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Lecturer Costin-Ciprian POPESCU, PhD
The Bucharest Academy of Economic Studies
Department of Applied Mathematics
E-mail: [email protected]
A FUZZY OPTIMIZATION MODEL
Abstract. A fuzzy optimization model, mainly aiming the portfolio selection
issue, is discussed. Instead of being modeled as piecewise linear fuzzy numbers
(triangular, trapezoidal etc.), the input data are presented as quasi S shape fuzzy
numbers. In this framework some theoretical and numerical results are given.
Key words: fuzzy number, possibilistic mean value, quasi S shape fuzzy
data, portfolio optimization.
JEL Classification: C020, C440, C610.
1. Introduction
The issue of fuzzy mathematical programming is a topic of great interest in
literature ([13], [14], [15], [16]). Also the financial market analysis, including
portfolio optimization, has numerous approaches ([3], [10], [14], [17], [19]). In
fact, the problem of portfolio optimization can be viewed as a problem of decision-
making. In this area, the theory of fuzzy sets is a very useful tool [20]. In this paper
we deal with both subjects mentioned above. More precisely, we discuss a fuzzy
optimization model from the theoretical point of view and we also show its
applications to the problem of portfolio selection. It is included in a certain class of
models, namely those based on some crisp characteristics of fuzzy data (such as
possibilistic mean value or possibilistic variance - these concepts are rigorously
defined in [1], [2]). Usually, this type of model has applications in the difficult
issue of portfolio selection but it also can be seen as a standalone optimization
problem with constraints. In this paper we use a generic model whose objective
function is based on possibilistic mean value of a certain fuzzy number. It is used
quite a lot in the literature but with various changes in the parameters or constraints
(see [4], [5], [9], [14], [18]). However, there is one thing in common: the
possibility of transforming the initial problem in some linear optimization
problems. Generally, the input data (in the case of portfolio optimization we
discuss about the possible return rates) are modeled as triangular or trapezoidal
numbers ([4], [5]). Also some generalization in the form of polygonal numbers was
given ([9], [18]). By contrast, in this work we propose the use of quasi S shape
fuzzy numbers, which have already proven useful in the field of statistical
regression (see also [13]). These numbers are partially linearized S shape ones (a
Costin Ciprian Popescu
presentation of the S shape membership functions can be found in [8]). The
linearization of some parts of the membership function is necessary for the model
proposed here. In this way the data fulfill the necessary conditions that allow us to
apply for our case some general results about binary operations (as sum, scalar
multiplication or even some comparisons between fuzzy numbers - see [6], [7],
[11], [12], [15], [16]). In section 2 these theoretical aspects are emphasized in a
more detailed presentation. We will also show the steps that must be taken in order
to find a solution to the model described here. In section 3 the applicability of the
proposed algorithm is verified for a set of numerical data.
2. Theoretical results
At first we give some results regarding quasi S shape fuzzy numbers [13],
which are derived from S shape numbers [8]. Because the membership function of
a S shape number has an asymptotic behavior regarding the lines 0y , 1y , it
appears to be difficult to make certain operations with them. However, their partial
linearization allows us to use the general results regarding sum or scalar (that is a
real number) multiplication ([6], [11], [16]). So consider the numbers R, ,
2
1,0, 2221 ll , 1,
2
1, 1211 ll and
11
1111
1ln
l
l,
21
2121
1ln
l
l,
12
1212
1ln
l
l,
22
2222
1ln
l
l.
