RETRACTED: Stiffness performance of polyethylene terephthalate modified asphalt mixtures estimation using support vector machine-firefly algorithm

Accepted Manuscript Stiffness performance of polyethylene terephthalate modified asphalt mixtures estimation using support vector machine-firefly algorithm Mehrtash Soltani, Taher Baghaee Moghaddam, Mohamed Rehan Karim, Shahaboddin Shamshirband, Ch Sudheer PII: DOI: Reference:

S0263-2241(14)00583-1 http://dx.doi.org/10.1016/j.measurement.2014.11.022 MEASUR 3141

To appear in:

Measurement

Received Date: Revised Date: Accepted Date:

1 October 2014 17 November 2014 27 November 2014

Please cite this article as: M. Soltani, T.B. Moghaddam, M.R. Karim, S. Shamshirband, C. Sudheer, Stiffness performance of polyethylene terephthalate modified asphalt mixtures estimation using support vector machinefirefly algorithm, Measurement (2014), doi: http://dx.doi.org/10.1016/j.measurement.2014.11.022

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Stiffness performance of polyethylene terephthalate modified asphalt mixtures estimation using support vector machine-firefly algorithm Mehrtash Soltani1 , Taher Baghaee Moghaddam1, Mohamed Rehan Karim1 , Shahaboddin Shamshirband2*, Ch Sudheer3 1

Center for Transportation Research, Department of Civil Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia. 2 Department of Computer Science, Chalous Branch, Islamic Azad University (IAU), 46615-397 Chalous, Mazandaran, Iran. 2 Department of Computer System and Technology, Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia 3 Department of Civil Engineering, Indian Institute of Technology Delhi, New Delhi, India

*Corresponding Author: Shahaboddin Shamshirband, Email: [email protected]

Abstract Predicting asphalt pavement performance is an important matter which can save cost and energy. To ensure an accurate estimation of performance of the mixtures, new soft computing techniques can be used. In this study, in order to estimate the stiffness property of Polyethylene Terephthalate (PET) modified asphalt mixture, different soft computing methods were developed, namely: support vector machine-firefly algorithm (SVM-FFA), genetic programming (GP), artificial neural network (ANN) and support vector machine. The Support Vector Machine-Firefly algorithm (SVM-FFA) is a metaheuristic search algorithm developed according to the socially dashing manners of fireflies in nature. To develop the models, experiments were performed. The process, which simulates the mixtures’ stiffness, was created with a soft computing method, the inputs being PET percentages, stress levels and environmental temperatures. The performance of the proposed system was confirmed by the simulation results. Soft computing methodologies show very good learning and prediction capabilities and the results obtained in this study indicate that the SVM-FFA contributed the most significant effect on stiffness performance estimation since the SVM-FFA model had a better correlation coefficient than the SVM, ANN and GP approaches. R2 and RMSE were utilized for making comparisons between the expected and actual values of SVM-FFA, GP, ANN and SVM. The proposed SVM-FFA methodology predicted the output values with 254.4743 (mm/day) and 0.9957 RMSE and R2 respectively. Keywords: Firefly Algorithm, Support Vector Machine; Pavement performance; PET modified asphalt mixtures; Environmental conditions; Estimation.

