On Hamacher-sum of triangular fuzzy numbers∗
Robert Fullér
rfuller@abo.fi
Abstract
This paper presents new results concerning the effective practical computation of the membership
function of the infinite sum (defined via the sup-Hamacher-norm convolution) of triangular fuzzy
numbers. Namely, we shall calculate the limit distribution of the Hγ -sum ã1 ⊕ ã2 ⊕ · · · ⊕ ãn ⊕ · · ·
of triangular fuzzy numbers ãi , i ∈ N, for γ = 0, 1, 2.
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 ã = 1 − (a − t)/α if a − α ≤ t ≤ a
1 − (t − a)/β if a ≤ t ≤ a + β and ã(t) = 0 otherwise, where a ∈ R is the centre and α > 0 isthe 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 and γ ≥ 0 a real number,
then their Hamacher-sum (Hγ -sum for short) is defined as
(ã ⊕ b̃)(z) = sup Hγ (ã(x), b̃(y)), x, y, z ∈ R
x+y=z
where
Hγ (u, v) =
uv
, u, v ∈ [0, 1].
γ + (1 − γ)(u + v − uv)
The support suppã of a fuzzy number ã is defined as
suppã = {t ∈ R|ã(t) > 0}.
2 Hγ -sum of triangular fuzzy numbers
In this section we shall determine the membership function of the Hγ -sum
ã1 ⊕ ã2 ⊕ · · · ,
where ãi , i ∈ N are fuzzy numbers of symmetric triangular form and γ ∈ {0, 1, 2}. Following Dubois
and Prade [1], the results of this paper can be regarded as derivation of exact calculation formulas for
adding triangular fuzzy numbers under weak noninteraction (when the fuzzy variable conjunction is
performed through Hamacher-operator). The proof of the following lemma directly follows from the
definition of Hγ -sum of fuzzy numbers (u, v 6= 0 → Hγ (u, v) 6= 0).
∗
The final version of this paper appeared in: R. Fullér, On Hamacher-sum of triangular fuzzy numbers, Fuzzy Sets and
Systems, 42(1991) 205-212. doi: 10.1016/0165-0114(91)90146-H
1
Lemma 2.1 Let ãi = (ai , α), i = 1, . . . , n, be fuzzy numbers of symmetric triangular form and let γ ≥ 0
be a real number. Then
supp(ã1 ⊕ · · · ⊕ ãn ) = suppã1 + · · · + suppãn = [a1 − α, a1 + α] + · · · + [an − α, an + α] =
[a1 + · · · + an − nα, a1 + · · · + an + nα], n ∈ N.
where ã1 ⊕ · · · ⊕ ãn is defined by (1).
It should be noted that Lemma 1 remains valid for non-symmetric triangular fuzzy numbers ãi =
(ai , α, β).
In the next two lemmas we shall calculate the exact membership function of Hγ -sum of two symmetric
triangular fuzzy numbers having common width α > 0 for each permissible value of parameter γ.
Lemma 2.2 Let 0 ≤ γ ≤ 2 and ãi = (ai , α), i = 1, 2. Then their Hγ -sum ã1 ⊕ ã2 (denoted by Ã2 ) has
the following membership function:
2
1 − |A2 − z|/(2α)
Ã2 (z) =
,
1 + (γ − 1)(|A2 − z|/(2α))2
if |A2 − z| < 2α and Ã2 (z) = 0, otherwise, where A2 = a1 + a2 .
Proof. By virtue of Lemma 1 we need to determine the value of ã2 (z) from the following relationships:
Ã2 (z) = (ã1 ⊕ ã) (z) =
sup
x+y=z
ã1 (x)ã2 (y)
, if |A2 − z| < 2α,
γ + (1 − γ)(ã1 (x) + ã2 (y) − ã1 (x)ã2 (y))
and Ã2 (z) = 0 otherwise.
According to the decomposition rule of fuzzy numbers into two separate parts [2], Ã2 (z), A2 − 2α <
z ≤ A2 , is equal to the optimal value of the following mathematical programming problem:
φ(x) :=
[1 − (a1 − x)/α][1 − (a2 − z + x)/α]
→ max
γ + (1 − γ){2 − (a1 + a2 − z)/α − [1 − (a1 − x)/α][1 − (a2 − z + x)/α]}
subject to a1 − α < x ≤ a1 ,
a2 − α < z − x ≤ a2 .
Using Lagrange’s multipliers method for the solution of the above problem we get that its optimal value
is
[1 − (A2 − z)/(2α)]2
1 + (γ − 1)[(A2 − z)/(2α)]2
and its unique solution is x = (a1 −a2 +z)/2 (where the derivative vanishes). Indeed, from the inequality
[1 − (A2 − z)/(2α)]2
A2 − z
≥1−
, for A2 − 2α < z ≤ A2 ,
1 + (γ − 1)[(A2 − z)/(2α)]2
α
2
1
and φ00 ( (a1 − a2 + z)) < 0 follows that the function φ attains its conditional maximum at the single
2
stationary point (a1 − a2 + z)/2.
If A2 ≤ z < A2 + 2α, then Ã2 (z) is equal to the optimal value of the the following mathematical
programming problem
[1 − (x − a1 )/α][1 − (z − x − a2 )/α]
→ max
γ + (1 − γ){2 − (z − a1 − a2 )/α − [1 − (x − a1 )/α][1 − (z − x − a2 )/α]}
(1)
subject to a1 ≤ x < a1 + α,
a2 ≤ z − x < a2 + α.
In a similar manner we get that the optimal value of (1) is
[1 − (z − A2 )/(2α)]2
1 + (γ − 1)[(z − A2 )/(2α)]2
and the unique solution of (1) is x = (a1 − a2 + z)/2 (where the derivative vanishes). Which ends the
proof.
The proof of the following lemma follows very much in the same fashion as the Lemma 2 and will,
therefore, be omitted.
Lemma 2.3 Let 2 < γ < ∞ and ãi = (ai , α), i = 1, 2. Then their Hγ -sum ã1 ⊕ ã2 (denoted by Ã2 )
has the following membership function

