A Study of SUOWA Operators in Two Dimensions

SUOWA operators are a new class of aggregation functions that simultaneously generalize weighted means and OWA operators. They areChoquet integral-based operatorswith respect to normalized capacities; therefore, they possess some interesting properties such as continuity, monotonicity, idempotency, compensativeness, and homogeneity of degree 1. In this paper, we focus on two dimensions and show that any Choquet integral with respect to a normalized capacity can be expressed as a SUOWA operator.


Introduction
The study of aggregation operators has received special attention in the last years.This is due to the extensive applications of these functions for aggregating information in a wide variety of areas.Two of the best-known aggregation operators are the weighted means and the ordered weighted averaging (OWA) operators (Yager [1]).Both classes of functions are defined by means of weighting vectors, but their behavior is quite different.Weighted means allow weighting each information source in relation to their reliability while OWA operators allow weighting the values according to their ordering.
A typical situation where both weightings are necessary is the following (Llamazares [9]): suppose we have several sensors to measure a physical property.On the one hand, sensors may be of different quality and precision, so a weighted mean type aggregation is necessary.On the other hand, to prevent a faulty sensor from altering the measurement, we might consider an OWA type aggregation where the maximum and minimum values are not taken into account.A similar situation occurs when a committee of experts has to assess several candidates or proposals.On the one hand, a weighted mean type aggregation is suitable for reflecting the expertness or the confidence in the judgment of each expert.On the other hand, an OWA type aggregation allows us to deal with situations where an expert feels excessive acceptance or rejection towards some of the candidates or proposals.
Different aggregation operators have appeared in the literature to deal with this kind of problems.A usual approach is to consider families of functions parameterized by two weighting vectors, one for the weighted mean and the other one for the OWA type aggregation, which generalize weighted means and OWA operators in the following sense.A weighted mean (or an OWA operator) is obtained when the other weighting vector has a "neutral" behavior; that is, it is (1/, . . ., 1/) (see Llamazares [10] for an analysis of some functions that generalize the weighted means and the OWA operators in this sense).Two of the solutions having better properties are the weighted OWA (WOWA) operator, proposed by Torra [3], and the semiuninorm based ordered weighted averaging (SUOWA) operator, introduced by Llamazares [9].

Mathematical Problems in Engineering
The good properties of WOWA and SUOWA operators are due to the fact that they are Choquet integral-based operators with respect to normalized capacities.In the case of SUOWA operators, their capacities are the monotonic cover of certain games, which are defined by using the capacities associated with the weighted means and the OWA operators and "assembling" these values through semiuninorms with neutral element 1/.
Because of their good properties, it seems interesting to analyze the behavior of SUOWA operators from different points of view.In this paper, we consider the case of two dimensions that, although simple, is attractive from a theoretical point of view, and we show that any Choquet integral with respect to a normalized capacity can be expressed as a SUOWA operator.
The remainder of the paper is organized as follows.In Section 2 we recall the concepts of semiuninorm and uninorm and give some interesting examples of such functions.Section 3 is devoted to Choquet integral, including some of the most important particular cases: weighted means, OWA operators, and SUOWA operators.In Section 4, we give the main results of the paper.Finally, some concluding remarks are provided in Section 5.
Semiuninorms are a class of necessary functions in the definition of SUOWA operators.They are monotonic and have a neutral element in the interval [0, 1].These functions were introduced by Liu [11] as a generalization of uninorms, which, in turn, were proposed by Yager and Rybalov [12] as a generalization of -norms and -conorms.
Before introducing the concepts of semiuninorm and uninorm, we recall some well-known properties of aggregation functions.Definition 1.Let  : R  → R be a function.
(5)  is homogeneous of degree 1 (or ratio scale invariant) if (x) = (x) for all x ∈ R  and for all  > 0.
SUOWA operators are defined by using semiuninorms with neutral element 1/.Moreover, they have to belong to the following subset (see Llamazares [9]): Obviously, U 1/  ⊆ Ũ1/ .Notice that the smallest and the largest elements of Ũ1/ are, respectively, the following semiuninorms: max (, ) otherwise. ( Mathematical Problems in Engineering 3 In the case of idempotent semiuninorms, the smallest and the largest elements of U 1/  are, respectively, the following uninorms (which were given by Yager and Rybalov [12]): max (, ) otherwise. ( In addition to the previous ones, several procedures to construct semiuninorms have been introduced by Llamazares [13].One of them, which is based on ordinal sums of aggregation operators, allows us to get continuous semiuninorms.Some of the most relevant continuous semiuninorms obtained are the following: Notice that the last two semiuninorms are also idempotent.The plots of all these semiuninorms are given, for the case  = 4, in Figures 1-8.

Choquet Integral
The notion of Choquet integral is based on that of capacity (see Choquet [14] and Murofushi and Sugeno [15]).The concept of capacity resembles that of probability measure but in the definition of the former additivity is replaced by monotonicity (see also fuzzy measures in Sugeno [16]).A game is then a generalization of a capacity where the monotonicity is no longer required.(2) A capacity (or fuzzy measure)  on  is a game on  satisfying () ≤ () whenever  ⊆ .In particular, it follows that  : A straightforward way to get a capacity from a game is to consider the monotonic cover of the game (see Maschler and Peleg [17] and Maschler et al. [18]).( Some basic properties of υ are given in the sequel.

