On product-sum of triangular fuzzy numbers ∗
Robert Fullér
rfuller@abo.fi
Abstract
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)?
Keywords: triangular fuzzy number, product-sum
1
Definitions
A fuzzy number is a convex fuzzy subset of the real line R with a normalized membership function. A
triangular fuzzy number ã denoted by (a, α, β) is defined as

a−t



if a − α ≤ t ≤ a
1−


α



 1
if a ≤ t ≤ b
ã(t) =

t−b


1
−
if a ≤ t ≤ b + β



β



0
otherwise
where a ∈ R is the centre and α > 0 is the left spread, β > 0 is the right spread of ã.
If α = β, then the triangular fuzzy number is called symmetric triangular fuzzy number and denoted by
(a, α).
If ã and b̃ are fuzzy numbers, then their product-sum ã + b̃ is defined as in [2]
(ã + b̃)(z) = sup ã(x)b̃(y).
x+y=z
The support suppã of a fuzzy number ã is defined as
suppã = {t ∈ R|ã(t) > 0}.
∗
The final version of this paper appeared in: R. Fullér, On product-sum of triangular fuzzy numbers, Fuzzy Sets and Systems,
41(1991) 83-87. doi: 10.1016/0165-0114(91)90158-M
1
2
Product-sum of triangular fuzzy numbers
In this paper we shall calculate the membership function of the product-sum ã1 + ã2 + · · · + ãn + · · ·
where ãi , i ∈ N are fuzzy numbers of triangular form.
The results of this paper are linked with those presented in [1,p.933] and extend them. The following
theorem can be interpreted as a central limit theorem for mutually product-related identically distributed
fuzzy variables of symmetric triangular form (see [3]).
Theorem 2.1 Let ãi = (ai , α), i ∈ N. If
A :=
∞
X
ai
i=1
exists and it is finite, then with the notations
Ãn := ã1 + · · · + ãn , An := a1 + · · · + an , n ∈ N,
we have
lim Ãn (z) = exp(−|A − z|/α), z ∈ R.
n→∞
Figure 1: Product-sum of two triangular fuzzy numbers.
Proof. It will be sufficient to show that
 n

 1 − |An − z|
if |An − z| ≤ nα
nα
Ãn (z) =


0
otherwise
(1)
for each n ≥ 2, because from (1) it follows that
|An − z| n
lim Ãn (z) = lim 1 −
= exp(− | lim An − z | /α) =
n→∞
n→∞
n→∞
nα
exp(− | A − z | /α), z ∈ R.
From the definition of product-sum of fuzzy numbers it follows that
suppÃn = supp(ã1 + · · · + ãn ) = suppã1 + · · · + suppãn =
[a1 − α, a1 + α] + · · · + [an − α, an + α] = [An − nα, An + nα], n ∈ N.
We prove (1) by making an induction argument on n. Let n = 2. In order to determine Ã2 (z), z ∈
[A2 − 2α, A2 + 2α] we need to solve the following mathematical programming problem:
|a1 − x|
1−
α
|a2 − y|
1−
→ max
α
subject to |a1 − x| ≤ α,
|a2 − y| ≤ α, x + y = z.
2
By using Lagrange’s multipliers method and decomposition rule of fuzzy numbers into two separate
parts (see [2]) it is easy to see that Ã2 (z), z ∈ [A2 − 2α, A2 + 2α] is equal to the optimal value of the
following mathematical programming problem:
a1 − x
a2 − z + x
1−
1−
→ max
(2)
α
α
subject to a1 − α ≤ x ≤ a1 ,
a2 − α ≤ z − x ≤ a2 , x + y = z.
Using Lagrange’s multipliers method for the solution of (2) we get that its optimal value is
|A2 − z| 2
1−
2α
and its unique solution is
x = 1/2(a1 − a2 + z)
(where the derivative vanishes).
Indeed, it can be easily checked that the inequality
A2 − z
|A2 − z| 2
≥1−
1−
2α
α
holds for each z ∈ [A2 − 2α, A2 ].
In order to determine Ã2 (z), z ∈ [A2 , A2 + 2α] we need to solve the following mathematical programming problem:
a1 − x
a2 − z + x|
1+
1+
→ max
(3)
α
α
subject to a1 ≤ x ≤ a1 + α, a2 ≤ z − x ≤ a2 + α.
In a similar manner we get that the optimal value of (3) is
|z − A2 | 2
.
1−
2α
Let us assume that (1) holds for some n ∈ N. By similar arguments we obtain
Ãn+1 (z) = (Ãn + ãn+1 )(z) =
|An − x|
|an+1 − y|
sup Ãn (x) · ãn+1 (y) = sup 1 −
1−
=
nα
α
x+y=z
x+y=z
|An+1 − z| n+1
, z ∈ [An+1 − (n + 1)α, An+1 + (n + 1)α],
1−
(n + 1)α
and
Ãn+1 (z) = 0, z ∈
/ [An+1 − (n + 1)α, An+1 + (n + 1)α],
This ends the proof.
Figure 2 gives a graphical illustration of the limiting distribution.
The proof of the following theorem carried out analogously to the proof of the preceding theorem.
3
Figure 2: The limit distribution of the product-sum ã1 + · · · + ãn + · · · .
Theorem 2.2 Let ãi = (ai , α, β), i ∈ N be fuzzy numbers of triangular form. If A :=
and it is finite, then with the notations of Theorem 1 we have

|A − z|



if z ≤ A
 exp −
α
lim Ãn (z) =
n→∞

|A − z|


 exp −
if z ≥ A
β
P∞
i=1 ai
exists
Figure 3: Product-sum of n triangular fuzzy numbers.
3
Question
Let ãi = (ai , α, β)LR , 1 ≤ i ≤ n be fuzzy numbers of LR-type. On what condition will the membership
function of the product-sum Ãn have the following form

An − z


n

if An − nα ≤ z ≤ An ,
 L
nα
Ãn (z) =
(4)

z
−
A
n

n

if An ≤ z ≤ An + nβ
 R
nβ
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4
Follow ups
The results of this paper have been extended and improved in the following papers.
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convex additive generator function and fuzzy numbers with concave shape functions,
which is the generalization of Fullér and Keresztfalvi’s result [4]. The purpose of this
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5
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
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

n

L
if An − nα ≤ z ≤ An ,

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Ãn (z) =

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
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
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