110 3
, ,
1 ,
, 5
. 2
, ,
1 1
1 1
1 3
1 1
1 2
1 1
1 1
2 1
= =
= =
= =
λ λ
λ λ
p x
g p
x g
p x
g x
x
Since
, ,
1 1
1
≠ λ
p x
g
, we calculate the new values of Lagrangemultipliers
, ,
2 2
2 2
1 2
c u
u u
by using Step 2 of the modified subgradient method. The solutions of the second iteration are obtained as
8 2
2 2
3 8
2 2
2 2
7 2
2 2
1 8
2 1
10 83
. 7
, ,
10 2
. 8
, ,
10 28
. 3
, ,
1832159 .
10 8
. 7
45804 ,
1
− −
− −
× −
= ×
= ×
= =
× =
=
λ λ
λ λ
p x
g p
x g
p x
g x
x
Since
x g
is quite small, by Theorem 5.4,
10 8
. 7
, 45804
. 1
8 2
1
≅ ×
= =
x x
and
= λ
0.1832159 are optimal solutions to the problem 5.106. This means that, the vector
,
2 1
x x
is a solution to the problem 5.105 which has the best membership grade
λ
. Note that, the optimal value of
λ
found at the second iteration of the modified subgradient method is approximately equal to the optimal value of
λ
calculated at the twenty first iteration of the fuzzy decisive set method.
5.8 Portfolio problem with fuzzy multi-objective
The Fuzzy Multiple Objective Decision Model FMODM studied by Lai and Hwang [93] states that the effectiveness of a decision makers’ performance in a decision process
can be improved as a result of the high quality of analytic information supplied by a computer. They propose an Interactive Fuzzy Multiple Objective Decision Model
IFMODM to solve a specific domain of Multiple Objective Decision Model MODM.
111
In this section we consider this approach for 4.2-4.4 portfolio model. Thus we have
the following problem
,..., x
f x
f Max
q i
5.107 subject
to
it jt
n j
j i
s r
x y
≤ −
∑
=1
,
T t
m i
, 1
, ,
1 =
=
, 5.108
;
2 1
i it
T t
t
y Y
F s
p ≤
∑
=
,
, ,
1 m i
=
5.109
≥
it
s
,
T t
m i
, 1
, ,
1 =
=
, 5.110
≥ x
,
X ∈
x
. 5.111 where
q k
x c
x f
n i
i ik
k
, 1
,
1
= =
∑
=
. Let us now consider the case of a decision-maker who has a fuzzy goal such as “the
objective function
x f
k
should be much greater than
min
k
p
”. Further, let us assume that the degree of satisfaction of the decision-maker with respect to achieving the objective
does not change beyond the level
max k
p
. Then the corresponding linear membership function that characterises the fuzzy goal of the decision-maker is given by:
⎪ ⎪
⎩ ⎪⎪
⎨ ⎧
≤ −
− ≤
= .
; 1
, ;
, ;
] [
max max
min min
max min
min min
k k
k k
k k
k k
k k
k k
k
p x
f p
f p
p p
p x
f p
f x
f μ
5.112
Given the membership functions for the various objectives of the decision-maker, the maximizing decision can be computed by solving the following optimization problem:
] [
min maximize
, 1
x f
k k
q k
μ
=
5.113 subject to
it jt
n j
j i
s r
x y
≤ −
∑
=1
,
m i
, 1
=
, 5.114
;
2 1
i it
T t
t
y Y
F s
p ≤
∑
=
,
m i
, 1
=
5.115
112 ≥
it
s
,
T t
m i
, 1
, ,
1 =
=
5.116
≥ x
,
X ∈
x
. 5.117 By introducing the auxiliary variable
λ
, the above optimization problem can be reduced to the following conventional linear programming problem :
λ Maximize
5.118 subject
to
q k
x f
k k
, 1
, ]
[ =
≥
λ μ
, 5.119
it jt
n j
j i
s r
x y
≤ −
∑
=1
,
T t
m i
, 1
, ,
1 =
=
, 5.120
;
2 1
i it
T t
t
y Y
F s
p ≤
∑
=
,
m i
, 1
=
5.121
≥
it
s
,
T t
m i
, 1
, ,
1 =
=
5.122
≥ x
,
X ∈
x
,
R ∈
λ
,
1 ≤
≤
λ
. 5.123 Let us consider the case of a fund manager who has to choose a structured portfolio from
an investment universe of n assets with
j
l
and
j
l
,
n j
, 1
=
being the minimum and maximum weight of the ith asset in the portfolio. In order to select the structured
portfolio, the fund manager may examine k potential market scenarios, and for each of these scenarios the decision maker may wish to maximize the portfolio return. To achieve
the return objective the fund manager could formulate the following optimization problem:
maximize
,...,
1
x R
x R
q
5.124 subject to
it jt
n j
j i
s r
x y
≤ −
∑
=1
,
T t
m i
, 1
, ,
1 =
=
, 5.125
;
2 1
i it
T t
t
y Y
F s
p ≤
∑
=
,
, ,
1 m i
=
5.126
1
1
=
∑
= m
i i
x
, 5.127
j j
j
l x
l ≤
≤
,
n j
, 1
=
, 5.128
113 ≥
it
s
,
T t
m i
, 1
, ,
1 =
=
, 5.129
≥ x
,
X ∈
x
. 5.130 where
q k
x R
x R
n j
j k
j k
, 1
,
1
= =
∑
=
In equation 5.76,
k j
r