Remark 5.
Let  be a game on .Then, one has the following: (1) υ is a capacity.(2) If  is a capacity, then υ = .
It is worth noting that the Choquet integral has several properties which are useful in certain information aggregation contexts (see, for instance, Grabisch et al. [20, pages 192-193 and page 196]).

Remark 7.
Let  be a capacity on .Then, C  is continuous, monotonic, and homogeneous of degree 1.Moreover, it is idempotent and compensative when  is a normalized capacity.
Notice that the Choquet integral can also be represented by using decreasing sequences of values (see, for instance, Torra [21] and Llamazares [9]): where where we use the convention  [0] = ⌀.

Weighted Means and OWA Operators.
Weighted means and OWA operators (Yager [1]) are well-known functions in the field of aggregation operators.Both families of functions are defined in terms of weight distributions that add up to 1. Definition 8.A vector q ∈ R  is a weighting vector if q ∈ [0, 1]  and ∑  =1   = 1.
The set of all weighting vectors of R  will be denoted by W  .Definition 9. Let p be a weighting vector.The weighted mean associated with p is the function  p : R  → R given by Definition 10.Let w be a weighting vector.The OWA operator associated with w is the function  w : R  → R given by It is well known that weighted means and OWA operators are a special type of Choquet integral (see, for instance, Fodor et al. [22], Grabisch [23,24], or Llamazares [9]).
Remark 11. (1) If p is a weighting vector, then the weighted mean  p is the Choquet integral with respect to the normalized capacity  p () = ∑ ∈   .
(2) If w is a weighting vector, then the OWA operator  w is the Choquet integral with respect to the normalized capacity  |w| () = ∑ || =1   .
So, according to Remark 7, weighted means and OWA operators are continuous, monotonic, idempotent, compensative, and homogeneous of degree 1.Moreover, in the case of OWA operators, given that the values of the variables are previously ordered in a decreasing way, they are also symmetric.

SUOWA Operators.
SUOWA operators were introduced by Llamazares [9] in order to consider situations where both the importance of information sources and the importance of values had to be taken into account.These functions are Choquet integral-based operators where their capacities are the monotonic cover of certain games.These games are defined by using semiuninorms with neutral element 1/ and the values of the capacities associated with the weighted means and the OWA operators.To be specific, the games from which SUOWA operators are built are defined as follows.
Definition 12. Let p and w be two weighting vectors and let  ∈ Ũ1/ .
(1) The game associated with p, w, and  is the set function   p,w : 2  → R defined by if  ̸ = ⌀ and   p,w (⌀) = 0. (2) υ p,w , the monotonic cover of the game   p,w , will be called the capacity associated with p, w, and .
Notice that   p,w () = 1.Moreover, since  ∈ Ũ1/ , we have   p,w () ≤ 1 for all  ⊆  (see Llamazares [9]).Therefore, according to the third item of Remark 5, υ p,w is always a normalized capacity.Definition 13.Let p and w be two weighting vectors and let  ∈ Ũ1/ .The SUOWA operator associated with p, w, and  is the function   p,w : R  → R given by where ) for all  ∈ , υ p,w is the capacity associated with p, w, and , and According to expression (7), the SUOWA operator associated with p, w, and  can also be written as By the choice of υ p,w , we have   p, =  p and   ,w =  w for any  ∈ Ũ1/ .Moreover, by Remark 7 and given that υ p,w is a normalized capacity, SUOWA operators are continuous, monotonic, idempotent, compensative, and homogeneous of degree 1.

The Results
The use of Choquet integral has become more and more extensive in the last years (see, for instance, Grabisch et al. [25] and Grabisch and Labreuche [26]).Although simple, the case  = 2 is interesting from a theoretical point of view.Thus, for instance, Grabisch et al. [20, page 204] show that, in this case, any Choquet integral with respect to a normalized capacity can be written as a convex combination of a minimum, a maximum, and two projections; that is, given a normalized capacity , there exists a weighting vector  belonging to W 4 such that In our case, we are going to show that any Choquet integral with respect to a normalized capacity can be written as a SUOWA operator.Notice that when  = 2,   p,w is always a normalized capacity for any weighting vectors p and w and for any semiuninorm .Therefore, given a normalized capacity , we need to prove that there exist weighting vectors p and w and a semiuninorm  such that where we use the notations  1 and  2 to denote the values ({1}) and ({2}), respectively.Firstly we are going to show that, in the case of the semiuninorms  ⊥ ,  ⊤ ,  min , and  max , there exist normalized capacities which cannot be expressed as SUOWA operators.For this, we will use the following lemma.Lemma 14.If  ∈ { ⊥ ,  ⊤ ,  min ,  max }, then (, ) = 0.5 if and only if  =  = 0.5.
In each of the following theorems we consider the semiuninorms   L ,   M ,  P, and   , respectively, and we show that any normalized capacity can be written as a SUOWA operator associated with appropriate weighting vectors p and w, which are given explicitly.

Definition 4 .
Let  be a game on .The monotonic cover of  is the set function υ given by υ () = max ⊆  () .
1], . . ., []}, and we use the convention  [+1] = 0. From the previous expression, it is straightforward to show explicitly the weights of the values  [] by representing the Choquet integral as follows: