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A Multi Expression Programming Application to High Performance Concrete

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Abstract
High performance concrete (HPC) is a class of concretes that provides superior performance than those of conventional types. The enhanced performance characteristics of HPC are generally achieved by the addition of various cementitious materials and chemical and mineral admixtures to conventional concrete mix designs. These additives considerably influence the compressive strength and workability properties of HPC mixes. To avoid testing several mix proportions to generate a successful mix and also to simulate the behaviour of strength and workability improvement efficiently that often lead to saving in cost and time, it is idealistic to develop predictive models so that the performance characteristics of HPC mixes can be evaluated from the influencing parameters. Accordingly, the main focus of the present study is to propose new formulations of compressive strength and slump flow of HPC mixes for the first time in the literature by means of a promising variant of genetic programming (GP) which is known as multi expression programming (MEP). The models are developed using an experimental database including compressive strength and slump flow of HPC test results obtained from the previously published literature. The results of proposed formulations are compared with other existing models and formulas found in the literature. The results demonstrate that the formulas obtained by the MEP method are able to predict the strength and slump flow to high degree of accuracy.
Figures
World Applied Sciences Journal 5 (2): 215-223, 2008
ISSN 1818-4952
© IDOSI Publications, 2008
Correspondig Author: H. Salehzade, College of Civil Engineering, Iran University of Science and Technology, Tehran, Iran
215
A Multi Expression Programming Application to High Performance Concrete
A.A.R. Heshmati, H. Salehzade, A.H. Alavi, A.H. Gandomi and M. Mohammad Abadi
1 1 12 3
College of Civil Engineering, Iran University of Science and Technology, Tehran, Iran
1
College of Civil Engineering, Tafresh University, Tafresh, Iran
2
Islamic Azad University, Gonabad Branch, Iran
3
Abstract: High performance concrete (HPC) is a class of concretes that provides superior performance than
those of conventional types. The enhanced performance characteristics of HPC are generally achieved by the
addition of various cementitious materials and chemical and mineral admixtures to conventional concrete mix
designs. These additives considerably influence the compressive strength and workability properties of HPC
mixes. To avoid testing several mix proportions to generate a successful mix and also to simulate the behaviour
of strength and workability improvement efficiently that often lead to saving in cost and time, it is idealistic to
develop predictive models so that the performance characteristics of HPC mixes can be evaluated from the
influencing parameters. Accordingly, the main focus of the present study is to propose new formulations of
compressive strength and slump flow of HPC mixes for the first time in the literature by means of a promising
variant of genetic programming (GP) which is known as multi expression programming (MEP). The models are
developed using an experimental database including compressive strength and slump flow of HPC test results
obtained from the previously published literature. The results of proposed formulations are compared with other
existing models and formulas found in the literature. The results demonstrate that the formulas obtained by the
MEP method are able to predict the strength and slump flow to high degree of accuracy.
Key words: High performance concrete multi expression programming Formulation Compressive strength
Workability
INTRODUCTION Genetic programming (GP) [2, 3] is a developing
Different factors influence the compressive strength evolution theory. GP may be defined generally as a
and workability properties of high performance concrete supervised machine learning technique that searches a
(HPC) mixes. A range of materials such as portland program space instead of a data space [3]. Recently, a
cement, silica fume, superplasticizer, fly ash, fine and particular variant of GP that uses a linear representation of
coarse aggregates and ground-granulated blast furnace chromosomes namely, multi expression programming
slag and combinations of these can be used in order to (MEP) have been proposed. MEP [4] has a special ability
obtain concrete mix designs with superior performance. to encode multiple computer programs of a problem in a
Using these material considerably influence the single chromosome. Based on the numerical experiments
compressive strength and workability properties of HPC MEP approach has the ability to significantly outperform
mixes. A deep knowledge of the nature of the relationship similar techniques and can be utilized as an efficient
between these interrelated parameters and the resulting alternative to the traditional Koza’s tree-based GP [5].
mix is required to provide an effective mix design Despite the significant advantages of MEP, there has
procedure [1]. The complex behavior of strength and been just some little scientific effort directed at applying
workability improvement and a need to avoid trying it to civil engineering tasks [6].
