Construction and Building Materials 42 (2013) 205–216

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Construction and Building Materials

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Modeling compressive strength of EPS lightweight concrete using regression,neural network and ANFIS

A. Sadrmomtazi a, J. Sobhani b,⇑, M.A. Mirgozar a

a Faculty of Engineering, University of Guilan, Rasht, Iranb Department of Concrete Technology, Road, Housing & Urban Development Research Center (BHRC), Pas Farhangian St., Sheikh Fazlollah Exp. Way, Tehran, P.O. Box 13145-1696, Iran

h i g h l i g h t s

" For the first time, models developed for prediction of the strength properties of EPS concrete." Robust ANN and ANFIS models proposed for predicting the compressive strength of EPS concrete." The overall performance of trained ANN is more accurate than ANFIS model." Such robust models could be easily utilized for EPS concrete mix proportioning as a problem with high complexities included." Higher accuracy of neural network is due to application of Levenberg–Marquardt backpropagation algorithm.

a r t i c l e i n f o

Article history:Received 1 August 2012Received in revised form 4 November 2012Accepted 12 January 2013Available online 27 February 2013

Keywords:EPS concreteSilica fumeCompressive strengthModelingRegressionNeural networkANFIS

0950-0618/$ - see front matter � 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.conbuildmat.2013.01.016

⇑ Corresponding author. Tel.: +98 21 88255942x6;E-mail addresses: ja_sobhani@yahoo.com, sobhani

a b s t r a c t

EPS concrete is an especial type of lightweight concrete made by partial replacement of concrete’s stoneaggregates with lightweight expanded polystyrene beads (EPSs). This type of concrete is very sensitive toits constituent materials which complicate the modeling process. Considering the involved complexities,this paper dealt with developing and comparing parametric regression, neural network (ANN) and adap-tive network-based fuzzy inference system (ANFIS) models for predicting the compressive strength of EPSconcrete for possible use in mix-design framework. The results emphasized that the elite ANN model con-structed with two hidden layers and comprised of three neurons in each layers, could be effectively usedfor prediction purposes. Moreover, ANFIS elite model developed by bell-shaped membership functionwas recognized as a proper model to this means; however, its prediction performances were evaluatedto be diluted than ANN model. On the other hand, the prediction results of second-order partial polyno-mial regression model as elite empirical one showed the weakness of this model comparing ANN andANFIS models.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

A unique attribute of concrete, which makes it truly versatile, isthat it consists of a family of materials with a large range in color,density, strength and durability characteristics [1]. It can be man-ufactured from a great number of materials, in many ways for dif-ferent applications. Structural concrete is now available with adensity range between 1800 and 3000 kg/m3 as lightweight, nor-mal weight and heavy weight concrete [2]. Lightweight concrete(LWC) is a multi-purpose material for construction, which offersa range of technical, economical and environment-enhancing andpreserving advantages and is destined to become a dominantmaterial for construction in the new millennium [1].

ll rights reserved.

fax: +98 21 88255941.@bhrc.ac.ir (J. Sobhani).

The first known use of lightweight concrete dates back over2000 years. There are several lightweight concrete structures inthe Mediterranean region, but the three most notable structureswere built during the early Roman Empire and include the Portof Cosa, the Pantheon Dome, and the Coliseum [3].

Lightweight concretes can be produced by replacing the normalaggregates in concrete either partially or fully, depending upon therequirements of density and strength [4]. Lightweight aggregatesare broadly classified in two main types: natural (pumice, diato-mite, volcanic cinders, etc.) and artificial (perlite, expanded shale,clay, slate, sintered PFA, and expanded polystyrene beads, etc.).Lightweight aggregates can be used to produce low density con-cretes required for building applications like cladding panels, cur-tain walls, composite flooring systems, and load-bearing concreteblocks [3,5,6]. When expanded polystyrene beads used for produc-tion of LWC, it is typically referred to EPS concrete.

Nomenclature

J Jacobin matrixl learning rateb delay rate (0 < b < 1)x (or y) input of the ANFIS nodeAi (or Bj) linguistic label (Fuzzy membership)l(x) (or l(y)) membership functionwi firing strength of a rulef1 and f2 fuzzy if–then rules in TSK systemf(�) regression model’s representative functiony regression model’s outputbi parameters of the regression modelxi independent variablesRMS root means square

CF correlation factorP total number of concrete samplesxci experimental result for compressive strengthxci predicted result for compressive strength;covðXc; bXcÞ covariance of experimental and predicted value for

compressive strengthum normalized input valuewRV rough values of input dataC, SF, W, FA, CA, EPS, PP weight per unit volume of concrete

respectively for Cement, Silica fume, water, fine aggre-gates, coarse aggregates, expanded polystyrene beadsand waste carpet polypropylene fibers

206 A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216

Lightweight EPS concrete has some distinguished advantageslike higher strength to weight ratio, better tensile strain capacity,lower coefficient of thermal expansion, and superior heat andsound insulation characteristics due to inclusion of air voids inthe lightweight aggregate. Beside the reduction in of the construc-tion materials led to a remarkable decrease in cross section ofconcrete structural elements (columns, beams, plates, and founda-tion), it is also possible to reduce steel reinforcement [5,7,8].