|A2 − z|
1


≤
< 2,
h1 (z) if 1 −



γ−1
α

Ã2 (z) =
1
|A2 − z|

<1−
,
h2 (z) if


α
γ−1



0
otherwise,
where
h1 (z) =
[1 − (A2 − z)/(2α)]2
,
1 + (γ − 1)[(A2 − z)/(2α)]2
h2 (z) = 1 −
|A2 − z|
α
and A2 = a1 + a2 .
The following theorems can be interpreted as central limit theorems for mutually Hγ -related fuzzy variables of symmetric triangular form (see [4]).
P
Theorem 2.1 Let γ = 0 and ãi = (ai , α), i ∈ N . Suppose that A := ∞
i=1 ai exists and it is finite, then
with the notation
Ãn = ã1 ⊕ · · · ⊕ ãn , An = a1 + · · · + an
we have
lim Ãn (z) =
n→∞
1
, z ∈ R.
1 + |A − z|/α
3
Theorem 2.2 (Product-sum) [3]. Let γ = 1 and ãi = (ai , α), i ∈ N. If
A :=
∞
X
ai
i=1
exists and it is finite, then with the notations of Theorem 1 we have
lim Ãn (z) = exp(−|A − z|/α), z ∈ R.
n→∞
Theorem 2.3 (Einstein-sum). Let γ = 2 and ãi = (ai , α), i ∈ N. If
A :=
∞
X
ai
i=1
exists and it is finite, then with the notations of Theorem 1 we have
lim Ãn (z) =
n→∞
2
, z ∈ R.
1 + exp(−2|A − z|/α)
Remark 2.1 According to the decomposition rule of fuzzy numbers into two separate parts, Theorem 1,2
and 3 remain valid for sequences of non-symmetric fuzzy numbers of triangular form ã1 = (a1 , α, β),
ã2 = (a2 , α, β), . . . with the difference that in the membership function of their Hγ -sum instead of α we
write β if z ≥ A.
Concluding remarks. In this paper we have calculated the membership function of Hγ -sum ã1 ⊕ ã2 ⊕
· · · for γ ∈ {0, 1, 2}, but to determine the exact limit distribution for arbitrary γ seems to be a very
complicated process. Similary, if the symmetric triangular fuzzy numbers ãi = (ai , αi ), i ∈ N have no
common width αi = α, then to summarize them (in Hγ -norm) seems to be a hopeless matter .
References
[1] D.Dubois and H.Prade, Additions of interactive fuzzy numbers,IEEE Transactions on Automatic
Control, 1981, Vol.26, 926-936.
[2] D.Dubois and H.Prade, Inverse operations for fuzzy numbers, in: E.Sanchez ed., Prococeedings of
IFAC Symp. on Fuzzy Information, Knowledge, Representation and Decision Analysis, Pergamon
Press,1983 399-404.
[3] R.Fullér, On product-sum of triangular fuzzy numbers, Fuzzy Sets and Systems 41(1991) 83-87.
[4] M.B.Rao and A.Rashed, Some comments on fuzzy variables, Fuzzy Sets and Systems, 6(1981)
285-292.
3
Follow ups
The results of this paper have been generalized or improved in the following works.
4
in journals
A30-c14 József Dombi and Norbert Győrbı́ró, Addition of sigmoid-shaped fuzzy intervals using the
Dombi operator and infinite sum theorems, FUZZY SETS AND SYSTEMS, 157(2006) 952-963.
2006
http://dx.doi.org/10.1016/j.fss.2005.09.011
Fullér has studied the sup-T sum with triangular fuzzy intervals [A32,A30] and in a
more general context [A24]. These results were developed further and extended by
Hong [11-13] and Mesiar [15]. (page 953)
A30-c13 Sato-Ilic M, Sato Y Asymmetric aggregation operator and its application to fuzzy clustering
model, COMPUTATIONAL STATISTICS & DATA ANALYSIS, 32 (3-4): 379-394, JAN 28 2000
http://dx.doi.org/10.1016/S0167-9473(99)00091-2
A30-c12 Sato-Ilic M, Sato Y A general fuzzy clustering model based on asymmetric aggregation operators IETE JOURNAL OF RESEARCH, 44 (4-5): 207-218 JUL-OCT 1998
A30-c11 S.Y.Hwang and D.H.Hong, The convergence of T-sum of fuzzy numbers on Banach spaces,
APPLIED MATHEMATICS LETTERS, vol. 10, No. 4, 129-134. 1997
http://dx.doi.org/10.1016/S0893-9659(97)00072-4
The purpose of this paper is to study the membership function of the t-norm-based sum
of fuzzy numbers on Banach spaces, which generalizes earlier results by Fullér [A30]
and Hong and Hwang [1]. (page 129)
A30-c10 D.H.Hong and S.Y.Hwang, The convergence of T-product of fuzzy numbers, FUZZY SETS
AND SYSTEMS, 85(1997) 373-378. 1997
http://dx.doi.org/10.1016/0165-0114(95)00333-9
Recently, t-norm-based addition of fuzzy numbers and its convergence have been studied [A32, A30, A28, A24, 6, 7, 9]. (page 373)
A30-c9 Mika Sato, Yoshiharu Sato A general fuzzy clustering model based on aggregation operators
BEHAVIORMETRIKA, 22: (2) 115-128 (1995)
http://dx.doi.org/10.2333/bhmk.22.115
A30-c8 D.H.Hong and S.Y.Hwang, On the convergence of T-sum of L-R fuzzy numbers, FUZZY SETS
AND SYSTEMS, 63(1994) 175-180. 1994
http://dx.doi.org/10.1016/0165-0114(94)90347-6
Abstract: This paper presents the membership function of infinite (or finite) sum (defined by the sup-t-norm convolution) of L-R fuzzy numbers under the conditions of
the convexity of additive generators and the concavity of L, R. As an application, we
shall calculate the membership function of the limit distribution of the Hamacher sum
(Hr -sum) for 0 ≤ r ≤ 2, which generalizes Fullér’s results [A32, A30] in the case
r ∈ {0, 1, 2}. (page 175)
5
Fullér [A32, A30] obtained the following result.
Theorem 1. Let r = 0, 1, 2 and let ãP
i = (ai , α, α)LR be L-R fuzzy numbers with
L(x) = R(x) = 1 − x such that A = ∞
i=1 exists and it is finite. Then the pointwise
limits of the partial product sums Ãn := ã1 + · · · = ãn for n → ∞ exist and are given
by

