A new systematic firefly algorithm for forecasting the durability of reinforced recycled aggregate concrete

Abstract

This study presents a new systematic algorithm to optimize the durability of reinforced recycled aggregate concrete. The proposed algorithm integrates machine learning with a new version of the firefly algorithm called chaotic based firefly algorithm (CFA) to evolve a rational and efficient predictive model. The CFA optimizer is augmented with chaotic maps and Lévy flight to improve the firefly performance in forecasting the chloride penetrability of strengthened recycled aggregate concrete (RAC). A comprehensive and credible database of distinctive chloride migration coefficient results is used to establish the developed algorithm. A dataset composite of nine effective parameters, including concrete components and fundamental characteristics of recycled aggregate (RA), is used as input to predict the migration coefficient of strengthened RAC as output. k-fold cross validation algorithm is utilized to validate the hybrid algorithm. Three numerical benchmark analyses are applied to prove the superiority and applicability of the CFA algorithm in predicting chloride penetrability. Results show that the developed CFA approach significantly outperforms the firefly algorithm on almost tested functions and demonstrates powerful prediction. In addition, the proposed strategy can be an active tool to recognize the contradictions in the experimental results and can be especially beneficial for assessing the chloride resistance of RAC.

Graphical abstract

Keywords

chloride penetrability / recycled aggregate concrete / machine learning / concrete components / durability

1 Introduction

Using waste materials from demolition and construction wastes as aggregate in concrete has secured widespread interest in recent decades. Recycled aggregate concrete (RAC) has some benefits such as reduction of demand for natural resources, as well as achieving sustainable evolution in the construction technology [1,2]. Despite the studies on RAC that have been ongoing for almost 70 years, using RAC is still restricted because of its poor properties [3]. The effect of the transition zone (ITZ) between the weak mortar and the aggregate, as well as the amount of attached mortar, are the core reasons for the inferior quality of RAC. This poor quality can be largely overcome either by eliminating or strengthening the adhered mortar [4]. Integrating pozzolanic materials is one of the most effective approaches in enhancing the durability performance of RAC [1,5].

Concrete durability is the capability to withstand different kinds of damage and keep the strength of concrete from exposure to the environment during its service life. The durability of RAC is usually less than that of traditional concrete because of the weak mortar on the recycled aggregate (RA) surface [3,6]. Thus, pozzolanic materials are used to ameliorate the durability performance of RAC. Chloride penetrability is considered one of the main factors which determine concrete durability, and it indicates the quantity of chloride ions that penetrate the concrete from the surrounding environment. Chloride penetration leads to corrosion in the concrete structure and so the study of chloride penetrability is a vital for the improvement of concrete durability [7]. The mechanism of chloride ion transmission is complex, and further knowledge of how the chloride ions can penetrate the concrete is becoming urgent [8]. Several laboratory tests are designed to study the chloride penetrability in hardened concrete, e.g., bulk diffusion test (ASTM C1556), salt bonding test (ASTM C1543), and rapid chloride migration test (NT Build 443 and ASTM C1202). Among these techniques, the rapid chloride migration test is the most extensively utilized migration method [9]. This test estimates the electric charge that passes through the concrete sample and estimates the resistance to chloride ion penetrability. Although the rapid chloride migration test is deemed to some extent a rapid test, at least 28 d of curing age is needed before carrying out the test. Thus, prediction of the chloride penetrability for strengthened RAC as a function of its components becomes essential as it can assist in scheduling operations at the initial design stages and reduce experimental requirements. Therefore, finding an appropriate method to forecast chloride penetrability for strengthened RAC can considerably reduce time and cost.

