Abstract
SUOWA operators are a new class of aggregation functions that simultaneously generalize weighted means and OWA operators. They are Choquet integral-based operators with 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.
1. 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.
Although both families of operators allow solving a wide range of problems, both weightings are necessary in some contexts. Some examples of these situations have been given by several authors (see, for instance, Torra [2–4], Torra and Godo [5, pages 160-161], Torra and Narukawa [6, pages 150-151], Roy [7], Yager and Alajlan [8], and Llamazares [9] and the references therein) in fields as diverse as robotics, vision, fuzzy logic controllers, constraint satisfaction problems, scheduling, multicriteria aggregation problems, and decision-making.
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 (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].
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 .
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.
2. Semiuninorms and Uninorms
Throughout the paper, we will use the following notation: ; given , denotes the cardinality of ; vectors are denoted in bold and denotes the tuple . We write if for all . For a vector , and denote permutations such that and .
Semiuninorms are a class of necessary functions in the definition of SUOWA operators. They are monotonic and have a neutral element in the interval . 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 be a function.(1) is symmetric if for all and for all permutation of .(2) is monotonic if implies for all .(3) is idempotent if for all .(4) is compensative (or internal) if for all .(5) is homogeneous of degree 1 (or ratio scale invariant) if for all and for all .
Definition 2. Let .(1) is a semiuninorm if it is monotonic and possesses a neutral element ( for all ).(2) is a uninorm if it is a symmetric and associative ( for all ) semiuninorm.
We denote by (resp., ) the set of semiuninorms (resp., idempotent semiuninorms) with neutral element .
SUOWA operators are defined by using semiuninorms with neutral element . Moreover, they have to belong to the following subset (see Llamazares [9]):
Obviously, . Notice that the smallest and the largest elements of are, respectively, the following semiuninorms:
In the case of idempotent semiuninorms, the smallest and the largest elements of are, respectively, the following uninorms (which were given by Yager and Rybalov [12]):
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 , in Figures 1–8.
3. 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.
Definition 3. (1) A game on is a set function, satisfying .
(2) A capacity (or fuzzy measure) on is a game on satisfying whenever . In particular, it follows that . A capacity is said to be normalized if .
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]).
Definition 4. Let be a game on . The monotonic cover of is the set function given by
Some basic properties of are given in the sequel.
Remark 5. Let be a game on . Then, one has the following:(1) is a capacity.(2)If is a capacity, then .(3)If for all and , then is a normalized capacity.
Although the Choquet integral is usually defined as a functional (see, for instance, Choquet [14], Murofushi and Sugeno [15], and Denneberg [19]), in this paper we consider the Choquet integral as an aggregation function over (see, for instance, Grabisch et al. [20, page 181]). Moreover, we define the Choquet integral for all vectors of instead of nonnegative vectors given that we are actually considering the asymmetric Choquet integral with respect to (on this, see again Grabisch et al. [20, page 182]).
Definition 6. Let be a capacity on . The Choquet integral with respect to is the function given bywhere , and one uses the convention .
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, 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 , and we use the convention .
From the previous expression, it is straightforward to show explicitly the weights of the values by representing the Choquet integral as follows: where we use the convention .
3.1. 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 is a weighting vector if and .
The set of all weighting vectors of will be denoted by .
Definition 9. Let be a weighting vector. The weighted mean associated with is the function given by
Definition 10. Let be a weighting vector. The OWA operator associated with is the function 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 is a weighting vector, then the weighted mean is the Choquet integral with respect to the normalized capacity .
(2) If is a weighting vector, then the OWA operator is the Choquet integral with respect to the normalized capacity .
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.
3.2. 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 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 and be two weighting vectors and let . (1)The game associated with , , and is the set function defined by if and .(2), the monotonic cover of the game , will be called the capacity associated with , , and .
Notice that . Moreover, since , we have for all (see Llamazares [9]). Therefore, according to the third item of Remark 5, is always a normalized capacity.
Definition 13. Let and be two weighting vectors and let . The SUOWA operator associated with , , and is the function given bywhere for all , is the capacity associated with , , and , and (with the convention that ).
According to expression (7), the SUOWA operator associated with , , and can also be written as
By the choice of , we have and for any . Moreover, by Remark 7 and given that is a normalized capacity, SUOWA operators are continuous, monotonic, idempotent, compensative, and homogeneous of degree 1.