1

1. Introduction Improving asphalt mixture properties is the aim of engineers and experts in order to increase the service life of asphalt pavement. Using additives such as various types of fibers and polymers is a common way of improving asphalt mixture characteristics [1]. Recycling asphalt pavement can save a lot of resources and protect the environment. Currently, more and more researchers are paying attention to the exploration and application of recycling technology. In this case, polymer modification offers the opportunity to conquer the deficiencies of asphalt and thereby improve the performance of bituminous mixtures. The construction of pavement structure with modified characteristics that can offer better performance as well as longer service life is one of the goals of road engineers and designers. Hence, many studies have been previously performed to evaluate asphalt pavement performance with modified characteristics [2-7]. However, although using virgin modifiers in road pavement can improve pavement properties, in many cases it incurs higher financial costs. Therefore, much research has focused on using modifiers obtained from waste materials to reduce costs imposed by using virgin modifiers and finding solutions to reuse post-consumed materials as secondary materials in road construction projects in environmentally friendly ways [8-14]. Stone matrix asphalt (SMA) is hot mix asphalt (HMA) with a coarse aggregate structure and high asphalt content. Stone matrix asphalt was developed in Germany in the 1960s and provides better resistance against permanent deformation [15]. SMA has several advantages over conventional densegraded asphalt mixture including: high rut resistance, high durability, improved resistance to reflective cracking, high skid resistance, better drainage conditions and reduced noise pollution [16, 17]. The prediction of asphalt pavement performance is a significant issue that can lead to saving cost and energy. To ensure an accurate estimation of the performance of mixtures based on environmental conditions, new soft computing techniques can be utilized [15-19]. Collecting input/output data pairs and learning the proposed network from these data is the main idea behind soft computing methodologies. Soft computing techniques are a popular means of exploring and presenting interactions between parameters affecting one phenomenon. The use of such methodologies in pavement engineering is increasing because they assist road engineers and designers to gain a better perspective about pavement performance parameters. The Support vector machine (SVM) is an exceptional soft computing learning technique, which has been used in different applications in the fields of environmental research, computing and hydrology [15-19]. Additionally, various applications have been found in classification and regression analysis, pattern recognition, and forecasting, which perform better than recently developed techniques, e.g. conventional neural network methodology and other statistical analysis models [20-25]. Recently, SVM has proved its performance in solving forecasting problems [26-28]. A study was recently performed [29] to estimate the flow number of dense asphalt-aggregate mixtures using the SVM. The investigation revealed that the SVM was capable of predicting the flow number of asphalt mixtures. Similarly, another study employed the SVM method to model the mechanical properties of hot-mix asphalt (HMA). This study showed that the SVM’s prediction performance is far better than multivariate regression-based models and comparable to the ANN [30]. Furthermore, in 2010 an efficient off-line nonlinear pavement back calculation system was introduced using Support Vector Machines (SVM). In this study, a comparison was performed with another common machine learning technique, the multi-layer perceptron (MLP). The results indicated 2

the effectiveness of the SVM method over other methods [31]. Moreover, a new road friction coefficient estimation method based on the SVM was proposed in the application of steering driving under a variety of road conditions. The results presented in the study showed that the SVM can accurately calculate the friction coefficient, which is important for controlling the stability of a vehicle that is over or under steering [32]. In another application of the SVM method, a signal processing algorithm based on the principle of the support vector machines characteristic of road surface malfunctions was developed. The test results proved that this algorithm can be used to detect pavement malfunctions with high efficiency [33]. In the current study, an estimating model is developed to predict the stiffness property of Polyethylene Terephthalate (PET) modified asphalt mixtures based on a series of environmental conditions using SVM coupled with the Firefly Algorithm (FFA). Subsequently, the performance of SVM-FFA for estimating the stiffness property of PET modified asphalt mixture is investigated. PET obtained from waste PET bottles served as a modifier. FFA was utilized to determine the SVM factors. Besides, it was aimed at analyzing the efficiency of SVM, ANN, SVM-FFA, and genetic programming in order to estimate the stiffness property of PET modified asphalt mixture based on a series of test conditions.

2. Background of soft computing techniques in pavement engineering This section presents an overview of studies which have used soft computing techniques in pavement engineering over the past few years. In this regard, neuro-fuzzy can be employed to calculate pavement moduli with present inputoutput data articulating the target behavior [34]. Correspondingly, an adaptive neuro-fuzzy inference system (ANFIS) can be utilized to back calculate asphalt mixture moduli. Fuzzy inference is not appropriate for substantial numbers of input-output patterns and input space partitioning. Nevertheless, it can be a good alternative for a small amount of training data relating to a considerable amount of uncertainty [35]. Training data may involve either in situ or synthetic pavement moduli. The training procedure can be carried out by either experimental records to describe particular test conditions or by synthetically collected records to inversely simulate the pavement response model. The primary advantages of the adaptive back calculation technique are the real-time back calculation capability and accurate results [36]. HMA modelling as a linear viscoelastic material helps to find the stress-strain performance as well as resilient modulus of HMA mixtures. The relative amount of rest period to loading time (R/D) of 4 can produce about 8% error in predicting the resilient modulus of HMA under square waveform. The theory of resilient modulus master curve could be used effectively for modelling the resilient modulus of asphalt concrete mixtures at different loading frequencies and elevated temperatures [37]. Alessandra Bianchini and Paola Bandini [38] evaluated a neuro-fuzzy model to estimate the performance of flexible pavement. They used parameters that were mostly obtained from falling weight deflectometer tests and were generally provided by agencies to evaluate the pavement condition. In their study, it was concluded that the neuro-fuzzy model led to satisfactory results, which represented the efficiency and potential of new modelling techniques. In a study performed by Ercan Özgan the Marshall stability of an asphalt mixture was modeled at elevated temperatures and exposure times 3

