FUZZY PROGRAMMING APPROACH FOR PORTFOLIO SELECTION
PROBLEMS WITH FUZZY COEFFICIENTS
H. A. Khalifa
[email protected]
Ramadan A. ZeinEldin
[email protected]
Operations Research Department, ISSR, Cairo University
Abstract
The portfolio selection problem (PSP) uses mathematical approaches to model stock exchange investments. Its aim is to find
an optimal set of assets to invest on, as well as the optimal investments for each asset. In this paper, a portfolio selection problem
(FPSP) with fuzzy objective function coefficient (FPSP) a multiple objective problem including uncertainties is investigated. The
FPSP is considered by incorporating fuzzy numbers into the coefficients of the objective functions. Through the use of the  -level
sets of the fuzzy numbers, the FDSP is converted into the corresponding ordinary from (  -PSP). And an extended Pareto optimality
concept called the  -Pareto optimality is introduced. A fuzzy programming approach is applied to solve the (  -PSP). Simulated
annealing and linear membership function is being used to find the compromise solution. Finally, a numerical example is given for the
sake of the paper for illustration.
Keywords: Portfolio selection problem; Multi-objective programming; Fuzzy numbers;
conditions; Fuzzy programming; Membership function; Simulated annealing.
 -Pareto
optimality; Kuhn-Tucker
model, a large number of extensions have been proposed,
Deng et al., (2005), Hirschberger and Pi (2007), Liu and
Wang (2003), Ammar (2008) and Several algorithms for
solving the problem of computation have also been studied
Lin and Liu (2008). Xu and Li (2002) formed an extended
portfolio selection model when liquidity is considered as
another objective to be optimized besides expectation and
risk. Ammar and Khalifa (2003) formulated fuzzy portfolio
optimization problem as a convex quadratic programming
problem. Hasuike and Ishii (2008) introduced multi-criteria
mathematical decision models with respect to portfolio
selection problems, particularly multi-scenario models to the
future return of each asset including ambiguity and the fuzzy
extension of mean-variance model and mean-absolute
deviation and model. Hassuike et al., (2009) proposed several
mathematical models with respect to PSP, particularly using
the scenario model including the ambiguous factors.
Aryanezhad et al., (2011) introduced a portfolio selection
problem where fuzziness and randomness appeared
simultaneously in optimization process. Bao et al., (2010)
studied portfolio models based on fuzzy interval numbers
under minmax rules. Ida (2004) considers PSP with interval
and fuzzy objective function coefficients as a kind of multiobjective problems including uncertainties where two kinds
1. INTRODUCTION
Recently, not only big companies and institutional
investors but also individual investors called Day-traders take
part in investment fields, and there exist various investments
such as stock, currency, proper, land, etc… Therefore, in
order to make an optimal investment fitted with each idea of
investor, the role of investment theory becomes more and
more important. Of course, it is easy to decide the most
suitable asset allocation of decision makers (DMs) can
receive reliable information with respect to future returns a
priori. However, there exist some cases that uncertainty from
social conditions have a great influence on the future returns.
In the real market, there are random factors derived from
statistical prediction based on historical data and ambiguous
factors derived from the mental point of investors and lack of
reliable information. Under such uncertainty situations, they
need to consider how to reduce a risk, and it becomes
important whether on investment makes profit greatest.
Since Markowitz (1952) published his path-break
work in the early mean-variance model has been a rather
popular subject in both theory and practice. Based on this
-1-
of efficient solutions are introduced: possibly efficient
solution as an optimistic solution, necessarily efficient
solution as a pessimistic solution. Wu and Liu (2012)
developed a robust method to describe fuzzy returns by
employing parametric possibility distributions. Gharakhani
and Jafor (2013) examined advanced optimization approach
for portfolio problem introduced by Blacu and Litterman
(1991).
Definition
3.
[ aL , aU ](  )[b L , bU ]  [ aL  b L  aL  bU  aU  b L  aU  bU ,
convex
fuzzy
set,
2
x 1, x 2  R , w [0, 1] .
is normal, i.e.,  x 0  R
[ a L , aU ](  )[b L , bU ]
PROBLEM
AND
SOLUTION
Let us give a brief description of Markowitz's
portfolio selection model (Markowitz (1959)). Assume that
type is denoted as R j ( j  1, 2, ..., n ) and the proportion
of total investment funds devoted to it is denoted as
n
x j ( j  1, 2, ..., n ) , i.e., 
x
j 1
since
j
 1 . In the real setting,
R j s vary due to uncertainties, those are assumed to be
random variables which can be represented by the pair of
average vector   ( 1, ...,  n )(  j : average rate for R j )
i.e.,
1
and covariance matrix
{ j k } (  j k : covariance between
R j and R k ,  j k   k j ). The total return associated
for all
for
which
with
a ( x 0 )  1.
the
R (x ) 
 -level set (or  -cut) of a fuzzy set a
x  ( x 1, ..., x n )
portfolio
n

