As I mentioned
in my previous article of “Fuzzy Delphi Method to Design a Strategic Plan”,
I would like to continue the debate on Distance Method.
But before going to the distance method, let me explain my story as
follows:
While I was working on discrepancy between basic and distance method on
driving forces, I encountered to a phenomena. Now, let me depict this
phenomenon in the framework of a theorem below cited:
Inequality Theorem in Fuzzy Logic
Assume,
there is the fuzzy subset A of X where X is a universal set. Then, we define
the fuzzy set of A by its membership function (MF=Membership Function) as
follows:
It
means that a real number MFA (x) in the interval [0, 1] is assigned to each
element x where x is a member of X and also the value of MFA (x) at x presents
the grade of membership of x in A.
We
consider below conditions for the fuzzy set A:
-Fuzzy
set A is a convex and normalized fuzzy set in which we can say
the fuzzy set A is a fuzzy number.
- Fuzzy
set A is a central triangular fuzzy number where we have:
For central triangular fuzzy number A= (a, b,
c): MFA
(x) = 2(x-a)/c-a If a <
x < b
MFA
(x) = 2(x-c)/a-c If b
< x < c
b = (a + c)/2
Now, we assume the set
of S is included all central triangular fuzzy numbers as follows:
S = [Ai], i = 1, 2, 3,…….n
In
fact, we have:
S =
[A1, A2, A3,…..An]
Or
S =
[(a1, b1, c1), (a2, b2, c2),
(a3, b3, c3),……(an, bn, cn)]
We
define the distance (di) between x1
and x2 into each central
triangular fuzzy number A
assigned to each alpha – cut level as follows:
di =
delta (x) If ai
< x < bi
bi
< x < ci, 0
< alpha-cut < 1 , i = 1,2,3,……
Theorem: If there is below inequality:
d1
< d2 < d3
< d4 …….< dn
Above inequality will be always the constant for
all alpha – cuts in the interval [0, 1].
I have two questions:
-Is this a new inequality
theorem in Fuzzy Numbers?
- If the answer is negative, could
you please introduce me the references?
To be continued……
