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2 Figures# A Multi Expression Programming Application to High Performance Concrete

**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.

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

slightly outperforms the other regarding its lower MSE REFERENCES

and MAE values. In this case, Eq. (8) has performed

poorer than the other models. As can be observed in 1. Malier, Y., 1992. High Performance concrete, from

Table 5, SLNN based formula with R, MSE and MAE Material to Structure. London: E & FN Spon.

values equal to 0.9795, 18.882 and 3.6533 has produced 2. Koza, J.R., 1992. Genetic programming: On the

better results on the testing data set followed by Eq. (8) programming of computers by means of natural

and Eq. (9). Considering the all element tests data, it can selection. Cambridge (MA): MIT Press.

be seen that the SLNN model has better performance 3. Banzhaf, W., P. Nordin, R. Keller and F. Francone,

followed by Eq. (9) and Eq. (8). It should be noted that in 1998. Genetic Programming – An Introduction: On the

spite of the better performance of SLNN models in Automatic Evolution of Computer Programs and Its

some of the aforementioned situations, the MEP based Application. Heidelberg/San Francisco: Morgan

prediction equations are really short, very simple and can Kaufmann.

be used facilitatory.Table 6 shows a comparative analysis 4. Oltean, M., 2002. Dumitrescu D. Multi expression

of results of the proposed MEP formulations and the programming, technical report. Babe -Bolyai

results obtained by SLNN model including compressive University, [UBB-01-2002].

strength and slump flow actual experimental values. 5. Oltean, M. and C. Gros an, 2003. A comparison of

CONCLUSIONS Complex Sys,14(4): 1-29.

This paper proposes a novel approach for the 2007. Prediction of compressive and tensile strength

prediction of compressive strength and slump flow of of limestone via genetic programming. Expert

HPC mixes using a variant of GP namely, MEP. Four Systems with Applications, Article in press.

formulations of compressive strength and slump flow, two 7. Hayati, M. and Z. Mohebi, 2007. Temperature

formulas for each of them, have been obtained by means Forecasting Based on Neural Network Approach.

of MEP and considering two different function sets. World Appl. Sci. J., 2(6): 613-620.

A reliable database including previously published 8. Shayanfar, M.A., S.R. Massah and H. Rahami, 2007.

compressive strength and slump flow of HPC test results An Inverse Reliability Method Using Neural

was used for training and testing the prediction models. Networks and Genetic Algorithms. World Appl. Sci.

The MEP based formulations results were compared with J., 2(6): 594-601.

the experimental results and the existing model proposed 9. Yeh, C., 2006. Exploring concrete slump model using

in the literature namely, SLNN (RBF). artificial neural networks. J. Comput. Civil Eng.,

Based on the values of performance measures for 20(3): 217-221.

the models it can be observed that the proposed MEP 10. Yeh, I., 1998. Modeling of strength of high-

models are able to predict the target values to an performance concrete using artificial neural networks.

acceptable degree of accuracy. The results of testing data Cem. Conc. Res., 28(12): 1797-808.

demonstrated that for the prediction of compressive 11. Kasperkiewicz, J., J. Racz and A. Dubrawski, 1995.

strength both of the formulae evolved by MEP outperform HPC strength prediction using artificial neural

the proposed formulation result of SLNN. Considering the network. J. Comput Civil Eng., 9(4): 279-84.

relevant results for the explicit formulation of slump flow 12. Rajasekaran, S. and R. Amalraj, 2002. Predictions of

it can be observed that SLNN has produced slightly design parameters in civil engineering problems

better results than the MEP based formulas. When the using SLNN with a single hidden RBF neuron.

performance of the MEP based prediction equations is Comput & Struc, 80(31): 2495-2505.

taken into consideration it can be seen that in addition to 13. Zhang, J. and AJ. Morris, 1988. A sequential learning

their considerable accuracy they are quite short and very approach for single hidden layer neural networks.

simple and seem to be more practical for use compared to Neural Networks, 11(1): 65-80.

the prediction equations produced by SLNN. However, 14. Rajasekaran, S., R. Amalraj and S. Anandakumar,

this investigation revealed that MEP is very promising 2001. Optimization of mix proportions for high

approach that can be utilized in order to produce explicit performance concrete using cellular genetic

formulations to be able to capture the underlying algorithms. Proceedings of National Seminar on

relationship between the different interrelated input and Concrete Technology for 21 Century.

output data for many of civil engineering tasks. Annamalainagar: Annamalai University.

several linear genetic programming techniques.

6. Baykaso lu, A., H. Güllü, H. Çanakç and L. Özbak r,

st

World Appl. Sci. J., 5 (2): 215-223, 2008

223

15. Ashour, A.F., L.F. Alvarez and V.V. Toropov, 2003. 19. Oltean, M. and C. Gros an, 2003. Evolving

Empirical modelling of shear strength of RC deep evolutionary algorithms using multi expression

beams by genetic programming. Comput & Struc, programming. In: W. Banzhaf et al. editors.

81(55): 331-338. Proceedings of the 7 european conference on

16. Javadi, A.A., M. Rezani and M. Mousavi Nezhad, artificial life. Dortmund: LNAI. pp: 651-658.

2006. Evaluation of liquefaction induced lateral 20. Aho, A., R. Sethi and J.D. Ullman, 1986. Compilers:

displacements using genetic programming. Comput Principles, Techniques and Tools. Reading (MA):

Geotech, 33(4): 222-233. Addison Wesley.

17. Baykaso lu, A., T. Dereli and S. Tan s, 2004. 21. Oltean, M., 2004. Multi Expression Programming

Prediction of cement strength using soft computing source code.

techniques. Cem. Con. Res., 34(11): 2083-2090.

18. Alavi, A.H., A.A. Heshmati, A.H. Gandomi, A.

Askarinejad and M. Mirjalili, 2008. Utilisation of

Computational Intelligence Techniques for Stabilised

Soil. In: M. Papadrakakis and B.H.V. Topping editors.

Proceedings of the 6 International Conference on

th

Engineering Computational Technology, Civil-Comp

Press, Scotland.

th

- CitationsCitations5
- ReferencesReferences11

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