Some Characterizations based on Double Aggregation Operators*

Transcription

1 Some Characterizations based on Double Aggregation Operators* T. Calvo A. Pradera Department of Computer Science Dept. of Experimental Sciences and Engineering University of Alcald Rey Juan Carlos University Alcala de Henares, Spain Mbstoles, Spain Abstract Double aggregation operators are introduced and several basic properties are studied, as neutral and annihilator elements, idempotency, etc., which are illustrated by some examples. The classes of additive and comonotone additive double aggregation operators are characterized. Keywords: aggregation operator, annihilator and neutral elements, idempotency, symmetry, additivity. 1 Introduction Standard aggregation operators deal with the inputs of one (n-dimensional) vector, not distinguishing (up to the index order) between the sources of data to be aggregated. However, in several real situations such approach may be inadequate. Take, for example, a global evaluation of a product which is evaluated from several technical aspects and from several esthetic aspects, where the parameters of each type of evaluating aspects are independent. Even in the case of applying simple arithmetic means on each step of aggregation, the final global evaluation differ from global arithmetic mean. The aim of this paper is to introduce double aggregation operators; these seem to be appropriate for modelling some kind of aggregation problems like the previous one. We discuss several properties of these operators and give some examples. Observe that we can easily extend the concept of double aggregation operators to triple aggregation operators, etc., and this allows us to accept any natural number of different data sources. 2 Double aggregation operators: definition and basic properties We assume the reader familiar with the basic concepts related to ordinary aggregation operators (see [2] for a recent overview). Operators that allow to combine two given lists of information coming from different sources will be called double aggregation operators, and are defined in terms of ordinary aggregation operators as follows: Definition 1 Let F, G : IJ [0, l.in + [O, I.] be nen two aggregation operators and H : [O, [O, 1.1 a binary aggregation operator. A double aggregation operator is a function with Id2]= m2 U ((0) x N) U (N x (0)), defined, for each x(,) = (xi,...,xn) E [0, lin, y(m) = (~1,--- l~m) E [O,:LIm, by: 'This work is partially supported by the projects BFN , TIC C03-01, E007/2001 (first author) and the projects PB C02-02, TIC C04 and 07T/0011/2000 (second author)

2 Note that double aggregation operators fulfill the two basic properties that define standard aggregation operators: they are non-decreasing with respect to the usual Cartesian order and the boundary values 0 and 1 are idempotent elements. Next subsections study some other important basic properties of double aggregation operators: symmetry, neutral and annihilator elements and idempotency. 2.1 Symmetry In the case of ordinary aggregation operators, the symmetry property establishes that the ordering of the inputs does not affect the result obtained. When performing a double aggregation, symmetry may be defined in the following way: Definition 2 A double aggregation operator A will be called symmetric when for all x(,) E [0, lin, y(,) E [0, lim, (n,m) E Id2], it is A(x(n) 7 ~(m)) = A(x(u(n)), ~(~(m)) ) for any Per- mutation a of (1,..., n) and any permutation 4 of {1,..., m). (ii) A(e, y(,)) = G(y(,)) for each m E N. The concept of right neutral element can be defined in a similar way, and an element will be called a neutral element when it is both a left and a right neutral element of A. Clearly, an element e E [O,1] will be a neutral element of A if and only if it is a neutral element of F, G and H. Finally, it is also interesting to remark that a double aggregation operator can not have different left and right neutral elements. 2.3 Annihilator In addition to the differentiation between left and right positions, the annihilator of a double aggregation operator may have either a weak effect (affecting exclusively the list it belongs to) or a global one (affecting the whole aggregation process) : Definition 4 Let A be a double aggregation operator. An element a E [O, I.] will be called: Note that it is also possible to require symmetry only in one of the two input lists, and we could then speak about left symmetry and right symmetry. All these concepts may be easily characterized in terms of the symmetry of the underlying ordinary aggregation operators F and G: A will be left symmetric (respectively right symmetric) if and only if F (resp. G) is symmetric, and, therefore, it will be symmetric if and only if both F and G are symmetric. 2.2 Neutral Element Regarding the existence of a neutral element, this may occur in any of the two input lists or in both of them: Definition 3 Let A be a double aggregation operator. An element e E [0, I.] will be called a left neutral element when for all x(,) E [0, lin, y(,) E [0, lim it is: = A((xl,-..,Xi-l,Xi+I,.--,~n),~(m)) for eachn > 2, i E (1,..., n) and m E NU (0). a weak left annihilator of A when for all X(n) E [O, lin, ~ (m) E [O, lim it is A((x1,. a. xi-17 a7 xi+l, -.., xn),y(m)) = A(a,y(,)) for each n 2 2, i E (1,...,n) and m E NU (0). a left annihilator of A when for all X(n) E [O,lIn,~(m) E [O,lIm it is A((xI,-..,X~-I,~~X~+I,..-,X~)~Y(~)) = a for each n E N, i E (1,..., n) and m E NU (0). The concepts of weak right annihilator and right annihilator are defined in a similar way; in addition, an element will be called a (weak) annihilator when it is both a (weak) left annihilator and a (weak) right annihilator. The following characterization may be obtained: Proposition 1 An element a E [O,1] is a weak annihilator of A if and only if it is an annihila- (i) A((xI,..-,x~-I~~,x~+~,-.-~x~),Y(~)) tor of F and G; it is an annihilator of A if, in addition, it is an annihilator of H. Note also that an operator A may have both a weak left annihilator and a weak right annihilator