Recently, several artificial intelligence (AI) based models such as machine learning and deep learning techniques have been successfully applied in solving issues related to model accuracy and design complexity [10–15]. For instance, Anitescu et al. [14] used artificial neural networks and adaptive collocation strategy to solve partial differential equations. Guo et al. [15] indicated that a deep neural network can be utilized to approximate any continuous function to analyze thin plate bending problems. Among several metaheuristic algorithms, the firefly algorithm (FA) represents an effective optimization method, which has multi-modality and rapid convergence properties and can be hybridized for optimizing AI models in several practices. In the past few years, scholars are beginning to improve the variants and characteristics of FA (see Tab.1). In a recent study, Bui et al. [26] optimized the ANN technique using FA which was verified to yield better prediction accuracy for compressive and tensile strengths of high-performance concrete compared with a set of non-hybrid models. Despite the significant performance of the neural network, it usually requires a large number of hidden neurons to account for the robust interaction between the influencing factors. In addition, a large number of hidden neurons can induce overfitting in the heterogeneous database. Therefore, there is a necessity to improve the current algorithms for avoiding the limitation of poor generalization and limited exploratory capacities. The main contributions of the present study are briefly given below.

• A novel fuzzy-systematic algorithm called ANFIS-CFA is proposed for predicting the chloride penetrability of strengthened RAC with pozzolanic materials at different curing ages, which results in comprehensive solutions with a great application in industry.

• The CFA optimizer is augmented with chaotic maps and Lévy flight approaches to fulfill a good equilibrium between exploration and exploitation phases. Credential and complete datasets have been assembled to use in the chloride penetrability predictive approach.

• Three numerical benchmark analyses have been applied to verify the proposed model. The k-fold cross-validation is applied to approve the prediction reliability and stability of the developed algorithm. Also, a comparison with classical AI models has been conducted.

• To detect the strengths of the systematic algorithm, several criteria based on the Friedman test, computational time, and sensitivity analysis have been presented. This study is the first to explore the possibility of applying a fuzzy-metaheuristic model for predicting chloride penetrability of strengthened RAC with pozzolanic materials that has yielded results that can give novel insights to direct engineers, forecasting chloride penetrability with good precision in a suitable estimation time.

2 Integrating pozzolanic materials

The ITZ between the virgin aggregate and the mortar generally represents the weakest points in RAC [32]. Thus, to avoid these limitations, several treatment methodologies have been developed. Integrating pozzolanic materials in RAC is one of the most effective methods in optimizing RAC properties. The pozzolanic materials optimize the RAC compactness by forming hydrated calcium silicate (H–C–S), which decreases the porosity of RAC by sealing the pores in the weak mortar and assists in enhancing durability [4]. Coating or submerging RA in pozzolan slurries also assists in sealing the cracks and pores, thus the ITZ becomes enhanced in the hardened concrete. In addition, the pozzolanic reaction in the adhered mortar between the calcium hydroxide (CH) and the pozzolanic materials strengthens the weak portions in the adhered mortar and forms a strengthening shell over RA (Fig.17) resulting in more effective RAC performance regarding durability [1,5,33].

Previous studies revealed that the RAC has lower durability than normal concrete but this can be ameliorated by integrating various pozzolanic materials [34,35]. Adding fly ash and slag into RAC decreases the chloride penetrability; this is due to the hydrated product (H–C–S) that is generated during the hydration process soaking up the chloride ions and obstructing the ingress path in RAC [36]. Furthermore, the addition of silica fume can be valuable in decreasing chloride penetrability. Integrating silica fume in RAC reduces the chloride penetration depth by approximately 22% [37]. The impact of integrating the pozzolanic materials on the chloride ions penetrability of RAC has been detected by several studies and is displayed in Fig.2. As can be seen, the pozzolanic materials have significantly reduced chloride penetrability compared with that in untreated RAC.

3 Mathematical tools

3.1 Adaptive neurofuzzy inference system

ANFIS is promising in different engineering scatters [40]. To achieve an effective technique, the type, the optimization approaches (hybrid, backpropagation, etc.), and the number of membership functions (MFs) must be adjusted in the ANFIS model. The structure of ANFIS is observed in Fig.3. To clarify the procedures of ANFIS, x and y are deemed as inputs and f as an output. In every layer, the description of the node function is as follows [41,42].