several mix proportions to produce a successful mix The main purpose of this paper is to utilize MEP
suggest the necessity to develop comprehensive technique to obtain formulas for the determination of
mathematical models to be able to evaluate the compressive strength and slump flow of HPC mixes. To
performance characteristics of HPC mixes with high our knowledge, this is the first time in the literature to
accuracy. utilize this approach to introduce explicit formulations of
subarea of evolutionary algorithms inspired from Darwin’s
1
( ) 8.2628 W
SLNN MPa e
=
2
( ) 12.2872 W
SLNN
S mm e
=
22
12
12
2 22
45
3
1[( 2.3357) ( 1.7148)
(2.6808) 2 4
(2 1.5897) ( 1.2737) ( 1.9704
)]
30 5
xx
W
xx
x
= + ++ +
−+−+
22
12
22
2 22
45
3
1[( 1.2086) ( 3.0106)
(1.41657) 2 2
(2 0.23047) ( 0.84016) ( 0.71905
)]
30 5
xx
W
xx
x
= − +− +
− +− +−
World Appl. Sci. J., 5 (2): 215-223, 2008
216
the performance characteristics of HPC mixes. In order to
evaluate the prediction quality, a comparison between the
proposed formulations results, as well as existing models (3)
found in the literature, was conducted considering the
influencing parameters such sand cement ratio, coarse
aggregate cement ratio, water cement ratio, percentage of
silica fume and percentage of superplasticizer. A reliable
database including previously published compressive (4)
strength and slump flow of HPC test results was utilized
to develop the models. where,
Review of Previous Studies: In composite materials like xcoarse aggregate cement ratio
HPC changes in constituent properties of the xwater cement ratio
cementitious materials and chemical admixtures may xsilica fume (%)
extremely influence the advantageous performance xsuperplasticizer (%)
characteristics of the mix. Numerous studies have
concentrated on assessing the performance and x , x ,..., x are the five input parameters to the
characteristics of HPC mixes using computational model. It should be noted that the required data used for
approaches in the literature. Artificial neural networks the training and testing of the SLNN model described
(ANNs) [7, 8] have a noteworthy quality of learning above were taken from [14] and have also been utilized in
the relationship between the input and output the present study. In this paper, a novel approach for the
parameters as a result of training with previously formulation of compressive strength and slump flow of
recorded data. ANNs have been applied to predict the HPC mixes using MEP is proposed.
compressive strength and slump flow of HPC mixes
several times [9-12]. There has been only limited research GeneticProgramming (GP): Genetic programming (GP) is
with the specific objective of opening ANN models one of the branches of evolutionary methods that creates
adequately and introducing explicit formulations of computer programs to solve a problem using the principle
compressive strength and slump flow of HPC mixes by of Darwinian natural selection.GP was introduced by Koza
means of them. as an extension of the genetic algorithms, in which
Rajasekaran and Amalraj [12] presented a sequential programs are represented as tree structures and expressed
learning approach (SLA) for single hidden radial basis in the functional programming language LISP [2]. A
function (RBF) neuron neural networks proposed by comprehensive description of GP is beyond the scope of
Zhang and Morris [13]. Their developed sequential this paper and can be found in [2, 3]. GP has been
learning neural network (SLNN) model was utilized for the successfully applied to some of the civil engineering
prediction of strength and workability of high problems [15-18].
performance concrete. The values for learning rate and
gamma have been respectively chosen as 0.6 and 0.000001 Multi Expression Programming (MEP): Multi expression
for the architecture of their proposed SLNN (RBF) model. programming (MEP) is a subarea of genetic programming
In that work two equations were introduced based on (GP) that was developed by [4Oltean and Dumitrescu
experimental results and by using the values of the (2002)]. MEP uses linear chromosomes for solution
weights obtained from neural network training to predict encoding and has a special ability to encode multiple
the compressive strength ( ) and slump flow (S) that are solutions (computer programs) of a problem in a single
given as follows: chromosome. According to the fitness values of the
(1) to represent the chromosome. Comparing MEP to other
(2) there are not increases in the complexity of the decoding
where, data is not a priori known [5]. The evolutionary steady-
xsand cement ratio
1
2
3
4
5
12 5
individuals, the best of the encoded solutions is chosen
GP variants that store a single solution in a chromosome,
process except on the situations where the set of training
1, 1
mi
n,
n
i
jj
imj
f EO
==


= −



World Appl. Sci. J., 5 (2): 215-223, 2008
217
state MEP algorithm starts by the creation of a random connecting the operands specified by the argument
population of individuals. In order to evolve the best
expression from a data file of inputs and outputs along a
specified number of generations, MEP uses the following
steps until a termination condition is reached [19]:
Selecting two parents by using a binary tournament
procedure [2] and recombining them with a fixed
crossover probability.