using the fuzzy logic model. Exposure times of 1.5, 3, 4.5 and 6 h were designated and temperatures of 30, 40 and 50˚C were considered. The results of this study showed good correlations between the experimental results and the outcome from the fuzzy logic model [39]. In a later investigation, artificial neural network (ANN) based modelling was developed for the Marshall stability of asphalt mixture. The same testing conditions and procedure were designated. According to the results, the ANN model could successfully model the Marshall stability of asphalt mixture [40]. In another investigation on calculating the stiffness modulus of asphalt concrete mixtures, 37 core samples were conditioned at temperatures of 17°C (reference temperature), 30, 40, and 50°C for 1.5, 3, 4.5, and 6 h respectively. The Sugeno type adaptive neuro-fuzzy inference system served as the prediction model. The specific gravity (g/cm3) of the mixture, temperature (°C), exposure time (h) and quantity of asphalt (g) were designated as input variables and the output parameter was the stiffness modulus (kg/cm2). Based on the results achieved in the study, the proposed model can successfully be used to evaluate the stiffness modulus of mixtures and the model’s correlation coefficient was found to be highly important with a value of 0.94 [41]. In 2012, the adaptive neuro-fuzzy inference approach was used in another study to predict the stiffness modulus of asphalt concrete mixtures. The results illustrated that the developed prediction models could successfully be employed for an unperformed situation that was difficult to carry out in an actual experiment [42]. In a related study, a neural network model was utilized to estimate the performance of an emulsified asphalt mixture. Residual asphalt content, cement addition level and curing time were the inputs and the output was the resilient modulus of emulsified asphalt mixture. It was shown that the neural network is an excellent way to estimate the resilience of emulsified asphalt mixtures with a high degree of accuracy [43]. The permanent deformation of asphalt mixture was modeled using soft computing techniques. Mohammad Reza Mirzahosseini et al. utilized ANN to estimate the permanent deformation (flow number) of dense graded asphalt mixture. In that study, the ratio of coarse aggregate to fine aggregate, the amount of filler and asphalt, the percentage of voids in mineral aggregate as well as the ratio of Marshall stability to Marshall flow (Marshall quotient) were considered the predictor variables. The study results showed that the proposed models successfully predicted the flow number and the nonlinear models performed better than the linear regression-based models [44]. In another investigation, models were developed to estimate the Marshall property of polypropylene (PP) modified asphalt mixture using neural networks. Different variables were designated to predict the Marshall property of mixtures, including: PP type, PP percentage, asphalt percentage, unit weight, specimen height, air voids, voids in mineral aggregate, and voids filled with asphalt. The final results of the study indicated there was a good agreement between the responses predicted by the neural network models and the experimental results [45].

3. Materials and methods 3.1.

Materials

Asphalt mixtures were fabricated using 80-100 asphalt penetration grade. Granite-rich aggregate particles were used in this investigation. For a better understanding of the materials applied, several tests were performed on the aggregate particles and asphalt cement, and the results are presented in Table 1. 4

Waste PET bottles were used in the present investigation. The bottles were turned into flakes suitable for use in the asphalt mixture. To use PET flakes in an asphalt mixture, the PET bottles were shredded into small pieces, which were crushed with a crushing machine. Thereafter, the crushed PET particles were sieved and particles smaller than 2.36 mm in size were used.

Table 1: Material properties

Property

Unit

Used specification

Value

Requirements

Penetration at 25°C

0.1mm

ASTM D 5

87.5

-

Softening point

°C

ASTM D 36

46.6

-

Flash point

°C

ASTM D 92

300

-

Fire point

°C

ASTM D 92

320

-

Specific gravity

(g/cm3)

ASTM D 70

1.03

-

L.A. Abrasion

%

ASTM C 131

19.45

<30

Flakiness index

%

BS 812 Part 105.1

2.72

<20

Elongation index

%

BS 812 Part 105.2

11.26

<20

Aggregate crushing value

%

BS 812 part 3

19.10

<30

Bulk specific gravity

(g/cm3)

ASTM C 127

2.60

-

Absorption

%

ASTM C 127

0.72

<2

Bulk specific gravity

(g/cm3)

ASTM C 128

2.63

-

Absorption

%

ASTM C 128

0.4

<2

Soundness loss

%

ASTM C 88

4.1

<15

Asphalt

Coarse aggregate

Fine aggregate

5

3.1.