To apply the ANFIS technique, the clustering algorithms need to be adapted. Fuzzy c-means (FCM) clustering is a robust algorithm that defines data clusters [42]. In this algorithm, the MF is used to define the degree of the database related to the cluster. The MF displays a great value for the data near the center of the cluster and a low value for the data far away from the center. The dataset (x) is divided into clusters (C) by reducing the error based on the distance of every point X_i to all centroids of the clusters C as follow [41,43]:

(1){\mu }_{sv}={\displaystyle \frac{1}{\sum _{k=1}^{\text{c}}{\left({\displaystyle \frac{d_{sv}}{d_{kv}}}\right)}^{\frac{2}{m-1}}}},

where d_sv = ||c_s − x_v|| denotes the Euclidean space between center of cluster sth and data point vth; µ_sv refers to the coefficient of matrix; m stands for the fuzzy index; C refers to the total numbers of clusters. The fitness function is assessed as follow:

(2)J\left(U,c_1,\dots ,c_2\right)=\sum _{s=1}^c{J}_s=\sum _{s=1}^c\cdot \sum _{v=1}^n{\mu }_{sv}^m{d}_{sv}^2.

The novel C fuzzy clusters center C_s (s =1,2,…,C) is determined as shown in Eq. (3):

(3)c_s=\frac{\sum _{sv}^n{\mu }_{sv}^m{x}_v}{\sum _{v=1}^n{\mu }_{sv}^m}.

3.2 Firefly based-algorithm

Firefly based-algorithm is a meta-heuristic algorithm inspired by a public performance of fireflies according to the following assumptions [44]:

(i) the total fireflies are assumed to be unisex and therefore, each firefly has the ability to attract another;

(ii) attractiveness is relative to the firefly’s light strength. A less attractive firefly would move randomly to the more attractive ones;

(iii) firefly light strength is selected through the landscape of the quantified target function.

In considering D-dimensional space, n refers to the number of the fireflies, while the position of the ith fireflies in the space is x_i=\left[x_i^1,x_i^2,\dots ,x_i^S\right]. The FA is adapted based on two actions: the variation of light strength and the attractiveness formulation. To simplify this model, the firefly attraction is represented based on the brightness.

The light strength I(r) of the firefly at a position x_i is formulated as follow [44]:

(4)I(r)=I_0{\text{e}}^{-\gamma r^2},

where γ is the coefficient of light absorption, I₀ is the original light intensity, and I is the light intensity. As a firefly’s attractiveness is proportional to the light intensity observed by adjacent fireflies, the attractiveness (\beta) of a firefly can be defined as follows:

(5){\beta }^t={\beta }_0{\text{e}}^{-\gamma r^2},

where {\beta }_0 refers to the attractiveness value at r = 0. Besides, the space among the firefly i and firefly j at the positions x_i and x_j is observed using Euclidian distance as follow:

(6)r_{ij}(t)=\Vert x_i(t)+x_j(t)\Vert =\sqrt{\sum _{k=1}^d\left(x_{id}-x_{jd}\right)},

where x_i and x_j denote the location of fireflies i and j in the Cartesian coordinate system. The schematic flowchart of the firefly algorithm is presented in Fig.4. Several efforts have been made to optimize the strategy of the firefly algorithm. For instance, Wang et al. [28] proposed a neighborhood attractiveness firefly algorithm in which every firefly moves to another brighter firefly chosen from a predetermined neighborhood. Wang et al. [45], however, suggested a random attraction firefly algorithm, where every firefly can attract to another randomly selected firefly. Yu et al. [46] introduced an optimized kind of the typical FA, denoted as a chaotic-based firefly algorithm (CFA) that adjusted the step of every firefly based on the local and global best positions. This optimized algorithm that depends on chaotic theory is a random search methodology. A chaos-based firefly algorithm can achieve comprehensive searches at higher speeds based on non-repeating chaos than randomized probability-dependent searches [47]. In addition, the CFA can increase the global search mobility for efficient global optima, relative to that provided by typical FA, by tuning the light absorption coefficient and attractiveness parameter of FA so that it can achieve the optimum performance of the model system.