Also consider two bijective functions:
1,01
,1
1: 11
11
21
21 llf , 1,0
1
1,
1: 22
22
12
12 llg
given by
11
11
1111
11
1121111
1121
2121
21
21
21
2122121
1, ,
1ln
, ,1
1
,1
1 ,
1ln
lxl
l
lxll
xe
lxl
l
lxll
xfx
and
A Fuzzy Optimization Model
. 1
1, ,
1ln
, ,1
1
,1
,1
ln
22
22
2222
22
2222222
2212
1212
12
12
12
1221212
lxl
l
lxll
xe
lxl
l
lxll
xgx
A quasi S shape fuzzy number, u~ , can be completely defined by its membership
function. Let 1,0:Ru be this function, as follows:
. 1
ln1
,1
ln1
,1
1ln
1
1,
1ln
1 ,
1ln
1,
1ln
1
1 ,
,1
ln1
11ln
1
1, ,0
12
12
1211
11
11
22
22
2212
12
12
11
11
1121
21
21
22
22
2221
21
21
l
l
ll
l
lx
l
l
ll
l
lxxg
l
l
ll
l
lxxf
l
l
ll
l
lx
xu
Figure 1. A quasi S shape fuzzy number
Costin Ciprian Popescu
The condition for the existence of such a number is [13]:
12
12
1211
11
11
1ln
11ln
1
l
l
ll
l
l
or
1211
1211
1211
1211 11lnll
ll
ll
ll.
In the case of equality we obtain a unique point, 0x , where the membership
function is equal to 1. This is equivalent to
1211
1211
1211
1211 11lnll
ll
ll
ll.
Remark that the parameters and control the left spreading and the right
spreading, respectively. The parameters 21l , 11l determine the length of the
linearization on the left side of the membership function. Similarly, the parameters
22l , 12l determine the length of the linearization on the right side of the
membership function. We denote the quasi S shape fuzzy number u~ by
22211211 ,,,,,~ llllu . It can also be defined as a triplet rruru ,, (like any
fuzzy number with certain properties, as it is shown in [6], [7], [11], [16]), where
R1,0:, ruru and [13]:
, 1, ,1
ln1
1
1
1
, ,1
ln
,0 ,1
ln1
1
1
1
11
11
11
111111
1121
21
21
21
212121
lrl
l
lr
ll
llrr
r
lrl
l
lr
ll
ru
. 1, ,1
ln1
1
1
1
, ,1
ln
,0 ,1
ln1
1
1
1
12
12
12
121212
1222
22
22
22
222222
lrl
l
lr
ll
llrr
r
lrl
l
lr
ll
ru
A Fuzzy Optimization Model
Some crisp characteristics of a general fuzzy number rraraa ,,~ are
introduced in [1]: the lower possibilistic mean value (
1
0
2~ drraraM ), the
upper possibilistic mean value (
1
0
2~ drraraM ) and the possibilistic mean
value (2
~~~ aMaMaM ). By applying these definitions to
22211211 ,,,,,~ llllu , we obtain the following results:
21
112111
11
21111
21
221
1
01
ln3
2
132~
l
lll
l
ll
l
ldrruruM ,
12
221222
12
21212
22
222
1
0
1ln
3
2
132~
l
lll
l
ll
l
ldrruruM
and (see also [13])
21
112111
11
21111
21
221
1ln
2
1
26
2
162
~
l
lll
l
ll
l
luM
12
221222
12
21212
22
222 1ln
2
1
26
2
162 l
lll
l
ll
l
l
12
21212
22
222
11
21111
21
221 2
1
2
16
1
2 l
ll
l
l
l
ll
l
l
2112
221112222111
1
1ln
2
1
2 ll
llllll.
For example, if we analyze the number 3.0,2.0,7.0,9.0,10,3~u (Figure 1) we
obtain: 0915.2~uM , 2571.9~uM and 5828.3~uM . We also remark that
if 1211 ll and 2221 ll (i.e. same linearization levels on both sides) we obtain
2
~uM . Taking into account some previous results from literature ([4],
[14], [18]), we consider the fuzzy optimization model S , as below. Notice that in
our problem the input data are modeled as quasi S shape numbers instead of
polygonal ones. So the initial problem is formulated as:
Costin Ciprian Popescu
nip
p
tup
upM
iii
n
i
i
n
i
ii
n
i
iin
,1 ,
1
~Pos
s.t.
~max
S
1
1
1Rp
where npp ,...,1p is a vector for which 11
n
i
ip , 1,0 , Rt and iu~ are
quasi S shape fuzzy numbers. When such a model is used for improving a
portfolio selection, the parameters gain some concrete meanings: for example p
show how an investor must share this capital between n (possibly risky) assets. In
order to solve S , in the next steps we change the form of this optimization model.
