Novel hybrid models of ANFIS and metaheuristic optimizations (SCE and ABC) for prediction of compressive strength of concrete using rebound hammer field test

Abstract

In this study, we developed novel hybrid models namely Adaptive Neuro Fuzzy Inference System (ANFIS) optimized by Shuffled Complex Evolution (SCE) on the one hand and ANFIS with Artificial Bee Colony (ABC) on the other hand. These were used to predict compressive strength (Cs) of concrete relating to thirteen concrete-strength affecting parameters which are easy to determine in the laboratory. Field and laboratory tests data of 108 structural elements of 18 concrete bridges of the Ha Long-Van Don Expressway, Vietnam were considered. The dataset was randomly divided into a 70:30 ratio, for training (70%) and testing (30%) of the hybrid models. Performance of the developed fuzzy metaheuristic models was evaluated using standard statistical metrics: Correlation Coefficient (R), Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). The results showed that both of the novel models depict close agreement between experimental and predicted results. However, the ANFIS-ABC model reflected better convergence of the results and better performance compared to that of ANFIS-SCE in the prediction of the concrete Cs. Thus, the ANFIS-ABC model can be used for the quick and accurate estimation of compressive strength of concrete based on easily determined parameters for the design of civil engineering structures including bridges.

Graphical abstract

Keywords

shuffled complex evolution / artificial bee colony / ANFIS / concrete / compressive strength / Vietnam

1 Introduction

Soft Computing (SC) techniques and optimization techniques are a generally recent trend in amelioration of the deficiencies in construction and design phases [1], particularly in estimation of the mechanical properties of concrete [2]. In addition, these techniques are used in the optimization of structural response of concrete [3–5]. The rapid estimation of the Compressive strength (Cs) of concrete is vital in almost all civil engineering works [6]. It is worth noting that SC techniques including Artificial Intelligence (AI) and Machine Learning (ML) methods are significantly robust in solving a variety of complex problems; however, such techniques still face issues such as the extraction of knowledge, interpretability of the formulated models, and uncertainty pertaining to the developed model [7–9]. Moreover, the evolutionary optimization algorithms, such as Artificial Bee Colony (ABC), exhibit superior performance in cases of intrinsically non-convex, multi-modal, and irregular real-world complexities, unlike the point-by-point searching criteria followed in conventional optimization algorithms [10–14]. Out of these, ABC has been extensively used to address constrained and unconstrained problems of optimization in various fields [15]. In geotechnical engineering a number of AI techniques have been efficiently employed for the determination of Cs and other properties of concrete [6]. However, further improvement in AI techniques is still continuing. In order to estimate the bond strength of concrete neural networks, algorithms including Artificial Neural Networks (ANN) and hybrid models such as ANN-ABC are being used besides for better results. The new hybrid model has performed robustly and with low errors (less than 10%) in comparison to ANN model [16]. In other studies Random Forest (RF) and Adaptive Neuro Fuzzy Inference System (ANFIS) methods performed well in predicting strength characteristics of a treated expansive soil [17]. Zhao et al. [6] employed the Shuffled Complex Evolution (SCE) alongside Teaching-Learning-Based Optimization (TLBO) to estimate the Cs of concrete and demonstrated that the formulated model using SCE optimizes relatively quickly. Naeini et al. [18] modified SCE as a global optimization algorithm in other fields such as hydrology in order to calibrate various developed models. Furthermore, to estimate the Cooling Load (CL), the SCE algorithm is found to surmount the computational demerits of the Multi-Layer Perceptron (MLP) algorithm in addition to outshining moth–flame optimization. Additionally, the deployed SCE-MLP exhibited relatively simple structure, thereby reducing computation time significantly [19]. Thus SCE method offers one of the best optimization techniques for many civil engineering fields.

