Prediction of falling weight deflectometer parameters using hybrid model of genetic algorithm and adaptive neuro-fuzzy inference system

Abstract

A falling weight deflectometer is a testing device used in civil engineering to measure and evaluate the physical properties of pavements, such as the modulus of the subgrade reaction (Y1) and the elastic modulus of the slab (Y2), which are crucial for assessing the structural strength of pavements. In this study, we developed a novel hybrid artificial intelligence model, i.e., a genetic algorithm (GA)-optimized adaptive neuro-fuzzy inference system (ANFIS-GA), to predict Y1 and Y2 based on easily determined 13 parameters of rigid pavements. The performance of the novel ANFIS-GA model was compared to that of other benchmark models, namely logistic regression (LR) and radial basis function regression (RBFR) algorithms. These models were validated using standard statistical measures, namely, the coefficient of correlation (R), mean absolute error (MAE), and root mean square error (RMSE). The results indicated that the ANFIS-GA model was the best at predicting Y1 (R = 0.945) and Y2 (R = 0.887) compared to the LR and RBFR models. Therefore, the ANFIS-GA model can be used to accurately predict Y1 and Y2 based on easily measured parameters for the appropriate and rapid assessment of the quality and strength of pavements.

Graphical abstract

Keywords

falling weight deflectometer / modulus of subgrade reaction / elastic modulus / metaheuristic algorithms

1 Introduction

Nondestructive testing (NDT) is required during and after the construction of structures to ensure their safety via an assessment of the quality of materials used. The falling weight deflectometer (FWD) is an important NDT device used to evaluate the quality of flexible asphalt pavements [1,2] and rigid pavements used in road construction [3,4]. An FWD test can be performed to assess the structural conditions of flexible roads for predicting the moduli of layers via the back-calculation method [5]. For a rigid pavement, the FWD test is performed to calculate the elastic modulus of the slab, the modulus of the subgrade reaction (k-value), and the load transfer at joints and cracks [3,4,6,7], thus facilitating the simulation of the displacement response at different traffic loads [8]. In addition, this test can be performed to obtain the structural information of pavement layers, including the subsurface conditions and subgrades [9,10]. The FWD test is a simple test that involves the release of a heavy mass from a specified height on mounted rubber buffers attached to the footplate (circular). Transducers of deflection sensors are used to record the responses as the deflection of the flexible pavement at every applied load, which is comparable to traffic loads [11]. FWD tests are conducted globally to predict the in situ stiffness of different pavement layers for the maintenance of roads and highways [8,12]. Although the FWD test is important for determining various essential parameters used for evaluating the quality of asphalt pavements, it is a complex, time-consuming, and costly test. Therefore, an alternative approach for predicting the parameters determined from this test must be identified. The application of the FWD test has facilitated the systematic management of airport pavement maintenance and rehabilitation activities [13,14].

In recent years, advanced artificial intelligence (AI) models have been used to solve prediction problems, such as the prediction of the properties of construction materials and structures based on training using previous data [15–18]. Meier [19] used artificial neural networks (ANNs) to analyze the elastic modulus of three-layer flexible pavement systems. Bayrak and Ceylan [20] used ANN models to predict the elastic modulus of the slab and the modulus of the subgrade reaction (k-value) values of rigid pavements using FWD-deflection data. Liu et al. [21] used an unsupervised machine-learning method to evaluate asphalt pavement structures using an FWD. Bellary and Suresha [22] utilized an ANN to predict the joint stiffness of white-topped pavements using FWD data. Karballaeezadeh et al. [23] used and compared various AI models, namely the single radial basis function (RBF) and multilayer perceptron (MLP) neural networks and their hybrid models, such as RBF-genetic algorithms (GAs), RBF-imperialist competitive algorithms, MLP-scaled conjugate gradient algorithms, and MLP-Levenberg-Marquardt algorithms, to predict the pavement condition index using surface deflection data obtained from an FWD. Han et al. [24] developed a hybrid AI model that combines a recurrent neural network, residual neural network, and wide and deep network to predict the road layer dynamic modulus using FWD data.