3 but not a weak annihilator (that is, left and right weak annihilators do not need to coincide): this is the case, for instance, of the operator AF,~,~, with F and G medians with two different parameters a and,6 and H the product t-norm; this operator has a as a weak left annihilator and,6 as a weak right annihilator. On the contrary, it Similarly to the ordinary case, it is not difficult to prove that the ptransform Aq of a double aggregation operator A = AF,~,~ is a double aggregation operator, Aq = AF,,~,,H,. may be proved that if a double aggregation ope- A distinguished example of ptransformation is rator has both a left and a right annihilator, then the duality, related to cpd(u) = 1 - u. Therefore, they need to coincide. the dual operator A ~= Aqd of a double aggrega- 2.4 Idernpotency tion operator A = AF,~,~ is also a double aggregation operator, = AFd,Gd,Hd. With regards to idempotency-related properties, the following concepts may be defined: Definition 5 Let A be a double aggregation operator. For each x E [O,l], n E N, x(~) will denote (2,...,x) E [0,].In. Then: Regarding self-duality (a double aggregation operator will be called self-dual if A~=A), the following result is obtained: Proposition 3 A double aggregation operator A is self-dual if and only if F, G and H are self-dual. A will be said to be left idempotent if A(X(~),~(,)) = A(X,~(~)) for each x E [O, 11, n > 2 and y(,) E [0, lirn, m E NU (0). A will be said to be idempotent if A(x(~), x(~)) = x for each x E [O,l], (n, m) E M21. The definition of right idempotency is analogous, and a double aggregation operator will be said to be double idempotent if it is both left and right idempotent. These definitions entail the following characterization: Proposition 2 A is double idempotent if and only if F and G are idempotent; it is idempotent if, in addition, H is idempotent. 3 Transformation of double aggregation operators One of the standard construction methods for aggregation operators is the transformation method, see, e.g., [2]. This concept can be straight forwardly extended to double aggregation operators: Definition 6 Let cp : [O, [O, 11 be a monotone bijection and let A be a given double aggregation operator. Then the ptransform Aq of A is defined by 4 Additive and comonotone additive double aggregation operators Two basic classes of standard aggregation operators are the additive and the comonotone additive aggregation operators. Recall that the additive aggregation operators are related to the Lebesgue integral and that they are just weighted means related to weighting triangles A = where n. win 2 0 and 2 win = 1. Similarly, comonotone i=l additive aggregation operators correspond to the Choquet integral ([3, 11) based aggregation related to systems (pn)nen of fuzzy measures on Xn = (1,..., n). These two concepts may be easily extended to the case of double aggregation operators: Definition 7 A double aggregation operator A is said to be i) additive, ii) comonotone additive, if for all x, u, x + u E [0, lin, y, v, y + v E [0, lim, n, m E NU{O), n+m > 0 it holds (for simplicity, we denote x(,) by x and so on): i) A(x + u, y + v) = A(x, y) + A(u, v), ii) A(x+u, y+v) = A(x, y)+a(u,v), whenever the couples x, u and y,v are comonotone, i.e., (xi - xj)(ui - uj) > 0 and (y, - ys)(v,-v,) 2 0 for alli,j E (1,..., n), r,s E {I,..,,m).