In recent years, several integrations of ANFIS model with other techniques—Genetic Algorithm (GA) [20], Ant Colony Optimization (ACO) [21], Particle Swarm Optimization (PSO) [22], Differential Evolution (DE) [23], Shuffled Frog-leaping algorithm [24], Imperialist Competitive Algorithm (ICA) [22], Differential Evolution (DE) [25], Simulated Annealing (SA) [26], Tabu search algorithm [27]—have been done to develop fuzzy metaheuristic models for accurate estimation of geotechnical parameters. The prediction accuracy of a model using an ANFIS on its own and with TLBO was evaluated by Ly et al. [28] for the prediction of Cs of Manufactured Sand Concrete (MSC), using Principal Component Analysis (PCA). Nazari and Sanjayan [29] studied Support Vector Machine (SVM) optimized individually with PSO, GA, ABC, ACO and ICO to predict the Cs of geopolymers. There is a scope for further improvement in the performance of the ANFIS model with good optimization algorithms such as SCE and ABC which have not been used so far in the prediction of Cs of concrete. Therefore, in this study we have developed fuzzy metaheuristic models namely ANFIS-SCE and ANFIS-ABC for the estimation of Cs of concrete. The data from 108 structural elements of 18 concrete bridges of the Ha Long - Van Don National Expressway, Vietnam was used in the models’ study. Performance of the models was evaluated using standard statistical metrics: Correlation Coefficient (R), Root Mean Square Error (RMSE), and Mean Absolute Error (MAE). For the data analysis and modelling, Matlab software was used.

2 Materials and methods

2.1 Study area

The research area encompasses 18 bridges of the Ha Long-Van Don Expressway, Viet Nam (Fig.1). The Expressway is 4-lane high speed, 24.5 m wide, connected with 18 simple span girder bridges. The bridge span lengths vary from 18 to 40 m. The types of the slab girder are I girder and Super-T girder. The bridge abutments are of U-shaped reinforced concrete abutments and the piers are of solid reinforced concrete.

We applied a non-destructive rebound hammer method to determine the Cs of concrete on the structural elements of bridges for model study following BS EN 12504-2: 2012 [30] and TCVN 9334: 2012 procedures [31]. Tested structural elements included piers, abutments and bridge girders. Experimental work was conducted from November 14, 2016 to March 17, 2018. The total number of tested structural elements was 108 which included 74 girders, 34 piers and abutments. The concrete strength tests were conducted after placing concrete in accordance with the project schedule, with the earliest date of concrete age being 17 d, and the oldest concrete age being 556 d. The structures were tested in dry weather conditions. The grades of concrete used were C35 in the bridge piers and abutments and C45 and C50 in the bridge beams. In the bridge structures composition of the cement concrete mixture was designed as per ACI 318M-11 [32].

2.2 Data used

The dataset consists of 14 attributes. This includes 13 input variables, namely: age of concrete, the quantity of cement, coarse aggregates, sand, water, admixture, aggregate to cement ratio, water-to-cement ratio, sand-to-aggregate ratio, admixture-to-cement ratio, strength (saturated) of stone (aggregate), modulus of sand, and Cs of Portland cement at an age of 28 d and 01 output namely the Cs of the concrete. Detailed descriptions of these attributes are given in following sections.

2.2.1 Compressive strength

The strength of concrete depends greatly on water-cement ratio. The water-cement ratio also greatly affects the durability, volumetric stability and many other properties related to concrete porosity. Therefore, the Cs of concrete is used for regulation, i.e., for controlling and evaluating the quality of concrete.

The Cs of concrete gives indirect information of shear strength and tensile strength, modulus of elasticity and flexural strength. Concrete with higher compressive strength can better withstand wear and tear due to impact of water with high velocity. Though there may be no exact quantitative relationships, compressive strength nevertheless gives sufficient general information of other related parameters and behavior of concrete to withstand stress and erosion [33].

Cs of concrete is generally determined in the laboratory on concrete samples but results may vary due to construction quality management. Therefore, samples are generally collected at site by drilling cores or tested directly by the application of rebound hammer to determine Cs. The advantages of the Rebound Hammer method over the Core Drilling method are: light and compact equipment, faster testing time, simpler testing method, lower cost, no sophisticated auxiliary equipment required and no damage caused to the structure. Thus, the Cs can be determined using Rebound Hammer without destruction at many locations on a structure, at different times of its life cycle (Fig.2).