j 1
( a )  { x : a ( x )  0} is the support of a
fuzzy set a .
Rj x
j
is
given
by
. The average and variance of R ( x )
are given as
supp
R is a non-fuzzy (or ordinary) set denoted by
DEFINITION
.
CONCEPT
a (w x  (1  w ) x )  min ( a ( x ), a x 2 ))
Definition 2. An
[ a L , aU ]()[b L , bU ]  [ a L  b L , aU  bU ]
a L / b L  a L / bU  a U / b L  a U / bU ].
following properties:
1. a ( x ) is upper semicontinuous membership
4.
2.
 [ a L / b L  a L / bU  a U / b L  a U / bU ,
Definition 1. Let R be the set of real numbers, the fuzzy
number a is a mapping a : R  [0, 1] , with the
a
[ a L , aU ]()[b L , bU ]  [ a L  b L , aU  bU ]
4.
In this section, some of the fundamental definitions
and concepts of fuzzy numbers initiated by Bellman and
Zadeh (1970) and interval of confidence introduced by
Kaufmann and Gupta (1988) are reviewed.
3.
1.
3.
1
[ a L , aU ] ,
that
a L  b L  a L  bU  aU  b L  aU  bU ]
2. PRELIMINARIES
2.
Suppose
[b L , bU ]  I ( R ) . We define:
In this paper, fuzzy portfolio selection problem is
presented as a multiple-objective problem including
uncertainties. Fuzzy programming approach is being used for
the auxiliary problem corresponding to the FPSP by defining
suitable membership functions to find the compromise
solution. The result of the paper is organized as in the
following manner; In section 2, some elementary concepts of
fuzzy numbers and interval confidence are introduced. In
section 3, fuzzy portfolio selection problem is presented as
basic definition and properties. In section 4, fuzzy
programming approach is applied to the auxiliary problem
corresponding to the FPSP. In section 5, numerical example is
given for illustration. Finally, some concluding remarks are
reported in section 6.
functions.
is
a
3.

E (R (x ))  E 


of
n
 Rj x j
j 1




n
n
j 1
j 1
 E (R j x j )    j x j
and

V ( R ( x ) ) V 


L ( a ) and
defined by:
L ( a )  { x  R : a ( x )   } .
n
 Rj x j
j 1




n
n

j 1 k 1
x
j
jk xk
.
Since the variance regarded as the risk of investment,
preferable investment is a solution of the following two
Let
I ( R )  {[ a , a ] : a , a  R  (  ,  ), a  a }
L
U
L
U
L
U
denoted the set of all closed intervals numbers on R.
-2-
objective linear-quadratic programming problem
n
max f m ( x )    j x j ,
j 1
min f v ( x ) 
1 n n
  x j jk xk ,
2 j 1 k 1
subject to
n


x  X   x  R n :  x j  1, x j  0, j  1, 2, ..., n 
j 1


Assuming that these fuzzy parameters are characterized by
fuzzy numbers, let the corresponding membership functions
be:
and
(  (  1 ), ...,  (  n ))






 (1






j
 j k (  j k
x
j 1
min f u ( x ) 
x
1 n n
  x j jk xk ,
2 j 1 k 1
subject to
x X ,
where
 j  (1, ...,  n )
and
jk
 11

  21


  n1
1n ( 1n ) 

 2 n (  2 n ) 
.

 n n (  n n ) 