The properties of LWC are very sensitive to its ingredientswhich intensify the complexities involved in the prediction of itsbehavior when compared with that of the normal weight concrete.Besides, for the means of concrete mix-design, it is a common prac-tice to produce the trial mixtures based on mix-design and projectrequirements. To aid in this process and minimize the experimen-tation tasks, mathematical free models, in the form of regressionformulas, are traditionally used to predict the strength behaviorof concrete mixes. These models could be applicable in many cases,however; if the problem contains many independent variables,regression methods cannot be used because of a loss of accuracyand increased number of variables in regression form (linear,non-linear, exponential, etc.). In the recent years, artificial intelli-gent-based modeling techniques like artificial neural networks(ANNs) [5,9], fuzzy systems [10–12], adaptive network-basedinference systems (ANFISs) [13,9], neuro-fuzzy systems [14], andgenetic fuzzy systems [15], have been utilized to approximatenon-linear and complex behavior for various properties of con-struction materials.

Among the aforementioned methods, ANN and ANFIS have beenwidely applied for various types of concrete. So, this paper aimedto design robust ANN and ANFIS models to predict the compressivestrength of and especial type of lightweight concrete named as EPSconcrete, incorporating the silica fume, rice husk ash, expandedpolypropylene, and waste carpet polypropylene fibers. Moreover,non-linear regression models were also proposed and comparedwith ANN and ANFIS models.

Fig. 1. Single input neuron model.

2. Modeling methods

2.1. Regression modeling

Regression modeling is a statistical tool for the investigation of relationships be-tween variables. Generally, regression is the process of fitting models to data. Theprocess depends on the model. In linear or non-linear parametric regression, itwas tried to develop empirical model(s) for system identification and experimentalstudies purposes. In such systems, estimation is based on search methods fromoptimization that minimize the norm of a residual vector. In this paper, generalparametric regression system proposed in the form of y ¼ f ðbi � xiÞ where f is themodel’s representative function, y is the model’s output, bi are the parameters ofthe model and xi are the independent variables.

2.2. Artificial neural network

Artificial neural network (ANN) is an artificial intelligence-based method fordealing with recognition of complex phenomenon and solution of problems whichcould not otherwise be solved through the existing algorithms. ANN is system ofsimple processing elements called neurons and is connected to a network by aset of weights (see Fig. 1). Generally speaking, the relationship between the ele-ments characterizes the rule of the nets. Nets could be trained to perform a specificjob, by setting the relationships between the elements (i.e., the weight and biasterms). Therefore, the trained nets would have a specific output response for a par-ticular input. A neural network, in each setting, compares the value of output out-come with the actual output and hence sets its parameters in such a way that theoutput response gets closer to the actual value. The trained nets could be utilized toact as an alternative replacement of the complex functions. Every neural networkconsists of an input, hidden and output layers, respectively. These layers are joinedtogether through connections with different weights. The duty of the hidden layer(HL) is to connect the input to output layers. A hidden layer enables the nets to ex-tract a non-linear correlation from the available dataset. The number of input, hid-den and output neuron layers highly depends on the number of input variables,number of output response variables and also to the application of the nets [16–18].

Fig. 1 demonstrates a typical single input neuron model of the nets. In this fig-ure, P stands for the number of inputs, W is the weight, b is the bias which employsthe result as the argument for a singular valued function, f the transfer function anda is the output neurons. In neural network, the term backpropagation refers to theprocedure in which the gradient is computed for non-linear multiple-layer net-works. Furthermore, to assess the performance of the neural network model, an er-ror measure like root mean square error (RMS) might be utilized. The choice of aspecific class of networks for the simulation of a non-linear and complex map isdependent upon a variety of parameters. The most popular ANN method is the feedforward multilayer perceptron scheme. In neural network, each neuron is com-posed of a transfer function which signifies the internal activation level. Generallyspeaking, the transfer functions are sigmoidal function, hyperbolic tangent and lin-ear function. Amongst the above transfer functions, the log-sigmoidal is the mostwidely used function for the non-linearity cases. To prevent numerical overflowin the case of a very large or small weights, normalization of the input datasetare unavoidable [16–18].