2.2.2 Compressive strength affecting factors

The concrete Cs depends on many factors such as quality and content of raw materials (aggregates, cement and admixtures), concrete mix design method, concrete mixing time; concrete production and work environment. The properties of the materials that affect the strength of concrete include quality of fine aggregates and coarse aggregates, and adhesion of cement slurry to aggregates (properties of cement slurry-aggregates transition zone).

More specifically, Abrams’ law (also called Abrams’ water-cement ratio law) [34,35] is a concept in civil engineering, which states that with hardened concrete the strength is inversely related to the mass ratio of water to cement [35,36]. Concrete strength is essentially dependent on the pore volume created by excess water. Empirical equation of this law is presented below:

(1)Cs=\frac{A}{B^{W/C}},

where the Cs is the strength of concrete; A and B are constants; W/C is the water-cement ratio.

Several studies have been conducted to establish relationship of aggregates of different sizes with the Cs of concrete [37]. They have also analyzed maximum coarse aggregate size for optimum compressive strength. It is generally considered that coarse aggregate below 12.5 mm produces high strength concrete (ACI 363-95). Further increase of size of aggregate gives lower strength [38,39].

Another important factor is the concrete age, in d. The French standard [40] gives the formula to calculate the compressive strength of the concrete at an age of j d (f_cj) based on the compressive strength of the concrete at an age of 28 d (f_c28) as follows:

(2)f_{\mathrm{c}j}=\frac{j}{4.76+0.83j}\ast f_{\mathrm{c}28},\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{f}_{\mathrm{c}28}⩽40 \mathrm{M}\mathrm{P}\mathrm{a},

(3)f_{\mathrm{c}j}=\frac{j}{1.40+0.95j}\ast f_{\mathrm{c}28},\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{f}_{\mathrm{c}28}>40 \mathrm{M}\mathrm{P}\mathrm{a}.

According to Fig.3, the Cs changes over time. In the first period up to 28 d, the concrete strength develops rapidly, thereafter the strength develops slowly.

The strength of concrete depends not only on ratios and types of cement mix but also on the physical properties of aggregates which in turn depend on the mineralogy and weathering conditions [41–43]. Texture, size and shape of aggregate affect the compressive and flexure strength of concrete [34,44]. Type of concrete mix also affects modulus of elasticity of concrete [41,45,46].

Attempts have been made to establish relationships of aggregate types with the strength of concrete [47]. The following equation shows the relationship between aggregate strength and concrete:

(4)fc=\frac{p\cdot {fc}_{\mathrm{m}}}{q\cdot {fc}_{\mathrm{m}}+1},

where fc is the Cs; fc_m is strength of the matrix; p and q are empirical constants that depend on the aggregate type.

The packing effect of aggregate also affects the strength of concrete [32]. Nowadays, there are many available additives which affect the strength and durability of concrete [48]. Reduced water volume leads to water-cement ratio reduced and concrete strength increased [34], so that content of admixture affects concrete strength. Fig.4 shows the distribution analysis of the data used in the model study. Initial statistical analysis of the parameters used in the modeling is presented in Tab.1.

2.2.3 Normalization of the data used for modeling

The numeric attribute data was normalized to a common scale to reduce data redundancy and improve data integrity without distorting differences in the ranges of values, using following equation:

(5)X^{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}=(x^{\mathrm{r}\mathrm{a}\mathrm{w}}-\beta ) /(\alpha -\beta ),

where α and β are the highest (maximum) and lowest (minimum) values of the parameter x.

The data was split randomly in a 70:30 ratio for generation of training dataset and testing dataset, respectively

2.3 Methods used

The different steps of the study conducted here are stated in Fig.5. The data points are randomly split first in a ratio of 70:30 into training and validation datasets, respectively. After that, the training dataset is subjected to ABC and SCE optimization techniques using ANFIS as a base algorithm. In addition, the test dataset is subjected to testing of hybrid ANFIS models. The process is repeated until the termination criteria is satisfied.