 -level sets of that the fuzzy numbers 
and  defined as the ordinary set (  ,  ) in which the
degree of their membership functions exceeds level  :
However, when formulating a mathematical
programming problem which closely describes and represents
a real-world decision situation, various decision situation,
various factors of the real-world system should be reflected in
the description of objective functions and constraints.
Naturally, those objective functions and constraints involve
many parameters whose possible values may be assigned by
experts. In the conventional approach, such parameters are
required to be fixed at some values in an experimental and /
or subjective manner through the expert's understanding of
the nature of the parameters in the problem formulation
process.
It must be observed here that, in most real-world situations,
the possible values of these parameters are often only
imprecisely or ambiguously known to the experts. With this
observation in mind, it would be certainly more appropriate to
interpret the expert's understanding of the parameters as fuzzy
numerical data which can be represented by means of fuzzy
sets of the real line known as fuzzy numbers. The resulting
mathematical programming problem involving fuzzy
parameters would be viewed as a more realistic version than
the conventional one (Sakawa (1993) and Sakawa and Yan,
(1990)). For this viewpoint, we assume that parameters
involved in the objective functions of (1) are characterized by
fuzzy numbers, As a result, a problem with fuzzy parameters
corresponding to (1) is formulated as
n
 11 ( 11 )

  21 (  21 )
)

  ( )
  n1 j k
We introduce the
)
max f m ( x )    j x j ,
n
(  ,  )  {(  ,  ) :  j (  j )   , j  1, 2, ..., n ;  j k ( j k )   ,
k
 1, 2, ..., n } .
(3)
Then such a degree  , problem (2) can be
interpreted as the following non fuzzy portfolio selection
problem which depends on a coefficient vector
(  ,  )  (  ,  ) (Sakawa (1993) and Sakawa and Yano
(1990)).
n
max f m ( x )    j x j ,
x
j 1
min f u ( x ) 
x
1 n n
  x j jk xk ,
2 j 1 k 1
subject to
x X .





 (4)




Observe that there exist an infinite numbers of such a
problem (4) depending on the coefficient vector
(  ,  )  (  ,  ) and the values of (  ,  ) are arbitrary





 (2)




for
(  ,  )  (  ,  ) in the sense that the degree of all of
the membership functions for the fuzzy numbers in problem
(4) exceeds level  . However, if possible, it would be
(  ,  )  (  ,  ) in problem
(4) so as to maximize and minimize the objective functions
f m and f u , respectively under the given constraints.
desirable for DM to choose
For such a point of view, for a certain degree  , it seems to
be quite natural to have understood the PSP with fuzzy
parameters as the following non fuzzy  -PSP (Sakawa
(1993) and Sakawa and Yano (1990).
1n 

 2n 


 n n 
are fuzzy parameters.
-3-




 (5)




n
max f m ( x ,  )    j x j ,
x,
j 1
1
min f u ( x ,  )   x j  j k x k ,
x,
2
subject to
x  X , (  ,  )  (  ,  ) ,   [0, 1].
f umin  f u ( x 0 ,  0 )
 min{f u ( x ,  ) : x  X ,  (  )0 } , (8)
Fmmax  Fm ( x 1,  1 )
 max{ Fm ( x ,  ) : x  X ,  (  )0 } , (9)
f umax  f u ( x 1,  1 )
 max{f u ( x ,  ) : x  X ,  (  )0 } , (10)
Definition 4. (  -Pareto-optimal Solution). x  X is said
to be an  -Pareto-optimal solution to the problem (5) if and
only if there close not exist on other x  X and
*
of the objective function are referred to when the DM elicits
membership function prescribing the fuzzy goal for the
objective functions f m ( x ,  ) and f u ( x ,  ) . The DM
(  ,  )  (  ,  ) such that f m ( x * ,  * )  f m ( x ,  )
f u ( x *,  * )  f u ( x ,  )
and
determines
and
f m ( x *,  * )  f m ( x ,  )
Fm ( x ,  ) ( f m ( x ,  ))
or
which
f u ( x * ,  * )  f u ( x ,  ) , where the corresponding values
of parameters
parameters.
*
x,
j 1
min f u ( x ,  ) 
1 n n
  x j jk xk ,
2 j 1 k 1
subject to
x  X , (  ,  )  (  ,  ) ,   [0, 1].
4.
FUZZY
PROGRAMMING
the values
Fm0 and f u0 of the objective functions for which
1. For the values under sired (larger 1than
defined
that
(smaller)
than
Fm0 and f u0 , it is
Fm ( x ,  ) ( Fm ( x ,  ))  0
f u ( x ,  ) ( f u ( x ,  ))  0 ,
Fm1
and
Fm ( x ,  ) ( Fm ( x ,  ))  1
and
also
and for the values desired
f u1 , it is defined that
and
also
f u ( x ,  ) ( f u ( x ,  ))  1 .
In this paper, we adopt membership functions, which
characterize the fuzzy goal of the DM. The corresponding
membership
functions
and
 ( Fm ( x ,  ))
FOR
 ( f u ( x ,  ))
It is natural that DM have fuzzy goals for his\her
objective functions when he\she takes fuzziness of human
judgments into consideration. For each of objective functions
Fm ( x ,  ) and f u ( x ,  ) , assume that the DM have
are defined as:
 0,