Obtaining two offspring by the recombination of two
parents.
Mutating the offspring and replacing the worst
individual in the current population with the best of
them (if the offspring is better than the worst
individual in the current population).
MEP is represented similar to the way in which C
and Pascal compilers translate mathematical expressions
into machine code [20]. The number of MEP genes per
chromosome is constant and specifies the length of the
chromosome. A terminal (an element in the terminal set T)
or a function symbol (an element in the function set F)
are encoded by each gene. A gene that encodes a
function includes pointers towards the function
arguments. Function parameters always have indices of
lower values than the position of that function itself in
the chromosome. The first symbol in a chromosome
must be a terminal symbol as stated by the proposed
representation scheme.
An example of a MEP chromosome can be seen
below. It should be noted that numbers to the left stand
for gene labels that do not belong to the chromosome.
Using the set of functions F= {+, *, /} and the set of
terminals T={x, x, x, x}, the example is given as
1234
follows:
0: x1
1: x2
2: * 0, 1
3: x3
4: + 2, 3
5: x4
6: / 4, 5
The translation of MEP individuals into computer
programs can be obtained by reading the chromosome
top-down starting with the first position. A terminal
symbol defines a simple expression and each of function
symbols specifies a complex expression obtained by
positions with the current function symbol [19]. In the
present example, genes 0, 1, 3 and 5 encode simple
expressions formed by a single terminal symbol. These
expressions are:
E = x,
01
E = x,
12
E = x,
33
E = x,
54
Gene 2 indicates the operation * on the operands
located at positions 0 and 1 of the chromosome. Therefore
gene 2 encodes the expression:
E = x *x .
212
Gene 4 indicates the operation + on the operands
located at positions 2 and 3. Therefore gene 4 encodes the
expression:
E = (x *x ) + x .
412 3
Gene 6 indicates the operation / on the operands
located at positions 4 and 5. Therefore gene 6 encodes the
expression:
E = ((x *x ) + x )/ x .
612 3 4
In order to choose one of these expressions (E ,...,E )
16
as the chromosome representer, multiple solutions in
a single chromosome are encoded. Each of MEP
chromosomes encodes a number of expressions equal to
the chromosome length (the number of genes). Each of
these expressions can be considered as a possible
solution of a problem. The fitness of each expression
encoded in a MEP chromosome is defined as the fitness
of the best expression encoded by that chromosome.
For solving symbolic regression problems the fitness of
a MEP chromosome may be computed by using the
formula [5]:
(5)
where nis the number of fitness cases, E is the
j
expected value for the fitness case j,O is the value
j
i
returned for the j fitness case by the i expression
th th
encoded in the current chromosome and mis the number
of chromosome genes.