2.3.1 Adaptive neuro fuzzy inference system

The ANFIS algorithm is one AI modeling technique and a Takagi–Sugeno–Kang or Sugeno type fuzzy inference system (FIS) which comprises the learning aptitude of neural networks, i.e., ANN, alongside the reasoning ability of fuzzy logic [49]. This computational intelligence approach involves hybrid evolutionary model projecting schemes exhibiting high prediction capability and acts as desired choice in evaluating complex problems [50,51]. However, the two main shortcomings attributed to ANFIS are: 1) complex models are generated owing to the “if–then” rules and the membership functions (m_f) that comprise the final solution; 2) inadaptability with the stochastic scenario [17]. According to Jalal et al. [7], ANFIS learns from the complicated training data in the same way as neural networks and the output is later mapped via FIS. The ANFIS tool is present in many application softwares including in the MATLAB interface which trains the entire set of data points so as to determine the input and output mapping. This resembles to the process of the ANNs.

In the ANFIS, the FIS transforms the input through fuzzification (according to linguistic rules activation) while considering pre-set rules and certain inferences, thus yielding the final output by performing defuzzification.

Moreover, the ANFIS architecture comprising four input parameters (in1, in2, in3 and in4) is depicted in Fig.6 wherein ‘fixed’ and ‘adaptive’ are illustrated in the form of circles and squares, respectively. In the ANFIS architecture the 2nd, 3rd and 5th layers are not adaptive. There are a total of six adaptable parameters, of which half are called premise parameters [17]. The remaining three are called consequent parameters [51] which are linked with the 1st order polynomial of the 4th layer [52].

2.3.2 Artificial bee colony

The ABC algorithm is a powerful tool which is inspired by honey bee foraging and considers the ‘group life of bees’ [53,54]. This algorithm was developed by Karaboga and Basturk [55] for the optimization of perplex engineering problems, and is an intelligent and comparatively new optimization technique. A significant merit of ABC is its robust convergence speed as well as small number of control parameters [56]. Additionally, the ABC algorithm is very suitable for addressing inverse analysis problems since it exhibits superior performance over the traditional GA and the PSO methods [54]. The optimization problem is addressed by finding the solution in the form of locating the food source, while the honey present in the source is associated to the quality of the corresponding solution. The theme of ABC comprises a multi‐factor setup (a group of hypothetical bees) which tends to find appropriate solutions for various hybrid optimization problems in accordance with the concept followed by the bees to collect honey. The hypothetical bees ameliorate the solutions collectively in a stepwise manner. The independent hypothetical bees search for the most viable solution via interacting and exchanging information, by exploring the available search space [16]. According to Najimi et al. [2], an individual cycle of this novel technique has three stages: 1) artificial bees are dispatched to various food sources; 2) nectar is assessed after information sharing in different sources of food; and 3) scout bees are identified and sent to new sources of the food. Furthermore, ABC consists of three imperative parts which are: employed, onlookers, and scouts [53]. It also employs four types of selection processes namely: general procedure, probabilistic local procedure, local greedy selection process, and random selection procedure [2]. The seven major phases of ABC can be summarized as: 1) first population is initialized; 2) re‐reading and evaluation; 3) the employed bees are sent to the sources of food; 4) onlooker bees are sent to food source based on available amount of honey; 5) scout bees are sent to region for detection of new food sources; 6) the most appropriate source of food is memorized whereas the undesired food source is abandoned; and 7) the above steps are repeated for fulfilling the required scenario.

2.3.3 Shuffled complex evolution

SCE is a well-known swarm based algorithm that was introduced by Duan et al. [57] for addressing a broad range of non-linear problems, having high-parameter dimensionality [19,58]. This algorithm encompasses the effects of four prominent principles for global optimization, namely: random search technique, Nelder-Mead simplex method, GA and complex shuffling method [6,11,59]. Furthermore, it consists of three different tiers, which include population, complex, and simplex [19]. According to Gopalakrishnan and Ceylan [11], the SCE is a robust and efficient evolutionary optimization technique which is attributed with tradeoff of exploration and exploitation. Vrugt et al. [60] argued that the main objective of the conventional SCE-UA (SCE-University of Arizona) algorithm is selection of the single most appropriate parameter set inside the feasible space. The seven major phases showing the mechanism of SCE can be summarized as [6,57]: 1) initialization (calculation of sample size); 2) generation of sample (function value assigned to both upper and lower bound); 3) ranking of points (function values arranged in ascending order); 4) partitioning process (division into n complex units); 5) evolving process (accomplished based on CCE technique); 6) shuffling of complexes (replacement of complex units stored in Q, i.e., Q = x_i, F_i, i = 1, 2, …, z); and 7) checking of convergence (Then if Number of Function Evaluations exceeds the maximum Number of Function Evaluations, algorithm terminates otherwise and goes back to step 4).