4. 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 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 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 , is always a normalized capacity for any weighting vectors and and for any semiuninorm . Therefore, given a normalized capacity , we need to prove that there exist weighting vectors and and a semiuninorm such that where we use the notations and to denote the values and , respectively.
Firstly we are going to show that, in the case of the semiuninorms , , , and , there exist normalized capacities which cannot be expressed as SUOWA operators. For this, we will use the following lemma.
Lemma 14. If , then if and only if .
Proof. Let . Since is the neutral element of , we have .
Conversely, suppose . In Table 1, where stands for a value that belongs to and stands for a value that belongs to , we show the values taken by the semiuninorms , , , and when . Therefore, if , then necessarily .
Theorem 15. Let be the normalized capacity on such that and . If , then there do not exist weighting vectors and such that .
Proof. Given , consider two weighting vectors and such that . By Lemma 14, we have . Therefore, and, consequently, is not possible.
In each of the following theorems we consider the semiuninorms , , , and , respectively, and we show that any normalized capacity can be written as a SUOWA operator associated with appropriate weighting vectors and , which are given explicitly.
Theorem 16. Let be a normalized capacity on and let and be two weighting vectors defined as follows:(1)If , then (2)If and , then (3)If , then Then, ; that is, .
Proof. Let be a normalized capacity on and recall that when , the semiuninorm is defined byWe distinguish the following cases:
(1) If , consider Then,(2) If and , consider We distinguish two cases:(a)If , then (b)If , then
(3) If , consider We distinguish three cases:(a)If , then (b)If , then (c)If , then
Theorem 17. Let be a normalized capacity on and let and be two weighting vectors defined as follows:(1)If , then (2)If and , then (3)If and , then (4)If , then Then, ; that is, .
Proof. Let be a normalized capacity on and recall that when , the semiuninorm is defined byWe distinguish the following cases:
(1) If , consider We distinguish three cases:(a)If , then (b)If , then (c)If , then
(2) If and , consider We distinguish two cases:(a)If , then (b)If , then
(3) If and , then the proof of this case is similar to that of the second item in Theorem 16.
(4) If , then the proof of this case is similar to that of the third item in Theorem 16.
Theorem 18. Let be a normalized capacity on and let and be two weighting vectors defined as follows: (1)If , then (2)If and , then (3)If and , then Then, ; that is, .
Proof. Let be a normalized capacity on and recall that when , the semiuninorm is defined by We distinguish the following cases:
(1) If , consider We distinguish two cases:(a)If , then (b)If , then
(2) If and , then notice that the case is not possible. Moreover, we have On the other hand, given that , we getand, consequently, . Now consider the following weighting vectors: We distinguish two cases:(a)If , then (b)If , then
(3) If and , then , and we also haveor, equivalently,On the other hand, since , we getand, consequently,Consider now the following weighting vectors: We distinguish three cases:(a)If , then (b)If , then (c)If , then
Theorem 19. Let be a normalized capacity on and let and be two weighting vectors defined as follows: (1)If and , then (2)If and , then (3)If and , then (4)If and , then Then, ; that is, .
Proof. Let be a normalized capacity on and recall that when , the semiuninorm is defined by We distinguish the following cases:
(1) If and , then notice that the case is not possible. Moreover, , and we also haveor, equivalently,On the other hand, since , we getand, consequently,Consider now the following weighting vectors: We distinguish three cases:(a)If , then (b)If , then (c)If , then
(2) If and , then notice that the case is not possible. Moreover, we have On the other hand, given that , we getand, consequently, . Now consider the following weighting vectors: We distinguish two cases: (a)If , then (b)If , then
(3) If and , then the proof of this case is similar to that of the second item in Theorem 18.
(4) If and , then the proof of this case is similar to that of the third item in Theorem 18.
5. Conclusion
SUOWA operators are a useful tool for dealing with situations where combining values by using both a weighted mean and an OWA type aggregation is necessary. Given that they are Choquet integral-based operators with respect to normalized capacities, they have some natural properties such as continuity, monotonicity, idempotency, compensativeness, and homogeneity of degree 1. For this reason, it seems interesting to analyze their behavior from different points of view. In this paper, we have shown that, in two dimensions, if we consider one of the following continuous semiuninorms: , , , and , then any Choquet integral with respect to a normalized capacity can be expressed as a SUOWA operator associated with the chosen semiuninorm and two weighting vectors and , which are given explicitly.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The partial financial support from the Ministerio de Economía y Competitividad (Project ECO2012-32178) and the Junta de Castilla y León (Consejería de Educación, Project VA066U13) is gratefully acknowledged.