Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations

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Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations J. Vicente Riera, Joan Torrens∗ Department of Mathematics and Computer Science, University of the Balearic Islands, E-07122 Palma de Mallorca, Spain

Abstract In this paper we propose a method to construct aggregation functions on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers from a couple of discrete aggregation functions. The interest on these discrete fuzzy numbers lies on the fact that they can be interpreted as linguistic expert valuations that increase the flexibility of the elicitation of qualitative information based on linguistic terms. Finally, a linguistic decision making model based on a pair of aggregation functions defined on discrete fuzzy numbers is given. © 2013 Elsevier B.V. All rights reserved. Keywords: Fuzzy connectives and aggregation operators; Decision making; Linguistic variable; Linguistic term; Discrete fuzzy number; Discrete aggregation function; Uninorm; Nullnorm

1. Introduction The aggregation of information, understood as a process of merging all collected data into a concrete representative value, has become a field of increasing interest because of the great quantity of applications which include many subjects not only from mathematics and computer sciences, but also from many applied fields like economics and social sciences (see for instance [1,12]). The process of fusioning some data into a representative output is usually carried out by the so-called aggregation functions that have been extensively investigated in the last few decades [1,2,28]. Decision making, subjective evaluations, optimization and control are, among others, examples of concrete application fields where aggregation functions become an essential tool. In all these fields, it is well known that the data to be aggregated vary among many different kinds of information, from quantitative to qualitative information. Moreover, many times some uncertainty is inherent to such information. Many different tools have been proposed for managing uncertainty, specially fuzzy logic and fuzzy sets. In this sense, some generalizations and extensions of fuzzy sets have also been considered like interval-valued fuzzy sets, intuitionistic fuzzy sets, type-2 fuzzy sets, multidimensional fuzzy sets. Recently, also Hesitant fuzzy sets have been introduced (see [29]), where the membership degree of an element can be any subset of [0,1]. All previous tools are mainly used in the management of imprecise quantitative information. However, experts deal in many problems with qualitative information usually expressed through linguistic terms whose meaning is imprecise and vague in general, leading to the fuzzy linguistic approach. ∗ Corresponding author. Tel.: +34 971173195; fax: +34 971173003.

E-mail addresses: [email protected] (J. Vicente Riera), [email protected] (J. Torrens). 0165-0114/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.09.001 Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Fig. 1. Graphical representation of a discrete fuzzy number whose support is the interval [3, 5]. In addition, note that this fuzzy set can be interpreted as expression “between Fair and Very Good”, after identifying the linguistic scale L with the chain L 6 .

Qualitative information is often interpreted to take values in a totally ordered finite scale like {E xtr emely Bad, V er y Bad, Bad, Fair, Good, V er y Good, E xtr emely Good}.

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In these cases, the representative finite chain L n = {0, 1, . . . , n} is usually considered to model these linguistic hedges and several researchers have developed an extensive study of aggregation functions on L n , usually called discrete aggregation functions (see [16,19,21]). Another approximation is based on assigning a fuzzy set to each linguistic term trying to capture its meaning. However, the modeling of linguistic information is limited because the information provided by experts for each variable must be expressed by a simple linguistic term. In most cases, this is a problem for experts because their opinion does not agree with a concrete term. On the contrary, experts’ values are usually expressions like “better than Good”, “between Fair and Very Good” or even more complex expressions. To avoid the limitation above, two approaches have recently appeared. 1 Both approaches are different in the form but very similar in the idea: they try to increase the flexibility of the elicitation of linguistic information. • In [25] the concept of Hesitant fuzzy linguistic term set (HFLTS) is introduced in order to perform linguistic expressions like those mentioned in the previous paragraph. Basically, an HFLTS is simply any subset of the selected linguistic scale (usually subintervals). Thus, the expression “between Fair and Very Good” can be interpreted by the subinterval of the scale L given in (1): [Fair, V er y Good] = {Fair, Good, V er y Good}.

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Thus, all these expressions can be manipulated through HFLTS in a similar way to common Hesitant fuzzy sets (see [29]). In particular, aggregation operators like Min-upper and Max-lower operators are used in this context and applied in multicriteria linguistic decision making problem. • The approach in [6,24] deals with the possibility of extending monotonic operations on L n to operations on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers contained in L n , usually denoted by A1L n . Here the idea lies on the fact that any discrete fuzzy number A ∈ A1L n can be considered (identifying the scale L given in (1) with L n with n = 6) as an assignation of a [0, 1]-value to each term in our linguistic scale. As an example, the above-mentioned expression “between Fair and Very Good” can be performed, for instance, by a discrete fuzzy number A ∈ A1L 6 , with support given by the subinterval stated in (2) (that corresponds to the subinterval [3, 5] in L 6 ). The values of A in its support should be described by experts, allowing in this way a complete flexibility of the qualitative valuation. A possible discrete fuzzy number A representing the expression mentioned above is given in Fig. 1 (note that only the values of A in its support are pictured there). Thus, aggregation functions on A1L n will allow us to manage qualitative information in a more flexible way. In [6] t-norms and t-conorms on A1L n are described and studied, as well as it is done for uninorms, nullnorms and general aggregation functions in [24]. In both cases, an example of application in decision making or subjective evaluation is included. 1 There are also other approaches that try to deal directly with linguistic expressions instead of single terms, see for instance [18,26].

Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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In this paper, we want to go further on this last approach based on discrete fuzzy numbers. Recall that the concept of discrete fuzzy number was introduced in [27] as a fuzzy subset of R with discrete support and analogous properties to a fuzzy number. It is well known that arithmetic and lattice operations between discrete fuzzy numbers defined using Zadeh’s extension principle [15] fail and some approaches have been introduced in order to avoid such a drawback [4,5,30]. In particular, it is proved in [5] that the set, A1L n , of discrete fuzzy numbers whose support is a set of consecutive natural numbers contained in L n is a distributive lattice. Thus, it becomes natural to study aggregation functions defined on A1L n equipped with the usual lattice order. In this way, one approach is based on extending monotonic operations defined on L n to monotonic operations defined on the set A1L n . This was done for t-norms and t-conorms in [6] and for different kinds of aggregation functions in [24]. Followed by this idea we want to study in this paper the possibility of constructing aggregation functions on A1L n from a pair of discrete aggregation functions F, G on L n with F ⱕ G. The special cases of uninorms and nullnorms are investigated proving that some properties and part of their structure are preserved under the presented construction method. At the end, we show an application of these aggregation functions in a group decision making problem, where some kind of compensation is required between the valuations of two different groups. The paper is organized as follows. After this introduction we devote Section 2 to give some preliminary results on discrete operations as well as on discrete fuzzy number that are necessary to understand the rest of the sections. In Section 3 the main results of the paper are proved, including a construction method of aggregation functions on A1L n from pairs of discrete aggregation functions on L n . The particular cases of uninorms and nullnorms are studied in detail. Finally, Section 4 includes a linguistic decision making model based on aggregation functions on A1L n constructed following the method presented in Section 3. A concrete example using idempotent uninorms is also included illustrating the above-mentioned decision making model. 2. Preliminaries In this section, we recall some definitions and results about some associative and monotonic operations on partially ordered sets [8] and on discrete settings [19,21], and, on the other hand, about discrete fuzzy numbers [4–6]. 2.1. Associative and monotonic operations on finite chains Let (P; ⱕ ) be a non-trivial bounded partially ordered set (poset) with 0 and 1 as minimum and maximum elements respectively. Aggregation functions can be defined in general on (P; ⱕ ) (see for instance [7] and also [9,10]). An important case is when we take as poset a finite chain L n with n + 1 elements. In such a framework only the number of elements is relevant (see [21]) and so it is usually considered the most simple one, that is, L n = {0, 1, . . . , n}. Operations on L n are usually called discrete operations and they have been studied by many authors [8,13,16,19,21]. Definition 2.1. A triangular norm (briefly t-norm) on L n is a binary operation T : L n × L n → L n which is associative, increasing in each place, commutative and such that T (x, n) = x for all x ∈ L n . Definition 2.2. A triangular conorm (t-conorm for short) on L n is a binary operation S : L n × L n → L n which is associative, increasing in each place, commutative and such that S(x, 0) = x for all x ∈ L n . Uninorms and nullnorms are generalizations of t-norms and t-conorms as follows. Definition 2.3. A uninorm on L n is a two-place function U : L n × L n → L n which is associative, increasing in each place, commutative, and such that there exists some element e ∈ L n , called neutral element, such that U (e, x) = x for all x ∈ L n . It is clear that the function U becomes a t-norm when e = n and a t-conorm when e = 0. Definition 2.4. A nullnorm on L n is a two-place function G : L n × L n → L n which is associative, increasing in each place, commutative, and such that there exists some element k ∈ L n , called absorbing element, such that G(k, x) = k Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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for all x ∈ L n , and satisfies G(0, x) = x for all x ⱕ k G(n, x) = x for all x ⱖ k. In this case nullnorms with k = 0 lead back to t-norms, while the case k = n leads back to t-conorms. In the study of discrete operations the following condition is usually considered. Definition 2.5. A function F : L n → L n is said to be smooth if it satisfies |F(x) − F(x − 1)| ⱕ 1 for all x ∈ L n with x ⱖ 1. Definition 2.6. A binary operation F on L n is said to be smooth when each one of its vertical and horizontal sections (F(x, ·) and F(·, y), respectively) are smooth. Remark 2.7. The importance of the smoothness condition lies on the fact that it is generally used as a discrete counterpart of continuity on the unit interval and it is equivalent to the Lipschitz condition and to the divisibility condition (a t-norm (t-conorm) T (S) : L n × L n → L n is said to be divisible if it satisfies the next condition: For all x, y ∈ L n with x ⱕ y, there is z ∈ L n such that x = T (y, z)(y = S(x, z)) [21]. Smooth discrete t-norms and t-conorms were characterized in [21]. For the case of nullnorms we have the following result. Proposition 2.8. A binary operation F : L n × L n → L n is a nullnorm if and only if there exist k ∈ L n , a t-conorm S on [0, k] and a t-norm T on [k, n] such that for all x, y ∈ L n , F is given by ⎧ 2 ⎪ ⎨ S(x, y) if (x, y) ∈ [0, k] F(x, y) = T (x, y) if (x, y) ∈ [k, n]2 ⎪ ⎩ k otherwise. Moreover, F is smooth if and only if T and S are smooth. With respect to uninorms on L n it is well known that (see [19]) • For any uninorm on L n we have U (n, 0) ∈ {0, n} and a uninorm U is called conjunctive when U (n, 0) = 0 and disjunctive when U (n, 0) = n. • Any discrete uninorm U on L n with neutral element 0 < e < n is always given by a t-norm T on the interval [0, e], by a t-conorm S on the interval [e, n] and it takes values between the minimum and the maximum in all other cases. There are no smooth uninorms on L n with neutral element 0 < e < n (that is, different from t-norms or t-conorms). However, when the underlying t-norm and t-conorm are smooth the corresponding uninorm can be smooth except in very restricted regions like for instance the e-sections. This is the case of uninorms in Umin and Umax . Definition 2.9 (Mas et al. [19]). A binary operation U : L n × L n → L n is a uninorm in Umin with neutral element 0 < e < n if and only if there is a t-norm T on [0, e] and a t-conorm S on [e, n] such that U is given by ⎧ 2 ⎪ ⎨ T (x, y) if (x, y) ∈ [0, e] U (x, y) = S(x, y) if (x, y) ∈ [e, n]2 ⎪ ⎩ min(x, y) otherwise Definition 2.10 (Mas et al. [19]). A binary operation U : L n × L n → L n is a uninorm in Umax with neutral element 0 < e < n if and only if there is a t-norm T on [0, e] and a t-conorm S on [e, n] such that U is given by ⎧ 2 ⎪ ⎨ T (x, y) if (x, y) ∈ [0, e] U (x, y) = S(x, y) if (x, y) ∈ [e, n]2 ⎪ ⎩ max(x, y) otherwise Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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In both cases, when T, S are smooth, the corresponding uninorm is also smooth except for certain points (x, e) or (e, y) of the e-sections. Moreover, De Baets et al. [8] characterized all idempotent uninorms defined on the finite chain L n . So, they showed that any discrete idempotent uninorm is uniquely determined by a decreasing function from the set of scale elements not greater than the neutral element to the set of scale elements not smaller than the neutral element, and vice versa. 2.2. Discrete fuzzy numbers By a fuzzy subset of R, we mean a function A : R → [0, 1]. For each fuzzy subset A and for any  ∈ (0, 1], let A = {x ∈ R : A(x) ⱖ } be its -level set (or -cut). By supp(A), we mean the support of A, i.e., the set {x ∈ R : A(x) > 0}. By A0 , we mean the closure of supp(A). Definition 2.11 (Voxman [27]). A fuzzy subset A of R with membership mapping A : R → [0, 1] is called discrete fuzzy number if its support is finite, i.e., there exist x1 , . . . , xn ∈ R with x1 < x2 < · · · < xn such that supp(A) = {x1 , . . . , xn }, and there are natural numbers s, t with 1 ⱕ s ⱕ t ⱕ n such that 1. A(xi )=1 for any natural number i with s ⱕ i ⱕ t (core). 2. A(xi ) ⱕ A(x j ) for each natural number i, j with 1 ⱕ i ⱕ j ⱕ s. 3. A(xi ) ⱖ A(x j ) for each natural number i, j with t ⱕ i ⱕ j ⱕ n. Remark 2.12. If the fuzzy subset A is a discrete fuzzy number then the support of A is a finite subset of real numbers and so it coincides with its closure, i.e., supp(A) = A0 . From now on, we will denote the set of discrete fuzzy numbers by DFN and the abbreviation dfn will denote a discrete fuzzy number. Moreover, let us also denote by A1L n the set of all discrete fuzzy numbers whose support is a subset of consecutive natural numbers of L n . Theorem 2.13 (Wang et al. [30], Representation of discrete fuzzy numbers). Let A be a discrete fuzzy number. Then the following statements (1)–(4) hold: 1. A is a nonempty finite subset of R, f or any  ∈ [0, 1]. 2. A2 ⊆ A1 for any 1 , 2 ∈ [0, 1] with 0 ⱕ 1 ⱕ 2 ⱕ 1. 3. For any 1 , 2 ∈ [0, 1] with 0 ⱕ 1 ⱕ 2 ⱕ 1, if x ∈ A1 − A2 we have x < y for all y ∈ A2 , or x > y for all y ∈ A 2 .  4. For any 0 ∈ (0, 1], there exist some real numbers 0 with 0 < 0 < 0 such that A0 = A0 (i.e., A = A0 for  any  ∈ [0 , 0 ]). Conversely, if for any  ∈ [0, 1] there are subsets A ⊂ R satisfying conditions (1)–(4) above, then there exists a unique A ∈ D F N such that its -cuts are exactly the sets A for all  ∈ [0, 1]. Let A, B be two dfn and A = {x1 , . . . , x p }, B  = {y1 , . . . , yk } their -cuts respectively. In [4] we consider the following sets: min(A, B) = {z ∈ supp(A) ∧ supp(B)| min(x1 , y1 ) ⱕ z ⱕ min(x p , yk )} max(A, B) = {z ∈ supp(A) ∨ supp(B)| max(x1 , y1 ) ⱕ z ⱕ max(x p , yk )} for each  ∈ [0, 1] where supp(A) ∧ supp(B) = {z = min(x, y)|x ∈ supp(A), y ∈ supp(B)} supp(A) ∨ supp(B) = {z = max(x, y)|x ∈ supp(A), y ∈ supp(B)}. Then we have the following result. Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Proposition 2.14 (Casasnovas and Riera [4]). Given A, B ∈ D F N , there exist two unique dfn that we will denote by M I N (A, B) and M AX (A, B), such that they have the above-defined sets min(A, B) and max(A, B) as -cuts respectively. Moreover, if A, B ∈ A1L n then also M I N (A, B) and M AX (A, B) belong to A1L n . Remark 2.15 (Casasnovas and Riera [5]). Using these operations, we can define a partial order on A1L n in the usual way: A B if and only if M I N (A, B) = A, or equivalently, A B if and only if M AX (A, B) = B for any A, B ∈ A1L n . We can also define the partial ordering in terms of -cuts: A B if and only if min( A , B  ) = A for all  ∈ [0, 1] A B if and only if max( A , B  ) = B  for all  ∈ [0, 1] On the other hand, from now on let us introduce the following notation. For any a ∈ L n we will denote by 1a the unique discrete fuzzy number whose support is given by the singleton {a}. The following result was proved in [6]. Theorem 2.16. The triplet (A1L n , M I N , M AX ) is a bounded distributive lattice where 1n ∈ A1L n and 10 ∈ A1L n represent the maximum and the minimum respectively. 3. Aggregation of discrete fuzzy numbers In this section we wish to investigate if it is possible to build aggregation functions on the bounded distributive lattice A1L n from a couple of aggregation functions on L n . Moreover, we will study some well known relevant special cases such as uninorms and nullnorms. Let us begin by recalling the following definition. Definition 3.1. An n-ary aggregation function on a bounded partially ordered set P with minimum element 0 and maximum element 1 is a function F : P n → P such that it is increasing in each component and it satisfies the boundary conditions F(0, . . . , 0) = 0 and F(1, . . . , 1) = 1. Of course, the number of inputs to be aggregated can be different in each case. Thus, aggregation functions are commonly defined not on P n , but on n ⱖ 1 P n and then they are usually called extended aggregation functions. An easy way to construct extended aggregation functions is from associative binary aggregation functions. For this reason, from now on, we will focus our study on the binary case and we will deal with special associative cases like uninorms and nullnorms. Let us begin with some notation. Let 2 L n be the set of all subsets of L n , and we consider any binary discrete aggregation on L n O : L n × L n −→ L n (x, y)  O(x, y) We will denote as well by O, the binary operation O : 2 L n × 2 L n −→ 2 L n (X, Y)  O(X, Y) where O(X, Y) = {O(x, y)|x ∈ X, y ∈ Y}, following the formalism provided by Moore in [22]. Note that aggregation functions on A1L n were already constructed from aggregation functions on L n in [6,24]. We recall here the main result in this sense. Theorem 3.2 (Casasnovas and Riera [6]). Let F be an aggregation function on L n . Then the function F : A1L n × A1L n −→ A1L n (A, B)  F(A, B) Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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is an aggregation function on A1L n , where F(A, B) is the discrete fuzzy number whose -cuts are the sets F(A, B) = {z ∈ L n | min F(A , B  ) ⱕ z ⱕ max F(A , B  )}. Now, we want to proceed in a similar way but from a pair of binary aggregation functions F and G on L n with F ⱕ G. Definition 3.3. Given a couple of binary aggregation functions F and G on L n with F ⱕ G and two discrete fuzzy numbers A, B ∈ A1L n , we define for each  ∈ [0, 1] the sets  (A, B) = {z ∈ L n | min F(A , B  ) ⱕ z ⱕ max G(A , B  )} C F,G