The mechanism of SCE resembles other metaheuristic methods, since it is also categorized as population-based technique. In order to determine a global solution, there is constant movement by the agents within the subject problem. ‘Complex’ refers to various sub-units of the main population and with the application of an apparent “Competitive Complex Evolution (CCE)” (referred to as a statistical reproduction process) these sub-units evolve [11]. Consequently, they form bigger communities that help the agents to perform information-sharing more accurately.

2.3.4 Validation methods

The results achieved by ANFIS-SCE and ANFIS-ABC algorithms for both the training and validation stages were quantified and evaluated using standard statistical measures: R, RMSE and MAE. Calculation of these indicators was carried out using following equations (Eqs. (6)–(8)):

(6)R=\frac{{\sum }_{i=1}^n\left(h_i-{\overline{h}}_i\right)\left(t_i-{\overline{t}}_i\right)}{\sqrt{{\sum }_{i=1}^n{\left(h_i-{\overline{h}}_i\right)}^2{\sum }_{i=1}^n{\left(t_i-{\overline{t}}_i\right)}^2}},

(7)RMSE= \sqrt{\frac{{\sum }_{i=1}^n{\left(h_i-t_i\right)}^2}{n}},

(8)MAE=\frac{{\sum }_{i=1}^n\left|h_i-t_i\right|}{n},

where n is the total number of data, h_i is the simulated data, t_i is the observational data, i is the time step, {\overline{h}}_i is the average of the h_i data and {\overline{t}}_i is the average of the t_i data.

In the regression analysis, evaluation of statistical metrics results is important to assess performance of the models. Correlation coefficient “R” is a measure of association of two variables. R values range from (–)1 to (+)1 indicating perfectly negative (inverse) correlation and perfectly positive correlation, respectively. “0” value indicates no correlation. RMSE and MAE values close to “0” indicate good predictive performance of the model.

3 Results and discussion

The two novel hybrid metaheuristic models namely ANFIS-SCE and ANFIS-ABC were used in the current study to estimate the Cs based on 13 strength-affecting parameters. In the development of these models, the data from 108 bridge locations were divided into a 70:30 ratio for model training and testing, respectively. However, it is true that a K-fold cross validation method is used in training and validating the ML models; it is often used in cases where the quantity of data is limited. In our study, we used holdout methods to divide the data into training and testing datasets for training and validating the models as this method has also been widely applied in many studies [61–63].

ABC and SCE algorithms were used as optimization techniques for optimizing the weights and bias of the base ANFIS predictor. Training data were used for constructing and training the models while testing data were used for validation of the models. Hyper-parameters of ANFIS-SCE and ANFIS-ABC models used in this study are presented in Tab.2. The results obtained in terms of evaluation parameters are presented in this section.

3.1 Performance of ANFIS-SCE model

The performance of the fuzzy metaheuristic ANFIS-SCE model was analyzed using variation in the value of R, RMSE, and MAE with respect to the number of iterations. Fig.7 depicts the convergence behavior of the corresponding ensemble for specific evaluation parameters. The highest value of R was obtained within the first 200 iterations. The maximum values of R obtained for training and testing data were 0.962 and 0.964, respectively. These values of R reflect a relatively high degree of accuracy in predicting the compressive strength and represent a close agreement of observed and predicted results. Fig.8 shows similar pattern of convergence for RMSE and MAE values. The convergence rate towards minimum error values beyond 200 iterations was very small as compared to initial convergence. The minimum values of RMSE obtained for training and testing data were 0.084633 and 0.08348 MPa, respectively (Tab.3 and Tab.4). Similarly, the minimum values of MAE obtained were 0.061924 and 0.067697 MPa, respectively. A high correlation coefficient R and a small magnitude of errors manifest the robustness of the developed metaheuristic model.

