A Study on Multiplication Operation on Triangular Fuzzy Numbers

The arithmetic operations on fuzzy number are basic content in fuzzy mathematics. But still the operations of fuzzy arithmetic operations are not established. There are some arithmetic operations for computing fuzzy number. Certain are analytical methods and further are approximation methods. In this paper we, compare the multiplication operation on triangular fuzzy number between α-cut method and standard approximation method and give some examples.

Fuzzy sets have been introduced by Lotfi.A. Zadeh(1965). Since its inception 50 years ago, the theory of fuzzy sets has advanced in a variety of ways and in many disciplines. Applications of this theory can be found, for example, in artificial intelligence, computer science, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, and robotics. Theoretical advances have been made in many directions. In application fuzzy set theory fuzzy number plays an important role. Arithmetic operations on fuzzy numbers have also been developed, and are based mainly on the extension principle or interval arithmetic. When operating with fuzzy numbers, the result of our calculations strongly depend on the shape of the membership functions of these numbers. Less consistent membership functions lead to more complicated calculations. Moreover, fuzzy numbers with simpler shape of membership functions often have more intuitive and more natural interpretation. Considering the interval arithmetic -based arithmetic operations on triangular fuzzy numbers, the product of two such fuzzy numbers is not of the same kind: the shape of these fuzzy numbers is not preserved. In many situations this problem is solved by approximation multiplication by a triangular or trapezoidal fuzzy number. In this paper we mentioned interval arithmetic operation as α-cut method and approximation multiplication as standard approximation and compare them over triangular fuzzy number.

A fuzzy set r
A is defined byÃ " tpx, µ A pxqq : x P A, µ A pxq P r0, 1su. In the pair px, µ A pxqq , the first element x belong to the classical set A, the second element µ A pxq, belong to the interval r0, 1s, called Membership function.

Definition 2.2 Support of Fuzzy Set
The support of fuzzy setÃ is the set of all points x in X such that µÃ ą 0. That is Support pÃq " tx | µÃpxq ą 0u Definition 2.3 α -cut The α -cut of α -level set of fuzzy setÃ is a set consisting of those elements of the universe X whose membership values exceed the threshold level α. That isÃ α " tx | µÃpxq ě αu.

Definition 2.4 Fuzzy Number A fuzzy set r
A on R must possess at least the following three properties to qualify as a fuzzy number, (i)Ã must be a normal fuzzy set; (ii)Ã α must be closed interval for every α P r0, 1s (iii) the support ofÃ,Ã 0`, must be bounded.

Definition 2.5 Triangular Fuzzy Number
It is a fuzzy number represented with three points as follows:Ã " pa 1 , a 2 , a 3 q. This representation is interpreted as membership functions and holds the following conditions (i) a 1 and a 2 is increasing function (ii) a 2 and to a 3 is decreasing function (iii) a 1 ď a 2 ď a 3 .

Figure 1: Triangular fuzzy number
Definition 2.6 α -cut of a triangular fuzzy number We get a crisp interval by α-cut operation; interval A α shall be obtained as follows @ α P r0, 1s. Thus Definition 2.7 Positive triangular fuzzy number A positive triangular fuzzy numberÃ is denoted asÃ " pa 1 , a 2 , a 3 q , where all a i 's ą 0 for all i " 1, 2, 3 Definition 2.8 Negative triangular fuzzy number A negative triangular fuzzy numberÃ is denoted asÃ " pa 1 , a 2 , a 3 q , where all a i 's ă 0 for all i " 1, 2, 3.
Definition 2.9 Partial Negative triangular fuzzy number A Partial Negative triangular fuzzy numberÃ is denoted asÃ " pa 1 , a 2 , a 3 q , where at least one a i is negative for all i " 1, 2, 3.

α´cut Method
The actual result is found by rewriting the membership function to define a set of closed intervals as in expression (1). Then the expressions defining the closed intervals are operated on using interval arithmetic. Case (i): For two positive fuzzy numbers A " xa 1 , b 1 , c 1 ąÑ rpb 1´a1 q α`a 1 ,´pc 1´b1 q α`c 2 s " rs 1 ,s 1 s r Say s B "ă a 2 , b 2 , c 2 ąÑ rpb 2´a2 q α`a 2 ,´pc 2´b2 q α`c 2 s " rs 2 ,s 2 s r Say s The product can be calculated, C "Ã bB " rmin ps 1 s 2 , s 1s2 ,s 1 s 2 ,s 1s2 q , max ps 1 s 2 , s 1s2 ,s 1 s 2 ,s 1s2 qs for α P p0, 1s Case(ii): When anyÃ andB is partial negative and other is positive fuzzy number, the product ofÃ bB can't obtain according to the expression (2). The interval of α P p0, 1s will be divided into two parts, according to the intersection point of the two minimum expression in the α´cut ofC. Let us this intersection point is α s . Theñ

C "Ã bB "
« min ps 1 s 2 , s 1s2 ,s 1 s 2 ,s 1s2 q , max ps 1 s 2 , s 1s2 ,s 1 s 2 ,s 1s2 q for α P p0, α s s min ps 1 s 2 , s 1s2 ,s 1 s 2 ,s 1s2 q , max ps 1 s 2 , s 1s2 ,s 1 s 2 ,s 1s2 q for α P pα s , 1s ff (3) Case (iii): WhenÃ is negative andB is positive fuzzy number. Then the multiplication ofÃ andB can be found as expression (2). Case (iv): WhenÃ andB both are negative fuzzy number. Then the multiplication ofÃ andB can also be found as expression (2).The expression (2) and (3) are known as analytical method for fuzzy arithmetic operation. The standard approximation and actual results are shown in figure p1, 2, 3, and 4q. In actual product the lines connecting the ends points are parabolic and in standard approximation lines connecting the ends points are triangular form.

Error analysis for α´cut Method
The error is the difference at a given α -level, between the approximated membership function and exact results in the expression (2) and (3). Each TFN can be separated in to left and right segments in accordance with the LR parametered representation. The actual product of (2) and (3) will have the value x at a given α defined as T L for the left segment, and T R for the right segment. The standard approximation (1) will have a value x, at a given α defined as P L and P R for the left and right segments respectively. This follows us to separately analyses the left and right portions of the membership curve. The left and right segment error are then.
E L " P L´TL and E R " P R´TR Graphically this is the horizontal distance between the two curves as shown in Figure   p1, 2, 3, and 4) and numerical error is shown in table p2, 3, 4 and 5q.

Conclusion
In this paper, we studied about the Product-sum of triangular fuzzy numbers and also calculated the membership function of the product-sumã 1`ã2`¨¨¨ãn`¨¨ẅ hereã i , i P N are fuzzy numbers of triangular form.