3.3 Chaotic-based firefly algorithm

Firefly search success largely depends on good balance between exploration and exploitation. Therefore, the firefly parameters need to be optimized to balance the exploitation with exploration. This work integrates metaheuristic mechanisms of chaotic maps and Lévy flight into the typical FA for improving its search ability. To optimize the ability of the typical FA, the present study utilizes Gauss/Mouse Map as a chaotic map (Eq. (7)) instead of the stochastic parameters that are utilized in the classical FA. The schematic flowchart of a chaotic firefly is shown in Fig.5. Gandomi et al. [48] illustrated that the Gauss/Mouse map represents the preferable model for adjusting the attractive variable (\beta). The Gauss/mouse map can be defined as follow [48]:

(7){\beta }_{\text{chaotic}}^{t+1}=\{\begin{array}{ll}0,& {\beta }_{\text{chaotic}}^t=0,\\ 1/{\beta }_{\text{chaotic}}^t,& \text{otherwise},\end{array}

where {\beta }_{\text{chaotic}}^{t+1} refers to the chaotic number at tth; t represents the iteration numbers, and {\beta }_{\text{chaotic}}^0 is randomly produced with a regular distribution in [0,1]. Thus, Eq. (5) is then modified as follow:

(8){\beta }^t=\left({\beta }_{\text{chaotic}}^t-{\beta }_0\right){\text{e}}^{-\gamma r^2}+{\beta }_0,

where \gamma is a coefficient of absorption. A motion of the firefly i that is attracted to new brighter firefly j is presented as follow:

(9)x_i^{t+1}=x_i^t+{\beta }^t\left(x_j^t-x_i^t\right)+{\alpha }^t\epsilon \times L(s),

where x_i and x_j represent the coordinates of the ith and jth fireflies; \alpha represents the trade-off coefficient for determining a stochastic performance; \epsilon represents the random number vector estimated from a uniform or Gaussian distribution, and can be defined as follow [26]:

(10)\epsilon =rand-1/2,

where rand represents a stochastic number produced through a regular distribution in [0,1]. To optimize the ability of FA to achieve the global optima, this research adjusts the variable \alpha with an adequate inertia weight. This determination can be \alpha through a feasible range:

(11){\alpha }^t={\alpha }_0{\theta }^t,

where {\alpha }_0 represents the coefficient of initial trade-off; {\alpha }^t represents the coefficient of trade-off at tth iteration, and \theta denotes randomness reduction (0 < \theta < 1).

The last part in Eq. (9) refers to the Lévy distribution that can be estimated as follow:

(12)L(s)\sim s=\frac{u}{|v{|}^{1/\tau }},

where L(s) represents the Lévy distribution for signal s, s refers to the power-law distribution, \tau is default value = 1.5. v and u are normally distributed as follow:

(13)u\sim N\left(0,{\sigma }_u^2\right),v\sim N\left(0,{\sigma }_v^2\right),

where

(14){\sigma }_{\text{u}}={\left\{\frac{\mathrm{\Gamma }(1+\tau )\sin (\text{π}\tau /2)}{\mathrm{\Gamma }[(1+\tau )/2]\tau 2^{(\tau -1)/2}}\right\}}^{1/\tau },{\sigma }_v=1,

where \mathrm{\Gamma }(z) represents the function of Gamma that is estimated as follow:

(15)\mathrm{\Gamma }(\mathit{\text{z}})={\int }_0^{\mathrm{\infty }}{t}^{z-1}{\text{e}}^{-t}\text{d}t.

The specific modeling procedure is presented as follows:

Step 1: a very diverse initial population of fireflies is generated: the chaotic map is utilized and the attractiveness parameter (β) is adapted based on the Gauss/mouse chaotic map;

Step 2: the stochastic parameter (α) is adapted to control exploitation and global exploration;

Step 3: a Lévy flight is utilized for generating novel solutions around the best setting to accelerate the exploitation;

Step 4: the light intensity is evaluated: the chaotic firefly performance can be assessed (prediction errors) in terms of the objective function f(x). In the present work, the root-mean square-error is utilized as an objective function;

Step 5: the stopping condition is considered: when the stopping criteria are achieved, the best answer can be presented as a solution using the firefly with the optimal light intensity; otherwise, repeat the abovementioned steps.