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 (A, B) can be Note that, from the monotonicity of the binary discrete aggregation functions F and G, the set C F,G written for each  ∈ [0, 1] as

{z ∈ L n |F(min A , min B  ) ⱕ z ⱕ G(max A , max B  )}. Proposition 3.4. Let us consider A, B ∈ A1L n and let F and G be a couple of binary aggregation functions on the  (A, B) finite chain L n with F ⱕ G. There exists a unique discrete fuzzy number whose -cuts are exactly the sets C F,G (defined in Definition 3.3) that will be denoted by [F, G](A, B). Moreover, [F, G](A, B) ∈ A1L n .  (A, B) satisfy the four conditions of Theorem 2.13. Proof. We only need to prove that the sets C F,G  (A, B) is a nonempty finite set, because A and B  are both nonempty finite sets (the 1. For each  ∈ [0, 1], C F,G discrete fuzzy numbers are normal fuzzy subsets).   (A, B) for any ,  ∈ [0, 1] with 0 ⱕ  ⱕ  ⱕ 1. Let us suppose that A , A , B  , B  are the 2. C F,G (A, B) ⊆ C F,G following sets of consecutive natural numbers included in L n : 







A = [x1 , x p ],

A = [x1 , xr ],

B  = [y1 , yk ],

B  = [y1 , yl ].

Then 



A ⊆ A implies x1 ⱕ x1 and xr ⱕ x p

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  B  ⊆ B  implies y1 ⱕ y1 and yl ⱕ yk

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Moreover, from the monotonicity of the aggregation function F and G and the relations (4) and (5) we obtain 







F(x1 , y1 ) ⱕ F(x1 , y1 ) ⱕ G(xr , yl ) ⱕ G(x p , yk ) 

 (A, B). Therefore, C F,G (A, B) ⊆ C F,G





 (A, B)−C 3. With the same notation as above, if x ∈ C F,G F,G (A, B) then x ∈ L n and x does not belong to C F,G (A, B). 









Hence either x < F(x1 , y1 ), which is the minimum of C F,G (A, B), or x > G(xr , yl ), which is the maximum of 

C F,G (A, B).

4. As A, B ∈ A1L n , according to Theorem 2.13, for each  ∈ (0, 1] there exist real numbers 1 and 2 with 0 < 1 <  and 0 < 2 <  such that for each r ∈ [1 , ], A = Ar . Moreover, B  = B r , for each r ∈ [2 , ]. Thus, if  = 1 ∨ 2 , we obtain for each r ∈ [ , ] F(min( Ar ), min(B r )) = F(min( A ), min(B  )) 

and



G(max( A ), max(B )) = G(max( A ), max(B )) r

r

 (A, B) = C r (A, B) for each r ∈ [ , ]. Hence, C F,G F,G  (A, B) fulfill for each  ∈ [0, 1] the conditions stated in Theorem 2.13, there exists a unique discrete As the sets C F,G fuzzy number that will be denoted by [F, G](A, B), such that its -cuts are exactly these sets. Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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 (A, B) is a set of consecutive natural numbers. Thus, [F, G](A, B) ∈ A L n . In addition, it is clear that each set C F,G 1 

The previous proposition allows us to define a binary operation [F, G] on A1L n from any couple of binary aggregation function F and G with F ⱕ G defined on the finite chain L n . Definition 3.5. Let us consider a couple of binary aggregation functions F, G on the finite chain L n with F ⱕ G. The binary operation on A1L n defined as follows: [F, G] : A1L n × A1L n −→ A1L (A, B)  [F, G](A, B) will be called the extension to A1L n of the couple of discrete aggregation functions F and G, [F, G](A, B) being the discrete fuzzy number whose -cuts are as in (3) for each  ∈ [0, 1]. Now we want to prove that function [F, G] defined above is in fact a binary aggregation function on A1L n . Firstly,  (A, B) considered in relation (3) for each  ∈ [0, 1]. from now on we will denote by [F, G](A, B) the sets C F,G Proposition 3.6. Let [F, G] : A1L n × A1L n → A1L n be the extension of the discrete aggregation functions F and G on L n to A1L n . Let 10 and 1n be the minimum and the maximum of A1L n respectively. Then the following properties hold: 1. [F, G] is increasing in each place. 2. [F, G](10 , 10 ) = 10 . 3. [F, G](1n , 1n ) = 1n . Proof. We only show the first and second conditions because the last condition is similar to the second one. 1. Now, we wish to see that [F, G] is an increasing mapping, i.e., for each A, B, C ∈ A1L n such that B C then [F, G](A, B) [F, G](A, C). By Remark 2.15 we only need to show that min([F, G](A, B) , [F, G](A, C) ) = [F, G](A, B)

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for all  ∈ [0, 1]. Let us suppose again that A , B  , C  are given by A = [x1 , x p ], B  = [y1 , yk ], C  = [w1 , wl ]. As B C then min(B  , C  ) = B  , that is, y1 ⱕ w1 and yk ⱕ wl for all  ∈ [0, 1]. Using these last relations and the monotonicity of F and G, we obtain that F(x1 , y1 ) ⱕ F(x1 , w1 ) and G(x p , yk ) ⱕ G(x p , wl ). From these inequalities relation (6) follows directly. 2. It is easy to see that [F, G](10 , 10 ) = 10 because [F, G](10 , 10 ) = {z ∈ L n |F(0, 0) ⱕ z ⱕ G(0, 0)} = {0} = 10 for all  ∈ [0, 1].  Thus, given any couple of binary aggregation functions F and G with F ⱕ G on L n , its extension [F, G] to A1L n is a binary aggregation function on A1L n . When F = G, it is clear from definitions that we obtain [F, F] = F, that is, extensions by a couple of aggregation functions generalize the original extension of aggregation functions given in [24] (see also Theorem 3.2). Moreover, in the general case, the extension of a couple (F, G) always leads to an aggregation function between F and G, as it is proved in the following proposition. Proposition 3.7. Let F and G be a couple of aggregation functions on L n with F ⱕ G and let [F, G] be its extension. Then F [F, G] G, that is, F(A, B) [F, G](A, B) G(A, B) for all A, B ∈ A1L n . Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Proof. Let us consider A, B ∈ A1L n where its -cuts sets A , B  are given by A = [x1 , x p ] and B  = [y1 , yk ] respectively. Thus, according to Remark 2.15 to prove the result is equivalent to show that min(F(A, B) , [F, G](A, B) ) = F(A, B)

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min([F, G](A, B) , G(A, B) ) = [F, G](A, B)

