Method of solving multi objective optimization problem in the presence of uncertainty

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 01 | Jan-2014, Available @ http://www.ijret.org 496
METHOD OF SOLVING MULTI-OBJECTIVE OPTIMIZATION
PROBLEM IN THE PRESENCE OF UNCERTAINTY
Mingliqulov Zafar
Centre for development of software and hardware - software systems at the Tashkent University of Information
Technologies, Uzbekistan
Abstract
Multi-objective formulations are realistic models for many complex engineering optimization problems. In many real-life problems
considered objectives conflict with each other and optimization of one target solutions can lead to unacceptable results for other
purposes. A reasonable solution of the multi-objective problem is to study the set of solutions, each of which satisfies the objectives at
an acceptable level with no dominance of any of the solutions. The article provides a brief overview of multi-objective optimization
methods (by Pareto criteria) and their improvement.
Keywords: A fuzzy set, forecasting, risk, weakly formalized process for Pareto multi-objective optimization.
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1. INTRODUCTION
The goal of this work is to present an overview of the methods
of multi-criteria optimization on Pareto criteria and its
improvement. In most cases objectives of the multi-objective
problem conflict with each other, thereby preventing
simultaneous optimization of each objective. Many, if not all,
real engineering problems actually have more than one
objective, that is, minimize costs, maximize productivity,
increase reliability, etc. They are challenging, but real
problems.
There are two approaches to the multi-criteria optimization.
One of them is the convolution of the individual objective
functions into a single component or to impose on all criteria
except one the set of restrictions. In the first case the
determination of the total objective function is possible with
the help of the theory of utility, the weighted sum, etc., but the
problem lies in the proper choice of weights or utility
functions that characterize the preferences of the customer.
This optimization method doesn’t provide a set, but a unique
solution, which can be examined to identify possible trade-
offs.
The second approach is to define a set of Pareto optimal
solutions, or a representative subset. Pareto solution is the set
of non-dominant to each other solutions. When moving from
one Pareto solutions to another, there are always some loss in
one solution, and certain improvements in others. Pareto-
optimal set of solutions are often preferred by single results
because they are more practical when considering the trade-
offs in the real life problems. The set of Pareto solutions may
be any size, but typically the size increases with increasing
number of purposes.
Note that the Pareto principle is not universal and applies only
if a number of axioms are met. And even if these axioms are
met, the construction of the Pareto set can cause significant
difficulties [1, 7-8].
2. STATEMENT OF THE PROBLEM OF MULTI-
OBJECTIVE OPTIMIZATION
In the formulation of multi-objective optimization problem
mandatory condition to meet all individual criteria and
constraints is introduced as a requirement for the optimality of
solutions, namely all of the functions belonging to the set of
optimal solutions must be different from zero in the optimum
point, the optimum criteria must be satisfied to the extent
possible. The increase in the value of the generalized criterion
should not happen while improving the value of some
indicators of quality at the expense of the deterioration of
others. In the terminology of decision theory, the latter
requirement is equivalent to the condition of belonging to a
Pareto optimum [1].
The problem of multi-criteria optimization has the following
form:
        
,
min,,...,, 21
Xx
xfxfxfxf T
q


(1)
Where

__________________________________________________________________________________________
 
 
 
.,1,
,0,|
,,...,2,1
,
1
1


















n
j
ijij
n
n
j
jkjk
mibxa
xbAxRxX
qQk
xcxf
The problem of multi-objective optimization with fuzzy goal
involves finding such х that satisfy the following constraints
[2-4]:
  ,,,...,2,1,~ Xxqkgxf kk 
(2)
Where kg~
- fuzzy set,
  
 
   
 












.,0
,,1
,,1
kkk
kkkk
k
kk
kk
kk
tgxf
tgxfg
t
gxf
gxf
xf
(3)
The solving the Fuzzy problem (2) can be reduced to the
solving the following unfuzzy problem:
  
.
,
max,
Xx
xfkk





(4)
3. FINDING THE PARETO OPTIMAL
SOLUTIONS
The solution Xx 0
is called Pareto optimal solution if
     0
xfyf kkkk  
for all y, and      0
xfyf SSSS  
for at
least one.
is called optimal solution on the Pareto
type criteria if there is no Xy better than
0
x on the Pareto
type criteria.
We introduce the concept the improvability of solution Xy
on the Pareto type criteria in fuzzy environment: we call
solution Xy improvable if there is the solution Xx 0
,
which is better than у on the Pareto type criteria.
is improvable in finding multi-objective
fuzzy solutions         xfxfxfxf p,...,, 21
if and only if there is a
vector
Q
R , for which the following inequalities are met:
      S
SS
k
kk cxfcxf  00
, 
For all  Qk ,...,1 and at least one  QS ,...,1 , where
k
k
cc 
,
   
