w k = S k - S k - 1, where S 0 = 0. Quantifier-Guided Aggregation As mentioned in the previous section, an important application of OWA operators is in the area of quantifier-guided aggregations [30]. Compensative connectives are characterized by a higher degree of satisfaction of one of the criteria, which can compensate for a lower degree of satisfaction of another criterion. Although in classic binary logic we have two quantifiers, " there exists " (7) and " for all " (6), in natural language, we use a huge amount of additional quantifiers (e.g., " few, " " many, " " or, " and " almost all " ). This is how the theory of approximate reasoning extends binary logic. According to Zadeh [40], quantifiers can be represented as fuzzy subsets of the unit interval (or the real line). Zadeh suggested the use of two kinds of quantifiers: those saying something about the number of elements (absolute quantities) and those saying something about the proportion of elements (relative quantities). Definition 8 Let Q be a function Q :[0, 1] " [0, 1] such that Q (0) = 0, Q (1) = 1 and Q (x) $ Q (y) for x > y. In this case, Q is called a BUM function or a RIM quantifier. With the help of this RIM quantifier, we get the associated OWA weights the following way: i i-1 w i = Qa n k - Qa n k. Let us face a decision problem with n criteria, A 1, ... A n, where A i (x) = a i stands for the degree to which alternative x satisfies criteria A i . If the decision maker desires that Q of the criteria be satisfied, then Q is an absolute quantity definable on L = [0, n] . For xeL, Q(x) indicates the degree to which the decision maker is satisfied with x criteria being solved. We can easily see the following: ◆◆ Q (0) = 0, i.e., the decision maker gets absolutely no satisfaction if he/she gets no criteria satisfied ◆◆ Q (n) = 1, i.e., he/she is completely satisfied if he/she gets all the criteria satisfied ◆◆ If r1 2 r2, then Q (r1) $ Q (r2), which means that if he/ she gets more criteria satisfied he/she will not become less satisfied The overall valuation of x is FQ (a 1, ..., a n), where FQ is an OWA operator. We can see the weighting vector as a manifestation of the quantifier underlying the aggregation process. If the decision maker wants Q of the objectives satisfied, then we obtain the following weighting vector: w k = Q (k) - Q (k - 1), where k = 1,...n and Q (0) = 0. Furthermore, if weights are obtained, we have k Q (k) = | w i . i=1 For instance, for " and " we get w n = 1, w i = 0, if i ! n, Q (k) = 0, i f k ! n, Q (n) = 1. For " or, " we obt a i n w 1 = 1, w i = 0, if i ! 1, Q (k) = 1, if k $ 1. For the pure averaging quantifier, w i = ^1/n h, Q (k) = ^ k/n h, which means it is a linear quantifier. If Q is a relative quantity, then it can be represented as a fuzzy subset of I such that for each reI, Q (r) indicates the degree to which r portion of the objects satisfies concept Q. For example, the quantifier " for all " can be represented by a fuzzy subset of I such that Q (1) = 1 and Q ^ r h = 0, if r ! 1. Other mentioned quantifiers can be expressed in the following way: Z 0, if t # a, ]] t-a Q a,b (t) = [ b - a , if a 1 t 1 b, ] 1, if t $ b. \ For instance, for " most, " we can choose pairs (a, b) = (0.3, 0.8); for " at least half, " (0, 0.5); and for " as many as possible, " (0.5, 1) . This means that, e.g., for " most, " with (a, b) = (0.3, 0.8) and n = 5, the weighting vector is w = (0, 0.2, 0.4, 0.4, 0) . Data-Based Methods Data-based methods share the common feature of eliminating nonlinearity due to a reordering of the components of a by restricting the domain to the simplex S 1 [0, 1] n defined by inequalities a 1 # a 2 # ... # a n . Thus, in that domain, the OWA operator is a linear function (it coincides with the arithmetic mean). By finding the coefficients of this function, the OWA operator can be computed on the whole [0, 1] n by using its symmetry. One can use least squares or least absolute deviation criterion, employing either quadratic or linear programming techniques. Filev and Yager [9] suggested a nonlinear change in variables to obtain an unrestricted minimization problem; however, the resulting nonlinear optimization problem was rather difficult because of the large number of local minimizers. For example, an approach that relied on quadratic programming was used in [2] and was shown to be numerically efficient and stable. Additionally, a desired value of the measurement of orness is often imposed. This requirement can also be incorporated into a quadratic programming or linear programming problem as an additional linear-equality constraint. Measurement-Based Methods Another approach for obtaining weights based on a simple specification of the measurement of orness was suggested by O'Hagan [21]. In this approach, a decision maker specifies one parameter, a, the attitudinal character of the aggregation procedure. O'Hagan developed a way to generate OWA weights that have a predefined degree of orness and that maximize the entropy, referring to them as maximal entropy OWA operators. The suggested approach was algorithmically based on the solution of a Ap ri l 2021 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 9

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