3.4 Hybrid model based on ANFIS-CFA

In the ANFIS modeling system, each input parameter commonly has various MFs and each membership in the ANFIS becomes maximum somewhere. To increase the ANFIS precision through the training process and optimize the position of MFs, CFA is utilized as an optimizer. During the training process, the CFA algorithm optimizes the antecedent and subsequent variables of the ANFIS method, consisting generally of the membership parameters. This integration is utilized for determining more precision results for optimizing the forecasting of the chloride penetrability for strengthened RAC. In this regard, the databases are divided into two sets including the training and testing parts. The training ratio is utilized to adapt the system of ANFIS, while the remaining set is utilized to evaluate the model performance. The process of the training set allows the structure to adjust the defined variables as output or input in the model. The training datasets are finished when the required criteria for terminating the model are achieved. Before training the model, a logistic map, introduced by May in 1976 [21], is utilized for generating the fuzzy inference system (FIS). Based on Eq. (16), the logistic map is utilized as an alternative to stochastic variables.

(16)\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{p}:x^{(i+1)}=\mu x^{(i)}\left(1-x^{(i)}\right),x^{(i)}ϵ(0,1),i=0,1,2,\dots ,

where x⁽ⁱ⁾ represents the number of the chaotic parameter x at the ith iteration, and\mu defined as the bifurcation parameter \mu ∈ [0,4]. In this hybrid model, the root mean square error is utilized for assessing the ANFIS-CFA model performance. Fig.6 displays the flowchart of the hybrid ANFIS-CFA.

3.5 k-fold cross validation

For validating the efficiency of the established model, several validation methods are commonly used, such as repeated random, substitution model, holdout model, and KFCV, etc. Among the abovementioned methods, the KFCV is the most common method [49, 50]. The KFCV method divides the data into k subgroups randomly. k − 1 subgroups are used as training sets whereas the residual datasets are the testing set. The process reiterates k times, and the performance of the model is evaluated using the average prediction error of k subsets, as shown in Eq. (17).

(17)Fitness=\frac{1}{k}\sum _{i=1}^{n}RMSE.

3.6 Performance assessment criteria

Considering the nature of chloride penetrability, we should not depend on single criteria when assessing the performance of the statistical approach. To evaluate the performance of the developed techniques, many statistical approaches, e.g., root mean square error (RMSE), mean absolute error (mae), and correlation coefficient (R²) are applied in the present research. These statistical analyses are presented as follows [41]:

(18)RMSE=\sqrt{\frac{\sum _{i=1}^n{\left(x_{\text{mea}}-x_{\text{pre}}\right)}^2}{n}},

(19)mae=\frac{\sum _{i=1}^n\left|x_{\text{mea}}-x_{\text{pre}}\right|}{n},

(20)R^2=1-\frac{\sum _{i=1}^n{\left(x_{\text{mea}}-x_{\text{pre}}\right)}^2}{\sum _{i=1}^n{\left(x_{\text{mea}}-x_{\text{m}}\right)}^2},

where n, x_mea, and x_pre represent the over-all data numbers, experimental, and predicted values, respectively; x_m refers to the mean values. Ideally, the predicted method with high accuracy is determined when the mae and RMSE are near to zero and R² is near to 1. For the suggested model in this study, the purpose of the developed approach is to achieve an accurate design for minimizing the mae and RMSE as well as maximizing R² simultaneously.

4 Data preparation

In this study, nine input parameters were used to establish the ANFIS-CFA model. The input parameters comprise the concrete constituents, e.g., cement content (c), water content (w), coarse recycled aggregate (RA), sand content (S), pozzolanic materials content (PM), water/cement ratio (w/c), and curing age (T). The fundamental RA properties, e.g., particle density (D) and water absorption (WA) are also considered as input parameters. These different parameters are deemed as the core parameters affecting the chloride penetrability of RAC [51]. The electric charge passing into concrete (EC) that represents the chloride penetrability for strengthened RAC with pozzolanic materials at different curing ages is the output value forecasted from this study. Tab.7 summarizes the statistical characteristics of the experimental database. The comprehensive database of 83 laboratory results gained from several literature sources is collected for training and checking the precision of the chloride penetrability method [34,38,39,52–57]. The graphical results for the electric charge influential factors in the datasets are shown in Fig.7.

5 Results and discussion

Results of the developed model for predicting the chloride penetrability of strengthened RAC with pozzolanic materials are discussed in this section. To build the database for the durability of reinforced RAC, we follow the following criteria: i) The datasets are screened and only complete and credential datasets are chosen; ii) data are collected from credible and reputable literature and any arbitrary source avoided. The dataset range for both output and input is important and cannot be ignored for several variables in the operating process. As a result, the datasets are normalized in the range of [0,1] for reducing the impact of various scales of parameters on the models’ performance by the following formula [17].

