision maker. From a generalized perspective of the reordering step, it is possible to distinguish between the descending FOWA (DFOWA) operator and the ascending FOWA (AFOWA) operator. By using a different manifestation in the weighting vector, we can obtain different families of FOWA operators such as the step- FOWA operator, the olympic-FOWA operator, the S-FOWA, the nonmonotonic-FOWA, the centered-FOWA operator, etc (Beliakov et al., 2007; Calvo et al., 2002; Casanovas and Merigó, 2007; Merigó, 2008; Merigó and Casanovas, 2007; 2008a; 2008b; Merigó and Gil-Lafuente, 2009; Xu and Da, 2003; Yager, 1993; 1996; 2007; Yager and Filev, 1994; Yager and Kacprzyk, 1997). 3. FUZZY OWAWA OPERATOR The fuzzy ordered weighted averaging – weighted averaging (FOWAWA) operator is a new model that unifies the FOWA operator and the FWA in the same formulation. Therefore, both concepts can be seen as a particular case of a more general one that includes the WA, the OWA operator and uncertain information represented in the form of FNs. This approach seems to be complete, at least as an initial real unification between FOWA operators and FWAs. It can also be seen as a unification between decision making problems under uncertainty (with FOWA operators) and under risk (with probabilities). Note that some previous models already considered the possibility of using OWA operators and WAs in the same formulation. The main models are the weighted OWA (WOWA) operator (Torra, 1997; Torra and Narukawa, 2007) and the hybrid averaging (HA) operator (Xu and Da, 2003). For the case with FNs, we would get the fuzzy WOWA (FWOWA) and the fuzzy HA (FHA) operator. Although they seem to be a good approach, they are not so complete than the FOWAWA because it can unify FOWAs and FWAs in the same model but they can not take in consideration the degree of importance of each case in the aggregation process. Moreover, in some particular cases we also find inconsistencies (Merigó, 2008). In the following, we are going to analyze the FOWAWA operator. It can be defined as follows. 6 Definition 3. A FOWAWA operator of dimension n is a mapping FOWAWA: Rn → R that has an associated weighting vector W of dimension n such that wj ∈ [0, 1] and ∑= = n jj w 11, according to the following formula: FOWAWA (a1, …, an) = ∑ = n jjjbv 1 ˆ (4) where bj is the jth largest of the ai, each argument ai has an associated weight (FWA) vi with ∑= n ii v 1 = 1 and vi ∈ [0, 1], jjj vwv )1( ˆ ββ −+= with β ∈ [0, 1] and vj is the weight (FWA) vi ordered according to bj, that is, according to the jth largest of the ai. Note that it is also possible to formulate the FOWAWA operator separating the part that strictly affects the FOWA operator and the part that affects the FWAs. This representation is useful to see both models in the same formulation but it does not seem to be as an unique equation that unifies both models. Definition 4. A FOWAWA operator is a mapping FOWAWA: Rn → R of dimension n, if it has an associated weighting vector W, with ∑= n jj w 1 = 1 and wj ∈ [0, 1] and a weighting vector V that affects the FWA, with ∑= n ii v 1 = 1 and vi ∈ [0, 1], such that: FOWAWA (a1, …, an) = ∑ −+ ∑== n iii n jjj avbw 11 )1( ββ (5) where bj is the jth largest of the arguments ai and β ∈ [0, 1]. Note that sometimes, it is not clear how to reorder the arguments. Then, it is necessary to establish a criterion for comparing FNs. For simplicity, we recommend the following method. 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. If there is still a tie, then, we recommend to use an average or a weighted average of the FN according to the interests of the decision maker. From a generalized perspective of the reordering step, it is possible to distinguish between the descending FOWAWA (DFOWAWA) and the ascending FOWAWA (AFOWAWA) operator by using wj = w*n − j+1, where wj is the jth weight of the DFOWAWA and w*n − j+1 the jth weight of the AFOWAWA operator. Note that different types of FNs can be used in the aggregation process according to the interests or necessities of the decision maker. For example, we could mention the following ones (Merigo, 2008): • Triangular FNs (Interval-valued, generalized, etc.). • Trapezoidal FNs (Interval-valued, generalized, etc.). • L-R FNs (Interval-valued, generalized, intuitionistic, Type 2 and n, etc.). • Interval-valued FNs (triplets, quadruplets, etc.). • Generalized FNs (simple, interval-valued, intuitionistic, Type 2 and n, etc.). • Intuitionistic FNs (simple, interval-valued, generalized, etc.). 