 Fm1 ( x ,  )  Fm0
 ( Fm ( x ,  ))  
,
1
0
 Fm  Fm
 1,

Fm ( x ,  ) and
f u ( x ,  ) should be substantially loss than or equal to
some value, say, Pi ".
Let   0 , then the individuals minimum
and
Fmmin  Fm ( x 0 ,  0 )
 min{ Fm ( x ,  ) : x  X ,  (  )0 } ,
Fm1 and f u1 of
the objective function for which the degree of satisfaction is
PORTFOLIO SECTION PROBLEM
fuzzy goals such as "the objective
u
the degree of satisfaction is 0 and the values





 (6)




APPROACH
u
monotone
decreasing
for
and f ( x ,  ) ( f u ( x ,  )) . The
[ Fmmin , Fmmax ] and [ f umin , f umax ] , and the DM specifies
For applying fuzzy programming approach, let us
rewrite the problem (5) as in the following form
max Fm ( x ,  )     j x j ,
strictly
membership
functions
and
f ( x ,  ) ( f u ( x ,  ))
domain of the membership functions are the intervals
Remark 1. It should be noted that the parameters (  ,  )
are treated as decision variables rather than constants.
n
are
f m ( x ,  ) ( f m ( x ,  ))
(  ,  ) are called  -level optimal
*
the
(7)
-4-



Fm1  Fm ( x ,  )  Fm0 ,  (11)


1
Fm ( x ,  )  Fm ,

Fm1 ( x ,  )  Fm0 ,
max 
 0,

 f (x ,  )  f 0
 ( f u ( x ,  ))   u 1 0 u ,
 Fm  Fm
 1,

x,


1
0 
f u  f u ( x ,  )  f u ,  (12)


1
fu (x ,  )  fu ,

f u ( x ,  )  f u0 ,
subject to
x X
Fm0 , f u0 , Fm1 and f u1 denote the value of the
objective functions Fm and f u such that the degree of
where
membership functions are 0, 0, 1 and 1, respectively, and it is
assumed that the DM subjectively assesses
Fm0 , f u0 , Fm1
Notice that
 ( Fm ( x ,  L ))
 ,
 ( f u ( x ,  L ))
0    1,
x  0.
 ,






 (15)






Fm*  f m* .
1
and f u .
5. SIMULATED ANNEALING
After eliciting the membership functions defined in
(11) and (12), suppose that applying the way suggested by
Zimmermann (1978), the following problem is obtained as
max 
x ,,, 
subject to
x X
 ( Fm ( x ,  ))   ,
 ( f u ( x ,  ))   ,
0    1,
x  0,
(  ,  )  (  ,  ) .
Simulated annealing (SA) is a trajectory-based, random
search technique for global optimization. It mimics the
annealing process in materials processing when a metal cools
and freezes into a crystalline state with minimum energy and
larger crystal sizes so as to reduce the defects in metallic
structures. The annealing process involves the careful control
of temperature and its cooling schedule. SA has been
successfully applied in many areas (Yang( 2014)).
Simulated annealing presents an optimization technique that






 (13)