In general, the studies above demonstrated the potential of AI models, particularly hybrid AL models, in predicting the parameters of the FWD test. However, studies regarding the prediction of two important FWD parameters, namely the elastic modulus and modulus of the subgrade reaction of the slab, are limited. Therefore, this study is conducted to predict these FWD parameters using a novel hybrid model, namely the adaptive neuro-fuzzy inference system (ANFIS)-GA model. Other benchmark machine learning models, namely, logistic regression (LR) and radial basis function regression (RBFR) algorithms, are selected for comparison and for selecting the best AI hybrid model. Standard statistical measures, such as the coefficient of correlation (R), mean absolute error (MAE), and root mean square error (RMSE), are used to evaluate the performance of the models. The FWD test data and other pavement parameters are obtained from the National Highway 18 highway, Quang Ninh Province, Vietnam. The MATLAB software was used for data and model analysis.

2 Materials and methods

Fig.1 shows the methodological flowchart used in this study; three different machine learning models (ANFIS-GA, LR, and RBFR) were applied using 13 input parameters (X1, X2, X3, X4, ..., X13) to obtain the modulus of subgrade reaction (Y1) and the elastic modulus of the slab (Y2) (output parameters). The input parameter included seven surface deflections (D₀, D₈, D₁₂, D₁₈, D₂₄, D₃₆, and D₆₀) and six other parameters, namely the surface temperature (T), surface load (P), slab thickness (hpcc), 8% cement-treated aggregate base course thickness (hb), rigid pavement thickness (h), and compressive strength of concrete slab (Rc). During the development of the models, training and testing data were randomly allocated at proportions of 70% and 30%, respectively. Subsequently, the models were validated and compared based on standard performance indices, i.e., the R, MAE, and RMSE, to evaluate the efficacy of the achieved results.

2.1 Data used

2.1.1 Outputs

In the modeling, two variables were used as outputs: the Y1 and Y2 of a Portland cement concrete (PCC) slab [6]. Whereas Y2 was used to evaluate the structural condition of the PCC slab, Y1 was used to evaluate the supporting layers. The values of these outputs were obtained from FWD tests performed along National Highway 18 in the road section extending from Ha Tu ward (Ha Long City) to Mong Duong ward (Cam Pha City, Quang Ninh province of Vietnam) (see Fig.2).

The highway road section included two lanes for cars. The width of the road surface was 11 to 12 m, and the width of the roadbed was 12 to 14 m. A PCC slab was designed according to the American Association of State Highway and Transportation Officials (AASHTO) standards [25]. The thickness of the PCC slab was 24 to 30 cm (Fig.3). The rigid pavement structure comprised a surface layer with a PCC slab and a base layer with an 8% cement-treated aggregate base course (Fig.3). The average slab size along route L_1 was 5.0 m, and the average slab size across route L_2 was 5.75 m (Fig.4). To minimize edge effects, the load (P) was applied at the center of the PCC slabs of the outer traffic lane (Fig.4) [3].

A total of 510 PCC slabs were evaluated via FWD tests to construct a database for modeling (Fig.4). Concrete cores were drilled to determine the compressive strength and Poisson’s ratio of the PCC slabs in the laboratory based on the ASTM C42/C42M standard test procedure [26–29]. Tab.1 lists the technical specifications of the FWD device used to determine Y1 and Y2.

2.1.2 Inputs

To predict Y1 and Y2, factors affecting quality and strength must be determined and used as inputs to the models. Hence, 13 factors, including the surface deflections (D_0,D_8,D_{12},D_{18},D_{24},D_{36},{\mathrm{a}\mathrm{n}\mathrm{d}D}_{60}) (Fig.5), surface load \left(P\right), surface temperature \left(T\right), rigid pavement thickness \left(h\right), 8% cement-treated aggregate base course thickness \left(hb\right), slab thickness \left(hpcc\right), and compressive strength of concrete slab \left(Rc\right) were selected.