2.2.1. Training algorithmFig. 2 shows ANN model to simulate the experimental results with backpropa-

gation (BP) algorithm. BP algorithm calculates the error, and then used to adjust theweights first in the output layer, and then distributes it backward from the outputto hidden and input nodes [9,16]. In this paper Levenberg–Marquardt-type BP(LMBP) algorithm is utilized [9,22].

The weigh update on the base of LMBP is as follows:

Fig. 2. Architecture of ANN [9].

Fig. 3. Schematic of ANFIS architecture [9].

A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216 207

Dw ¼ ½JT J þ lI��1JT e ð1Þ

where J is the Jacobin matrix, l is the learning rate which is to be updated using thedelay rate b (0 < b < 1) depending on the outcome as lnew = lold�b.

2.3. Adaptive network-based fuzzy inference system (ANFIS)

ANFIS is the famous hybrid neuro-fuzzy network for modeling the complex sys-tems [19–20]. ANFIS incorporates the human-like reasoning style of fuzzy systemsthrough the use of fuzzy sets and a linguistic model consisting of a set of If-Thenfuzzy rules. The main strength of ANFIS models is that they are universal approxi-mators [19] with the ability to solicit interpretable If-Then rules.

To illustrate the procedures of an ANFIS, for simplicity, we consider only two in-puts x, y and one output fout in this system. The framework of ANFIS is shown inFig. 3, and the node function in each layer is described below.

Layer 1: Every node in this layer is an adaptive node with node function as:

O1;i ¼ lAiðxÞ for i ¼ 1;2 ð2Þ

Table 1Chemical composition and properties of cement, silica fume and rice husk ash.

Chemical composition (%) Cement Silica fume (SF) Rice husk ash (RHA)

SiO2 21 91.1 91.62Al2O3 4.6 1.55 0.49Fe2O3 3.2 2.0 0.73CaO 64.5 2.42 2.51MgO 2.0 0.06 0.88SO3 2.9 0.45 –Na2O + 0.685K2O 1.0 – 2.39

O1;i ¼ lBi�2ðyÞ for i ¼ 3;4 ð3Þ

where x (or y) is the input of the node, Ai (or Bj) is the linguistic label, l(x) (or l(y)) isthe membership function, usually adopting the bell shape with maximum and min-imum equal to 1 and 0, respectively.

Layer 2: Every node in this layer is a fixed node, marked by a circle and labeledP, with the node function to be multiplied by input signals to serve as output signal

O2;i ¼ lAiðxÞ � lAi

ðxÞ ¼ wi for i ¼ 1;2 ð4Þ

The output signal wi represents the firing strength of a rule.Layer 3: Every node in this layer is a fixed node, marked by a circle and labeled

N, with the node function to normalize the firing strength by calculating the ratio ofthe ith node firing strength to the sum of all rules’ firing strength.

O3;i ¼wiP

wi¼ wi

w1 þw2for i ¼ 1;2 ð5Þ

Layer 4: Every node in this layer is an adaptive node, marked by a square, withnode function

O3;i ¼ �wi � fi for i ¼ 1;2 ð6Þwhere f1 and f2 are the fuzzy if-then rules as follows:

Rule 1: if x is A1 and y is B1 then

f1 ¼ p1xþ q1yþ r1

Rule 2: if x is A2 and y is B2 then

f2 ¼ p2xþ q2yþ r2

where {pi, qi, ri} is the parameters set, referred to as the consequent parameters.

Layer 5: Every node in this layer is a fixed node, marked by a circle and labeledR, with node function to compute the overall output by

O5 ¼X

�wi � fi ¼ fout for i ¼ 1;2 ð7Þ

The basic learning rule of ANFIS is the backpropagation gradient descent, whichcalculates error signals recursively from the output layer backward to the inputnodes. This learning rule is exactly the same as the backpropagation learning ruleused in the common feed-forward neural networks [21]. Recently, ANFIS adopteda rapid learning method named as hybrid learning method which utilizes the gra-dient descent and the least-squares method to find a feasible set of antecedentand consequent parameters [9,13,19,20]. Thus in this paper, the later method isused for constructing the proposed models.

3. Materials and data collection

3.1. Materials

Locally available ordinary Portland cement meeting the requirements of ASTMC150 [23], and two types of supplementary cementitious materials (CMs) includingsilica fume (SF) and rice husk ash (RHA) were used in this investigation. The chem-ical compositions of these binders are presented in Table 1.

The fine aggregate was natural siliceous river sand and the coarse aggregatewas crushed limestone aggregate. The physical and mechanical properties of thefine and coarse aggregates are reported in Table 2. The grading of these aggregatesis presented in Fig. 4 together with the ASTM C33 limits [24].

In addition to the natural aggregates, EPS beads were utilized as artificial light-weight aggregates in order to decrease the density of concrete and produce differ-ent strength grades of EPS concrete. The size of 85% of EPS particles were about3.5 mm and their density as measure of mass per volume was evaluated throughcontinuous water injecting-withdrawal technique as 0.0257 g/cm3. Moreover,

Table 2Aggregate properties.