(
, FA/C, CA/C, W/C, SF, SP sf=
World Appl. Sci. J., 5 (2): 215-223, 2008
218
Table 1: Database used in developing the models
Mix No. FA/C CA/C W/C SF (%) SP (%) test (MPa) Stest (mm)
Training
1 1.88 2.84 0.45 9.99 2 80 110
2 1.6 2.4 0.4 9.99 2 90.08 90
3 1.8 2.68 0.45 19.98 3 110.08 120
4 1.3 2.12 0.35 9.99 2 110.08 80
5 0.96 1.72 0.3 5.01 2 130.08 50
6 1.02 1.8 0.3 15.03 2.5 120 60
7 1.08 1.92 0.3 15.03 2.5 130.08 70
8 0.78 1.52 0.25 9.99 2 140 40
9 1.64 2.44 0.35 12 2 80 100
10 1.7 2.56 0.35 15.99 2.5 90.08 120
11 1.8 2.68 0.35 20.01 3 90.08 130
12 1.9 2.84 0.35 24 3.5 90.08 150
13 1.98 3 0.35 27.99 4 90.08 170
14 1.38 2.28 0.35 7.5 2 90.08 80
Testing
15 1.64 2.48 0.4 9.99 2 80 100
16 1.76 2.64 0.43 15 2.5 100 110
17 1.92 2.888 0.45 9.99 2 80 120
18 1.36 2.2 0.36 9.99 2 100 80
19 1.44 2.36 0.38 15 2.5 100 90
20 1.54 2.52 0.46 20.01 3 140 110
21 1.16 2.08 0.32 9.99 2 110.08 70
22 1.1 1.96 0.3 5.01 2 100 60
23 1.24 2.2 0.34 15 2.5 120 90
Table 2: The variables used in model development
Normalization
Parameters Range value Code
Inputs
Sand cement ratio FA/C 0.78–1.98 2 x1
Coarse aggregate cement ratio CA/C 1.52–3 4 x2
Water cement ratio W/C 0.25–0.46 0.5 x3
Silica fume SF (%) 5.01–27.99 30 x4
Superplasticizer (%) 2–4 5 x5
Outputs
Compressive strength (MPa) 80–140 160
Slump flow S(mm) 40–170 200
Table 3: Parameter settings for MEP
Settings
------------------------------------------------------------
Parameter s1 , S1 s1 , S1
Function set +, -, *, /, exp, sin, cos +, -, *, /
Population size 250-500 250-500
Chromosome length 50 genes 50 genes
Number of generations 250 250
Crossover probability 0.5,0.9 0.5,0.9
Crossover type Uniform Uniform
Mutation probability 0.01 0.01
Terminal set Problem inputs Problem inputs
Database: The database contains 23 compressive strength
and slump flow of HPC test results managed by
Rajaseraran et al. [14]. Table 1 shows the experimental
database used for proposed models. The other cited
information in Table 1 consists of sand (FA)/cement (C),
coarse aggregate (CA)/cement (C), water (W)/cement (C),
silica fume content (%SF) and superplasticizer content
(%SP) as the five input parameters to the models to
predict the compressive strength( )and slump flow (S) of
HPC mixes as the outputs. It is noteworthy that the input
and output parameters entering the models have been
normalized between 0 and 1 before the learning process.
The range of the samples, normalization values and
the format of the input data used in this study are given
in Table 2.
Model Development Using MEP Model: The main
goal is to obtain the explicit formulations of the
compressive strength and slump flow of HPC mixes as
functions of variables given as follows:
The five parameters are used for the MEP models as
the input variables. Two MEP models for single output
have been separately used, one for strength and the other
for the slump flow. In order to evaluate the capabilities of
the MEP model, the correlation coefficient (R), mean
squared error (MSE) and mean absolute error (MAE) are
used as the criteria between the actual and predicted
values. Various parameters are involved in MEP predictive
algorithm such as population size, chromosome length,
number of generations, tournament size and other
parameters that are shown in Table 3. The parameter
selection will affect the model generalization capability of
MEP. They were selected based on some previously
suggested values [6] and also after trial and error
approach. Four formulations of compressive strength and
slump flow, two formulas for each of them, have been
considered using two different function sets for runs. The
first function set consists of nearly all functions and the
latter includes just addition, subtraction, division and
multiplication in order to obtain short and very simple
formulas. Subsequently, the results obtained from these
equations were compared with each other, as well as the
results obtained by other researcher. For the analysis,
source code of MEP [21] in C++ was modified by the
authors to be utilizable for the available problems. For the
prediction of compressive strength and slump flow of
HPC the first 14 values of Table 1 are taken for training the
2
15
13
2
320
( ) cos(cos
2)
45
xx
MPa x
x

= −−


12
2 5 12 3
xx2
( ) 80 (x - x - x /2+ 4 x + 2
10
MPa −+

=


2
24 1
11/4
35
50 ( 15 )
()
15 30
x
xx x
S mm e xx
+
=++
12
2
3 35 2
500
()
40 8 5
xx
S mm x xx x
=−+
World Appl. Sci. J., 5 (2): 215-223, 2008
219
Fig. 1: Results of compressive strength prediction Fig. 3: Results of slump flow prediction for Eq. (8).
for Eq. (6).
Fig. 2: Results of compressive strength prediction for Eq. (6) has better performance than Eq. (7) for both of
Eq. (7). training and testing sets.
