Monday, February 6, 2012
Fuzzy Delphi Method to Design a Strategic Plan (CON). Is This a New Inequality Theorem in Fuzzy Set Theory?
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]
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……
Posted by Gholamreza Soleimani at 3:58 PM