Aggregate type Specific gravity Absorption (%) Fineness modulus

Fine (0–4.75 mm) 2.51 3.40 2.82Coarse (4.75–12 mm) 2.54 2.57 –

Table 3Characteristics of waste carpet polypropylene fibers.

Properties Description

Morphology Fibrillated or mono filamentSpecific gravity (g/cm3) 0.95Diameter (lm) 50Modulus of elasticity (GPa) 5Tensile strength (MPa) 450Ultimate strain (%) 5–15Elongation of fracture (%) �20Melting point (�C) 160Bonding with cement GoodStability in cement GoodAspect ratio (L/d) 120

Fig. 5. Distribution of EPS beads in concrete mixture.

208 A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216

polypropylene (PP) fibers obtained from waste carpets were utilized in this study toimprove the toughness of EPS concrete. The properties of these fibers are presentedin Table 3.

The water used for mixing and curing of all concrete mixes and specimens wasclean, fresh, and free from any impurities. Also, owing to the necessity of loweringwater to cement ratio for obtaining enough compressive strength and desired fluid-ity, a polycarboxylate based superplasticizer was incorporated in all mixtures.

3.2. Specimen preparation

As homogeneity is a main issue in EPS concrete, to prepare humongous speci-mens, the following steps adopted:

Step 1: EPS beads were wetted with a part of the mixing water and superplast-icizer, before adding the remaining materials.Step 2: The remaining materials were added to the mixer and the remainingwater was gradually added while the mixing was in progress.Step 3: Mixing was continued until a uniform and flowing mixture wasobtained.

The observation on specimens as depicted in Fig. 5 showed a uniforms distribu-tion of EPS beads in matrix.

3.3. Data collection

The mix proportions of the study are summarized in Table 4. As seen, three per-centages of EPS beads and four percentages of PP fiber (0.1%, 0.3%, 0.5% and 1%)were used in preparation of concrete specimens. Respectively 10% and 20% of silicafume and rice husk were replaced by weight of cement as supplementary cementi-tious material. Based on the mix proportions, EPS concrete specimens were pre-pared in the standard condition.

To gather the database for training and testing pairs of neural network or ANFISmodels, cube specimens based on the mix designs reported in Table 4 were made,cured for 28 days. Afterwards the compressive strength of these specimens deter-mined according to ASTM C39 [25]. The average compressive strength for EPS mix-tures without PP could be seen in this table.

A total number of 75 records of EPS concrete compressive strength at 28 dayswere gathered based on the aforementioned mix design to construct the train-ing–testing database. For training and testing of the proposed models, 64 and 11samples were randomly chosen respectively. Moreover, to monitor the training

Fig. 4. Grading of fine and coarse agg

process, 10 checking data were selected randomly from both training and testingdata. This idea might be utilized to avoid miss-training (over training, saturationproblem) [14,16].

The structure of the input–output of the modeler systems were schematicallyshown in Fig. 6. In this figure, the input parameters are (i) cement (C), (ii) silicafume (SF), (iii) water (W), (iv) fine aggregates (FA), (v) coarse aggregates (CA), (vi)

regates and limits of ASTM C33.

Table 4Mix proportion of the specimens.

Mixture Cement(kg/m3)

S.F (%) R.H (%) EPS(kg/m3)

PP Water(kg/m3)

W/(C + CM) Aggregate (size mm) Compressivestrengtha (MPa)

0–3(kg/m3)

3–6(kg/m3)

6–12(kg/m3)

1 400 – – – 0%, 0.1%, 0.3%, 0.5%, 1% By volume 180 0.45 666 118 957 432 400 – – 15 170 0.43 540 95 777 333 400 – – 25 165 0.41 431 76 620 16.74 400 – – 40 160 0.4 294 52 423 9.85 360 10 – – 190 0.48 652 115 940 47.66 360 10 – 15 175 0.44 524 93 755 27.87 360 10 – 25 175 0.44 422 75 607 24.48 360 10 – 40 170 0.43 282 50 406 10.29 320 – 20% – 210 0.52 620 110 895 29.5

10 320 – 20% 15 205 0.51 470 80 670 22.411 320 – 20% 25 205 0.51 385 68 555 10.612 320 – 20% 40 200 0.5 245 43 352 6.7

a Mixture without PP fiber.

C

SF

W

FA

CA

EPS

Neural network/ANFIS

Model

Compressive Strength (CS)

PP

Fig. 6. Schematic structure of modeler systems.

A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216 209

expanded polystyrene beads (EPS) and (vii) waste carpet polypropylene fibers (PP)by weight per unit volume of concrete. Moreover Table 5 summarizes the ranges ofinput and output of total data used for modeling purposes.

