A Study on Product-Sum of Triangular Fuzzy Numbers

We study the problem: if ãi, i ∈ N are fuzzy numbers of triangular form, then what is the membership function of the infinite (or finite) sum ã1 + ã2 + · · · (defined via the sup-product-norm convolution)


Introduction
A fuzzy number is a convex fuzzy subset of the real line R with a normalized membership function. A triangular fuzzy numberã denoted by (a, α, β) is defined as where a ∈ R is the center and α > 0 is the left spread, β > 0 is the right spread ofã. If α = β, then the triangular fuzzy number is called symmetric triangular fuzzy number and denoted by (a, α). Ifã andb are fuzzy numbers, then their product-sumã +b is defined as, The support suppã of a fuzzy numberã is defined as Let us take a setÃ, which is defined byÃ = {(x, µÃ(x)) : x ∈ A, µÃ(x) ∈ [0, 1]} . If in the pair (x, µÃ(x)) , the first one, x belongs to the classical set A and the second one µÃ(x) belongs to the interval [0,1], then setÃ is called a fuzzy set. Here µÃ(x) is called a Membership function.
. (λ and ω are the maximum value of upper and lower membership function, respectively) The upper and lower membership function of IVFN is defined by

Product-sum of triangular fuzzy numbers
In this section we shall calculate the membership function of the product-sum a 1 +ã 2 + · · · +ã n + · · · whereã i , i ∈ N are fuzzy numbers of triangular form. The following theorem can be interpreted as a central limit theorem for mutually product-related identically distributed fuzzy variables of symmetric triangular form.
exists and it is finite, then with the notations A n :=ã 1 + · · · +ã n , A n := a 1 + · · · + a n , n ∈ N we have lim n→∞Ã n (z) = exp(−|A − z|/α), z ∈ R proof: It will be sufficient to show that for each n ≥ 2, because from (1) it follows that lim n→∞Ã n (z) = lim From the definition of product-sum of fuzzy numbers it follows that suppÃ n = supp (ã 1 + · · · +ã n ) = suppã 1 + · · · + suppã n = [a 1 − α, a 1 + α] + · · · + [a n − α, a n + α] = [A n − nα, A n + nα] , n ∈ N We prove (1) by making an induction argument on n. Let n = 2. In order to determineÃ 2 (z), z ∈ [A 2 − 2α, A 2 + 2α] we need to solve the following mathematical programming problem: By using Lagrange's multipliers method and decomposition rule of fuzzy numbers into two separate parts, it is easy to see thatÃ 2 (z), z ∈ [A 2 − 2α, A 2 + 2α] is equal to the optimal value of the following mathematical programming problem: Using Lagrange's multipliers method for the solution of (2) we get that its optimal value is 1 − |A 2 − z| 2α 2 and its unique solution is X = 1/2 (a 1 − a 2 + z) (where the derivative vanishes). Indeed, it can be easily checked that the inequality In order to determineÃ 2 (z), z ∈ [A 2 , A 2 + 2α] we need to solve the following mathematical programming problem: In a similar manner we get that the optimal value of (3) is Let us assume that (1) holds for some n ∈ N . By similar arguments we obtaiñ This ends the proof.

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 +ã 2 + · · ·ã n + · · · whereã i , i ∈ N are fuzzy numbers of triangular form.