k
kkk
y
yfc   minmax
.
Let required inequalities are satisfied, then according to the
definition of
k
c Xy  , for which the inequality
   kkk yfc  
, and therefore
        yfcxfyfc SS
S
SSkk
k
  0
,
,      yfcxf kk
k
SS  0
are
met for all  Qk ,...,1 or at least one  QS ,...,1 , exists. These
inequalities show that the solution Xx 0
is improvable.
Let solution Xx 0
is improvable and let Xy  is the solution,
which is better than the solution
0
x on Pareto criteria. We
assume that      yfyf kkSSk  
for all  Qk ,...,1 , where
     .: 0
xfyfS SSSS  
Then
        .][min][max yfyfyf SSkkk
y
kkk
k
 
Considering
   cyf kkk
k
 ][min 
for all  form Q
R , we get
          ][max0
kkk
k
kkkkkk yfyfxf 
   cyf kkk
y
 ][min 
For all  Qk ,...,1 and
      cyfxf SSSSSS   0
for at least one  QS ,...,1 . Thus
the validity of these inequalities is proved.
Solution Xx 0
is improvable in situation of making
multipurpose decisions if and only if there exists a vector γ
from the set
      ,minmax:R{ Q
k
y
kk
y
yfyfГ   
 kQk   ,,...,1,
such that inequalities
      S
SS
k
kk cxfcxf  00
, 
are met. The validity of these
inequalities follows from the following considerations
                ,
minmax0
cyfyfyfxf kkk
k
kkkkkk  


for
every  Qk ,...,1,  ,         cyfxf SSSSSS   0
.
It follows that if the evaluation functionals
   Q
kkk yf 1
are
obtained after the application of the natural normalization,
then the domain Г has following form:
 kQkГ k    ;,...,1,,1;RQ
.

__________________________________________________________________________________________
Thus, the solution to the problem about improvability, Pareto
optimality of multi-purpose solution Xx 0
on Pareto
criterion is reduced to the existence (absence) of a vector
 for which the inequalities       S
SS
k
kk cxfcxf  00

are
met.
Thus the following inequalities should be met (be
incompatible) for the solution Xy to be improvable
(optimal on Pareto):
        .,...,1,
minmaxmax Qkzfyf k
z
kk 










 

The validity of these inequalities follows from the following
considerations:
             
        ,minmaxaxminmax
minmax0

















zfmzf
zfzfyfxf
zz
kkk
z
kkkkkk
           










 zfyfxf SS
z
SSSSSS minminmax0
for all  kQk   ,,...,1,
or at least one  QS ,...,1
.
Let   yfkk
be the membership function of  yfk , which is
defined as (3).
0
x is optimal solution of improvable problem
   .0,,,...,1max, *
1

kkkk
Q
k
k XxQkxf 
(5)
Then solution Xx 0
is Pareto optimal solution of the problem
(1).
We suppose inverse case. Let Xx 0
be Pareto optimal solution
of (1). Then there exists such solution Xy that    0
xfyf kk 
for
all ( Qk ,...,1 ) and    0
xfyf SS 
for some  .,..,1 QS  Since the vector
Qkk ,...,1, 
is positive and satisfies the following equations:
   Qkxf kkk ,...,1,*0
 
And
    

Q
k
Q
k
kkk Qxf
1 1
*0
.
Nevertheless, there is    0
xfyf kk 
for all ( Qk ,...,1 ) and
   0
xfyf SS 
for some .S
This leads to the following inequality:
        
      .
1
*
1
*00
1 1
*0

 



 


Q
Sk
k
SSkk
Q
Sk
k
SSkk
Q
k
Q
k
kkk
Qyfyf
QxfxfQxf


This means that the solution (5) Xx 0
is not optimal.
Finding Pareto optimal solution (5) in condition of (3) leads to
solving following problem of linear programming:



















.,...,1,0
,,...,1,
max,cR
1
q
1k
k
njx
mqkbxa
x
j
n
j
kjkj
k
We solve this problem using recurrent neural networks.
For calculating the model with recurrent neural networks
moving to the opposite function


q
kk xcR
1k and to write the
value of kc
in the appropriate table with opposite sign is
needed.
When vector is given to the input of the network, the state of
neurons are determined, but then because the outputs of
neurons are the feedbacks, to their inputs again a new vector is
entered and the state changes again. The concept of stability is
connected with recurrent networks [6]. The network is
considered to be stable if, after a finite number of iterations,
the neurons receive state that no further changes. When a
vector is given to the input of stable recurrent networks, the
output signals of neurons are generated, which are then
applied to the inputs again, generating a new state vector
again, but while the network is not set to the final state, with
the increasing number of iterations, the number of changes of
nodes’ states is reduced. Network without feedback is always
stable, since when one vector is given to the input, nodes of
the network change their states only once due to the constancy
of input neurons.

