132
quantitative or qualitative Dubois and Prade, [44] according to the range of these measures which may be the real interval [0, 1], or a finite linearly ordered scale as well.
6.3.1 Possibilistic theory. Some preliminari
We consider the possibilistic theory proposed by Zadeh [216]. Let
a ~
and
b ~
be two fuzzy numbers with membership functions
a ~
μ
and
b ~
μ
respectively. The possibility operator Pos is defined as follows Dubois and Prade [44].
⎪ ⎩
⎪ ⎨
⎧ ∈
= =
∈ =
≤ ∈
= ≤
}. ,
sup{min ~
~ ,
} ,
, sup{min
~ ~
}, ,
, sup{min
~ ~
~ ~
~ ~
~ ~
R R,
R, x
x x
b a
Pos y
x y
x y
x b
a Pos
y x
y x
y x
b a
Pos
b a
b a
b a
μ μ
μ μ
μ μ
6.9
In particular, when
b ~
is a crisp number b, we have
⎪⎩ ⎪
⎨ ⎧
= =
∈ =
≤ ∈
= ≤
. ~
, }
, sup{
~ },
, sup{
~
~ ~
~
b b
a Pos
b x
x x
b a
Pos b
x x
x b
a Pos
a a
a
μ μ
μ R
R
6.10
Let
R R
R →
× :
f
be a binary operation over real numbers. Then it can be extended to the operation over the set of fuzzy numbers. If we denoted for the fuzzy
numbers
b a
~ ,
~
the numbers
~ ,
~ ~
b a
f c
=
, then the membership function
c~
μ
is obtained from the membership function
a ~
μ
and
b ~
μ
by
} ,
, ,
, sup{min
~ ~
~
y x
f z
y x
y x
z
b a
c
= ∈
= R
μ μ
μ
6.11 for
R ∈
z
. That is, the possibility that the fuzzy number
~ ,
~ ~
b a
f c
=
achives value
R ∈
z
is as great as the most possibility combination of real numbers x,y such that z = fx,y, where the value of
a ~
and
b ~
are x and y respectively.
133
6.3.2 Triangular and trapezoidal fuzzy numbers
Let the rate of return on security given by a trapezoidal fuzzy number
, ,
, ~
4 3
2 1
r r
r r
r =
where
4 3
2 1
r r
r r
≤
. Then the membership function of the fuzzy number
r~
can be denoted by:
⎪ ⎪
⎪ ⎩
⎪⎪ ⎪
⎨ ⎧
≤ ≤
− −
≤ ≤
≤ ≤
− −
=
. ,
, ,
, 1
, ,
, 4
3 4
3 4
3 2
2 1
1 2
1
otherwise r
x r
r r
r x
r x
r r
x r
r r
r x
x μ
6.12
We mention that trapezoidal fuzzy number is triangular fuzzy number if
3 2
r r
= .
~
x
b
μ
~
x
r
μ
1
δ
0 b
1
b
2
r
1
b
3
x
δ
r
2
r
3
b
4
r
4
Figura 6.1: Two trapezoidal fuzzy number
r~
and
b ~
. Let us consider two trapezoidal fuzzy numbers
, ,
, ~
4 3
2 1
r r
r r
r =
and
= b
~
, ,
,
4 3
2 1
b b
b b
, as shown in Figure 6.1. If
3 2
b r
≤
, then we have
{ }
y x
y x
b r
Pos
b r
≤ =
≤ }
, min{
sup ~
~
~ ~
μ μ
{ }
{ }
, 1
1 ,
1 min
, min
3 ~
2 ~
= =
≥ b
r
b r
μ μ
which implies that
1 ~
~ =
≤ b r
Pos
. If
3 2
b r
≥
and
4 1
b r
≤
then the supremum is achieved at point of intersection
x
δ
of the two membership function
~
x
r
μ
and
~
x
b
μ
. A simple computation shows that
134 ~
~
1 2
3 4
1 4
r r
b b
r b
b r
Pos −
+ −
− =
= ≤
δ
and
δ δ
1 2
1
r r
r
x
− +
=
. If
4 1
b r
, then for any
y x
, at least one of the equalities
,
~ ~
= =
y x
b r
μ μ
hold. Thus we have
~ ~
= ≤ b
r Pos
. Now we summarize the above results as
⎪⎩ ⎪
⎨ ⎧
≥ ≤
≥ ≤
= ≤
. ,
, ,
, ,
, 1
~ ~
4 1
4 1
3 2
3 2
b r
b r
b r
b r
b r
Pos
δ
6.13
Especially, when
b ~
is the crisp number 0, then we have
⎪⎩ ⎪
⎨ ⎧
≥ ≤
≤ ≤
= ≤
, ,
, 1
~
1 2
1 2
r r
r r
r Pos
δ
6.14
where
2 1
1
r r
r −
=
δ
. 6.15 We now turn our attention the following lemma.
Lemma 6.1 Dobois and Prade [42] Let
4 3
2 1
, ,
, ~
r r
r r
r =
be a trapezoidal fuzzy number. Then for any given confidence level
α
with
α α
≥ ≤
≤ ≤
~ ,
1 r
Pos
if and only if
1
1 r
α
−
+
2
≤ r
α
. The
λ
level set of a fuzzy number
4 3
2 1
, ,
, ~
r r
r r
r =
is a crisp subset of R and denoted by
} ,
{ ]
~ [
R x
x x
r ∈
≥ =
λ μ
λ
, then according to Carlsson et al [26], we have
] ,
[ }
, {
] ~
[
3 4
4 1
2 1
r r
r r
r r
R x
x x
r −
− −
+ =
∈ ≥
=
λ λ
λ μ
λ
. Given
] ,
[ ]
~ [
2 1
λ λ
λ
a a
r =
, the crisp possibilistic mean value of
4 3
2 1
, ,
, ~
r r
r r
r =
is
∫
+ =
1 2
1
~ ~
λ λ
λ λ
d a
a r
E
.
135
where
E ~
denotes fuzzy mean operator. We can see that if
4 3
2 1
, ,
, ~
r r
r r
r =
is a trapezoidal fuzzy number then
6 3
~ ~
4 1
3 2
1 3
4 4
1 2
1
r r
r r
d r
r r
r r
r r
E +
+ +
= −
− +
− +
=
∫
λ λ
λ λ
. 6.16
6.3.3 Construction efficient portfolios