Mixture fabrication

To fabricate the asphalt mixture, 1100 g of mixed aggregate and filler was heated in an oven at 160˚C for 3 hours. Asphalt cement was also heated at 130˚C to be suitable for mixing with the aggregate particles. All materials were mixed at 160˚C. PET particles with different percentages (0%, 0.2%, 0.4%, 0.6%, 0.8% and 1% by weight of mixed aggregate) were added directly to the mixture as the dry process means. The loose mixture was compacted using a Marshall Compactor and 50 blows of compaction efforts were applied on each side of the mixture. It is worth mentioning that the specimens were constructed using optimum asphalt content.

3.2.

Indirect tensile stiffness modulus test

Asphalt mixture stiffness is a fundamental design parameter for flexible pavement. A correlation was found between stiffness and other mixture properties, such as rutting and fatigue, and thus, it can serve as a criterion to evaluate Asphalt Concrete (AC) mixture performance [46]. In addition, according to the Strategic Highway Research Program (SHRP), AC mixture stiffness values are very susceptible to environmental temperature and loading conditions [47]. The Indirect Tensile Stiffness Modulus (ITSM) test gives the relationship between stress and strain of an asphalt mixture and is used to evaluate the stiffness of an asphalt mixture in specific environmental conditions (temperature and stress). The ITSM test was performed in accordance with AASHTO TP31. ITSM testing can be carried out using a Universal Testing Machine (UTM) among the important testing equipment in a pavement laboratory. The UTM is a computer controlled system which operates automatically. During the test, compressive haversine waveform loads were applied across the vertical section of the specimen’s thickness. Also, by utilizing Linear Variable Differential Transducers (LVDTs) installed along a diametrical section of specimen, the displacement of the asphalt mixture was measured. The horizontal tensile stress and stiffness modulus of the asphalt mixtures was calculated using the following equations:
 max =  =

2×P 1 × ×

P × ν + 0.27 2 H×t

where σ max is the maximum horizontal tensile stress in the middle of the specimen; S is the stiffness modulus; P is the applied vertical peak load, H is the amplitude of horizontal deformation; t represents the average thickness of the specimen; d is the average diameter of the specimen; and ν is Poisson’s ratio. The ITSM test was conducted at stress levels of 200, 300 and 400 kPa and temperatures of 10˚C, 25˚C and 40˚C for each PET percentage.

6

3.3.

Support vector machine

For this research, the SVM-FFA model was established to estimate the stiffness of PET modified asphalt mixtures in relation to the input parameters. Training and checking data for the SVM-FFA network are extracted from a series of laboratory tests performed under different testing conditions. The theory of the support vector machine (SVM) technique was presented by Vapnik [48, 49]. The SVM was developed according to statistical machine learning development as well as structural risk minimization to reduce upper bound generalization error compared to local training error. It is a common technique previously used in machine learning methodologies [49]. This technique proved it has several advantages over other soft computing learning algorithms. Additional advantages provided by this methodology include: (1) applying a high dimensional spaced set of kernel equations, which discreetly include non-linear transformation; hence, there is no assumption of functional transformation, which makes linearly separable data indispensable; and (2) it is a unique solution due to the convex nature of the optimal problem. The SVM functions, according to Vapnik’s theory, are represented in Eqs. (3-6).  =  ,  !" is used to assume a set of data points, where  indicates the input space vector of the data sample, and the target value and data size are defined as  and n respectively. The SVM estimates the equation as: #  = $%   + &

(3)

'()* + = ‖$‖- + + ∑"1, 0  ,   , -

,

"

(4)

In Eq. (3) %  indicates the high dimensional space characteristic that maps the input space vector x

while w and b are a normal vector and scalar, respectively. In addition, + " ∑"1, 0  ,   stands for the ,

empirical error, risk. Factors b and w are measured by minimizing a regularized risk equation followed by the introduction of positive slack variables 2 and 2∗ that indicate the upper and lower excess deviation [50]: Minimize '()* 4$, 2 ∗ 5 = - ‖$‖- + + ∑"1, 2 + 2∗  ,