(21)X_{\text{n}}=\frac{\left(X-X_{\text{min}}\right)}{\left(X_{max}-X_{min}\right)},

where X_n and X, X_max and X_min denote the normalized, measured, maximum and minimum values of X, respectively.

In the present study, the 83 datasets are randomly spilt into two sets, of which about 80% of all datasets are used for training the models and the other 20% are utilized for verifying the models, based on the recommendation of Swingler [17]. In order to discipline the datasets, the K-FCV procedure is used. Kohavi [20] observed that utilizing 10 fold cross-validation represents the optimum number of folds, which can achieve a satisfactory result in an adequate time frame. Therefore, 10 fold cross-validation are utilized during the analyses of this study. To adapt the process for forecasting the chloride penetrability of strengthened RAC, all samples in the datasets are randomly divided into 10 featured folds. In the first validation stage, the initial fold is utilized to test the model whereas, for the training process, the remaining nine folds are utilized. Then, the second fold is utilized to test the subsequent validation and this stage is iterated till all validation numbers are finished. The measurements of each fold in the 10-fold cross validation are observed in Fig.17. This figure shows that the greatest values of statistical metrics appear in the cross validation sets 8 and 9, indicating the good capability of the developed approach.

5.1 Numerical benchmark analyses

A benchmark function is widely utilized for evaluating performance in applied mathematics. This approach utilizes numerical functions with identified optimal solutions to test the developed algorithm. On the other hand, the developed algorithm tries to achieve an optimum solution of the benchmark functions and then compares the detected solutions with that achieved from the classical algorithms. In this study, three benchmark functions are applied to evaluate the CFA performance. The CFA performance is compared with the classical FA based on three benchmark functions with the same condition (see Tab.7). Three functions with 50 levels of dimensions are applied to solve the high-dimensional problems and evaluate the performance of the CFA and FA models. Twenty independent runs are executed for every benchmark function, to eliminate any random discrepancy. The number of iterations and the size of the population are taken as 1000 and 100, respectively. Results obtained by CFA using three numerical benchmark analyses are presented in Tab.7.

As can be seen, the CFA is the preferable algorithm for all three functions. The convergence curves of CFA and FA in the run of three benchmark functions are observed in Fig.17. Based on the convergence curves of the three functions, it can be concluded that the CFA converges in very few iterations; FA converges after many iterations. Furthermore, in the early optimization stage, the nonlinear term is highly correlated with an attractiveness or brightness term with a higher possibility of being chosen. Regarding the convergence curves of the three functions, it can be concluded that the strong point of the CFA is the faster convergence in the primary phase of the search process than that of FA. The results illustrate a major improvement of the CFA relative to FA in all numerical cases.

5.2 ANFIS based FCM

ANFIS based FCM technique is utilized to forecast the flowing electric charge of concrete specimens. To perform this model, nine effective parameters (w/c, w, c, S, RA, PM, T, D, and WA) are chosen as inputs, and EC is chosen as the target. To design the ANFIS model, there are two key phases: pattern formation and pattern generation with the corresponding input vector and target vector [19].

To design the structure of the ANFIS technique, the number and type of MFs and the epoch number need to be considered. Based on previous studies, there are no obvious approaches to forecast the required MFs. Thus, the trial-and-error method is used to determine the MF. The best design is determined from the Gaussian membership (Tab.7). The Takagi-Sugeno approach is selected based on its high efficiency and good ability for performing the systematic model to determine fuzzy rules parameters. The FIS is initially produced based on a FCM clustering approach. Tab.5 summarizes the results of the computational analyses for the training and testing datasets for quantifying the influence of varying the number of clusters on the network. Among the different models, the third ANFIS model with 10 clusters achieves the best forecasting precision. Thus, this model (i.e., ANFIS 3) is selected for predicting the chloride penetrability of strengthened RAC. MATLAB software is applied with the function of genfis3 for adjusting the initial FIS. Fig.17 displays the correlation coefficient for the testing dataset and training dataset for EC values. It can be noted that the relation between the observed output of the electric charge passing into concrete and the predicted values are applicable.