MEP algorithm and the next 9 values were used for Explicit Formulation of Slump Flow and Analysis Using
testing the generalization capability of MEP based MEP Model: Similar to compressive strength, for the
models. The details of the compressive strength and slump flow five parameters are considered in the
slump flow predictive models are highlighted in next formulation process, namely FA/C (x1), CA/C (x2), W/C
sections. (x3), %SF (x4) and %SP (x5). These values have been
Explicit Formulation of Compressive Strength and prediction equations for the best result by the MEP
Analysis Using MEP Model: Formulation of compressive algorithm are given in Eq. (8) and Eq. (9) for the
strength in functional form in terms of the independent aforementioned function sets.
variables, sand cement ratio (FA/C = x1), coarse aggregate
cement ratio (CA/C = x2), water cement ratio (W/C = x3),
percentage of silica fume (%SF = x4) and percentage of (8)
superplasticizer (%SP = x5) as presented in Table 1, for the
best result by the MEP algorithm are given in Eq. (6) and
Eq. (7) for two different function sets. (9)
(6)
(7)
The comparison of MEP prediction and actual
compressive strength of HPC for Eq. (6) is shown in
Fig. 1. It can be seen from Fig. 1 that Eq. (6) generated by
MEP model yielded high R values for training and testing
data equal to 0.9663 and 0.924, respectively. Fig. 2 shows
the relevant results obtained by Eq. (7). It can be
observed from this figure that Eq. (7) yielded R values for
training and testing data equal to 0.9465 and 0.9083,
respectively. It can be concluded from these figures that
chosen from Table 1 as inputs for the MEP model. The
World Appl. Sci. J., 5 (2): 215-223, 2008
220
Fig. 4: Results of slump flow prediction for Eq. (9). Figures 5 and 6 represent the results for all element test
The comparison of MEP prediction and actual slump respectively. Statistical performance of MEP based
flow of HPC for Eq. (8) is shown in Fig. 3. It can be seen formulations, as well as SLNN model [12], in terms of their
from Fig. 3 that Eq. (8) yielded high R values for training prediction capabilities are summarized in Tables 4 and 5.
and testing data equal to 0.992 and 0.9761, respectively. The results for compressive strength, presented in
Fig. 4 demonstrates that Eq. (9) has produced results with
very high R values for training and testing data equal to
0.9965 and 0.9699, respectively. It can be seen from these
figures that while Eq. (8) outperforms Eq. (9) on the
testing data, better results are obtained by Eq. (9) for the
training set.
DISCUSSION
Four formulations of compressive strength and
slump flow of HPC in functional form in terms of FA/C,
CA/C, W/C, %SF and %SP were obtained by using MEP
and given in Eqs. (6)-(9). As mentioned previously, R,
MSE and MAE are selected as the target statistical
parameters to evaluate the performance of the models.
data for compressive strength and slump flow,
Fig. 5: Compressive strength relative comparison for all element tests data.
Fig. 6. Slump flow relative comparison for all element tests data.
World Appl. Sci. J., 5 (2): 215-223, 2008
221
Table 4: Statistical performance of models for compressive strength prediction
Training Testing All elements
------------------------------------------------- --------------------------------------------------- --------------------------------------------------
Models R MSE MAE R MSE MAE R MSE MAE
MEP (Eq. (6)) 0.9663 28.273 4.403 0.924 74.545 7.038 0.9441 46.379 5.434
MEP (Eq. (7)) 0.9465 41.886 5.521 0.9083 69.180 6.861 0.9229 52.566 6.046
SLNN 0.9546 34.126 4.651 0.8977 61.212 6.898 0.9351 44.725 5.53
