80
Step 13 Ask the decision maker to specify the refined p
, and then go to Step 0. It is rather reasonable to ask the decision maker
p at this step, because he has a
good idea about p
now figure 5.1. For implementing the above IFLP, we need only two solution-finding techniques,
the simplex method and parametric method. Therefore, the IFLP approach proposed here can be easily programmed in a PC system for its simplicity, Lai and
Hwang [92].
5.4 Portfolio problem
The process of selecting a portfolio may be divided into two stages. The first stage starts with observation and experience and end with beliefs about the future performance of
available securities. The second stage starts with the relevant beliefs about future performances and ends with the choice of portfolio by Markowitz [114]. This chapter is
concerned with the second stage. The problem of standard portfolio selection is as follows. Assume
a n securities, b an initial sum of money to be invested,
c the beginning of a holding period, d the end of the holding period,
and let x
1,…,
x
n
be the investors investment proportion weights. These are the proportions of the initial sum to be invested in the n securities at the beginning of the holding period
that define the portfolio to be held fixed until the end of the holding period. The standard view is that there is only one purpose in portfolio selection, and that is to maximize
portfolio return, the percent return earned by the portfolio over the course of the holding period.
Now we consider the problem 4.11-4.12 given in Chapter 4 ,
for
1 =
υ
. Thus we have
max x
f
5.23 subject
to
Y x
R
SSD
f
, 5.24
X ∈
x
. 5.25
81
Here
R →
X :
f
is a concave continuous functional. Also in particular, we may use
] [
x R
x f
E =
and this will still lead to nontrivial solutions, due to the presence of the dominance constraint.
Yes No
Yes No
Figure 5.1 Flow chart decision support system Werner’s, 1987 Using the Chapter 4, for
1 =
υ
, we get the following proposition.
Proposition 5.1 Assume that
Y
has a discrete distribution with realizations
m i
y
i
, 1
, =
. Then relation 5.24 is equivalent to
] [
] [
+ +
− ≤
− Y
y x
R y
i i
E E
,
m i
, 1
= ∀
. 5.26 Model formulation
Efficient Extreme solution
Compromise Solution Local Information
Solution Acceptable ? “Best”
Compromise STOP
Modification of membership functionstion
Local consequences ?
82
Let us assume now that the returns have a discrete joint distribution with realizations
jt
r
,
T t
, 1
=
,
n j
, 1
=
, attained with probabilities
t
p
,
T t
, 1
=
. The formulation of the stochastic dominance relation 5.24 respectively 5.26 simplifies event further.
Introducing variables
it
s
representing shortfall of Rx below y
i
in realization t,
m i
, 1
=
and
T t
, 1
=
, we obtain the following proposition.
Proposition 5.2 The problem 5.23-5.25 is equivalent to the problem
max x
f
5.27 subject
to
i it
j n
j jt
y s
x r
− ≤
− −
∑
=1
,
m i
, 1
=
,
T t
, 1
=
, 5.28
;
2 1
i it
T t
t
y Y
F s
p ≤
∑
=
,
m i
, 1
=
5.29
≥
it
s m
i ,
1 =
,
T t
, 1
=
, 5.30
∑
=
≤
n j
j
x
1
1
, 5.31
∑
=
− ≤
−
n j
j
x
1
1
, 5.32
≥
j
x
,
n j
, 1
=
, 5.33
and problem 5.27-5.33 can be written as
max X
ϕ
=
∑
= n
j j
j
X c
1
, 5.34
subject to:
i mT
n j
j ij
b X
a ≤
∑
+ =1
,
2 ,
1 +
+ =
m mT
i
, 5.35
≥
j
X
,
mT n
j +
= ,1
, 5.36 where,
,..., ,...,
,..., ,
,..., ,
,...,
1 2
21 1
11 1
mT m
T T
n
s s
s s
s s
x x
X =
.
83 ⎪
⎩ ⎪
⎨ ⎧
+ −
+ +
= −
= +
+ =
− −
= +
+ =
= −
= otherwise
i T
n n
j and
T K
m K
Km i
T K
m K
Km i
n j
r a
ij ij
, 1
1 ,
1 ,
1 ,
, 1
, 1
, 1
1 ,
, 1
, 1
, ,
1 ,
⎩ ⎨
⎧ +
= =
= otherwise
mT i
n j
a
ij
, 1
, ,
1 ,
1
⎩ ⎨
⎧ +
= =
− =
otherwise mT
i n
j a
ij
, 2
, ,
1 ,
1
⎩ ⎨
⎧ +
+ +
= =
+ +
− +
= =
− −
−
otherwise m
mT mT
i m
K TK
n K
T n
j p
a
K T
n j
ij
, 2
, 3
, ,
1 ,
, 1
1 ,
1
In the next section we extended this result to fuzzy decisions theory.
5.5 Case of fuzzy technological coefficients and fuzzy right-hand side numbers