3.4. Measures for evaluation of models

To evaluate the performance of models, root means square (RMS), and correla-tion factor (CF) are utilized as follows:

RMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXP

i¼1

ðxci � xciÞ2.

P

vuut ð8Þ

where P is the total number of concrete samples, xci is experimental result and xci thepredicted result:

CFðxc ; xcÞ ¼ covðXc ; bXcÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficovðbX c ; bXcÞ � covðXc ;XcÞ

q�ð9Þ

where

Xc ¼ ðxc1; xc2; :::; xcpÞ; bXc ¼ ðxc1; xc2; . . . ; xcpÞ ð10Þ

Table 5Ranges of input/output variables.

Variable Range

Min Max

InputCement (kg/m3) 320 400Silica fume (kg/m3) 0 40Water (kg/m3) 160 230Fine aggregate (kg/m3) 97 784Coarse aggregate (kg/m3) 118 958Expanded polystyrene beads (kg/m3) 0 55Waste carper propylene fiber (kg/m3) 0 9.1

OutputCompressive Strength (MPa) 0.9 47.6

and

covðXc ; bXcÞ ¼ E½ðXc � lcÞ � ðbXc � lcÞ� ð11Þ

where

lc ¼ EðXcÞ; lc ¼ EðbXcÞ ð12Þ

where E is the mathematical expectation.

3.5. Data preprocessing

To avoid the saturation problem and consequently the low rate of the training[9,14,16], in particular to avoid the saturation region of log-sigmoid activation func-tion, normally used in the neural networks with backpropagation algorithm, it isnecessary to normalize the real rough data into a suitable range. In this paper, a lin-ear normalization adopted to map the rough data range to the range of [0.1, 0.95] asfollows:

um ¼ 0:1þ 0:85� wRV �wRVmin

wRVmin �wRV

maxð13Þ

where um is the normalized value, wRV is the rough values of data, wRVmax and wRV

min arethe maximum and minimum of rough values respectively. Obviously inverse map-ping could be then applied to draw the real values for the purposes of representationand possible practical applications.

4. Results and discussion

Modeling with regression analysis, neural network and ANFIS,consist of three stages: (a) preprocessing of data, (b) designingthe model (architecture), (c) training, and (d) testing of regression,neural network or ANFIS models. For implementing the proposedmodels, MATLAB software is utilized.

4.1. Regression models

Table 6 summarizes the proposed regression models for pre-dicting the compressive strength of EPS concrete. Table 7 showsthe b-parameters of the proposed models. The performance of pro-posed models were evaluated and presented in Table 8 in terms ofcorrelation factor and root means square. Moreover, Fig. 7 demon-strates a comparison between results of prediction of EPS concretewith experimental observation. Considering results of model’s per-formance summarized in Table 8 and illustrated in Fig. 7, it couldbe deduced that second order polynomial model (NRM2) is thebest model for the purpose of compressive strength prediction.This model predicts the compressive strength of EPS concrete withCF of 0.9663 and 0.9879 respectively for training and testing pairsand RMS of 21.4631 and 21.7042 for training and testing pairsrespectively. As seen this model has the minimum value of RMSfor testing data set with an acceptable correlation factor. In justifi-

Table 6Proposed regression models for predicting compressive strength of EPS concrete.

Model Type Equation

NRM1 1st Polynomial b0 + b1C + b1SF + b1W + b1FA + b1CA + b1EPS + b1PPNRM2 Partial 2nd Polynomial b0 þ b1C þ b2SF þ b3W þ b4FAþ b5CAþ b6EPSþ b7PP þ b8C2 þ b9SF2 þ b10W2 þ b11FA2 þ b12CA2 þ b13EPS2 þ b14PP2

NRM3 Power-fractionalb0 þ b1 W

b2 Cþb3SF

� �2þ b4Cþb5 SF

b6 FAþb7CAþb8 EPSþb9PP

� �2

NRM4 Power-fractionalb0 þ b1

WCþSF

� �b2 þ b3CþSF

FAþCAþEPSþPP

� �b4 þ b5W

EPSþPP

� �b6

Table 7b-Parameters of the proposed models.

Model b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14

NRM1 .246 .065 .099 �.057 2.07 �1.53 �.2641 �.09 – – – – – – –NRM2 .341 .176 �.238 �.288 �4.198 3.541 .493 �.038 �.16 .284 .176 5.641 �4.368 �.587 �.054NRM3 �.136 .079 1.669 �.807 .021 .01 �.031 .049 .033 .003 – – – – –NRM4 �43.637 2.349 .205 �38.737 .012 84.756 .005 – – – – – – – –

Table 8Performance of proposed regression models.