__________________________________________________________________________________________
For solving the problem (1)-(2) recurrent neural network,
which is defined with the following differential equation, is
proposed:























t
cbxa
t
tu
k
n
j
kjkj
k
exp
)(
1
,
Where )exp(1
1
)()),((
u
uftufx kk


. As in the Hopfield network,
it uses neurons matrix with the size of nn , but the neurons
interact not on the principle “with each other”, but by the rows
and columns.
The difference version of this equation has the following form:



























 



t
cbxatuu k
n
j
kjkj
t
k
t
k exp
1
1
(6)
Where t - step by time. Parameters  ,,,,t
are selected
experimentally and they significantly impact to the speed of
problem solving and to the quality of solution.
To speed up the solving of equations (6) the principle of
«Winner takes all» is proposed [2-4]:
Matrix
0
kx
of random values ]1,0[0
kx
is generated.
Iteration (6) is continued until following inequality is met:

n
j
kjkj bxa
1
,
Where  - the specified accuracy of the constraints
Steps 1 and 2 are repeated.
One of the main properties of artificial neural networks is the
reliability of the neural network models. This property may
allow building fo neural network systems for areas where high
reliability is required.
No less important is the learning property of neural networks.
Because of this property, not only they can recognize patterns
applied to their input, but also adjust their parameters with the
help of the appropriate procedure to carry out correct
recognition as much as possible. Thus, the neural networks
can operate in two modes or phases: a learning mode and the
recognition mode.
The ability of neural networks to generalize is also very
important property. Due to this property not only the network
can show the imaged given during the training, but also build
new ones. This increases the “competence” of systems based
on neural networks.
4. COMPUTATIONAL EXPERIMENT
We consider an example taken from [5]. For solving multi-
criteria problem
















8
10
5
3
min2)(
min64)(
1
21
2
21
212
211
x
xx
x
xx
xxxz
xxxz
(7)
With the following fuzzy objectives:
9;20 21  gg
And tolerances
2;2 21  tt
We form following membership functions:
 








rxz
rxz
xzr
xz
2)(0
2)(
2
)(2
)(
1
1
1
11
 








7)(0
7)(
2
)(7
)(
1
1
2
11
xz
xz
xz
xz
Then model (7) takes the following form:
  
  
























8
10
5
3
27
2
1
6422
2
1
max
1
21
2
21
21
21
x
xx
x
xx
xx
xx




__________________________________________________________________________________________
The solution of this model is
   ,0;5,4, *
2
*
1
*
 xxx
Optimal value is 1*
 , the values of the objective function
are
    .9,18 **
 xzxz
We consider the existence of the vector  21,  
, which
may improve the solution
*
x .
For this we solve the following problem:


















8
10
5
3
922
20264
max
1
21
2
21
221
121
21
x
xx
x
xx
xx
xx



(8)
The solution of the model (8) is
  )5;2(, **
2
**
1
**
 xxx
Optimal values are:
0;21 21  
, the values of the objective function are
;22)( **
1 xz .9)( **
2 xz
Thus there is a vector
 ,, 21  
where the Pareto-optimal
solution is improvable, i.e.
   *
1
**
1 xzxz 
и
   *
2
**
2 xzxz 
5. CONCLUSIONS
Thus, the solving the problem of Pareto optimality of the
multi-objective solution
n
Rx 0
on Pareto criteria or its
improvability is reduced to establishing the existence or
absence of a vector that satisfies the inequalities
      S
ss
k
kk cxfcxf  00

.
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relative importance of the criteria in the case of fuzzy
preference relations // Journal of Comp. Mathematics
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[3] Rotshtein A.P. Fuzzy multicriteria choice among
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[4] Rotshtein A. P., Intelligent Identification Technologies:
Fuzzy Sets, Neural Networks, Genetic Algorithms
(Universum, Vinnitsa, 1999) [in Russian], 320 p.
[5] N.N.Vilyams. Parametric programming in the economy.
Moscow. [in Russian] 1976 y. 96 p.
[6] Yazenin A.V. The problem of vector optimization with
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Mathematical methods of optimization and control in
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BIOGRAPHIES
He got his BSc and MSc degrees from
"Software engineering for Computer
Systems, networks and complexes"
department, Tashkent University of
Information Technology in 2005 and 2007
respectively. He published several papers on
the topic of Data mining, such as fuzzy sets
theory and neural networks in many international conferences
and journals. Currently, He is working as senior researcher at
Centre for development of software and hardware - software
systems at the Tashkent University of Information
Technologies.

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Method of solving multi objective optimization problem in the presence of uncertainty