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
Let
j
x
the proportional of the total amount of money devoted to security j,
j
M
1
and
j
M
2
represent the minimum and maximum proportion respectively of the total amount of money devoted to security
j
. The trapezoidal fuzzy number of
ji
r
is
4 3
2 1
, ,
, ~
ji ji
ji ji
ji
r r
r r
r =
where
4 3
2 1
ji ji
ji ji
r r
r r
≤
. In addition, we denote the VaR
i
level by the fuzzy number trapezoidal
4 3
2 1
, ,
, ~
i i
i i
i
b b
b b
b =
,
q i
, 1
=
. Using this approach we see that the model given by 6.5-6.8 reduces itself to the form
from the following theorem .
Theorem 6.1 The possibilistic mean VaR portfolio selection for the vector mean VaR
efficient portfolio model 6.5-6.8 is
⎪⎭ ⎪
⎬ ⎫
⎪⎩ ⎪
⎨ ⎧
− ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∑ ∑
∑ ∑
= =
= =
∈ n
j j
jq n
j j
jq n
j j
j n
j j
j R
x
x c
x r
E x
c x
r E
n
1 1
1 1
1 1
~ ~
,..., ~
~ max
6.17
i n
j i
j ji
b x
r Pos
to subject
β
≤ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛
∑
=
1
~ ~
,
q i
, 1
=
, 6.18
∑
=
=
n j
j
x
1
1
, 6.19
n j
M x
M
j j
j
, 1
,
2 1
= ≤
≤
. 6.20
In the following using Section 6.3 we obtain the efficient portfolios given by the Theorem 6.1.
Theorem 6.2. If
q i
i
, 1
, =
λ
, then an efficient portfolio for possibilistic model is an optimal solution of the following problem:
136 ⎥
⎥ ⎦
⎤ ⎢
⎢ ⎣
⎡ −
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∑ ∑
∑
= =
= ∈
n j
j ji
n j
j ji
q i
i R
x
x c
x r
E
n
1 1
1
~ ~
max λ
6.21
i n
j i
j ji
b x
r Pos
to subject
β
≤ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛
∑
=
1
~ ~
,
q i
, 1
=
, 6.22
∑
=
=
n j
j
x
1
1
, 6.23
n j
M x
M
j j
j
, 1
,
2 1
= ≤
≤
. 6.24
Using the fact that rate of return on security
, 1
n j
j =
is given by trapezoidal fuzzy number, then we get the following results.
Theorem 6.3 Let rate of return on security
, 1
n j
j =
by the trapezoidal fuzzy number
4 3
2 1
, ,
, ~
ji ji
ji ji
ji
r r
r r
r =
where
4 3
2 1
ji ji
ji ji
r r
r r
≤
and addition
4 3
2 1
, ,
, ~
i i
i i
i
b b
b b
b =
is trapezoidal fuzzy number for VaR level and
i
λ
,with
q i
, 1
=
. Then using the possibilistic mean VaR portfolio selection model an efficient portfolio is
an optimal solution for the following problem:
⎥ ⎥
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎢ ⎢
⎣ ⎡
− +
+ +
∑ ∑
∑ ∑
∑ ∑
= =
= =
= =
∈
n j
j ji
n j
j ji
n j
j ji
n j
j ji
n j
j ji
q i
i R
x
x c
x r
x r
x r
x r
n
1 1
4 1
1 1
3 1
2 1
6 3
max λ
6.25
: .
to ubject
s
1
1 3
2 1
4 1
≥ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
+ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
−
∑ ∑
= =
n j
i j
ji i
n j
i j
ji i
b x
r b
x r
β β
,
q i
, 1
=
, 6.26
∑
=
=
n j
j
x
1
1
, 6.27
n j
M x
M
j j
j
, 1
,
2 1
= ≤
≤
. 6.28
Proof : Really, from the equation 6.16, we have
137
6 3
~ ~
1 4
1 1
1 3
1 2
1
∑ ∑
∑ ∑
∑
= =
= =
=
+ +
+ =
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
n j
j ji
n j
j ji
n j
j ji
n j
j ji
n j
j ji
x r
x r
x r
x r
x r
E
,
p i
, 1
=
. From Lemma 6.1, we have that, for any
q i
, 1
=
,
i n
j i
j ji
b x
r Pos
β
≤ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛
∑
=1
~ ~
, is equivalent with
1
1 3
2 1
4 1
≥ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
+ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
−
∑ ∑
= =
n j
i j
ji i
n j
i j
ji i
b x
r b
x r
β β
. Furthermore, from 6.25-6.28 given by Theorem 6.2, we get that this problem is of the
following form :
⎥ ⎥
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎢ ⎢
⎣ ⎡
− +
+ +
∑ ∑
∑ ∑
∑ ∑
= =
= =
= =
∈ n
j j
ji n
j j
ji n
j j
ji n
j j
ji n
j j
ji q
i i
R x
x c
x r
x r
x r
x r
n
1 1
4 1
1 1
3 1
2 1
6 3
max λ
6.29
to subject
1
1 3
2 1
4 1
≥ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
+ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
−
∑ ∑
= =
n j
i j
ji i
n j
i j
ji i
b x
r b
x r
β β
,
q i
, 1
=
6.30
∑
=
=
n j
j
x
1
1
, 6.31
n j
M x
M
j j
j
, 1
,
2 1
= ≤
≤
. 6.32 This completes the proof.
□
Problem 6.29-6.32 is a standard multi-objective linear programming problem. For optimal solution we can used some algorithms of multiobjective programming Slowinski
and Teghem [175] and White [209].
138
6.4 A Weighted possibilistic mean value approach