Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 271491, 12 pages

http://dx.doi.org/10.1155/2015/271491

## A Study of SUOWA Operators in Two Dimensions

Departamento de Economía Aplicada, Instituto de Matemáticas (IMUVA), Universidad de Valladolid, Avenida Valle de Esgueva 6, 47011 Valladolid, Spain

Received 17 March 2015; Accepted 14 June 2015

Academic Editor: Eckhard Hitzer

Copyright © 2015 Bonifacio Llamazares. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.