Model Training (Interpolation) Testing (Extrapolation)

CF RMS CF RMS

NRM1 0.9990 21.7081 0.9937 22.8693NRM2 0.9663 21.4631 0.9879 21.7042NRM3 0.9051 20.8990 0.9165 25.2815NRM4 0.9287 21.1673 0.9186 23.0578

Fig. 7. Comparison of regression models with experimental observation

210 A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216

cation on the goodness of NRM2 as the best model, comparisonsmade with the experimental observation depicted in Fig. 7a andb are of great importance. These figures show unfeasible resultsof NRM1, NRM3 and NRM4. It should be note that Fig. 7a and bare related to the prediction results of training and testing dataset (or formally interpolation and extrapolation in terms of regres-sion problem) respectively. These figures confirm our justificationon proposing NRM2 as elite regression model.

: (a) Training set (interpolation) and (b) testing set (extrapolation).

Fig. 8. Schematic of NNM architecture.

Table 9General properties of ANN models.

Name Type Training method/algorithm

Activation functionin HLs

Activation function inoutput layer

No. of PE inHL

Layersnumber

HLsnumber

NLMBP Feed-forward backpropagationnetwork

Supervised/LMBP Log-sigmoid Linear transfer function Variable 4 2

Table 10Summary of ANN models for prediction of 28-days compressive strength.

Name No. of neurons in Training set Testing set Checking set

HL1 HL2 CF RMS CF RMS CF RMS

NLMBP11 1 1 0.9804 2.5754 0.9828 2.6113 0.9745 2.7121NLMBP22 2 2 0.9976 0.9069 0.9798 3.6138 0.9968 1.2211NLMBP33 3 3 0.9990 0.5718 0.9937 1.9302 0.9979 1.0782NLMBP44 4 4 0.9998 0.2272 0.9805 3.1545 0.9953 1.1763

A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216 211

4.2. Neural network models

The schematic structure and general properties of used ANN areshown in Fig. 8 and Table 9 respectively.

Different topologies could be utilized for developing ANN mod-els. As discussed earlier, the architecture of ANNs are composed ofinput/output/hidden layers and some neurons in each layers. Thenumbers of neurons in input and output layers are constant andconstrained to the number of input and output parameters respec-tively. However, number of hidden layers and neurons in corre-sponding layers are variable and these numbers are governingfactor of ANN model’s performance. If too few hidden neuronsare used, the network will be unable to model complex data;resulting in a poor performance. If too many hidden neurons areused; then training will become excessively long and the networkmay over fit. Over fitting occurs when the network begins to modelrandom noise contained within the data, resulting in a failure toconverge on a generalized solution. So, one practical method todeal with this issue is to exploit the structured trial and error ap-proach. The process generally used to determine the number ofhidden layers and number of neurons. In this approach, the goalis try to quickly find the network that converges and simulatethe goal of ANN. In this paper, three-hidden-layer structure is sup-posed for the final ANN model with variable number of neurons ineach layer. Moreover, equal number of neurons is supposed to bein each hidden layers. Afterwards, ANN models constructed by thismethod are trained and their prediction measures were evaluatedas correlation factor and root means square. To decide on the suit-able model, it is tried to find an architecture which yield the best

correlation factor and root means square concurrently occurredfor training and testing pairs. Eventually, this approach proposesANN model with higher performance for both estimation (interpo-lation) and generalization (extrapolation) properties concurrently.

In this regard, Table 10 summarizes the performance of 6 eliteneural network models trained via the aforementioned procedure.Fig. 9a–c illustrates the performances of trained networks for train-ing, testing and checking data sets respectively. In these figures,the horizontal axis represents the number of neurons in hiddenlayers while two vertical axes dedicated to evaluated performanceindexes. Left vertical axis presents the correlation factor and theright vertical axis shows the root means square. As depicted inFig. 9a the performance indexes (correlation factor and root meanssquare) for training data set were improved by increasing the num-ber of neurons in hidden layers showing the effectiveness of higherorder neural networks for the prediction purposes. Fig. 9b showsthe performance indexes for testing pairs which confirms the bestnumber of neurons in hidden layers as three neurons for general-ization purposes. These findings could be verified by the resultsgained for checking data sets as seen in Fig. 9c. Thus NLMBP33which contains three neurons in its layers considered to be thesuitable ANN model for prediction of the compressive strength ofEPS concrete.

4.3. ANFIS models

ANFIS models could be composed of different architectureswhich govern the output and prediction performances. Numbersof input/output units are constant and basically limited to the

Fig. 9. Performance of ANN models with different number of neurons in hiddenlayers: (a) Training, (b) testing and (c) checking data set.

Table 11Utilized membership functions.

