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:

MFA:  X         [0, 1]     

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……