5.3 ANFIS-CFA model

To forecast the EC passing into concrete with high precision, an intelligent ANFIS-CFA model is proposed. The ANFIS provides the search space and uses CFA to find the best solution by tuning the MFs required for achieving an acceptable value. The idea of the hybrid model is proposed to forecast the passing electric charge that generates nonlinear relations among the input and output parameters. Therefore, concrete components and RA characteristics are set as input parameters, and EC is chosen as the output parameter. The proposed method is analyzed in the MATLAB program and applied in a PC with Intel i⁷-4790 CPU at 3.60 GHz with 32 GB installed RAM. The form of the Takaji Sugeno model is applied for integrating the best features of the FIS in the proposed ANFIS-CFA model. Parameters play a vital role in optimizing the performance of the algorithm and different parameters can cause various effects. The user-defined CFA model including γ, β₀, and α are considered, as indicated in Tab.6. In our developed models, two main parameters need to be set with an appropriate initial value: β₀ and γ, so the experiments with these values are tested for analyzing what value of these two parameters should be set in CFA. According to these trials, the size of the population is selected as 100 and the maximum iterations number is chosen as 1000. Meanwhile, the values of optimal absorption and attractiveness that could be utilized for enhancing the accuracy of the proposed technique are 1.2 and 2.0, respectively (Fig.11). Furthermore, the execution time of the proposed model is observed in Fig.12. At a glance, a linear increase in the execution time can be detected as the population size increases. For 100 population sizes, the executed ANFIS-CFA model required nearly 460 s.

Fig.13 shows the convergence graph of the CFA and classical FA algorithms in optimizing the objective function. As shown, the CFA has higher accuracy than FA for optimizing the ANFIS technique. Specifically, from approximately 200 iterations onwards, CFA displays a higher convergence speed in minimizing the objective function compared with FA. Hence, the ANFIS-CFA method can be used as it performs well in estimating the chloride penetrability of strengthened RAC in a relatively short time.

Fig.14 displays the correlation between the experimental and forecasted values assessed from the ANFIS-CFA technique for the training and testing data in ten-cross folds. In addition, the results are compared with the classical ANFIS-FA model. Findings indicate that the experimental EC values are less scattered and near to the forecasting values for ANFIS-CFA than ANFIS-FA, as is indicated by its closeness to the dashed line (line of equality). To provide a visual sense for the introduced techniques, Fig.15 is added to illustrate the relation between the measured and predicted EC for all databases. A strong agreement between the experimental and the forecasting electric charge passing into concrete is confirmed for almost all of the data.

To provide further clarification, the relative error of outputs observed from ANFIS-CFA and ANFIS-FA models are displayed in Fig.16. Evidently, the relative error of the electric charge changes around zero, more frequently in a smaller range (±10%) for ANFIS-CFA than in the larger range (±20%) of the ANFIS-FA method. This reveals that the proposed ANFIS-CFA technique has better precision in the forecasting of passing electric charge in the concrete specimens than the ANFIS-FA method.

5.4 Comparison with current classical models

In this section, the computational performance of the proposed ANFIS-CFA model is compared with the current state-of-art models including genetic algorithm (GA), particle swarm optimization (PSO), ANFIS, and ANFIS-FA. During this comparison, the main parameters used for current traditional PSO and GA are taken from previous literature [12,19,22,24]. For PSO algorithm, the main parameters including accelerating rate (c₁) and social coefficient (c₂) are 1, 2, respectively, and for GA, the crossover and mutation rates are 0.8, 0.02, respectively [12]. The R² and RMSE are chosen as assessment criteria within the experimental and predicted electric charge of the concrete specimens. Fig.17 displays the statistical analysis of the proposed ANFIS-CFA compared with the classical techniques. According to testing sets, the proposed ANFIS-CFA technique has R² = 0.933, and RMSE = 0.092, indicating that the developed model is more accurate than the others. According to the smaller RMSE and greater values of R², the ANFIS-CFA technique performs better than the other techniques. Overall, this study uses the CFA as an optimization tool, which is highly superior relative to the classical FA algorithm, in optimizing the predicting precision of EC, by a substantial margin. For the practical applications, we can use the concrete components and RA properties, e.g., w/c, w, c, S, RA, PM, T, D, and WA as inputs for forecasting the electric charge. It is noteworthy that the developed ANFIS-CFA technique can, in a limited time, offer initial estimations of the laboratory tests for forecasting the electric charge of the concrete specimens.