Type Formula

Triangular max min x�ab�a ;

c�xc�b

� �;0

� �Trapezoidal max min x�a

b�a ;1;d�xd�c

� �;0

� �Bell-shape 1

1þjx�ca j

2bð ÞGaussian e

�ðx�cÞ22r

Fig. 10. Training error tre

Table 12Summary of ANFIS predictions.

ANFIS model MF Training set

CF RMS

ANFTRI Triangular 1.0000 0.0079ANFTRA Trapezoidal 0.9998 0.2335ANFBEL Bell-shape 1.0000 3.9 � 10�4

ANFGUS Gaussian 1.0000 7.6 � 10�4

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space of concerned problem. Similar to the ANN models, in ANFIScase, number of input/output units is constant. Units specified inlayer I (schematically represented in Fig. 3) are related to the num-ber of fuzzy sets (labels or membership function) utilized fordecomposing the input space of each input variable. Other impor-tant option of this architecture is the type of fuzzy membershipfunction (MF) used for such decompositions. It was practicallystudied by author that such membership function is an importantfactor affect the prediction performance [13]. So, considering theconstraints of utilized computer’s CPU, three fuzzy sets used fordecomposition of each input space with variable fuzzy label types.In this regard, similar to that of neural network models, the struc-ture of proposed ANFIS networks was consisted of seven inputvariables (i.e., C, SF, W, FA, CA, EPS and PP) and CS as its output var-iable. As said, the input space is decomposed by three fuzzy labels.In this study, for comparison purposes, four types of MFs includingthe triangular, trapezoidal, bell-shape, and Gaussian functions (seeTable 11) were utilized to construct the suggested models.

ANFIS models were then trained in a similar manner for neuralnetwork. 100 epochs were specified for training process to assuregaining the minimum error tolerance. To identify the parametersof Sugeno-type fuzzy inference system, a hybrid learning algorithmwas utilized [22]. Fig. 10 compares the error trends of ANFIS mod-els during 100 epochs regarding checking data set which under-score reaching the equilibrium state after completion of trainingprocess. The performance of ANFIS models are examined by RMSand CF and the results summarized in Table 12. As seen, the corre-lation factors of training data set for all ANFIS models are near 1 as

nd for ANFIS models.

Testing set Checking set

CF RMS CF RMS

0.8333 12.0251 0.7033 12.22710.9505 5.0968 0.9634 3.52160.9783 3.4053 0.9836 2.43270.9098 8.3303 0.8546 7.6508

Table 13Parameters of bell-shaped convex fuzzy sets as vector of [a,b,c] (see Fig. 11 and Table11).

Variablea Small Medium Big

NSF [0.137,2,0.07] [0.126,2,0.518] [0.138,2,0.979]NW [0.20,2,0.097] [0.01,2,0.526] [0.2,2,0.952]NFA [0.174,2,0.063] [0.23,2,0.47] [0.19,2,0.94]NCA [0.12,2,0.065] [0.058,2,0.42] [0.224,2,0.9]NEPS [0.09,2,0.029] [0.115,2,0.435] [0.16,2,0.927]NPP [0.094,2,0.043] [0.127,2,0.458] [0.174,2,0.942]NCS [0.13,1.99,0.012] [0.142,2,0.429] [0.318,2,0.837]

a Prefix N stands for normalized data.

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the optimum value; however this norm for testing data set demon-strated different values. Of all, ANFBEL exhibited the best correla-tion of 0.9783. Correlation factor gained for ANFTRI, ANFTRA, andANFGUS are as 0.8333, 0.9505 and 0.9098 respectively. RMS valuesfor ANFTRI, ANFTRA, ANFBEL, and ANFGUS calculated as 0.0079,0.2335, 3.9 � 10�4, and 7.6 � 10�4 for training data set and12.0251, 5.0968, 3.4053, and 8.3303 for testing pairs respectively.As seen, the RMS values for all of models are satisfactory for train-ing set; however, this norm is elevated for testing pairs. Again,ANFBEL exhibited the best performance; however, other modelshave a great amount of error in comparison. It should be noted thatthese findings could be confirmed by performance norms of check-ing data set. Thus ANFBEL which consisted of bell-shaped fuzzy

Fig. 11. Fuzzy domain decomposition using bell-shaped linguistic variables.

Fig. 12. Comparison of NLMBP33 and ANFBEL with experimental observations: (a) Training set and (b) testing set.

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membership functions is nominated as the best model for predic-tion of the compressive of EPS concrete. Fig. 11, with respect to Ta-ble 13, shows the membership functions for input variables ofANFBEL after completion of training process.