7 • Type 2 and n FNs (simple, generalized, etc.). • Etc. If B is a vector corresponding to the ordered arguments bj, we shall call this the ordered argument vector and WT is the transpose of the weighting vector, then, the FOWAWA operator can be expressed as: FOWAWA (a1, …, an) = BW T (6) Note that if the weighting vector is not normalized, i.e., W = ∑≠ = n jj w 11, then, the FOWAWA operator can be expressed as: FOWAWA (a1, …, an) = ∑ = n jjjbv W1 ˆ 1 (7) The FOWAWA is monotonic, commutative, bounded and idempotent. It is monotonic because if ai ≥ ui, for all ai, then, FOWAWA(a1, a2,…, an) ≥ FOWAWA(u1, u2…, un). It is commutative because any permutation of the arguments has the same evaluation. That is, FOWAWA(a1, a2,…, an) = FOWAWA(u1, u2,…, un), where (u1, u2,…, un) is any permutation of the arguments (a1, a2,…, an). It is bounded because the FOWAWA aggregation is delimitated by the fuzzy minimum and the fuzzy maximum. That is, Min{ai} ≤ FOWAWA(a1, a2,…, an) ≤ Max{ai}. It is idempotent because if ai = a, for all ai, then, FOWAWA(a1, a2,…, an) = a. Another interesting issue to analyze are the measures for characterizing the weighting vector W. Following a similar methodology as it has been developed for the FOWA operator (Merigó, 2008; Yager, 1988) we can formulate the attitudinal character, the entropy of dispersion, the divergence of W and the balance operator. Note that these measures affect the weighting vector W but not the WAs because they are given as some kind of objective information. 4. FAMILIES OF FOWAWA OPERATORS First of all we are going to consider the two main cases of the FOWAWA operator that are found by analyzing the coefficient β . Basically, if β = 0, then, we get the FWA and if β = 1, the FOWA operator. Note that if vi = 1/n, for all i, then, we get the unification between the fuzzy arithmetic mean (or simple fuzzy average) and the FOWA operator. Another interesting result is when the FNs are reduced to the usual exact numbers and to the interval numbers (Moore, 1966). Theorem 1. If the FNs are reduced to the usual exact numbers, then, the FOWAWA operator becomes the OWAWA operator (Merigó, 2008). 8 Proof. Assume a TpFN = (a1, a2, a3, a4). If a1 = a2 = a3 = a4, then (a1, a2, a3, a4) = a, thus, we get the OWAWA operator. Remark 1. In a similar way, we could develop the same proof for all the other types of FNs available in the literature. Theorem 2. If the FNs are reduced to the usual exact numbers, then, the FOWAWA operator becomes the uncertain OWAWA (UOWAWA) operator (Merigó, 2008). Proof. Assume a TpFN = (a1, a2, a3, a4). If we only consider the points (a1, a2, a3, a4), then, the FN becomes an interval number (a quadruplet). Therefore, the FOWAWA operator becomes the UOWAWA operator. Remark 2. In a similar way, we could develop the same proof for all the other types of FNs. Remark 3. Note that similar analysis could be developed for considering situations when the FNs are representing linguistic variables, etc. By choosing a different manifestation of the weighting vector in the FOWAWA operator, we are able to obtain different types of aggregation operators. For example, we can obtain the partial fuzzy maximum, the partial fuzzy minimum, the partial fuzzy average and the partial fuzzy weighted average. Remark 4. The partial fuzzy maximum is found when w1 = 1 and wj = 0 for all j ≠ 1. The partial fuzzy minimum is formed when wn = 1 and wj = 0 for all j ≠ n. More generally, the step-FOWAWA is formed when wk = 1 and wj = 0 for all j ≠ k. Note that if k = 1, the step-FOWAWA is transformed to the partial fuzzy maximum, and if k = n, the step-FOWAWA becomes