Table 5: Statistical performance of models for slump flow prediction
Training Testing All element tests data
------------------------------------------------ --------------------------------------------------- -------------------------------------------------
Models R MSE MAE R MSE MAE R MSE MAE
MEP (Eq. (8)) 0.992 28.311 4.427 0.9781 24.974 4.162 0.9893 26.998 4.323
MEP (Eq. (9)) 0.9965 17.780 3.455 0.9699 61.920 6.692 0.9903 35.052 4.722
SLNN 0.9965 9.129 2.471 0.9795 18.882 3.653 0.9936 12.945 2.934
Table 6: Comparative analysis of proposed MEP based formulae with experimental and SLNN results
Eq.(6) Eq.(7) S S Eq.(8) S Eq.(9) S
Test 1 2 SLNN TESAT 1 1 SLNN
Mix No. (Mpa) (Mpa) /(MPa) /(MPa) /(mm) (mm) S / S (mm) S /S (mm) S /S
Test 1Test 2Test SLNN TESAT 1 TESAT 2 Test SLNN
Training
1 80 80.74 0.991 83.11 0.963 76.96 1.040 110 108.44 1.014 106.78 1.030 113.40 0.970
2 90.08 92.69 0.972 98.56 0.914 91.84 0.981 90 89.98 1.000 88.89 1.013 88.80 1.014
3 110.08 107.44 1.025 114.24 0.964 105.44 1.044 120 121.50 0.988 117.09 1.025 119.40 1.005
4 110.08 102.43 1.075 108.32 1.016 106.56 1.033 80 74.46 1.074 72.53 1.103 73.40 1.090
5 130.08 124.49 1.045 129.63 1.004 120.64 1.078 50 43.86 1.140 52.25 0.957 48.20 1.037
6 120 129.38 0.927 127.21 0.943 132.64 0.905 60 68.20 0.880 61.20 0.980 63.20 0.949
7 130.08 119.43 1.089 117.73 1.105 127.04 1.024 70 73.59 0.951 66.46 1.053 69.60 1.006
8 140 134.30 1.042 134.21 1.043 135.68 1.032 40 46.52 0.860 43.59 0.918 45.00 0.889
9 80 82.67 0.968 79.19 1.010 88.96 0.899 100 101.24 0.988 97.13 1.030 99.60 1.004
10 90.08 84.82 1.062 80.20 1.123 91.36 0.986 120 114.34 1.050 109.90 1.092 116.80 1.027
11 90.08 87.35 1.031 82.24 1.095 91.46 0.985 130 129.05 1.007 126.95 1.024 132.80 0.979
12 90.08 89.31 1.009 83.96 1.073 91.65 0.983 150 144.73 1.036 146.63 1.023 152.00 0.987
13 90.08 91.93 0.980 87.05 1.035 88.72 1.015 170 159.50 1.066 166.85 1.019 167.00 1.018
14 90.08 93.48 0.964 96.39 0.935 98.24 0.917 80 74.72 1.071 79.45 1.007 79.00 1.013
Testing
15 80 88.83 0.901 92.95 0.861 88.96 0.899 100 93.48 1.070 92.44 1.082 93.80 1.066
16 100 95.56 1.046 100.35 0.996 90.40 1.106 110 110.57 0.995 106.57 1.032 112.20 0.980
17 80 78.59 1.018 79.65 1.004 74.88 1.068 120 111.19 1.079 109.84 1.092 117.00 1.026
18 100 99.17 1.008 104.79 0.954 103.04 0.970 80 78.11 1.024 76.17 1.050 77.60 1.031
19 100 104.62 0.956 106.27 0.941 106.40 0.940 90 93.97 0.958 87.59 1.028 94.20 0.955
20 140 122.08 1.147 127.64 1.097 126.40 1.108 110 106.58 1.032 97.21 1.132 102.20 1.076
21 110.08 102.01 1.079 106.65 1.032 111.84 0.984 70 69.77 1.003 66.73 1.049 70.40 0.994
22 100 105.90 0.944 109.60 0.912 110.08 0.908 60 53.39 1.124 63.41 0.946 59.00 1.017
23 120 108.68 1.104 108.35 1.108 116.48 1.030 90 84.58 1.064 76.63 1.174 84.32 1.067
Table 4, show that the best performance is achieved by by MEP approach outperform the SLNN formulation on
Eq. (6) for both of the training (R = 0.9663, MSE=28.273, the testing data set. The results for all element tests data
MAE = 4.403) and testing data (R = 0.924, MSE=74.545, demonstrate that Eq. (6) has better performance followed
MAE = 7.038). Comparing the results of the SLNN based by SLNN and Eq. (7).
formula and Eq. (7) for the training set, it can be seen that Considering the slump flow, it can be concluded from
the former performs superior than the latter. It can be Table 5 that while Eq. (9) and SLNN formula yielded R
observed from Table 4 that both of the formulae obtained values equal to 0.9965 for the training data set, SLNN
World Appl. Sci. J., 5 (2): 215-223, 2008
222
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