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for all  ∈ [0, 1], where F(A, B) = {z ∈ L n |F(x1 , y1 ) ⱕ z ⱕ F(x p , yk )} [F, G](A, B) = {z ∈ L n |F(x1 , y1 ) ⱕ z ⱕ G(x p , yk )} G(A, B) = {z ∈ L n |G(x1 , y1 ) ⱕ z ⱕ G(x p , yk )} Now, as F ⱕ G it is evident that F(x1 , y1 ) ⱕ G(x1 , y1 ) and F(x p , yk ) ⱕ G(x p , yk ) for all  ∈ [0, 1]. So, from these inequalities the relations (7) and (8) follow straightforward.  Proposition 3.8. Let F and G be aggregation functions on L n with F ⱕ G. When we restrict [F, G] to crisp numbers of A1L n we obtain the discrete fuzzy number whose -cut sets are the interval of the finite chain L n , [F(a, b), G(a, b)] for all  ∈ [0, 1]. Proof. Indeed, [F, G](1a , 1b ) = {z ∈ L n |F(min 1a , min 1b ) ⱕ z ⱕ G(max 1a , max 1b )} = {z ∈ L n |F(a, b) ⱕ z ⱕ G(a, b)} = [F(a, b), G(a, b)] for all  ∈ [0, 1].  Proposition 3.9. Let [F, G] : A1L n × A1L n → A1L n be the extension of the aggregation functions F and G on L n to A1L n . Then the following properties hold: 1. [F, G] is a commutative aggregation function if and only if F and G are commutative as well. 2. [F, G] is an associative aggregation function if and only if F and G are associative as well. Proof. Suppose that F and G are commutative (associative). Then the proof that [F, G] is also commutative (associative) is similar to the proof of Theorem 4.9 in [6]. The converses are immediate consequences of Proposition 3.8.  Theorem 3.10. Let [F, G] : A1L n × A1L n → A1L n be the extension of the couple of aggregation functions F and G to A1L n . Then, the following properties hold: 1. e is a common neutral element of F and G if and only if 1e is a neutral element of [F, G]. That is, [F, G](A, 1e ) = [F, G](1e , A) = A for all A ∈ A1L n . 2. k is a common absorbing element of F and G if and only if 1k is an absorbing element of [F, G]. That is, [F, G](A, 1k ) = [F, G](1k , A) = 1k for all A ∈ A1L n . 3. [F, G] is idempotent if and only if F and G are idempotent. Proof. Let us consider A ∈ A1L n 1. If e is a common neutral element of F and G then [F, G](A, 1e ) = A because of [F, G](A, 1e ) = {z ∈ L n | min F(A , 1e ) ⱕ z ⱕ max G(A , 1e )} = {z ∈ L n |F(min A , min 1e ) ⱕ z ⱕ G(max A , max 1e )} = {z ∈ L n |F(min A , e) ⱕ z ⱕ G(max A , e)} Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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= {z ∈ L n | min A ⱕ z ⱕ max A } = A for all  ∈ [0, 1]. Reciprocally, if 1e is a neutral element of [F, G], then for all a ∈ L n we have [F, G](1a , 1e ) = 1a and the result follows from Proposition 3.8. 2. Analogous to the item described above. 3. If F and G are idempotent aggregation functions we have [F, G](A, A) = {z ∈ L n | min F(A , A ) ⱕ z ⱕ max G(A , A )} = {z ∈ L n | min A ⱕ z ⱕ max A } = A for all  ∈ [0, 1] and s A ∈ A1L n . Therefore, [F, G](A, A) = A for all A ∈ A1L n and so [F, G] is idempotent. The converse is again a consequence of Proposition 3.8.  It is well known that uninorms and nullnorms are two special cases of binary associative aggregation functions on L n . So, in next sections, we will use the results above in order to construct uninorms and nullnorms on A1L n from a couple of uninorms and nullnorms of L n . 3.1. Aggregations based on a couple of discrete uninorms Let us deal in this section with the case of uninorms. Definition 3.11. Let us consider a couple of discrete uninorms U and U  on the finite chain L n with the same neutral element e ∈ L n and U ⱕ U  . The binary operation on A1L n defined as follows [U, U  ] : A1L n × A1L n −→ A1L n (A, B)  [U, U  ](A, B) will be called the extension of the couple of discrete uninorms U and U  to A1L n . Theorem 3.12. Let U and U  be a couple of discrete uninorms on L n with U ⱕ U  and e ∈ L n as common neutral element and let [U, U  ] : A1L n × A1L n −→ A1L n (A, B)  [U, U  ](A, B) be the extension of the couple of U and U  to A1L n , defined according to Definition 3.11. Then, [U, U  ] is a uninorm on A1L n with neutral element 1e . Moreover, U and U  are idempotent uninorms if and only if so is its extension [U, U  ]. Proof. Direct from the previous results.  Remark 3.13. Note that, in particular, a couple of discrete t-norms on L n leads to a t-norm on A1L n and a couple of discrete t-conorms on L n leads to a t-conorm on A1L n . Example 3.14. Consider the discrete idempotent uninorms [8]  min(x, y) if y ⱕ 6 − x U (x, y) = max(x, y) otherwise and

 