To evaluate the generalization ability and exploratory capacity of the proposed model, a non-parametric test is utilized to compare the performance of the proposed model with other classical algorithms. Lately, the non-parametric tests such as Friedman test and Wilcoxon signed-rank test are commonly applied in comparing the performance among some algorithms [18]. In this study, the Friedman test, performance metric, is used for this comparison. The statistical analysis is performed in a Minitab software. According to Friedman test analysis, the null hypothesis is analyzed as there are no variance among results. Ranks are ranged from 1 (least error) to k (highest error) and represented by r_i^j (1 ≤ j ≤ k). Therefore, four forecasting methods have been analyzed and the mean ranking \left(R_j\right) achieved in all datasets is simulated as follow [23].

(22)R_j=\frac{1}{N}\sum _{i=1}^N{r}_i^j,

where r_i^j denotes the model ranking j on dataset i, N is the number of datasets. The mean ranking estimated according to the Friedman test is shown in Tab.7. The results confirm that the ANFIS-CFA has the best performance relative to classical models. Therefore, it can be noted that our proposed model achieved better forecast accuracy than other models.

5.5 Sensitivity analysis

To investigate the generalization ability of the developed method, sensitivity analysis (SA) is used. The SA is used for calculating the magnitude of connection weights to study the relative effect of each input parameter on the model output according to Garson [16].

(23)SA=\frac{\sum _{j=1}^{n_{\text{hid}}}\frac{w_{ji}}{\sum _{l=1}^{n_{\text{inp}}}\left|w_{jl}\right|}\times w_{\text{o}j}}{\sum _{k=1}^{n_{\text{inp}}}\left(\sum _{j=1}^{n_{\text{hid}}}\left|\frac{w_{jk}}{\sum _{l=1}^{n_{\text{inp}}}\left|w_{jl}\right|}\times w_{\text{o}j}\right|\right)},

where SA denotes the importance of the input parameter; n_inp refers to the number of inputs; n_hidd refers to the number of hidden units; w is the weight; j is the input unit; o is the output unit.

Fig.18 displays the impact of every input parameter on the model prediction. Results observed that the water/cement, water content, and coarse recycled aggregate are closely correlated to the chloride penetrability prediction of the strengthened RAC with mean values of 12.12, 12.01, and 12.01, respectively. In addition, the other concrete parameters make a satisfactory contribution in forecasting chloride penetrability.

6 Conclusions

This study introduces an intelligent fuzzy ensemble approach based on the ANFIS-CFA model for predicting the chloride penetrability of strengthened RAC. The main conclusions of this work are outlined as follows.

• The proposed ANFIS-CFA model can predict the chloride penetrability of strengthened RAC with pozzolanic materials in terms of the electric charge, with good match with the experimental values for the training and testing data. The proposed ANFIS-CFA technique utilizes computation parameters tailored to predict the electric charge.

• To assess the possibility of the fuzzy-ensemble technique optimized by the chaotic based firefly algorithm, its performance is compared with that of the classical firefly algorithm. The performance of the numerical benchmark results showed that the CFA approach has the higher precision and flexibility in estimating the chloride penetrability of strengthened RAC than the typical firefly algorithm.

• Since metaheuristic models have an undeniable impact on their performance, a sensitivity analysis is performed for specifying the most effective parameters. The relative error of the proposed ANFIS-CFA technique is in a suitable range of ±10%, while the typical ANFIS-FA technique shows a vaster error range of ±0.20. This verifies the accurate forecasting of the approach developed herein in predicting the electric charge passing into the concrete.

• The developed model can be part of a set of technologies utilized in making a decision by construction engineers, with small experimental batches and less time-expense for assessment than experimental tests. This hybrid model is adaptable and versatile due to its easy extention to comprise a wide set of experimental data and various characteristics related to other concrete components and RA properties.

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