Page 72 - IJOCTA-15-4

B. Shiri et.al. / IJOCTA, Vol.15, No.4, pp.610-624 (2025)
Finally, we recall the shape function from Obviously, Definition Equation (10) is not prac-
23
Ref. . Let tical. Therefore, we try another approach. We
−1 + aim to make it consistent with Zadeh’s extension
S ={L : R → [0, 1] : L is upper semi-continous
principle and previous definitions for as large a
+
on R , L(0) = 1, L(1) = 0}.
class of fuzzy numbers as possible.
Any fuzzy number µ LR with Supp(µ LR ) = [a, b] Let µ ∼ (d, f 1 , f 2 ) and η ∼ (e, g 1 , g 2 ). Then,
can be described by L-R functions as
µ ⊕ η ∼ (d + e, f 1 + g 1 , f 2 + g 2 ). (11)
d−w
L( ), w ≤ d,
µ LR (x) = d−a (7) This definition aligns with Equation (10) for tri-
w−d
R( ), w ≥ d, 23
b−d angular fuzzy numbers. Dubois et al. explic-
−1 itly defined arithmetic operations for fuzzy num-
where R, L ∈ S .
For any such representation, there exist bers of the same type using (L-R) shape functions
˜ ˜
f 1 , f 2 ∈ S such that based on Zadeh’s extension theory. They derived
e+d−w
˜
˜
f 1 (L(w)) = f 2 (R(w)) = w. (8) µ LR (w) ⊕ η LR (w) = L( e−c+d−a ), w ≤ e + d,
R( w−e+d ), w ≥ e + d,
Thus, h−e+b−d
˜
˜
˜
˜
µ ∼ (d, (d − a)f 1 , (b − d)f 2 ). ∼ (e + d, (e − c + d − a)f 1 , (h − e + b − d)f 2 )
Furthermore, if r = µ LR (x), then = (d + e, f 1 + g 1 , f 2 + g 2 ),
˜
w = d − (d − a)f 1 (r), x ≤ d, (12)
˜
˜
˜
and where f 1 = (d−a)f 1 , f 2 = (b−d)f 2 , g 1 = (e−c)f 1 ,
˜
˜
w = d + (b − d)f 2 (r), x ≥ d. g 2 = (h − e)f 2 , and
e−w
This forms the foundation for converting the re- L( e−c ), w ≤ e,
sult from Ref. 23 to a symmetric representation. η LR (x) = R( w−e ), w ≥ e. (13)
h−e
Defining arithmetic operations for fuzzy num-
In the given mathematical context, the con-
bers is complex, relying on Zadeh’s extension cept of subtraction presents a thorny issue due
principle. 24 Each arithmetic operation on two
to inconsistent definitions. Suppose we define the
fuzzy numbers requires solving an optimization operation ζ = µ ⊖ η as ζ = µ ⊕ (−η). A simple
problem. 25–27 Considerable research has aimed
verification shows that ζ ⊕η ̸= µ, immediately in-
to derive explicit formulas for arithmetic opera- dicating a deviation from the expected behavior
tions. For example, Ref. 27 reviews arithmetic def- of subtraction.
initions for triangular fuzzy numbers consistent The complexity escalates with Zadeh’s exten-
with Zadeh’s extension principle. In this paper, sion theory, which defines
we adopt the classical definitions based on the
symmetrical representation. 21,22 µ ⊖ η ∼ (d − e, f 1 + g 2 , f 2 + g 1 ).
Curiously, this still leads to ζ = µ ⊕ (−η), recre-
ating the same paradox.
2.1. Fuzzy operation
In interval theory, Hukuhara and generalized
First, we recall the operations between fuzzy num- Hukuhara differences (H and GH differences) have
bers and scalars. been proposed to address this problem. 28 How-
ever, a major drawback of these definitions is that
Definition 2. Let λ ∈ R and µ ∈ R F . Then,
they may not be well-defined for arbitrary fuzzy
• Scalar Multiplication/Division: (λµ)(w) := numbers.
µ(w/λ). Equivalently, We propose using the term “differenceable”

(λd, λf 1 , λf 2 ), λ ≥ 0, when the difference of two fuzzy numbers ex-
λ(d, f 1 , f 2 ) = (9)
(λd, |λ|f 2 , |λ|f 1 ), λ < 0. ists. In the study of discrete dynamic systems,
H-differenceable fuzzy numbers appear to play a
• Scalar Summation and Difference: (λ ± 21
µ)(w) := µ(w ± λ). Equivalently, significant role. Clearly, the H-difference µ ⊖ η
is differenceable if and only if f i −g i ≥ 0 and thus
λ ± (d, f 1 , f 2 ) = (λ ± d, f 1 , f 2 ).
ζ = µ ⊖ η ∼ (d − e, f 1 − g 1 , f 2 − g 2 ). (14)
Suppose µ and η are two fuzzy numbers.
Theorem 5. If η ⊕ ζ = µ, then the H-differences
Then, according to Zadeh’s extension principle,
µ ⊖ ζ and µ ⊖ η are differenceable and equal to η
the elementary operations of these two fuzzy num-
and ζ, respectively.
bers are defined by
Proof. The proof follows directly from Equations
(µ⊕⊖⊗η)(w) = sup min{µ(x), η(y)}. (10)
x+−×y=w (11) and (14). □
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