U (x, y) =

min(x, y) if (x, y) ∈ [0, 3]2 max(x, y) otherwise

Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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defined on the finite chain L 6 = {0, 1, 2, 3, 4, 5, 6} (it is obvious that U ⱕ U  for all (x, y) ∈ [0, 6]2 ). Consider the discrete fuzzy numbers A = {0.3/0, 0.5/1, 1/2, 0.3/3}, B = {0.3/2, 0.5/3, 1/4, 0.8/5} ∈ A1L 6 . Then, [U, U  ](A, B) = {0.3/0, 0.5/1, 1/2, 1/3, 1/4, 0.8/5}. Note that, since U and U  are idempotent so is [U, U  ]. Thus, for instance, [U, U  ](A, A) = A and [U, U  ](B, B) = B. Lemma 3.15 (Riera and Torrens [24]). Let U be a uninorm on the finite chain L n with e as a neutral element. Then, (i) For A ∈ A1L n we have A 1e if and only if supp(A) ⊆ [0, e]. (ii) For A ∈ A1L n we have 1e A if and only if supp(A) ⊆ [e, n]. Proposition 3.16. Let U and U  be a couple of uninorms on the finite chain L n with the same neutral element e ∈ L n and U ⱕ U  . Then its extension [U, U  ] : A1L n × A1L n −→ A1L n (A, B)  [U, U  ](A, B) satisfies 1. [U, U  ](A, 1n ) = 1n for all A  1e and [U, U  ](A, 10 ) = 10 for all A 1e . 2. [U, U  ](10 , 1n ) ∈ {10 , 1n , L} where 1n and 10 denote the maximum and the minimum of the bounded distributive lattice A1L n respectively and L is the discrete fuzzy number whose -cuts are the proper finite chain L n = [0, n] for all  ∈ [0, 1]. Proof. The first item is clear from the increasingness of the uninorm [U, U  ]. The second item is a consequence of Proposition 3.8. From this result we obtain • [U, U  ](10 , 1n ) = 10 when both U and U  are conjunctive, • [U, U  ](10 , 1n ) = 1n when both U and U  are disjunctive, • [U, U  ](10 , 1n ) = L when U is conjunctive and U  is disjunctive.  Proposition 3.17. Let us consider the uninorms ⎧ T (x, y) if (x, y) ∈ [0, e]2 ⎪ ⎪ ⎨ U (x, y) = S(x, y) if (x, y) ∈ [e, n]2 ⎪ ⎪ ⎩ min(x, y) otherwise ⎧  T (x, y) if (x, y) ∈ [0, e]2 ⎪ ⎪ ⎨ U  (x, y) = S  (x, y) if (x, y) ∈ [e, n]2 ⎪ ⎪ ⎩ min(x, y) otherwise on L n with neutral element 0 < e < n, T ⱕ T  being a pair of t-norms on [0, e] and S ⱕ S  a pair t-conorms on [e, n]. Let [U, U  ] be the extension of the couple U and U  to A1L n according to Definition 3.11. (i) If A, B 1e then [U, U  ](A, B) = [T , T  ](A, B) where [T , T  ] is the extension of the couple T and T  to A1[0,e] following a similar method to Definition 3.11. (ii) If A, B  1e then [U, U  ](A, B) = [S, S  ](A, B) where [S, S  ] is the extension of the couple S and S  to A[e,n] 1 according to Definition 3.11. (iii) If A 1e B then [U, U  ](A, B) = M I N (A, B) = A. Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Proof. Let us consider [U, U  ] the extension to A1L n of the discrete uninorms U and U  . (i) Suppose that A, B 1e . To show [U, U  ](A, B) = [T , T  ](A, B) is enough to see that [U, U  ](A, B) = [T , T  ](A, B) for all  ∈ [0, 1]. According to Definition 3.11 [U, U  ](A, B) = {z ∈ L n | min U (A , B  ) ⱕ z ⱕ max U  (A , B  )} = {z ∈ L n |U (min A , min B  ) ⱕ z ⱕ U  (max A , max B  )} (As A, B 1e from Lemma 3.15 supp(A), supp(B) ⊆ [0, e] thus) [U, U  ](A, B) = {z ∈ L n |T (min A , min B  ) ⱕ z ⱕ T  (max A , max B  )} = {z ∈ L n | min T (A , B  ) ⱕ z ⱕ max T  (A , B  )} = [T , T  ](A, B) for all  ∈ [0, 1] and A, B ∈ A[0,e] 1 (ii) Similar to the previous step. (iii) Suppose that A 1e B. To show [U, U  ](A, B) = A is equivalent to see that [U, U  ](A, B) = A for all  ∈ [0, 1]. According to Definition 3.11 [U, U  ](A, B) = {z ∈ L n | min U (A , B  ) ⱕ z ⱕ max U  (A , B  )} = {z ∈ L n |U (min A , min B  ) ⱕ z ⱕ U  (max A , max B  )}. According to Lemma 3.15, since A 1e B we have supp(A) ⊆ [0, e] and supp(B) ⊆ [e, n]. Thus, uninorms U and U  are given by the minimum in all points (x, y) with x ∈ supp(A) and y ∈ supp(B). Consequently, [U, U  ](A, B) = {z ∈ L n | min(min A , min B  ) ⱕ z ⱕ min(max A , max B  )} = {z ∈ L n | min A ⱕ z ⱕ max A } = A for all  ∈ [0, 1], A ∈ A[0,e] and B ∈ A[e,n] .  1 1 Analogously, for uninorms in Umax we obtain the following result. Proposition 3.18. Let us consider a uninorm ⎧ 2 ⎪ ⎨ T (x, y) if (x, y) ∈ [0, e] U (x, y) = S(x, y) if (x, y) ∈ [e, n]2 ⎪ ⎩ max(x, y) otherwise and

⎧  2 ⎪ ⎨ T (x, y) if (x, y) ∈ [0, e] U  (x, y) = S  (x, y) if (x, y) ∈ [e, n]2 ⎪ ⎩ max(x, y) otherwise

on L n with neutral element 0 < e < n, T ⱕ T  being t-norms on [0, e] and S ⱕ S  t-conorms on [e, n]. Let [U, U  ] be the extension of the couple U and U  to A1L n according to Definition 3.11. (i) If A, B 1e then [U, U  ](A, B) = [T , T  ](A, B) where [T , T  ] is the extension of the couple T and T  to A[0,e] . 1 (ii) If A, B  1e then [U, U  ](A, B) = [S, S  ](A, B) where [S, S  ] is the extension of the couple S and S  to A[e,n] . 1 (iii) If A 1e B then U(A, B) = M AX (A, B) = B. Remark 3.19. Note that the previous theorems do not give the complete structure of uninorms in A1L n that are extensions of uninorms in Umin and Umax on L n . Since the order in A1L n is not total there are elements A, B ∈ A1L n not comparable with 1e (those whose support is contained neither in [0, e] nor in [e, n] by Lemma 3.15). Thus if A or B is one of these elements only the general expression of U(A, B) given in Definition 3.11 works. Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Example 3.20. Let us consider the finite chain L 5 = {0, 1, 2, 3, 4, 5}, the dfn A = {0.2/0, 0.3/1, 0.5/2, 1/3, 0.8/4, 0.8/5}, B = {0.1/1, 0.7/2, 0.8/3, 1/4, 0.6/5} and the discrete uninorm ⎧ 2 ⎪ ⎨ max(0, x + y − 3) if (x, y) ∈ [0, 3] U (x, y) = min(5, x + y − 3) if (x, y) ∈ [3, 5]2 ⎪ ⎩ min(x, y) otherwise and

