The Basic Concepts of the Fuzzy Set Theory - Expert Fuzzy Information Processing - page 30

There are some algorithms of a fuzzy logic conclusion. Let us consider most
known of them, using system of two fuzzy conclusion rules:

A and Y is

C ;

B , then Z is

if X is

1

1 1

B , then Z is

A and Y is

C .

if X is

2

2

2

and having denoted membership functions of linguistic values of variables X , Y ,
Z through

μ

μ μ

~

i = 1 2 , , , accordingly.

~ ~

i

Algorithm of Mamdani
1. Fuzzification. Membership functions

μ

μ

, , are applied to

~

~ i = 1 2

x ,

y of variables X , Y .

physical (real) values

2. Fuzzy conclusion

[ ]

() ()

α = min μ x , μ y ;

~ ~

1 0 0

A B

1 1

[ ]

() ()

α =
Definition of the truncated membership functions of variable Z :

min μ x , μ y .

~ ~

2 0 0

A B

2 2

[ ]

()

μ = min α , μ

z

;

~ ~

1

1

C C

1

[ ]

() μ = min α , μ z .

~

~ 1

2

2

C C

2

3. A composition (union of the truncated membership functions)

[ ]

() () () μ z = max μ z , μ z .

~ ~

Σ

1
1

1
2

C C

4. Defuzzification:
For a continuous case

()

∫

z μ z dz

Σ

z =

U

;

()

0

∫

μ z dz

Σ

U

For a discrete case

()
()

∑

z μ z

i Σ i

z =

i

.

0 ∑

μ z

Σ i

i