In [1] it is proved that n
i
i
n
i
i aMaM11
~~ and abMabM ~~ (where Rb
and 1~a ,…, na
~ , a~ are fuzzy numbers). Applying these results for our case, we
conclude that the objective function can be transformed as follows: n
i
iii
n
i
ii pupM11
2
1~
n
i
ipl
ll
l
l
l
ll
l
l
112
21212
22
222
11
21111
21
221 2
1
2
16
1
n
i
i
n
i
i pll
llpllll
12112
2211
1
122221111
1ln
2
1
2
1.
Moreover, by using the substitutions
12
21212
22
222
11
21111
21
221
1
2
1
2
16
1
l
ll
l
l
l
ll
l
lL ,
1222211122
1llllL ,
2112
22113
1
1ln
2
1
ll
llL
and
2
iii (for ni ,1 )
A Fuzzy Optimization Model
we can write:
n
i
ii
j
j
n
i
ii pLupM1
3
11
~ .
Taking into account the definition of the possibility of ba~ (where a~ is a fuzzy
number and Rb ) [15], we can find some characteristics of a quasi S shape
fuzzy number u~ . So, for Rt , we have:
1) 0~Pos tu
when
21
21
21
1ln
1
1
l
l
lt ;
2) 21
2121212121212121
1ln111~Pos
l
lllllltlltu
when
21
21
21
21
21
1ln
1ln
1
1
l
lt
l
l
l;
3) te
tu1
1~Pos
when
11
11
21
21 1ln1
lnl
lt
l
l;
4) 11
1111111111111111
1ln111~Pos
l
lllllltlltu
when
11
11
1111
11 1ln11
lnl
l
lt
l
l;
5) 1~Pos tu
when
11
11
11
1ln
1
l
l
lt .
Remark that the subset of quasi S shape fuzzy numbers, with the usual two binary
operations (addition and scalar multiplications), is not necessarily closed. For a
convenient conversion of the first constraint, we need to discuss the inequality
tupn
i
ii
1
~Pos . We see that there are several possibilities, as follows:
Costin Ciprian Popescu
1) If n
i
ii pl
l
lt
1 21
21
21
1ln
1
1
then
0~Pos1
tupn
i
ii .
2) If n
i
ii
n
i
ii pl
ltp
l
l
l1 21
21
1 21
21
21
1ln
1ln
1
1
then
n
i
i
n
i
iin
i
ii
p
pl
l
ltll
tup
1
1 21
21
21
22121
1
1ln
1
1
~Pos .
Thus
tupn
i
ii
1
~Pos
is equivalent to
tpl
l
lll
n
i
ii
1 21
21
2212121
1ln
1
1.
3) If n
i
ii
n
i
ii pl
ltp
l
l
1 11
11
1 21
21 1ln1
ln
then
n
i
i
n
i
ii
p
pt
n
i
ii
e
tup
1
1
1
1~Pos1
.
Thus
tupn
i
ii
1
~Pos
is equivalent to
tpn
i
ii
1
1ln .
A Fuzzy Optimization Model
4) Assume that n
i
ii
n
i
ii pl
l
ltp
l
l
1 11
11
111 11
11 1ln11
ln .
It follows that
tupn
i
ii
1
~Pos
is equivalent to
tpl
l
lll
n
i
ii
1 11
11
2111111
1ln
1
1.
5) If n
i
ii pl
l
lt
1 11
11
11
1ln
1
then
1~Pos1
tupn
i
ii .
Taking into account the previous lines, we can conclude that the solution for the
initial model can be obtained by solving three linear models (that will be denoted
by (S1), (S2), (S3)), which have the same objective function. The solution for (S) is
chosen such that the common objective function reaches its maximum. In other
words, if 1Sp , 2Sp and 3Sp are the solution vectors for (S1), (S2), (S3) then the
solution for the initial problem is one of these three but so that the objective
function takes the greatest possible value. Therefore 1Sp , 2Sp and 3Sp can be
viewed only as preliminary results for the initial model, only one being the optimal
solution. On the other hand, there is the possibility that one of these problems has
no admissible solutions, as we shall see in the next section. In this case we
compare only the existing solutions. We have:
Costin Ciprian Popescu
, ,1 ,
1
1ln
1ln
1
1
1ln
1
1
s.t.
