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

lim (A ∩w
B)(x) = A(x) ∧ B(x) .
w→0
At the other end, i.e., the limits as w
→
∞ , generate the drastic union and intersection, defined by
⎧A(x)
if B(x) = 0
⎪
( A ∪d B)(x) = ⎨B(x)
if A(x) = 0 and
⎪
⎩ 1 else
⎧A(x)
if B(x) = 1
⎪
( A ∩d B)(x) = ⎨B(x)
if A(x) = 1.
⎪
⎩ 0 else
Besides their use in what is to come, fuzzy operators have been used extensively in multicriteria decision
making [Bellman and Zadeh, 1970; Yager, 1988; Yager, 2004]. We end this section with an example of
the use of fuzzy set operators in multicriteria decision making.
Example X.1. As a particularly simplistic illustration, consider a decision tree, as in Figure X.3, to assess
cancer risk based on the following observations. Suppose we decide that cancer risk should be high if
either internal factors or environmental factors are high. This is modeled by a union operator (OR). We
define the internal factors to be the conjunction (AND) of genetic predisposition and genetic test results.
In this particular example, the rationale for using a conjunction might be that for some types of cancer, the
test might be subject to a high rate of false positives, and so, these results can be offset by low genetic
propensity. Environmental factors are aggregated as the disjunction (OR) of amount of smoking,
hazardous work risk and the negation, or complement (NOT), of good nutrition. Clearly, this is a gross
oversimplification and is included to demonstrate the utility of fuzzy operators in multicriteria decision
making more than focusing on reality. In Figure X.3, we model each of the operators with the
corresponding Yager connective (Equation X.1). The parameters for these four connectives will be
labeled w 1 for the top disjunction, w 2 for the conjunction of internal factors, w 3 for the disjunction of
external factors and w 4 for the complement.
Given the tree in Figure 2.4, suppose that we have determined the following fuzzy membership values for
the leaf nodes: propensity = 0.2, test results = 0.8, smoking risk = 0, job risk = 0.1, and good nutrition =
0.8. That is, a particular patient has a low genetic propensity but a reasonably high likelihood from a test,
while living a good lifestyle. If the logical operators are the classical binary ones, then the risk of cancer
would be zero for this set, assuming that the fuzzy memberships are turned binary at a 0.5 threshold. The
advantage of fuzzy set theory is that the operators that govern complement, disjunction and conjunction