The rules of combining membership functions discussed above
are known as the minmax rule for conjunctive (AND) and disjunctive
(OR) reasoning. These rules have a major drawback: They are not robust
at all. If we try to imitate the way humans reason, the minmax
rule is definitely not the way.
Many researchers have proposed different rules of combining conjunctive
or disjunctive clauses: for example, instead of taking the minimum or the
maximum of the membership functions, they take the arithmetic or the geometric
mean. These rules are arbitrary, and there are lots of them. It is possible,
if we have enough training data, i.e. conditions and class assignments
by the experts, to train our system so that it chooses the best rule that
fits the way of reasoning of the expert that did the classification.
Another disadvantage of the rules discussed earlier is that they give
the same importance to all factors that are to be combined. For example,
it is possible that the role of soil depth or rock permeability is not
of the same importance to soil erosion as the role of slope. This issue
can be resolved if we do not insist on all membership functions taking
values between 0 and 1.
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