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Abstract— This article is devoted to the development of method
of solution for multicriteria problems of decision-making in
conditions of uncertainty. A model of fuzzy synthetic evaluation
method is proposed. The method is realized by a neural network.
Weights of the method and the network coincide.
Keywords— decision making, method of an fuzzy synthetic
evaluation, neural networks, multi-criteria tasks.
I. INTRODUCTION
The method of synthetic fuzzy evaluation is a method of the
solution of multicriteria problems of decision-making in
conditions of uncertainty. It is used for solving various
problems when a complete assessment to some object with
diverse properties is required [1], [2].
When using the method of synthetic fuzzy evaluation, the
most important task is to define a quantitative assessment of
the importance of various criteria: weights. Weights are
mostly defined by experts, variously set weights lead to
different results of estimation. The project involves the
modification of the method of synthetic fuzzy evaluation
realized by neural network where the method weights are
defined at setting weight vectors of network.
Let’s find out a determination for the method of synthetic
fuzzy evaluation [1]. Previously we will remind necessary
concepts and definitions.
Definition. [3] Let be arbitrary set and
We say that F is a fuzzy subset of the set M.
Here is a membership function. Sometimes we will
write instead of F. The value is interpreted as a
degree of membership of the element to a set F.
Let be a finite set of assessed objects. The
objects are vectors of dimension of n.
We say that is defined by n properties or attributes. The
value expresses the quantitative value of the th property of
the object .
Let’s describe the method.
According each property an object belongs to one of
classes. The object memebership to the one of the classes is
determined as follows.
J.A. Tussupov is from L.N. Gumilyev Eurasian National University,
Astana, Kazakhstan (corresponding author to provide phone: +7 701 789 27
10; e-mail: tussupov@mail.ru ).
L.L. La is from L.N. Gumilyev Eurasian National University, Astana,
Kazakhstan (e-mail: lira_la@hotmail.com)
A.A. Mukhanova is from L.N.Gumilyov Eurasian Natinal University,
Astana, Kazakhstan (corresponding author to provide phone: +7 775 530 59
77; e-mail: ayagoz198302@mail.ru )
Let the object be set by properties. By
each property of we can determine fuzzy sets
(we will designate them as ) corresponding to
classes as follows.
Let . breaks intervals
. Then the membership
function is defined as follows:
The values turn diverse values of properties of ,
of the object into homogenious, belonging to the
segment of [0,1].
Let . The first level of the model of the
synthetic fuzzy evaluation is described by the equation
where , is a weight vector,
, , ,
, . The weight vector is defined by
experts of the subject domain of the problem.
Now let’s describe the second level of the model of
synthetic fuzzy evaluation. It is supposed that at the first level
we estimate the object by one factor, let be given factors
on which the evaluation of is made and
be the equations describing the first level of the
model of the synthetic fuzzy evaluation on the factor ,
where is a weight vector of th factor, is
a matrix made of values of membership functions for fuzzy
sets for th factor, is a resultant
vector of the first level of model for th factor,
.
Let ,
Then the second level of the model of synthetic fuzzy
evaluation is defined by equation , where
is a weight vector of the second level of the
model of synthetic fuzzy evaluation, ,
is a resultant vector of the second level.
Similarly the 3rd, the 4th, etc. model levels are defined.
Let is a resultant vector of the last level of
the model of synthetic fuzzy evaluation,
. Then we believe that the object belongs to th
class.
A model of fuzzy synthetic evaluation method
realized by a neural network
J. Tussupov, L. La, A. Mukhanova
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Volume 8, 2014
ISSN: 1998-0140
103
II. THE MODIFICATION OF A METHOD OF SYNTHETIC FUZZY
ASSESSMENT
The given work involves one model of the modified method
of synthetic fuzzy evaluation realized by a neural network,
where weights of method are defined by network weights.
Let’s describe the model and review some definitions.
Definition. [5] The artificial neural systems or neural
networks are systems physically organized as system of cells
which can do requests, store and use the empirical knowledge
gained as a result of operation.
Let be a finite set of evaluated objects,
be a finite lattice,
where if
Definition. [4] Map is called a fuzzy subset of
the set M.
It is believed that expresses a degree of membership of
an element to a fuzzy set or, in our case, we suppose that
with a degree of possesses the estimated property.
The task of assessment of object will consist in
determination of
We suppose that accept values of the natural language,
expressing some quality, for example, good, bad, etc.
Determination of happens at some stages called
model levels. Depending on quantity of levels we will
distinguish one, two, etc. level models.
Let’s try to describe the first level of model.
Let
be segments of a set of the real numbers . The variable
takes values in . Function is defined by experts
of subject domain of the problem.
Further, we suppose that functions are either increasing
or decreasing, i.e. whereas (or
) which corresponds to the
majority of real tasks of objects assessment.
The first level of model is described by the equation of
where is
weight vector, , is output
vector,
Finally, we determine the resultant vector )
where .
The object assessment at the level occurs
according to the following scheme.
1. Factors for evaluation S are defined.
2. To each factor corresponds properties for an object
assessment.
3. At each level, starting from the second, factors of the
previous level appear as properties of the current level.
Now we will describe the second level of model. Let
be factors on which the assessment of object is
made, be equations describing the
first level of model on a factor , where is
a weight vector, is an resultant vector of the
first level of model for factor ,
Let , ,
Form a matrix as follows. Let be closest
integer to then and for all
Then the second level of model is defined
by the equation
where is weight vector of the second
level of model, weight of factor ,
, normalized vector is the
resultant vector of the second level.
The 3rd, the 4th, etc. model levels are defined by the same
way.
Let be an resultant vector of the last level of
model of synthetic fuzzy evaluation, let k be closest integer to
. Then we suppose that the object of with a degree
of possesses estimated property.
III. DEFINITION OF WEIGHT VECTORS
Determination of weights is an important task when using
the method of synthetic fuzzy evaluation. Weights are usually
defined by experts, variously defined weights lead to different
results of assessment. The article also offers to realize the
given method of assessment by neural network where the
weights of the method are defined by the network weights.
Let’s give the network determination on the example of two
factor, two-level models where the objects are estimated on
three properties of on the first factor, on the second,
on two properties of . The lattice of
contains two elements, with . The network looks as
follows (see figure below).
Fig. 1 Network on the example of two factor, two-level
models
To calculat the weights of the method of synthetic fuzzy
evaluation we’ll break the network into two parts
corresponding to the first and second levels of model. In figure
1 this splitting is represented by the vertical line. Each of the
parts is used for formation of two new networks. The first
network realizes one-level model of synthetic fuzzy
evaluation, corresponds to the first level of initial model and is
applied to calculate weights of the first level. It also consists
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
Volume 8, 2014
ISSN: 1998-0140
104
of two subnets, each of which estimates an object on one of
the factors. Let’s describe it.
0 layer. The signals corresponding to the values of
properties of an estimated object are given to the input layer
for the first factor and for the second one.
Weights from the 0 layer to the first are absent.
1 layer. calculation, , , . It
contains 5 neurons corresponding to the input signals of
for the first factor, for the second. The values
for the first factor, , ,
are calculated on this layer. The vector is
formed for each neuron as follows. If , then
, for . The signals , ,
go to neuron of the second layer. The similar
calculations are made for the second factor.
2 layer. Contains 2 neurons for each factor. Neurons
correspond to values of the lattice .
These are calculated in the second layer
, and normalized vector where
, , , is the weight of
line going from th neuron of the first layer to th neuron of
the second layer of the th factor. The weights from the second
layer to the third are absent.
3 layer. The third layer is output for the first level and
corresponds to the calculation of a matrix of . The layer
contains one neuron for each factor. The vector of is
formed for the factor of , as follows: such is
calculated that be closest integer to so,
and for , .
The second network consists of the third layer of the first
network and the second part of the general network. It will be
used for finding weights of the second level. Let’s try to
describe it.