 − $%  + & ≤ 9 + 2 Subject to 6$%   + & −  ≤ 9 + 2∗ > 2 , 2∗ ≥ 0, ; = 1, … , =

(5)

where ‖$‖- is the regularization term, + represents the error penalty feature utilized to control the , -

trade-off between the empirical error (risk) and regularization term, 9 represents the loss function associated with the approximation accuracy of the trained data point, and the number of factors in the training data set is defined as =. Optimality constraints and the Lagrange multiplier can be used to solve Eq. (3) and are consequently obtained using the following generic function: #, ? ?∗ = ∑"1, ? − ?∗ @,   + &

(6)

7

In this equation, @ ,   = % %4A 5 where @4 , A 5 is defined as the kernel function, which is dependent on the two inner vector  and A in the feature space %  and %4A 5 respectively. The most important aim of SVMs is to complete data correlation using non-linear mapping. It could be possible to develop a non-linear learning machine known as a direct calculation method of a kernel equation, denoted by K, if a method of calculating the inner product in a feature space is available in a straight line as a function of the original input points. The flexibility of the SVM is recognized through using kernel functions by adopting the data of a higher-dimensional feature space. The results in the higher-dimensional feature space stand for the results of the original, lowerdimensional input space. Sigmoid, lineal, polynomial and radial basis functions are the four basic kernel functions provided by the SVM, among which radial basis function (RBF) is considered the best kernel feature due to its computational effectiveness, reliability, simplicity, and ease of adaption to optimization as well as its adaptability in handling factors that are more complicated [50-52]. Only the solution of a set of linear functions is required for training the RBF kernel equation rather than the lengthy and complicated, demanding quadratic programming problem [53]. Accordingly, the radial basis equation with parameter σ is adopted. The non-linear radial basis kernel function is defined as: -

@ , A  = BC D−EF − A F G

(7)

where  and A are vectors in the input space, (i.e. vectors of features computed from training or test samples). In addition, the prediction accuracy using an RBF kernel function depends on the selection of its three factors E, 9 and +. In this study, the optimal values of these factors are established by using the Firefly Algorithm.

3.4. Firefly Algorithm The Firefly Algorithm (FFA) is one of the metaheuristic search algorithms [53–55]. It is developed according to the socially dashing behavior of fireflies in nature. The difference in the formulation of attractiveness and light intensity are the two main concerns of the FFA. The objective function is proportional to the brightness, or light intensity, emitted by a firefly for the optimal design considering the maximization of the objective function. Eq. (8) gives the Gaussian form of light intensity I with varying distances: (8) In this equation, I0 is the distance r = 0 from the firefly, the light absorption coefficient is defined as γ, and I represents the light intensity at distance r from the firefly. As a firefly's attractiveness is proportional to the light intensity seen by adjacent fireflies, the attractiveness ω at a distance r from the firefly can be defined as:

(9) 8

where, ω shows the light absorption coefficient; it can be considered as a constant when it locates between 0.1 and 10. The attractiveness at r = 0 is defined as ω0. Cartesian distance is the distance between two fireflies i and j at xi and xj, respectively:

(10) In Eq. (10), xi,k is the kth element of the spatial coordinate xi of the ith firefly. The change in location by which a firefly i is attracted to a more attractive (brighter) firefly j is defined as:

(11) where is due to attraction, is the randomized term with α as the randomization parameter whose value lies between 0 and 1, and εi is the vector of random numbers drawn from a Gaussian distribution or uniform distribution. A fitness criterion function of the Root Root Mean Square Error (RMSE) is utilized to determine suitable factors for the SVM model. Equation 12 is employed to calculate the RMSE value.

3.5. Evaluation of model performance In this study, in order to assess the performance of the GP, ANN, SVM-FFA and SVM, the subsequent statistical indicators were designated:

1) Root Mean Square Error (RMSE)

(12)

2) Coefficient of determination (R2)

(13)

In these equations, Oi and Pi are the models’ predicted stiffness values and measured values, respectively. The total number of test data is n.