 U  (x, y) =

min(x, y) if (x, y) ∈ [0, 3]2 max(x, y) otherwise

Then [U, U  ](A, B) = {0.3/0, 0.5/1, 0.7/2, 0.8/3, 1/4, 0.8/5}. Note that the underlying t-norm of U is the Łukasiewicz t-norm TL on [0, 3] and the underlying t-norm on [0, 3] of U  is the minimum t-norm min(x, y) (which is the largest t-norm on [0, 3]). Thus, for A0 = {0.2/0, 1/1, 0.8/2} and B0 = {0.6/1, 1/2, 0.7/3} we have by Proposition 3.17 that [U, U  ](A0 , B0 ) = [TL , min](A0 , B0 ) = {1/0, 1/1, 0.8/2}, where [TL , min] denotes the extension of the couple Łukasiewicz t-norm TL and the minimum t-norm on [0, 3] to A[0,3] . 1 3.2. Aggregation based on couple of discrete nullnorms Let us deal in this section with the case of nullnorms. Definition 3.21. Let us consider a couple of discrete nullnorms F and F  on the finite chain L n with F ⱕ F  and with the same absorbing element. The binary operation on A1L n defined as follows [F, F  ] : A1L n × A1L n −→ A1L n (A, B)  [F, F  ](A, B) will be called the extension of the couple of the discrete nullnorms F and F  to A1L n , [F, F  ](A, B) being the discrete fuzzy number whose -cuts are the sets {z ∈ L n | min F(A , B  ) ⱕ z ⱕ max F  (A , B  )} for each  ∈ [0, 1]. Theorem 3.22. Let F and F  be a pair of discrete nullnorms on L n with the same element k ∈ L n as absorbing element and F ⱕ F  . And, let [F, F  ] : A1L n × A1L n −→ A1L n (A, B)  [F, F  ](A, B) be the extension of the couple F and F  to A1L n . Then [F, F  ] is a nullnorm on A1L n with absorbing element 1k . Proof. Straightforward from previous results.  Similar to the case of uninorms we can easily prove the following result showing part of the structure of the extensions of nullnorms. Proposition 3.23. Let F and F  be a couple of nullnorms on L n with absorbing element k, F ⱕ F  and underlying t-norms T ⱕ T  and t-conorms S ⱕ S  respectively. Then its extension [F, F  ] to A1L n satisfies the following properties: (i) If A, B 1k then [F, F  ](A, B) = [S, S  ](A, B) where [S, S  ] is the extension to A[0,e] of the couple of t-conorms 1 S and S  . Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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(ii) If A, B  1k then [F, F  ](A, B) = [T , T  ](A, B) where [T , T  ] is the extension to A[e,n] of the couple of t-norms 1 T and T  . (iii) If A 1k B then [F, F  ](A, B) = 1k . Proof. Similar to the proof of Proposition 3.17.  Example 3.24. Consider the discrete nullnorms defined on the finite chain L 6 = {0, 1, 2, 3, 4, 5, 6} ⎧ if (x, y) ∈ [0, 3]2 ⎪ ⎨ min(x + y, 3) F(x, y) = max(3, x + y − 6) if (x, y) ∈ [3, 6]2 and ⎪ ⎩ 3 otherwise ⎧ 2 ⎪ ⎨ min(x + y, 3) if (x, y) ∈ [0, 3] F  (x, y) = min(x, y) if (x, y) ∈ [3, 6]2 ⎪ ⎩ 3 otherwise and A = {0.3/0, 0.5/1, 1/2, 0.3/3}, B = {0.3/2, 0.5/3, 1/4, 0.8/5} ∈ A1L 6 . Then [F, F  ](A, B) = {0.3/2, 1/3}. 4. Decision making based on representable uninorms on A1L n In recent years, the issue of aggregation operators has been developed mainly from two points of view. On one hand, the theoretical study of these operators and their properties, for example [1,3,7,12,13,15,19]. And, on the other hand, the possible applications of them in several fields of knowledge such as social science (e.g. decision making [14,23,31,32]), applied sciences (e.g. image processing [11,17]), and educational sciences [24]. In particular, it is well known the use of uninorms defined on the unit interval or on a finite chain as a useful tool in decision making problems [20,31,32]. In the previous section, we have discussed a method to build uninorms and nullnorms on A1L n from a couple of uninorms and nullnorms defined on the finite chain L n . Thus, we propose to use these operators obtained from a couple of uninorms 2 on L n , in order to get the final decision of two groups of experts. Suppose that a company engages the services of two expert groups to evaluate a possible investment in a foreign country. The first one expert group is usually hired by the company for such decisions. The second one is specifically hired in this foreign country only to assess the viability of the investment. The proposed method is presented as follows: Step 1: Establishing the expert groups N E G = {O1 , . . . , Or } and F E G = {(F O)1 , . . . , (F O)k } who carry out the evaluation process of the parameters P = {P1 , . . . , Ps }. Step 2: Choose the linguistic chain L used to make the evaluation process that we will identify with the corresponding finite chain L n where n denotes the total number of linguistic terms in L. Step 3: Each expert O j ∈ N E G (with j = 1, . . . , r ) performs an assessment O jPi ∈ A1L n of all parameters Pi ∈ P chosen. And, analogously each expert (F O) j ∈ F E G (with j = 1, . . . , k) takes into action an assessment (F O) Pj i ∈ A1L n of all parameters Pi ∈ P chosen. Step 4: For each parameter Pi ∈ P, the FEG chooses a uninorm FU i (built following Theorem 3.2) to calculate the aggregation of all valuations {(F O)1Pi , . . . , (F O)kPi }. According to the previous election, the NEG chooses another uninorm Ui (also built from Theorem 3.2) to calculate the aggregation of all valuations {O1Pi , . . . , OrPi } and fulfilling the order relation Ui FU i . 3 These aggregations will be denoted by D(N O, Pi ) = Ui (O1Pi , . . . , OrPi ) D(FO, Pi ) = FU i ((F O)1Pi , . . . , (F O)kPi ) 2 We use uninorms only because they are specially studied in this paper, but of course any other kind of aggregation function on L could be used n

instead of idempotent uninorms. 3 This order is interpreted as a favorable point of view that the Foreign Expert Group can show on the proposal about receiving potential investments of this company. Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Table 1 National Expert Group sheet. Expert

National Expert Group P1

···

Ps

O1 O2 . . . Or

O1P1 O2P1 . . . OrP1

··· ··· . . . ···

O1Ps O2Ps . . . OrPs

NO

D(N O , P1 )

···

D(N O, Ps )

Foreign Expert Group P1

···

Ps

(F O)1 (F O)2 . . . (F O)k

(F O)1P1 (F O)2P1 . . . (F O)kP1

··· ··· . . . ···

(F O)1Ps (F O)2Ps . . . (F O)kPs

FO

D(FO , P1 )

···

D(FO , Ps )

Table 2 Foreign Expert Group sheet. Expert

Table 3 Final opinion sheet. Company D(N O, P) D(FO , P) D(O, P)

Final opinion P1

···

Ps

D(N O, P1 ) D(FO , P1 ) D(O, P1 )