More specifically, the surface deflections recorded by the Strategic Highway Research Program (SHRP) sensor at various distances (D_0 determined at the center of the test load, D_8 determined at a distance of 203 mm from the center of the test load, D_{12} determined at a distance of 305 mm, D_{18} determined at a distance of 457 mm, D_{24} determined at a distance of 610 mm, D_{36} determined at a distance of 914 mm, D_{60} at a distance of 1524 mm from the load) are important variables affecting Y1 and Y2 [30]. Additionally, the slab thickness significantly affects Y1 and Y2, as slight changes in the thickness can result in considerable differences in the back-calculated elastic modulus of the layers [20]. The surface temperature is vital to the measurement of PCC pavement deflection, as the joints and cracks may be locked at warmer temperatures owing to the expansion of slabs, and thinner slabs are affected more significantly by the temperature gradient [31]. In addition, the compressive strength of the concrete slab should be determined from the cores based on ASTM C469 [28] and ASTM C42 [26,27] such that the variability in the elastic modulus of the existing concrete can be ascertained [6]. Other factors, such as the surface load (P) and 8% cement-treated aggregate base course thickness (hb), are equally vital to the prediction of Y1 and Y2 [3].

In the modeling, X1 to X13 were used as input variables, whereas Y1 and Y2 were used as model outputs. Tab.2 presents the initial statistical analysis of the data used in this study. Fig.6 shows the correlation analysis of the variables used in the model.

2.2 Methods used

2.2.1 Adaptive neuro-fuzzy inference system

The ANFIS is an AI-based model that merges the reasoning capabilities of an ANN with those of fuzzy logic. It exhibits high prediction capability and is regarded as an excellent system for solving nonlinear complex mathematical problems accurately [32–34]. The ANFIS tool in MATLAB was used to train all the parameters (both inputs and outputs) to determine the optimal input and output mapping, as in the ANN. A typical ANFIS model comprises five main layers [35].

The first layer, which is also known as the fuzzification layer wherein the output is yielded, explains the fuzzy membership functions (MFs) of the inputs alongside the initial fuzzy rule base (Eqs. (1) and (2)).

(1)U_i^1={\mu }_{\mathrm{A}i}\left(V_1\right),i=1,2,

(2)U_i^1={\mu }_{\mathrm{B}i-2}\left(V_2\right),i=3,4,

where μ refers to the weight achieved in the linkage of the MF, and μ_Ai(V_1) and μ _Bi−2(V_2) are used to differentiate the method by which a fuzzy MF is applied. For a bell-shaped MF, μ_Ai(V_1) is expressed as

(3){\mu }_{\mathrm{A}i}\left(V_1\right)=\frac{1}{1+{\left({\displaystyle \frac{V_1-C_i}{A_i}}\right)}^{B_i}},

where A_i, B_i, and C_i are parameters that affect the MF.

In the second layer, the output refers to the firing strength of the preset guidelines for a specified input form. The output is expressed as follows:

(4)U_i^2=w_i={\mu }_{\mathrm{A}i}\left(V_1\right)\cdot {\mu }_{\mathrm{B}i}\left(V_2\right),i=1,2.

In the third layer, the nodes are once again fixed, as in the second layer, while the firing strengths of the previous layer are normalized. Thus, the output is expressed as

(5)U_i^3={\overline{w}}_i=\frac{w_i}{w_1+w_2},i=1,2.

In the fourth layer, the adaptive nodes alongside the output can be regarded as a product of the normalized firing strength and the first-order polynomial when the first-order Sugeno model is considered. Thus, the output can be expressed as

(6)U_i^4={\overline{w}}_i{f}_i={\overline{w}}_i(p_i\left({XV}_1\right)+q_i\left(V_2\right)+r_i).

In the fifth (final) layer, a single fixed node (Ʃ) comprising the summation of the weighted consequence of the previously attained rules is accomplished such that the output can be expressed as follows:

(7)U^5={\text{∑}}_{i=1}^2{\overline{w}}_i{f}_i=\frac{{\sum }_{i=1}^2{w}_i{f}_i}{w_1+w_2}.