9

4. Experimental data The SVM-FFA model stiffness estimation of PET modified asphalt mixtures based on three inputs was analyzed. According to the experiments, the input parameters (PET percentage, stress level and temperature) and output (stiffness) were collected and defined for the learning techniques. Fifty percent of the data served to train the samples for the experiments, and the other fifty percent were to test the samples. The performance of soft computing models in estimating the stiffness property of PET modified asphalt mixtures was evaluated according to statistical criteria, such as RMSE and the coefficient of determination R2. The estimated stiffness is represented in Fig. 1 in the form of scatter plots for testing phase. As seen from the fit line equations, the equations were y=a0x+a1 in the scatter plots. The a0 coefficient for the SVM-FFA model was closer to 1 with a very high R2 value. Support vector machine - Firefly

12000

12000

10000

10000 Predicted values

Predicted values

Support vector machine

8000 6000 y = 0.9981x + 345.53 R² = 0.9834

4000 2000

8000 6000 y = 1.0014x + 12.625 R² = 0.9957

4000 2000

0

0 0

5000

10000

15000

0

Actual values

12000

10000

10000

8000 6000 y = 0.9557x + 354.42 R² = 0.9821

Predicted values

Predicted values

Artificial neural network

12000

8000 6000 4000

2000

2000

0

0 5000

15000

(b)

Genetic programming

0

10000

Actual values

(a)

4000

5000

10000

Actual values

15000

y = 0.9846x + 497.28 R² = 0.9828

0

5000

10000

15000

Actual values

(c) (d) Figure 1: Stiffness estimation by a) SVM, b) SVM-FFA, c) GP and d) ANN versus PET measurements 10

4.1.

Performance analysis

According to the experiments, the input parameters (PET percentage, stress level and temperature) and output (mixture’s stiffness) were collected and defined for the learning technique. The developed SVM-FFA model was compared with traditional models like artificial neural networks (ANN), Genetic programming (GP) and support vector machines. In this section, a brief description of the development of various artificial intelligence tools and the parameters used while developing the models are initially presented. Then the results obtained from these traditional models are compared with those obtained from the proposed SVM-FFA model. A multilayer feedforward backpropagation ANN was developed using the Demuth Neural Network toolbox in MATLAB R2010a. The ANN model developed consists of a single input layer, a hidden layer and an output layer. The parameters used while developing the ANN model are summarized in Table 2. Table 2: properties of the model Parameters Value Learning algorithm Levenberg–Marquardt algorithm Architecture of ANN 7-39-1 Transfer function Sigmoid Number of neurons in hidden layer 39 R2 and RMSE are utilized for making a comparison between the expected and actual values of SVM-FFA, GP, ANN and SVM. Table 3 depicts the performance indices of different features in order to estimate the stiffness in the testing and training phases. Results indicate that in the training phase, the performance of all three models is more or less similar. Furthermore, it is observed that the RMSE values are significantly lower in the training phase for all methods except SVF-FFA. However, in testing phases SVM-FFA has the lowest RMSE. Schunn and Wallach (2005) identified there is no concrete criterion for how high R2 should be to consider a fit as “good.” For RMSE, the situation is even less clear, because the measure depends on the dependent variable measure. It should just be as low as possible, but there is no standard for what is sufficiently low, because the values are dependent on the experiment. Lacking any formal criteria, it is often assumed that a model with higher R2 values and lower RMSE value is preferable. On the basis of these criteria, the SVM-FFA model is superior to all other models.

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Table 3: Performance indices of various approaches for PET estimation Training Testing Method RMSE RMSE R2 R2 (mm/day) (mm/day) GP 611.8989 0.9873 542.0659 0.9821 ANN

545.4903

0.9879

656.8885

0.9828

SVM-FFA

574.5119

0.9815

254.4743

0.9957

SVM

517.1737

0.9884

599.753

0.9834

5. Conclusions The main purpose of this study was to estimate the stiffness property of PET modified asphalt mixtures by utilizing different soft computing techniques (GP, ANN, SVM-FFA and SVM). The significance of these methods is highlighted in their estimation of stiffness. Statistical measures were conducted to analyze the selected soft computing methods. The SVM implements structural minimization. However, the other common, traditional methodologies consider the process of error minimization. The FFA provides more reliable and accurate predictions by exploring the optimal parameters for SVM. Based on the results achieved in this study the following conclusions can be derived: (1) Generally, the SVM-FFA method can estimate the stiffness of PET modified asphalt mixture performance with high estimation accuracy. (2) The soft computing methodologies showed very good learning and prediction capabilities. (3) The prediction model overcomes the main weakness of artificial neural networks without defining network structure or trapping in the local optimum.

Acknowledgement The authors express their sincere thanks for the funding support they received from University of Malaya grant no: RP010A-13SUS.

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Highlights • • • •

Stiffness property of Polyethylene Terephthalate (PET) modified asphalt mixture. To achieve this aim experiments were performed. Support vector machine with firefly algorithm (SVM-FFA) application. Inputs were PET percentages, stress levels and temperatures.

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