3.2 Performance of ANFIS-ABC model

A similar evaluation criterion was used to analyze the performance of the ANFIS-ABC metaheuristic fuzzy ensemble (Fig.8). Unlike ANFIS-SCE, ANFIS-ABC showed a rapid convergence in attaining the highest value of R. The highest value of R of 0.962 and 0.972 for training and testing data, respectively, was achieved in the first few iterations. Moreover, the testing data set represents the closer agreement of observed and predicted values as compared to the training data set. Similarly, the convergence towards the least values of RMSE and MAE represents fast convergence as compared to ANFIS-ABC fuzzy ensemble. The evaluation of the ANFIS-ABC fuzzy ensemble also manifests high correlations and robustness of the developed model.

3.3 Comparison of performance of developed fuzzy metaheuristic models

The statistical evaluation of both of the developed models revealed a high degree of accuracy in estimating the Cs of the concrete. However, ANFIS-ABC showed slightly better performance indicators as compared to ANFIS-SCE. For instance, Fig.9 and Fig.10 represent the correlation analysis between actual and predicted values for ANFIS-SCE and ANFIS-ABC. The training data set for both of the developed models can be seen to have similar values of R, whereas the value of R for the testing data was more in the case of ANFIS-ABC. This reflects better reliability of ANFIS-ABC in predicting the Cs. For similar values of R in training data, a larger value of R in testing data for ANFIS-ABC represents the least overfitting of the developed model. The higher correlation in the case of testing data for ANFIS-ABC is also reflected in Fig.11 and Fig.12 in which the testing data in the case of ANFIS-ABC shows a close following of actual and predicted trends. Tab.3 and Tab.4 illustrate the comparison of calculated errors for the developed ensembles. The values of RMSE and MAE for ANFIS-SCE and ANFIS-ABC are almost similar for the training and testing data sets. In addition, the calculation of R for the first iteration in the case of ANFIS-SCE starts from 0.64, whereas it starts from 0.85 in the case of ANFIS-ABC. A similar pattern of calculation was observed in calculating the magnitudes of errors. This also manifests a relatively high accuracy and fast convergence of ANFIS-ABC as compared to ANFIS-SCE.

In general, the statistical evaluation metrics of the models show both SCE and ABC as promising optimizing algorithms in improving ANFIS performance as a reliable prediction technique for estimating the compressive strength of concrete from other simple easily determined parameters. This may be attributed to the hybrid approach of combining neural networks with fuzzy logic supplemented by the advantages of ABC and CSE. The value of R greater than 0.96 and smaller magnitudes of errors for training and testing data set manifest a relatively high degree of accuracy in prediction for both ANFIS-ABC and ANFIS-SCE models. However, ANFIS-ABC attains greater accuracy in comparatively less time than ANFIS-CSE, reflecting improved convergence characteristics and less computation time for ABC algorithm [64]. Thus ABC optimization technique is better as it is faster in convergence than SCE in identifying “best parameter set” [65].

4 Conclusions

In this study, two capable metaheuristic techniques of SCE and ABC optimization were employed in developing novel hybrid models with ANFIS as a base algorithm to predict the Cs of concrete from the thirteen easily determined Cs affecting parameters including types and characteristics of raw material and age of concrete. Both the applied metaheuristic ML techniques were shown to have a satisfying ability to optimize the ANFIS for predicting the Cs. Statistical analysis results (R > 0.96, RMSE < 0.08, MAE < 0.06) of models indicated that both the ANFIS-SCE and ANFIS-ABC hybrid models performed well in predicting the correct Cs of concrete. However, ANFIS-ABC model showed slightly better accuracy in terms of correlation in testing data as compared to ANFIS-SCE and also exhibited a relatively fast convergence rate. Therefore, ANFIS-ABC model can be considered suitable for the accurate prediction of the Cs of concrete for the design and construction of not only bridges but also other civil engineering structures. Development of hybrid models with different individual models is a continuous process for better predictive results. Therefore, it is proposed that other optimization techniques, such as PSO and GA with ANFIS should be used as base algorithms for the comparison of results. The developed hybrid models for refining the prediction performance may then be selected, depending on the aggregate used in the concrete as per local geo-environmental conditions.

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