Examples of the Implementation of Fuzzy Models in Tourism in the South Moravian Region - Geoinformatics for Intelligent Transportation - page 160

3.2 Compositional Rule of Inference

Practically we need to interpret verbal values of sets A, B mathematically and de ne

the rule of fuzzy relation R between variables X, Y. We use the compositional rule of
inference for assignment value B ′ of variable Y, which corresponds with value A ′

of

variable X.
We can get term, where the set B ′ is the sup-min composition of the fuzzy set A ′

and the fuzzy relation R, written as B 0 ¼ A 0 R with the membership [ 6 ]

l B 0 ð y Þ¼ sup
x 2 X

min ð l A 0 ð x Þ ; l R ð x ; y ÞÞ standard intersection

or generally

l B 0 ð y Þ¼ sup
x 2 X

T ð l A 0 ð x Þ ; l R ð x ; y ÞÞ union based on t-norm T

Rule (X, Y)isR(A, B)

X is A ′

Observing compositional rule of inference on t-norm T

, B 0 ¼ A 0 T R ð A ; B Þ Conclusion Y is B ′

We have to keep generalized modus ponens during relational reasoning, too, i.e.
A T R ð A ; B Þ¼ B.
The fuzzy relations can be modelled by a logical implication or by a cartesian
product T based on t-norm. We con ne to the second possibility and we get:

l R ð A ; B Þ ð x ; y Þ¼ T ð l A ð x Þ ; l B ð y ÞÞ
l B 0 ð y Þ¼ sup
x 2

ð l A 0 ð x Þ ; T l A ð x Þ ; l B ð y ÞÞÞ min

X

We can generalize the properties to t-norm T:

ðÞ ; T l A ðÞ ; l B ðÞ

l B 0 ðÞ¼ sup
x 2 X T l A 0

ð ð Þ Þ

If we choose T ¼ T ¼ T M we get:

l B 0 ðÞ¼ sup
x 2X

min ð l A 0 ðÞ ; min ð l A ðÞ ; l B ðÞ Þ Þ Mamdani's method

For T ¼ T M and T ¼ T P , it is: [ 5 ].

l B 0 ðÞ¼ sup

x 2X

min ð l A 0 ðÞ ; l A ðÞ l B ðÞ Þ Larsen's method