Notably, the first and fourth layers are adaptive in the architecture of the ANFIS. The adaptable parameters (A_i, B_i, and C_i) are defined as “premise parameters”, which are associated with the input MFs in the first layer. The other three adaptable factors (p_i, q_i, and r_i) are known as the “consequent parameters”, which are associated with the first-order polynomial and presented in the fourth layer [36,37]. The premise and consequent parameters considered are the most important parameters that must be optimized to identify the best ANFIS model. In this study, they were optimized using a GA.

2.2.2 Genetic algorithm

The GA is a heuristic-based search technique that encompasses the concepts of natural genetics and is inspired by the survival of the fittest Darwin theory [38,39]. It offers various advantages; for example, it can solve optimization problems in two continuous and discrete scenarios using stochastic operators [40]. The GA has been used as a robust solver to evaluate combinational optimization problems in the previous decade [41]. In the GA, the initial population, which contains various chromosomes, is generated first. Subsequently, the fitness of the chromosomes within the population is evaluated, after which a new population is generated at each stage. In the new population, mutation (which introduces modifications in the children via the application of various operators) and crossover (where children are formed after they are applied to the parents), in addition to accepting, replacing, and testing, are performed to achieve the viable output value [42]. If the criterion is not satisfied, then the loop is reiterated, whereas all the steps are re-performed in the form of a new generation such that the criterion to achieve the best model is satisfied [43,44].

2.2.3 Logistic regression

LR is one of the most well-established classic AI-based models used for solving classification and regression problems. LR coefficients are generally optimized using iterative techniques [45]. LR is applied extensively in data mining [46], statistics [47], medical science [48], social sciences [49], pattern recognition [50], and machine learning [51]. Furthermore, it is used to explain the association between the sample data and their attributed binary response based on a conditional probability [52].

The LR model allows some of the parameters to be regulated conveniently [45], for instance, a) penalty—in cases involving high variance, it is used to add bias; b) C—greater values of C indicate the generalization of the LR model, whereas lower values indicate its constraining degree; c) max iteration—it is performed to achieve model convergence; d) dual—a type of objective function; e) fit intercept—it regulates the effect of an intercepted value; and f) solver—a type of algorithm that facilitates solving.

2.2.4 Radial basis function regression

RBFR constitutes the probabilistic neural network [53] and comprises two different layers with a typical architecture, as illustrated in Fig.1. The network comprises input layers, one hidden layer, and an output layer. Based on the supervised learning technique and using a defined learning algorithm, the residual error between the original and estimated values was reduced based on the available optimization criteria [54].

The basis function (also known as the hidden layer) nonlinearly transforms the input information into a new space using the RBFs of the aforementioned layer. RBFR is more advantageous than the MLP network as it guarantees a robust and accurate solution via the linear optimization [55]. The RBF is typically employed after a feedforward ANN. These networks can be successfully implemented by selecting suitable central locations for the Gaussian functions.

2.2.5 Validation methods

In this study, the performance of the models was validated using standard validation statistical indicators, namely the MAE, R, and RMSE, which are expressed as follows [37,56]:

(8)R=\frac{{\sum }_{i=1}^N\left(E_i-{\overline{E}}_i\right)\left(M_i-{\overline{M}}_i\right)}{\sqrt{{\sum }_{i=1}^N{\left(E_i-{\overline{E}}_i\right)}^2{\sum }_{i=1}^N{\left(M_i-{\overline{M}}_i\right)}^2}},

(9)MAE=\frac{{\sum }_{i=1}^N\left|E_i-M_i\right|}{N},

(10)RMSE=\sqrt{\frac{{\sum }_{i=1}^N{(E_i-M_i)}^2}{N}},

where E_i and M_i are the ith actual and predicted outputs, respectively; {\overline{E}}_i and {\overline{M}}_i are the means of the actual and predicted outputs, respectively; and N is the number of samples.

