IEEE Systems, Man and Cybernetics Magazine - April 2021 - 9

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

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IEEE Systems, Man and Cybernetics Magazine - April 2021

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