Evaluation of Fuzzy IF-THEN Rules Assignment Help

Fuzzy Rules And Their Operations - Evaluation of Fuzzy IF-THEN Rules

Evaluation of Fuzzy IF-THEN Rules

Look at the problem of evaluating a fuzzy IF-THEN rule as:

μA B  (a, b) namely μA  (a) ⇒ μB  (b), a ∈ A, b ∈ B

For classical two valued logic, this evaluation is easy as:

μB(b) = 1 if μA  (a) = 1,

And     0  if μA(a) = 0.

That is,

"a ∈ A ⇒ b ∈ B," and

μ (b) = 1 if μA  (a) = 1,

And       0 if μA (a) = 0.

As is, "a ∉ A ⇒ b ∈ B ."

Though, for fuzzy logic, according to the alternative of particular logical system, there are some options for the IF-THEN rule

" μA(a) ⇒ μA(a) "

 As like:

  • μA B (a, b) = min {μA (a), μB (b)} ;
  • μA B (a, b) = μA (a), μB(b) ;
  • μA B (a, b) = min {1, 1 + μB(b) - μ A (a)} ;
  • μA B (a, b) = max {min {μ A(a), μB (b)}, 1 - μ A(a)} ;
  • μA B (a, b) = max {1 - μ A(a), μB (b)} ;
  • Goguen's formula is:

                  34_Evaluation of Fuzzy IF-THEN Rules.png

Each these evaluation formulas are valid for the fuzzy logic inference purpose, provided one employs consistently the similar formula for the implication relation ⇒. Obviously, different formulas give various resulting values, which merely imply various degrees of inference based on various logical systems, however not the validity of the answers. Of course, formulas (a) and (b) are extremely simple to employ; but they are the similar as the logical AND operation ∧, formulas (d) and (f) appear too complicated. The most general one is formula (c).

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