denotes the return from the jth asset for the kth market scenario at the end of the investment period and
x R
k
the portfolio return for the kth scenario. Since the above optimization problem has multiple objective functions, one has to
compute a Pareto optimal solution for the problem see Sakawa [164]. Also we can use the model of Chapter 2 for instance, one could characterize the set of Pareto optimal
solutions using the weighted minimax method and select one solution from this set. The set of Pareto optimal solutions to the above optimization problem is characterized by:
Maximize
λ
5.131 subject
to
λ ≥
x R
w
k k
,
p k
, 1
=
, 5.132
it jt
n j
j i
s r
x y
≤ −
∑
=1
,
T t
m i
, 1
, ,
1 =
=
, 5.133
;
2 1
i it
T t
t
y Y
F s
p ≤
∑
=
,
m i
, 1
=
5.134
1
1
=
∑
= n
j j
x
, 5.135
j j
j
l x
l ≤
≤
,
, ,
1 n j
=
5.136
≥
it
s
,
T t
m i
, 1
, ,
1 =
=
5.137
≥ x
,
X ∈
x
,
R ∈
λ
. 5.138
In above relation,
λ
is an auxiliary variable and
q k
w
k
, 1
, =
are any arbitrarily chosen nonnegative weights. Given any suitable weighting vector, one can determine the Pareto
optimal solution. Here, we assume without loss of generality that
, x
R
k j
j j
l x
l ≤
≤
,
n j
, 1
=
. If this is not the case, the objective functions can be rewritten as
q k
C R
x R
k k
, 1
, ˆ
= =
, 5.139
114
where C is a suitable constant that ensures
k x
R
k
∀ ,
ˆ
. Incorporating this change in equation 5.132, one can compute the Pareto optimal solution.
The optimization problem formulated above is a linear programming problem and can be easily solved using standard algorithms. However, finding a satisfactory Pareto optimal
solution requires one to define the a priori probabilities of various scenarios that incorporate the market views. In the face of uncertainty these a priori probabilities are not
computable, and hence it is difficult to compute a Pareto optimal solution that can be characterised as being satisfactory. Moreover, the fund manager may like to structure the
portfolio such that the return targets are different for each market scenario, for instance with those scenarios that heshe considers more likely to occur although no experimental
evidence is available being targeted to achieve greater return. Transforming such goals into suitable weights
q k
w
k
, 1
, =
for the various scenarios is not obvious from the fund manager’s perspective.
Let us now consider a fund manager structuring a portfolio based on p potential market scenarios. For each such scenario, the fund manager may have a target range for
the expected return over the investment period. We will denote by
min k
p
and
max k
p
the minimum and maximum expected return for the jth market scenario. Note that it is quite
easy for the fund manager to provide information on the expected target range of return for various scenarios rather than to define the a priori probabilities for different scenarios.
Using the linear membership function given in equation 5.112 it is possible to compute the degree of satisfaction
x R
k k
μ
for any given portfolio x for the kth market scenario. Given that the degree of satisfaction to the fund manager for the kth market
scenario is
x R
k k
μ
, the structured portfolio can be computed by solving the following optimization problems, for
q k
, 1
= λ
Maximize
5.140 subject
to
λ μ
≥ x
R
k k
, 5.141
it jt
n j
j i
s r
x y
≤ −
∑
=1
,
T t
m i
, 1
, ,
1 =
=
, 5.142
115
;
2 1
i it
n j
t
y Y
F s
p ≤
∑
=
,
m i
, 1
=
5.143
1
1
=
∑
= n
j j
x
, 5.144
j j
j
l x
l ≤
≤
,
, ,
1 m i
=
5.145
≥
it
s
,
T t
m i
, 1
, ,
1 =
=
5.146
≥ x
,
X ∈
x
,
R ∈
λ
. 5.147
It is easy to show that the solution to the above optimization problem if one exists will be Pareto optimal, Sakawa [164]. It is again useful to remind that we can interpret the
membership function
x R
k k
μ
for the kth market scenario in 5.136 as modelling the fuzzy utility of the investor for the given scenario. In this case, the structured portfolio
computed by solving the above optimization problem maximizes the fuzzy utility of the investor.
5.9. Multiobjective fractional programming problems under fuzziness