1



if r = 0,



 1 + |A − z|/α
lim Ãn (z) =
n→∞
exp(−|A − z|/α)
if r = 1,




2


if r = 2.

1 + exp(−2|A − z|/α)
In this paper, we determine the exact limit distributions of T-sum Ãn under very mild
conditions, which generalize earlier results by Fullér [A32, A30]. (page 176)
A30-c7 M.F.Kawaguchi and T.Da-te, Some algebraic properties of weakly non-interactive fuzzy numbers, FUZZY SETS AND SYSTEMS, 68(1994) 281-291. 1994
http://dx.doi.org/10.1016/0165-0114(94)90184-8
A30-c6 M.F.Kawaguchi and T.Da-te, Properties of fuzzy arithmetic based on triangular norms, Journal of Japan Society for Fuzzy Theory and Systems, 5(1993) 1113-1121 (in Japanese). Japanese
Journal of Fuzzy Theory and Systems, 5(1993) 677-687 (English Translation version). 1993
in proceedings and in edited volumes
A30-c5 M. Sato-Ilic, Fuzzy clustering for uncertainty data, in: Proceedings of the 1999 IEEE International Conference on Systems, Man, and Cybernetics, [doi 10.1109/ICSMC.1999.814117], vol.1,
pp. 359-364. 1999
A30-c4 M. Sato and Y. Sato, A generalized fuzzy clustering model based on aggregation operators and
its applications, in: B. Bouchon-Meunier ed., Aggregation and Fusion of Imperfect Information,
(Studies in Fuzziness and Soft Computing, ed., J. Kacprzyk. Vol. 12) Physica-Verlag, 1998 261278. 1998
A30-c3 M. Sato and Y. Sato, A generalized fuzzy clustering model based on fuzzy aggregation operators, in: Proceedings of Sixth IFSA World Congress, July 22-28, 1995, Sao Paulo. 1995
in books
A30-c2 Jorma K. Mattila, Text Book of Fuzzy Logic, Art House, Helsinki, [ISBN 951-884-152-7], 1998.
A30-c1 R. Lowen, Fuzzy Set Theory, Kluwer Academic Publishers, [ISBN: 0-7923-4057-4], 1996.
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