For R values exceeding 0.8, the strength of the correlation between the experimental and measured values is considered high [57,58]. Meanwhile, the RMSE is advantageous owing as it can address greater residual errors more precisely (notably, RMSE ≈ 0 signifies the least error) [43,59,60]. However, the RMSE is not suitable for achieving a higher degree of accuracy in certain cases; in this regard, the MAE is determined because as it is suitable for both smooth and continuous data [61,62]. Moreover, greater R values and lower RMSE and MAE values signify the robust performance and accurate calibration of the model [37,63].

3 Results and discussion

3.1 Performance of genetic algorithm-optimized adaptive neuro-fuzzy inference system

The result of R with respect to an increase in the number of iterations for the prediction of Y1 and Y2 is depicted in Fig.7. The MAE and RMSE were calculated in a similar manner for the two output parameters above. In this study, the optimal R indicator was used as a stopping criterion for establishing the ANFIS-GA model. A total of 1500 iterations were performed for each evaluation. The training data were optimized after 700 iterations for each correlation case and error for Y1. The convergence toward the highest correlation and the lowest error was initially rapid and persisted until 500 iterations, beyond which the convergence rate decreased, as shown in Fig.7(a), Fig.7(c), and Fig.7(d). Meanwhile, the optimized training data for Y2 resulted in a faster convergence compared to that achieved by Y1. The convergence toward the highest correlation plateaued beyond 800 iterations, whereas for the MAE and RMSE, the convergence curve plateaued beyond 400 iterations, as shown in Fig.7(d)–Fig.7(f). Moreover, the optimization of Y1 and Y2 resulted in R values exceeding 0.9 and 0.88, respectively, for both the training and testing data sets.

The high correlation obtained using the training data for both output parameters is indicated by lower values of errors (MAE and RMSE) for the training data as compared to those for the testing data. The minimum MAE and RMSE values for the training data of Y1 were 5.609 and 8.375, respectively, whereas those of Y2 were 2.289 and 3.078, respectively.

Fig.8(a) and Fig.8(b) show a comparison between the actual and predicted results for output Y1 based on the training and testing data. The results show that the predicted results are similar to the actual values. The relatively accurate predictions based on the training and testing data sets is similarly reflected in the R value, i.e., 0.975 and 0.945, respectively (see Tab.3). The R value exceeding 0.8 signifies a strong correlation between the predicted and actual values [37,64–68]. Meanwhile, the MAE values for the predictions based on the training and testing data sets were 5.609 and 7.549 kPa/mm, respectively. Fig.9 shows a comparison between the actual and predicted results. The experimental and predicted values are plotted on the x- and y-axis, respectively [37,65]. The slope of the regression line is similar to the ideal fit, which indicates a strong correlation. The error values are plotted against the Y1 data from the developed model (Fig.10), and the graph shows that the variation is presented primarily in the vicinity of the zero radial line for both the training and testing data sets. The RMSE for the training and testing data sets were 8.375 and 12.458 kPa/mm, respectively.

Fig.8(c) and 8(d) show a comparison between the actual and predicted results for output Y2 based on the training and testing data sets. As shown, the predicted results are similar to the actual values. This is similarly confirmed by the R values, i.e., 0.945 and 0.887 for the training and testing data sets, respectively (Tab.3). Meanwhile, the MAE values were 2.289 and 3.415 GPa for the training and testing data sets, respectively. The R values for Y2 were lower than those for Y1. Fig.9 shows a comparison between the actual and predicted results. The regression line slope of the observed vs. predicted values is similar to the ideal fit, which signifies a strong correlation. The error values are plotted against the Y2 data (Fig.10) of the developed model, and the graph shows that the variation is presented primarily in the vicinity of the zero radial line for both the training and testing data sets. The RMSE values for training and testing data sets were 3.078 and 4.848 GPa, respectively.

In general, the ANFIS-GA performed well in predicting Y1 and Y2; however, the ANFIS-GA predicted Y1 better than Y2.