the partial fuzzy minimum operator. Remark 5. The partial fuzzy average is obtained when wj = 1/n for all j, and the partial fuzzy weighted average is obtained when the ordered position of i is the same as the ordered position of j. Remark 6. For the median-FOWAWA, if n is odd we assign w(n + 1)/2 = 1 and wj* = 0 for all others. If n is even we assign for example, wn/2 = w(n/2) + 1 = 0.5 and wj* = 0 for all others. For the weighted median-FOWAWA, we select the argument bk that has the kth largest argument such that the sum of the weights from 1 to k is equal or higher than 0.5 and the sum of the weights from 1 to k − 1 is less than 0.5. Remark 7. Another interesting family is the S-FOWAWA operator. It can be subdivided into three classes: the “or-like,” the “and-like” and the generalized S-FOWAWA operators. The generalized S-FOWAWA operator is obtained if w1 = (1/n)(1 − ( α + β )) + α , wn = (1/n)(1 − ( α + β )) + β , and wj = (1/n)(1 − ( α + β )) for j = 2 to n − 1, where α , β ∈ [0, 1] and α + β ≤ 1. Note that if α = 0, the generalized S-FOWAWA operator becomes the “andlike” S-FOWAWA operator, and if β = 0, it becomes the “or-like” S-FOWAWA operator. 9 Remark 8. The olympic-FOWAWA is generated when w1 = wn = 0, and for all others wj* = 1/(n − 2). Note that it is possible to develop a general form of the olympic-FOWAWA by considering that wj = 0 for j = 1, 2, …, k, n, n − 1, …, n − k + 1, and for all others wj* = 1/(n − 2k), where k < n/2. Note that if k = 1, then this general form becomes the usual olympic-FOWAWA. If k = (n − 1)/2, then this general form becomes the median-FOWAWA aggregation. That is, if n is odd, we assign w(n + 1) / 2 = 1, and wj* = 0 for all other values. If n is even, we assign, for example, wn/2 = w(n / 2) + 1 = 0.5 and wj* = 0 for all other values. Remark 9. Note that it is also possible to develop the contrary case, that is, the general olympic-FOWAWA operator. In this case, wj = (1/2k) for j = 1, 2, …, k, n, n − 1, …, n − k + 1, and wj = 0, for all other values, where k < n/2. Note that if k = 1, then we obtain the contrary case for the median-FOWAWA. Remark 10. Another family of aggregation operator that could be used is the centered-FOWAWA operator. We can define a FOWAWA operator as a centered aggregation operator if it is symmetric, strongly decaying and inclusive. Note that these properties have to be accomplished for the weighting vector W of the FOWAWA operator but not necessarily for the weighting vector V of the WA. It is symmetric if wj = wj+n − 1. It is strongly decaying when i < j ≤ (n + 1)/2 then wi < wj and when i > j ≥ (n + 1)/2 then wi < wj. It is inclusive if wj > 0. Note that it is possible to consider a softening of the second condition by using wi ≤ wj instead of wi < wj, and it is also possible to remove the third condition. We shall refer to it as a non-inclusive centered-FOWAWA operator. Remark 11. Another type of aggregation that could be used is the E-Z FOWAWA weights. In this case, we should distinguish between two classes. In the first class, we assign wj* = (1/q) for j* = 1 to q and wj* = 0 for j* > q, and in the second class, we assign wj* = 0 for j* = 1 to n − q and wj* = (1/q) for j* = n − q + 1 to n. Remark 12. A further interesting type is the non-monotonic-FOWAWA operator. It is obtained when at least one of the weights wj is lower than 0 and ∑= = n jj w 11. Note that a key aspect of this operator is that it does not always achieve monotonicity. Therefore, strictly speaking, this particular case is not an FOWAWA operator. However, we can see it as a particular family of operators that is not monotonic but nevertheless resembles an FOWAWA operator. Remark 13. Note that other families of FOWAWA operators could be used following the recent literature about different methods for obtaining the FOWAWA weights such as (Beliakov et al., 2007; Calvo et al., 2002; Casanovas and Merigó, 2007; Merigó, 2008; Merigó and Casanovas, 2007; 2008a; 2008b; Merigó and Gil- Lafuente, 2009; Xu and Da, 2003; Yager, 1993; 1996; 2007; Yager and Filev, 1994; Yager and Kacprzyk, 1997).