Chapter X: Introduction to Fuzzy Set Theory Uncertainty is universal ...

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{ A(x ) ∧ B(x ) y x # } ( A#B)(y) = sup 1 2 = 1 x2 . This is tedious at best, and in the continuous case involves solving a nonlinear program for each value of y. However, for fuzzy numbers and the basic arithmetic operators, we have that α (A#B) = ( α A) # ( α B) . Since the α-cuts of a fuzzy number are closed intervals, computing the a-cut of the extended arithmetic operation reduces to interval arithmetic, something that is easy to do. Then using the decomposition theorem, we finish this off by noting that ⎛ ⎞ ⎜ ⎟ A#B= ⎜ α ( A#B) ⎟ ⎝α∈[0,1] ⎠ ⎧ ⎪ if x - 4 or x 3 x 0 < > ⎪ + 4 ⎨ if - 4 x 0 Example X.xx: Let A(x) = - x 4 ≤ ≤ ⎪ 3 if 0 < x ≤ 3 ⎪ + ⎩ 3 and B(x) = ⎧ 0 ⎪ ⎨x + 2 ⎪ ⎩ - x if x < - 2 or x > 0 if - 2 ≤ x ≤ -1 if -1< x ≤ 0 Find 0.5 A and 0.5 B. Compute 0.5 A + 0.5 B = 0.5 (A+B) From the Extension Principle, give a detailed explanation of how the single value of MAX(A,B)(-3/2) is calculated. NEED PICTURE and EXAMPLE HERE R X.4.3 Compensatory operators
Of course, the meaningfulness of the results of an analysis as in example X.1 depends on the faithfulness of the model and the accuracy of assessing the input values. This problem is not specific to fuzzy set theory, but is inherent to all computational paradigms. The discussion here makes no overt claims to accurately model cancer risk, but is only used to demonstrate the flexibility of fuzzy connectives in decision processes. In Figure X.3, we might conjecture that the internal and external factors might better be combined in a compensative manner, i.e., more like an average than a union. Besides modeling negation (NOT), disjunction (OR) and conjunction (AND), fuzzy set theory admits mechanisms to model compensatory connections, i.e., aggregation operators where a high value in matching one criterion can compensate to some extent for a low value for another criterion. The simplest of these is called the generalized mean. If a1,a2,..., an are the degrees of satisfaction of n criteria, the generalized mean is defined as: 1/ α ⎛ α α α ⎞ ⎜ a + + + ⎟ α = 1 a 2 an h (a1,a2, ,an ) ⎜ n ⎟ (X.2) ⎝ ⎠ where α is a fixed real number. For α = 1, this equation implements the arithmetic average, for α = -1, we have the harmonic average, and for α converging to 0, Equation X.2 produces the geometric mean, the nth root of the product of the values. All instantiations of the generalized means produce values between the minimum and maximum of the degrees of satisfaction of the individual criteria. Additionally, { a , , } lim hα (a1, ,an ) = min 1 an α→−∞ { a , , } lim hα (a1, ,an ) = max 1 an α→∞ . This high degree of coverage makes fuzzy set connectives appealing for multicriteria decision making. In [Krishnapuram and Lee, 1992a, 1992b], Yager unions and intersections, along with generalized means, were used in hierarchical decision networks, and a gradient descent-based training algorithm was created to learn the parameters of the connectives in the network from a set of input/output training data. However, there were fairly cumbersome tests to decide if a node should be a union, intersection or mean (and to flip between them). A more general class of connectives, called fuzzy hybrid operators, combine all three types of linguistic connectives into a single equation. The typical arithmetic and multiplicative hybrid operators are given by: A ⊕ B = (1 − γ)(A ∩ B) 1−γ γ + γ(A ∪ B) (X.3) and
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