··· ··· ···

D(N O, Ps ) D(FO , Ps ) D(O, Ps )

F.Opinion: [U , U  ](D(O, P1 ), . . . , D(O, Ps ))

Step 5: Now, the company (based on their experience) gets for each parameter Pi the aggregation [Ui , FU i ](D(N O, Pi ), D(FO, Pi )) which will be denoted by D(O, Pi ). Finally, the company computes the aggregation of all these valuations D(O, Pi ) using a representable uninorm [U, U  ], according to the expression [U, U  ](D(O, P1 ), . . . , D(O, Ps )), in order to obtain a final decision to assess the viability of the investment. Tables 1, 2 and 3 describe the procedure explained previously. Remark 4.1. Other possible situation is to consider when the Foreign Expert Group expresses some reluctance to foreign investment. In this case, if Ui and FU i denote the extension of the uninorms assigned by the company to expert groups NEG and FEG respectively to assess the parameter Pi , these two uninorms would be selected to fulfill the order relation FU i Ui . Remark 4.2. Note that from Proposition 3.7 the discrete fuzzy number D(O, Pi ) = [Ui , FU i ](D(N O, Pi ), D(FO, Pi )) can be interpreted as a mean of the discrete fuzzy numbers D(N O, Pi ) and D(FO, Pi ). Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Example 4.3. Assume that the National Expert Group and the Foreign Expert Group are made up of three experts (N E G = {O1 , O2 , O3 }) and two experts (F E G = {(F O)1 , (F O)2 }) respectively. Let P = {P1 , P2 , P3 } be the set of parameters which will be evaluated, where P1 = Risk of the investment, P2 = Social–Political Impact Analysis and P3 = Tax Benefits. Consider the nine linguistic hedges L = {E B, V B, B, M B, F, M G, G, V G, E G} where the letters refer to the linguistic terms Extremely Bad, Very Bad, Bad, More or Less Bad, Fair, More or Less Good, Good, Very Good and Extremely Good and they are listed in an increasing order: E B ≺ V B ≺ B ≺ M B ≺ F ≺ M G ≺ G ≺ V G ≺ E G. It is obvious that we can consider a bijective application between this ordinal scale L and the finite chain L 8 = {0, 1, 2, 3, 4, 5, 6, 7, 8} of natural numbers which keep the order. Furthermore, each normal convex fuzzy subset defined on the ordinal scale L can be considered like a discrete fuzzy number belonging to A1L 8 , and vice versa. Suppose that O1P1 = {0.6/2, 1/3, 0.8/4, 0.7/5} O1P2 = {0.3/3, 0.6/4, 1/5, 0.7/6} O1P3 = {0.7/2, 0.8/3, 1/4, 0.5/5} O2P1 = {0.8/6, 0.9/7, 1/8} O2P2 = {0.6/5, 0.7/6, 1/7, 0.7/8} O2P3 = {0.5/4, 0.7/5, 1/6, 0.7/7, 0.4/8} O3P1 = {0.4/0, 0.6/1, 1/2, 0.4/3} O3P2 = {0.5/3, 0.7/4, 1/5} O3P3 = {0.6/2, 0.7/3, 1/4, 0.8/5} represent the assessments of the National Expert Group corresponding to the chosen parameter Pi . Now, suppose that (F O)1P1 = {0.4/2, 0.7/3, 1/4, 0.7/5} (F O)1P2 = {0.4/3, 0.8/4, 1/5, 0.8/6} (F O)1P3 = {0.6/2, 0.9/3, 1/4, 0.6/5} (F O)2P1 = {0.7/6, 1/7, 0.9/8} (F O)2P2 = {0.7/5, 0.8/6, 1/7, 0.8/8} (F O)2P3 = {0.6/4, 0.8/5, 0.9/6, 1/7, 0.7/8} represent the assessments of the Foreign Expert Group corresponding to the chosen parameter Pi . Suppose that the National Expert Group (N E G) uses the extension U of the uninorm  max(0, x + y − 4) if (x, y) ∈ [0, 4]2 U (x, y) = max(x, y) otherwise and the foreign expert group uses the extension FU of the uninorm ⎧ ⎪ if (x, y) ∈ [0, 4]2 ⎨ min(x, y)  U (x, y) = min(8, x + y − 4) if (x, y) ∈ [4, 8]2 ⎪ ⎩ max(x, y) otherwise both of them defined on the finite chain L 8 = {0, 1, 2, 3, 4, 5, 6, 7, 8}. Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Fig. 2. Aggregations of the evaluations corresponding to each one of the parameters Pi of Example 4.3. The blue cross represent the parameter P1 , the red diamonds represent the parameter P2 , and the green triangles represent the third parameter P3 . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Fig. 3. Graphical representation of the discrete fuzzy number which represents the final decision corresponding to Example 4.3.

It is easy to show that U and FU fulfill the order relation U FU because the uninorms U and U  satisfy the inequalities U (x, y) ⱕ U  (x, y) for all (x, y) ∈ L 8 .2 Easy calculations show that D(N O, P1 ) = {0.8/6, 0.9/7, 1/8} D(N O, P2 ) = {0.6/5, 0.7/6, 1/7, 0.7/8} D(N O, P3 ) = {0.5/0, 0.5/1, 0.5/2, 0.5/3, 0.5/4, 0.7/5, 1/6, 0.7/7, 0.4/8} and D(FO, P1 ) = {0.7/6, 1/7, 0.9/8} D(FO, P2 ) = {0.7/5, 0.8/6, 0.8/7, 1/8} D(FO, P3 ) = {0.6/2, 0.6/3, 0.6/4, 0.8/5, 0.9/6, 1/7, 0.7/8} Based on the step 5 we can compute the final opinion of each parameter Pi (all of them jointly depicted in Fig. 3): F(O, P1 ) = {0.7/6, 0.9/7, 1/8} F(O, P2 ) = {0.6/5, 0.7/6, 0.8/7, 1/8} F(O, P3 ) = {0.5/0, 0.5/1, 0.5/2, 0.5/3, 0.5/4, 0.7/5, 0.9/6, 1/7, 1/8} Finally, we obtain the discrete fuzzy number which expresses the final opinion of the two groups of experts: [U, FU](F(O, P1 ), F(O, P2 ), F(O, P3 )) = {0.7/6, 0.8/7, 1/8} where U and FU are the extensions of the uninorms U and U  defined above. The final discrete fuzzy number is depicted in Fig. 3. Thus, according to the previously obtained result, the company considers suitable to invest in this foreign country. Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001

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Please cite this article as: J. Vicente Riera, J. Torrens, Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.09.001