1 layer. Consists of two neurons, one for each factor. The
vectors , calculated on the third layer of the
first network are input vectors of the neurons first layer. The
signals , move to th neuron of the second layer.
2 layer. Corresponds to calculation of the output vector of
the second level and normalized vector
The 2nd layer consists of two neurons on which
are calculated, where are weights
corresponding to network lines going from th neuron of the
1st layer to th neuron of the 2nd layer. Weights from the
second layer to the third are absent.
3 layer. Finds out the k that is the closest integer to
. The 3rd layer contains one neuron. The number k is
output of the network. We suppose that the object with a
degree of possesses the estimated property.
IV. NETWORK TRAINING
The modification of generalized -rule is used for training
of the both networks [5]. Let’s consider training of the first
network. The training set consists of pairs ,
where is an estimated object, is an integer.
It is supposed that for the pair the object
with a degree of possesses the estimated
property.
Let’s attach to all weights of a network some values close to
"0". Let l be value of an output for the input
from the pair . Let’s determine
Then the weights of a network change as follows:
for each of factors, where
, is a step of training,
, is a value of output signal
from th neuron of the layer 1, is a coefficient of training
speeds which allows to control the average size of weights
change, is a correction connected with an output from th
neuron of the layer 1, is a value of weight after the
correction, is the value of weight before the
correction.
Note the following.
1. At each step of training if there is a pair on the input
, it only changes the weights corresponding to
the lines directed to th neuron and on which nonzero signals
from neurons of the first layer flow.
2. Upon the completion of the process of training let’s put
for ,
. Let’s suppose that ,
is a weight of th property of a method of
synthetic fuzzy evaluation.
Let be a a training set. From 1 it follows
that in the received trained network on input we
will get output i.
Training of the second subnetwork is performed similarly.
After the completion of the training process both
subnetworks are united so that the output layer of the first
subnetwork, which consists of neurons on which ,
are calculated will coincide with the first layer of the
second subnetwork and form the 3rd layer of the network.
The received network realizes offered two-level model of
the method of synthetic fuzzy evaluation.
V. AN APPLICATION FOR THE METHOD OF SYNTHETIC
FUZZY EVALUATION TO EVALUATION OF AIR QUALITY
In this section one aplication of the method of synthetic
fuzzy evaluation is offered: evaluation of air quality. We will
give the description of the method on the example of two-level
model.
Our model consists of three factors. The first and third
factors include 15 properties each, the second factor only one
property. As a result of an evaluation the object will belong to
one of five classes corresponding to the following degrees of
purity of air Very Pure, Pure, Average, Dirty, Very Dirty. That
is latice L consists of 5 elements that
corespond to these degrees of purity.
The data have been taken according to the State Standards
of the Republic of Kazakhstan No. 14-5-2244/I from
31.03.2011 about "Sanitary and epidemiologic requirements to
atmospheric air", and interstate standard of pure rooms and
related controlled environments, part 1. Classification of air
purity" State Standard 14644-1-2002, the appendices of
hygienic standards 2.1.6.2177-07 "The Maximum Permissible
Concentration (MPC) of microorganisms - producers, bacterial
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ISSN: 1998-0140
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preparations and their components in atmospheric air in the
settlements". Let’s describe the factors.
I. Content of harmful chemicals (mg/m3). This factor is
characterized by the following 15 properties: content of
epichlorohydrin, toluol, phenol, aniline, formaldehyde, butyl
alcohol, methyl anhydride, isopropyl alcohol, acetone, CO
carbon oxide, dioxide of sulfur, hydrogen sulfide, oxide of
nitrogen, nitrogen dioxide.
II. Concentration of particles (particles/m3)
III. Concentration of microorganisms producers, bacterial
preparations and their components (mg/ ). This factor is
characterized by the following 15 properties: maintenance of
Acetobacter methylicum, Actinomyces roseolus, Alcaligines
denitrificans, Aspergillus awamori, Bacillus amyloliquefaci
ens, Bacillus licheniformis, Bacillus licheniformis, Bacillus
polymyxa, Bacillus subtilis, Brevibacterium flavum, Candida
tropicalis, Clostridium acetobutilicum, Penicillium canescens,
Trichoderma viride, Yarrowia lipolytica.
At the first level we form matrixes according
to formulas given in first section. For example, for the first
factor it will be the matrix of calculated according to
table below. We will show only some part of the table.
Table. Criteria of an evaluation of air purity of the room
according to the content of harmful chemicals
Criteria
Attributes
Linguistic assessment
VP
P
A
D
VD
Concrnytation of harmful chemicals
(мг\ )
Epichloroh
ydrin
[0-
0,05)
[0,05-
0,1)
[0,1-
0,2)
[0,2-
0,3)
[0.3-
0,5]
Toluene
[0-
0,2)
[0,2-
0,4)
[0,4-
0,6)
[0,6-
0,8)
[0,8-
1,0]
Phenol
[0-
0,00
1)
[0,001-
0,002)
[0,002-
0,003)
[0,003-
0,004)
[0,004-
0,005]
Aniline
[0-
0,01)
[0,01-
0,02)
[0,02-
0,03)
[0,03-
0,06)
[0,06-
0,1]
Formaldehy
de
[0-
0,00
1)
[0,001-
0,002)
[0,002-
0,003)
[0,003-
0,004)
[0,004-
0,005]
Alcohol
butyl
[0-
0,02)
[0,02-
0,05)
[0,05-
0,1)
[0,1-
0,2)
[0,2-
0,5]
Metil
anhydride
[0-
0,01)
[0,01-
0,03)
[0,03-
0,05)
[0,05-
0,1)
[0,1-
0,2]
Alcohol
isopropyl
[0-
0,2)
[0,2-
0,4)
[0,4-
0,6)
[0,6-
0,8)
[0,8-
1,0]
The vector of an evaluation is calculated by a formula
, and correspondig normalized vectors .
At the second level the matrix B consists of three lines
corresponding to vectors of evaluation for three factors. The
resulting vector of the second level can be obtained by
normilizing , , here are weight vectors. As a
result of evaluation we get number , and air purity
will be estimated by value , that is as Very Pure or Pure or
Average and so on.
VI. CONCLUSION
This article is devoted to the problem of automatic
determination of weights for method of fuzzy synthetic
evaluation. The modification of the method of synthetic fuzzy
evaluation realized by neural network is offered. Weights of
the method and the network coincide. The proposed method is
multilevel and multifactor one.
The two-level, multiple-factor model modified for air
quality assessment is developed.
REFERENCES
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Evaluation and Optimal Classification for Fire Risk Ranking of Buildings //
Neural Computing and Application. – 2009. – No 2.
[2] Ni-Bin Chang, H. W. Chen and S. K. Ning Identification of river water
quality using the Fuzzy Synthetic Evaluation approach // Journal of
Enviromental Management. – V.63, I.3. – P. 293-305.
[3] Заде Л.А. Размытые множества и их применение в распознавании
образов и кластер анализе. – Москва: Мир, 1980 – С.208-247.
[4] Аверкин, А.H., Батыршин, И.З., Блишун, А.Ф., Силов, В.Б.,
Тарасов, В.Б. Hечеткие множества в моделях управления и
искусственного интеллекта. – Москва: Hаука, 1986. – 312 с.
[5] Уоссермен Ф. Нейрокомпьютерная техника: Теория и практика. –
СПб.: Питер, 1992.
Tussupov Jamalbek is Doctor of physic and mathematical science, full
professor, chair of information systems department, L.N.Gumilyov Eurasian
Natinal University, Astana, Kazakhstan
La Lira is associated professor, L.N.Gumilyov Eurasian Natinal
University, Astana, Kazakhstan
Mukhanova Ayagoz is third year doctoral student, master of Information
systems, L.N.Gumilyov Eurasian Natinal University, Astana, Kazakhstan
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
Volume 8, 2014
ISSN: 1998-0140
106