3.2 Comparison of genetic algorithm-optimized adaptive neuro-fuzzy inference system against benchmark ML models

To confirm the predictive capability of the novel ANFIS-GA model, two other benchmark AI-based models, i.e., LR and RBFR, were used for comparison using the same data sets and validation criteria (R, RMSE, and MAE); the comparison results are shown in Tab.3. In terms of Y1, the ANFIS-GA yielded the highest R values of 0.975 and 0.945 for the training and testing data sets, respectively, followed by the RBFR model (with R values of 0.898 and 0.887 for the training and testing data sets, respectively) and the LR model. A similar order in R values was observed for the prediction of Y2, except that the R values of the LR model were greater than those of the RBFR model. However, the correlation R values of the LR and RBFR models were lower than those of ANFIS-GA.

In addition, the ANFIS-GA yielded the lowest MAE values for Y1 and Y2 based on both the training and testing data sets, as listed in Tab.3. For the Y1 models, the training data of the ANFIS-GA yielded the lowest MAE value of 5.609 kPa/mm, followed by those of the RBFR (11.01 kPa/mm) and LR (11.435 kPa/mm). Similarly, the testing data set of the ANFIS-GA yielded the lowest MAE value (7.549), followed by those of the RBFR (11.572) and LR. In the case of Y2, the MAE values of the training and testing data sets for the ANFIS-GA were 2.289 and 3.415 GPa, respectively; 7.307 and 8.081 GPa, respectively, for the LR model; and 7.069 and 8.096 GPa, respectively, for the RBFR model.

In terms of the RMSE of Y1, the ANFIS-GA showed the lowest values of 8.375 and 12.458 kPa/mm for the training and testing data sets, respectively, followed by LR with 11.237 and 18.187 kPa/mm, respectively, and RBFR with 16.426 and 17.848 kPa/mm, respectively. For Y2, the variation in the RMSE showed a similar pattern, except that RBFR yielded lower values than LR. The error histogram shown in Fig.11 clearly indicates the variation among the three developed models.

In general, the comparison results show that the novel model ANFIS-GA outperformed other benchmark AI-based models, namely LR and RBFR, in predicting both Y1 and Y2. This is reasonable as the developed ANFIS-GA model utilizes the advantage of conventional back-calculating iteration-based programs, such as the consideration of complex material properties and the elimination of seed moduli, which causes its performance to be comparable to those of other methods reported in the literature [69,70]. Compared to the ANN model used for predicting asphalt deflections as a supplement to experimental measurements using an FWD [71], the ANFIS-GA model performed slightly better. This is attributable to the fact that the hybrid model (ANFIS-GA) is superior to classical ANN models in terms of two aspects. 1) The ANFIS model combines an ANN and fuzzy logic principles, which allows it to offer the benefits of both in a single framework [72]. In particular, a fuzzy inference system uses a set of fuzzy IF-THEN rules, which offers significant learning capability for approximating nonlinear functions. 2) The ANFIS-GA that uses a GA to optimize the hyperparameters used in training the ANFIS; thus, it can improve the performance of the ANFIS [73].

4 Conclusions

The FWD test is an NDT field test that has been performed widely to determine the modulus of the subgrade reaction (Y1) and the elastic modulus of the slab (Y2) for assessing the quality and strength of pavements. In this study, we developed a novel hybrid AI-based model, named ANFIS-GA, to predict Y1 and Y2 based on 13 easily determined input parameters. The performance of the ANFIS-GA model was compared to that of two benchmark AI models, namely LR and RBFR, using standard statistical measures, i.e., R, MAE, and RMSE. The results indicated that the developed ANFIS-GA model performed well in predicting Y1 and Y2; additionally, it performed better than the other two models (LR and RBFR) on both the training and testing data sets. Hence, the developed ANFIS-GA model can facilitate the rapid assessment and evaluation of pavement quality and strength.

Notably, the application range and accuracy of the ANFIS-GA model developed in the current study are limited to the range of parameters considered in the database obtained and training process. Model development is a continuous process. Therefore, to enhance the capability of the developed models in the future, the training process should be performed using a wider range of input parameters. In addition, feature selection methods can be used to improve the performance of the model by removing irrelevant and unimportant input parameters.

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