4.4. Comparison of non-linear regression, ANN and ANFIS

Fig. 12a and b represents the prediction results of NLMBP33 andANFBEL as elite models of ANN and ANFIS models respectively incomparison with the experimental observations. In these figures,the horizontal axis is representative of experimental results andthe vertical one is related to the results of model’s prediction forcompressive strength of EPS concrete. Error lines of +25% and�25% are also plotted to visualize the prediction performances.Based on these figures, it is evident that the proposed models arecapable to predict the compressive strength of EPS concrete fortraining data set. As seen in Fig. 12b, using ANFBEL, two samplesof testing data were predicted beyond 25% error line; however,all of samples could be successfully predicted using NLMPB33.NRM2 as elite regression model is justified to be the worst modeldue to wrong predictions made by this model depicted inFig. 12a and also numerous predictions beyond 25% error lineshown in Fig. 12b.

It should be noted that the ANFIS model might be improved byintroduction of more fuzzy sets (or fuzzy labels) to each of input

Fig. 13. Effect of PP content on compressive strength: (a

variables. For example in the current modeling system with ANFIS,input space is decomposed by three labels defined here as Small,Medium and Big (Table 13). By adding up the number of fuzzy setsfor decomposing the input space, for example four fuzzy sets, thenumber of rules for estimation procedures would be increasedand thus a better estimation performance might be gained. How-ever, such task increase the complexity of the system and accord-ingly more memory needed to complete the training process. Thisissue is unfeasible in computers with limited memory and CPUspeed for problems comprising numerous input variables. On theother hand, the neural network operates in the parallel processingstrategy and needed less memory for training process. One anotherreason might be related to the utilized training algorithm of ANNand ANFIS. It should be noted that ANN trained by so-called Leven-berg–Marquardt algorithm while ANFIS trained by gradient des-cent backpropagation combined with least-squares approach. Itis well established that prior one is more rapid and robust thanother later method utilized in ANFIS training and optimizationprocess.

4.5. Effects of PP and EPS content on compressive strength

To evaluate the effects of EPS and PP content on the compres-sive strength of EPS concrete, trained models of ANN (NLMBP33)and ANFIS (ANFBEL) were utilized. Figs. 13a and 10b demonstrate

) NLMBP33 estimation and (b) ANFBEL estimation.

A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216 215

the interactive effects of PP fiber content on the compressivestrength of EPS concrete regarding the cement content estimatedby elite model of ANN and ANFIS respectively in view of 3-D plots.X, Y, and Z axis of these plots are PP fibers content, cement contentand compressive strength respectively. As seen, by increasing thecement content, both model proposed an increase of compressivestrength while increasing the PP content lead to a reductive effecton the compressive strength of EPS concretes. It should be notedthat increasing effects of cement content on compressive strengthis more evident for PP content below 0.8%. Moreover, Fig. 14a and billustrate the interactive effects of EPS and cement contents on thecompressive strength of EPS concrete in a similar manner moldedby ANN and ANFIS elite models. It should be noted that, in theseplots, X-axis are the EPS content. Considering the response surface,it could be concluded that the more cement content used, the morecompressive strength could be gained and vice versa, by applica-tion of higher amount of EPS, the compressive strength of suchconcretes reduced substantially. Meanwhile, the 3-D surface ofinteractive effects of PP, EPS and cement content might be assistedin the EPS-concrete mix designer to have a general view on howthese parameters affects the compressive strength and accordinglythese surfaces might be utilized in a framework of concrete mix-design system with capability of optimization task. Another pointcould be drawn is that the effect of PP fibers is more tangible inthe mixtures with EPS content of below 20%.

Fig. 14. Effect of EPS content on compressive strength: (

5. Conclusion

Concrete is a highly non-homogenous material, so modeling itsbehavior is a difficult task. The artificial intelligent-based modelsare known to be robust tools to model complex systems. In this pa-per, the application of two types of such systems including ANN andANFIS in the estimation of 28 days compressive strength of an espe-cial type of LWC mixtures made by EPS beads have been outlined.Based on an experimental program, a database was collected forstrength of EPS concretes and then four regression models, ANNand ANFIS models were designed and trained by training data setrandomly chosen from the whole database. The performances ofthe proposed models were evaluated by correlation factor and rootmeans square to assess the best regression, ANN and ANFIS models.The results showed that NRM2 as elite regression model in the formof partial second order polynomial, NLMBP33 as elite ANN modelconstructed by two hidden layers having three neurons in eachand ANFBEL as elite ANFIS one constructed by bell-shaped fuzzymembership functions are proposed as the suitable models. Promis-ing results were obtained using both ANN and ANFIS models; how-ever, comparing the prediction results of elite ANN and ANFISmodels for training and testing data sets revealed the higher accu-racy and generalization capabilities of NLMBP33 as elite neural net-work model. The regression model was found to be unable to predictthe compressive strength. In general, materials and civil engineers

a) NLMBP33 estimation and (b) ANFBEL estimation.

216 A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216

may use the proposed model to predict the stability of EPS concretemixtures and avoid conducting costly experimental tests that re-quire specialized equipments and expertise.

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