can:
(a) process cost functions possessing quite arbitrary degrees
of nonlinearities, discontinuities, and stochasticity;
(b) process quite arbitrary boundary conditions and
constraints imposed on these cost functions;
(c) be implemented quite easily with the degree of coding
quite minimal relative to other nonlinear optimization
algorithms;
(d) statistically guarantee finding an optimal solution.
SA is used to solve complex problems in different areas;
Oliveira and Petraglia (2013) proposed a fuzzy adaptive
simulated annealing which they called a stochastic global
optimization for finding solutions of arbitrarily nonlinear
systems of functional equations. Zarandi et al., (2013)
proposed a new hybrid clustering algorithm for model
structure identification and presented a new fuzzy functions
model and its main parameters are optimized with simulated
annealing approach. Jolai, et al., (2013) and Xhafa, et al.,
(2011) described the steps of SA.
By solving problem (13), we obtain a solution maximizing a
smaller satisfactory degree for the DM. To solve problem
(13), we introduce the set-valued functions: (Sakawa (1993),
and Sakawa and Yano (1990))
S (  )  {( x ,  ) :  ( Fm ( x ,  )   },
S ( )  { x ,  ) :  ( f u ( x ,  )   },
Proposition 1. If
1   2,
then
(14)
S (  1 )  S (  2 ) and if
 1   2 , then S ( 1 )  S ( 2 ) .
From the properties of the  -level set of the fuzzy numbers
 and  , it should be noted that the feasible regions for 
and  can be denoted, respectively, by the closed intervals
[  L ,  U ] and [ L ,  U ] .
6. SOLUTION PROCEDURE
The solution procedure of the linear-quadratic programming
problem (6) is summarized in the following steps:
Step 1: Calculate the individual minimum and maximum of
each objective function under the given constraints for α = 1.
Step 2: Ask the DM to select the initial value of α (0 ≤α≤ 1).
Step 3: For the degree of α, solve the problem (15) using
simulated annealing.
Step 4: The DM is supplied with the corresponding
compromise optimal solution. If the DM is satisfied with the
current objective function values of the compromise solution,
Therefore, through the use of proposition 1, we can
obtain an optimal solution to problem (13) by solving the
following programming problem
-5-
stop. Otherwise, the DM must update the degree α and return
to step 2.
And the individual assumes that these payments are
indicative of what can be expected in the future. Through the
use of the α-level sets of the fuzzy numbers, the fuzzy
numbers plotted in table 1 can be rewritten in table 2 in nonfuzzy form.
7. NUMERICAL EXAMPLE
Consider an individual has identified three mutual
funds as attractive opportunities. Over the last five years,
dividend payments (in Piaster per L. E. invested) has been
shown in table 1.
Table 2: Investment stock with interval valued return
Years
Funds
1
2
3
4
5
Table 1: Investment stock with fuzzy return
Years
Investment
[α+9,-
[α+3,-
[α+11,-
[α+12,-
[α+5,-
1
α+11]
α+5]
α+13]
α+14]
α+7]
Funds
Investment
[α+5,-
[α+8,-
[α+5,-
[α+4,-
[α+8,-
Investment 1
2
α+7]
α+10]
α+7]
α+6]
α+10]
Investment 2
Investment
[α+16,-
[α+0,-
[α+10,-
[α+18,-
[α+1,-
Investment 3
3
α+18]
α+2]
α+12]
α+20]
α+3]
1
2
3
4
5
At α = 0, table 3 shows the mean-variance estimation
Table 3: Mean-Variance estimation
X1n
X2n
X3n
X1n X2n
X1n X3n
X2n X3n
1
[9,11]
[5,7]
[16,18]
[81,121]
[25,49]
[256,324]
[45,77]
[144,198]
[80,126]
2
[3,5]
[8,10]
[0,2]
[9,15]
[64,100]
[0,4]
[24,50]
[0,10]
[0,20]
3
[11,13]
[5,7]
[10,12]
[121,169]
[25,49]
[100,144]
[55,91]
[110,156]
[50,84]
4
[12,14]
[4,6]
[18,20]
[144,196]
[16,36]
[324,400]
[48,84]
[216,280]
[72,120]
5
[5,7]
[8,10]
[1,3]
[25,49]
[64,100]
[1,9]
[40,70]
[5,21]
[8,30]
Sum
[40,50]
[30,40]
[45,55]
[380,560]
[194,334]
[681,881]
[212,372]
[475,665]
[210,380]
cooling rate = 0.95, and number of iterations at each
temperature is 20. The best solution found for this problem is:
,
and
.
8. CONCLUSION
In this paper, the portfolio selection problem with
fuzzy coefficients has been presented as linear-quadratic
programming problem. A fuzzy programming approach has
been applied to the problem by defining a linear membership
function where the optimal compromise solution has been
obtained. Simulated annealing is used to get the optimal
compromise solution.
At the first step, the individual maximum and minimum are
(α = 1)
At α = 0, the problem (15) becomes:
Max λ
s.t.
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