max
S1
1
1 21
21
1 21
21
21
1 21
21
2212121
1
3
1
nip
p
tpl
l
tpl
l
l
tpl
l
lll
pL
iii
n
i
i
n
i
ii
n
i
ii
n
i
ii
n
i
ii
j
jnRp
nip
p
tpl
l
tpl
l
tp
pL
iii
n
i
i
n
i
ii
n
i
ii
n
i
ii
n
i
ii
j
jn
,1 ,
1
1ln
1ln
1ln
s.t.
max
S2
1
1 11
11
1 21
21
1
1
3
1Rp
and
. ,1 ,
1
1ln
1
1ln
1ln
1
1
s.t.
max
S3
1
1 11
11
11
1 11
11
1 11
112111111
1
3
1
nip
p
tpl
l
l
tpl
l
tpl
l
lll
pL
iii
n
i
i
n
i
ii
n
i
ii
n
i
ii
n
i
ii
j
jnRp
3. Numerical test
We will study the case of a virtual portfolio with five assets. Assume that
the return rates (including the possibility of some losses) are modeled as five quasi
S shape fuzzy data (see Fig. 2), as follows: 3.0,2.0,7.0,9.0,10,3~1u ,
3.0,2.0,7.0,9.0,9,1~2u , 3.0,2.0,7.0,9.0,6,2~
3u , 3.0,2.0,7.0,9.0,7,5.0~
4u ,
3.0,2.0,7.0,9.0,5.6,5.1~5u . For each number iu~ ( 5,1i ) we consider the pair
A Fuzzy Optimization Model
RRii , , which gives the left point and the right point where the
membership function intersects the horizontal axis. For our numbers, we obtain:
276.12,6363.5 , 276.11,6363.3 , 2759.8,6363.4 , 2759.9,1363.3 and
7759.8,1363.4 , respectively. The level Rt is taken equal to ii 5,1max i.e.
1363.3t . The other parameters can be chosen depending on the level of risk
acceptable to the investor. So we consider 5.0 and 1.0i , 5.0i , for all
5,1i . Taking into account those stated in section 2, we proceed directly to the
formulation of problems (S1), (S2) and (S3). After calculations, we obtain:
5,1 ,5.01.0
1
1363.38863.28863.13863.33863.23863.4
1363.31363.41363.36363.46363.36363.5
1363.30113.101129.05113.151129.05113.2
s.t.
5828.23328.30828.20828.45828.3max
S1
54321
54321
54321
54321
543215
ip
ppppp
ppppp
ppppp
ppppp
ppppp
i
Rp
5,1 ,5.01.0
1
1363.36972.06972.11972.01972.18028.0
1363.38863.28863.13863.33863.23863.4
1363.35.15.023
s.t.
5828.23328.30828.20828.45828.3max
S2
54321
54321
54321
54321
543215
ip
ppppp
ppppp
ppppp
ppppp
ppppp
i
Rp
5,1 ,5.01.0
1
1363.38083.18083.23083.13083.23083.0
1363.36972.06972.11972.01972.18028.0
1363.37472.37472.22472.42472.32472.5
s.t.
5828.23328.30828.20828.45828.3max
S3
54321
54321
54321
54321
543215
ip
ppppp
ppppp
ppppp
ppppp
ppppp
i
Rp
The solution for (S1) is 1.0,1.0,1.0,5.0,2.01Sp , for which the maximum of the
objective function is 5578.3 . The solution for the problem (S2) is
1.0,1.0,1.0,375.0,325.02Sp , for which the maximum of the common objective
Costin Ciprian Popescu
function is 4953.3 . The problem (S3) has no admissible solutions. Due to the fact
that 5578.34953.3 , the final result for the initial model (S) is 1Sp .
Figure 2. Five data of quasi S shape type
4. Conclusion
From the general set of fuzzy numbers, the subset of quasi S shape
elements was considered. Some theoretical features (such as possibilistic mean
value, some binary operations) of these elements were emphasized. An
optimization model in which the data are presented in the form of quasi S shape
fuzzy numbers was discussed. A numerical application, in order to test the
applicability of the model, was given.
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A Fuzzy Optimization Model
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