Wednesday, May 8, 2013

The New Theorems to Measure the Distance among Triangular Fuzzy Numbers

In the reference with the article of “Fuzzy Method for Decision Making: A Case of Asset Pricing Model” posted on below link:
I remind you about the final part of above article which described new definitions and laws of the distance measurement among fuzzy numbers. The purpose of this article is to demonstrate the new theorems for measuring the distance among triangular fuzzy numbers where these theorems have been inferred from the method modified by Hsieh and Chen (1999).

Before releasing these theorems, let me divide the debate of the distance among triangular fuzzy numbers into two types as follows:

1)      All triangular fuzzy numbers have been limited into interval [x1, x2] in which “xm referred to membership function µ(x) is the same for all fuzzy numbers and equal to 1.   For instance, if we consider the fuzzy numbers of “A”, “B”, “C” and “D”, we will have:
A = (x1, xm, x2)
B = ((x1< x < xm), xm, (xm < x < x2))
C = ((x1< x < xm), xm, (xm < x < x2))
D = ((x1< x < xm), xm, (xm < x < x2))
Fig (1) shows us the concept of type (1):

2)      In this case, there is not the limitation for independent variable of “x” and also “x” assigned to membership function µ(x) = 1 is not the same for all fuzzy numbers in which fuzzy numbers A, B, C and D can be considered as follows:
      A = (x1, x2, x3)
B = (x4, x5, x6)
C = (x7, x8, x9)
D = (x10, x11, x12)
Fig (2) illustrates the concept of type (2):


The method Modified by Hsieh and Chen (1999)
According to Fu (2006), there are many approaches to measure the distance between two triangular fuzzy number such as Bortolan and Degani (1985), Liou and Wang (1992) and Heilpern (1997) who applied a geometrical distance measuring and his method was modified by Hsieh and Chen (1999). This modified method is described as follows:
If we have two triangular fuzzy numbers below cited:

Then, we can calculate the distance by below function:

The New theorems for measuring the distance among triangular fuzzy numbers

In the reference with above function, new theorems can be listed as follows:
Theorem (1)
Assume, we have three triangular fuzzy numbers A, B and C where:

Theorem (2)
According to theorem (1), we can define theorem (2) as follows:

Theorem (3)
If A = (a1, a2, a3) and B = ((n + a1), (n + a2), (n + a3)) are triangular fuzzy numbers so that “n” is real numbers (n ϵ R), Then we can define the distance between A and B as follows:
d (A, B) = n
Theorem (4)
If triangular fuzzy number “Ж” is the sum all triangular fuzzy numbers A, B, C, D, E…
We have:
Ж = A + B+ C + D + E+…    and    Д = B+ C + D + E+…   
Then we will have:
Д = Ж – A
Therefore, in the reference with Theorem (1), we can define below mentioned:
d ((Ж – A), Ж) = d (A, 2A)
d (Д, Ж) = d (A, 2A)
To be continued…….


- Fu, Guangtao. (2006). A fuzzy optimization method for multicriteria decision making:
An application to reservoir flood control operation. Expert Systems with Applications, 34, 145- 149.