4 In the following we characterize the resulting double aggregation operators classes. Proposition 4 Let A = AF,~,~ be a double aggregation operator. Then A is additive if and only if all operators F, G, H are additive. Therefore, each comonotone additive double aggregation operator A can be represented as a Choquet integral with respect to a system of fuzzy measures (p,,), A where ptm = pe, pto = pr, and if n,m E N, then p&, acts on Xn+, and for I c Xn+, is defined by Because of this proposition, F and G will be associated, respectively, with weighting triangles AF = (w5)nen and ng = (~z)m~n, while k.p[(in{l,...,n))+(l-k)ape({j I j+n E I)). H(u,v) = ku + (1- k)v for some k E [0, 11. Then additive double aggregation operators AF,~,~, Remark 1 OWA operators [lo] are just symmewhich will be called double weighted means, will be tric Choquet integral based aggregation operarelated to triplets (k, AF, AG). This means that tors [7, 61. This characterization allows us to AF,~,~ can be associated to a 3-pieces weighting define a double OWA operator as a symmetric triangle AA = (~$,),,,~~[2~ = w, A given by: comonotone additive double aggregation operaw$, = wl; wto = w; and, if n, m E N, w&, = tor. Due to proposition 5, each double OWA opek n..., w F 1 - k)w&,...,(1- k) WE,). rator A can be represented in the form (n, m E IN) A((xI,...,xn),(YI,.--,~rn)) = ~F(xI,..- 7%) + Note that, then, a double weighted mean can be (1- k)g(yl,..., y,), where both F and G are represented by means of a simple weighted mean OWA operators. So, for example, we can put W with weighting triangle Aw only in two ex- k = 112, F = min, G = max, i.e., tremal situations. Namely, if either F = G = H = Pj is the projection of the first coordinate, ( min xi, ifm=o; then also A = Pj (i.e. k = l,af = AG = A(x(n) 1 ~ (rn)) = ifn=o; min zi+max yj otherwise. APf = (win) with wl, = 1,n E IN), or, similarly, F = G = H = Pi is the projection associated with the last coordinate, and then A = Pl. 5 Conclusions Example 1 Taking H, F and G equal to the arithmetic mean, i.e., k = 112 and AF = AG = (win) with win = lln, n E N,i E {I,...,n). 1 1 Then (k, AF, AG) is given by wio, =, wino =, and Similarly to the previous case, we can characterize all comonotone additive double aggregation operators. Proposition 5 Let A = AF,~,~ be a double aggregation operator. Then A is comonotone additive if and only if H is a weighted mean, H(u,v) = ku + (1- k)v for some k E [O, 11, and F and G are Choquet integral based aggregation operators related to some fuzzy measures systems (pc)ne~ and respectively, where pr and pz are fuzzy measures defined on Xn = (1,..., n). We have introduced double aggregation operators, which look convenient to aggregate inputs coming from two different sources. Several properties have been presented, related to the standard properties of aggregation operators. We have characterized the class of additive double aggregation operators which, up to the case of Pj and Pl, cannot be reduced to simple weighted means. Similarly, the class of comonotone additive double aggregation operators was characterized, including double OWA operators. Future work will include alternative approaches to the monotonicity of these operators, as well as the study of their behaviour when working with ordinal scales. References [I] P. Benvenuti and R. Mesiar: Integrals with respect to a general fuzzy measure. In: M. Grabisch, T. Murofushi and M. Sugeno, eds.

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