Simulation of foamed concrete compressive strength prediction using adaptive neuro-fuzzy inference system optimized by nature-inspired algorithms

Abstract

Concrete compressive strength prediction is an essential process for material design and sustainability. This study investigates several novel hybrid adaptive neuro-fuzzy inference system (ANFIS) evolutionary models, i.e., ANFIS–particle swarm optimization (PSO), ANFIS–ant colony, ANFIS–differential evolution (DE), and ANFIS–genetic algorithm to predict the foamed concrete compressive strength. Several concrete properties, including cement content (C), oven dry density (O), water-to-binder ratio (W), and foamed volume (F) are used as input variables. A relevant data set is obtained from open-access published experimental investigations and used to build predictive models. The performance of the proposed predictive models is evaluated based on the mean performance (MP), which is the mean value of several statistical error indices. To optimize each predictive model and its input variables, univariate (C, O, W, and F), bivariate (C–O, C–W, C–F, O–W, O–F, and W–F), trivariate (C–O–W, C–W–F, O–W–F), and four-variate (C–O–W–F) combinations of input variables are constructed for each model. The results indicate that the best predictions obtained using the univariate, bivariate, trivariate, and four-variate models are ANFIS–DE– (O) (MP= 0.96), ANFIS–PSO– (C–O) (MP= 0.88), ANFIS–DE– (O–W–F) (MP= 0.94), and ANFIS–PSO– (C–O–W–F) (MP= 0.89), respectively. ANFIS–PSO– (C–O) yielded the best accurate prediction of compressive strength with an MP value of 0.96.

Keywords

foamed concrete / adaptive neuro fuzzy inference system / nature-inspired algorithms / prediction of compressive strength

Introduction

Concrete is a fundamentally important material in the construction industry [¹,²]. Various types of concrete have been studied over the years, such as reinforced concrete, air-entrained concrete, lightweight, and high-strength concrete [³,⁴]. This study focuses on air-entrained concrete, which is made using foam with bubbles added to the concrete mixture. Foamed concrete is synthesized by adding a foaming agent to the cement-based mortar and thoroughly mixing them, or by aerating the added foaming before it is added to the mortar [⁵]. Foamed concrete possesses attractive properties. It is light for void filling, attenuates sound, and can be used in non-load-bearing walls, insulation, and roof decks. Its mechanical properties have been investigated extensively such that it can be used for various structural purposes [⁶,⁷].

The mechanical properties of concrete, including its compressive strength, tensile strength, flexural strength, shrinkage, creep, and modulus of elasticity, are used to evaluate its performance and quality [⁸]. Among these properties, compressive strength is typically regarded as the most significant, and it is considered as the main factor in defining the quality of concrete [⁹]. The typical parameters that affects its compressive strength include water content, cement content, cement type, curing period, aggregate content, and aggregate type [¹⁰,¹¹].

In addition, for foamed concrete, additives, air-void size, dry density, and shape parameters are important [¹²]. Several researchers have investigated the properties of foamed concrete via laboratory experiments [¹³–¹⁹]; additionally, numerous mathematical models have been developed [⁶,²⁰–²⁴]. Computer aid for simulating materials has received more attention over the past two decades [²⁵,²⁶]. Soft computing can potentially reduce laboratory experimental costs and time consumption, as well as provide the accurate characteristics of related materials [²⁷].

Empirical formulations have been developed for compressive strength simulation [²⁸,²⁹]. However, those empirical formulations are associated with the behavior limitations of each experimental data set used and cannot be implemented in a generalized form. Hence, the use of machine learning in this context has been investigated [³⁰]. Modeling techniques to predict concrete properties have received significant attention owing to their potential in reducing the time and cost of designing concrete mixtures [³¹–³³].

Nehdi et al. [⁶] investigated the use of an artificial neural network (ANN) to predict the performances of cellular concrete mixtures. They used four input parameters, including the ratio of water to cementitious material, ratio of sand to cementitious material, cement content, and ratio of foam to cementitious material. The results revealed that the production yield, foamed and un-foamed density, and compressive strength were accurately predicted using the proposed ANN model compared with parametric methods.

Recently, Abd and Abd [²⁰] proposed multivariable nonlinear regression (MNR) and revolutionary support vector machine (SVM) models for predicting the compressive strength of foamed concrete from known proportions of mixed elements. The results revealed an excellent agreement between the observed and predicted values for the used data set, although the SVM model proved to be superior to the MNR model.

Yaseen et al. [⁷] proposed an extreme learning machine (ELM) model to predict the compressive strength of lightweight foamed concrete. The input variables included the oven dry density, water binder ratio, cement content, and foamed volume. The potential of the ELM model was verified by comparing the results yielded by it to those of the multivariate adaptive regression spline, M5 tree, and support vector regression models. All four models achieved excellent results; however, the ELM performed the best among them.

Studies involving the use of mathematical tools to predict the compressive strength of lightweight foamed concrete have indicated the significant potential of such techniques. Based on previous studies and from the engineering perspective, the compressive strength of concrete is highly affected by several physical variability attributes. Hence, providing an accurate intelligent modeling strategy is the main focus of material engineering scientists, as it can facilitate material modeling and fabrication [³⁴].

Over the last two decades, a new subfield of intelligent systems that combine the advantages of the ANN and fuzzy logic (FL) has been developed. Jang proposed an adaptive neuro-fuzzy inference system (ANFIS) that produces input–output mapping from human knowledge and input–output information using a hybrid algorithm [³⁵]. The ANFIS model has been applied successfully in a various science and engineering applications [³⁶–³⁹].

In a comparative study regarding the application of three data-driven models, i.e., the ANN, ANFIS, and multiple linear regression (MLR) to predict a preliminary mix design and the compressive strength of concrete, it was discovered that the ANN and ANFIS outperformed MLR [⁴⁰]. Sadrmomtazi et al. [⁴¹] modeled the compressive strength of lightweight concrete using an ANN, an ANFIS, and parametric regression. The results indicated that the ANN and ANFIS models yielded superior results.

The feasibility of the ANFIS model was investigated for predicting the compressive strength of mortars containing metakaolin at different ages [⁴²]. The results demonstrated the potential of the ANFIS model for predicting the compressive strength of mortars. Several studies have been conducted that involved concrete compressive strength simulations using the potential of the ANFIS model [⁴³–⁴⁷].

Based on the previous studies mentioned above, the ANFIS offers attractive properties that can be harnessed in the mathematical modeling of concrete properties. However, the main disadvantage of the ANFIS model is that the member function must be optimized efficiently for its internal parameters, rendering training and learning difficult and time consuming [⁴⁸].

The performances of artificial intelligence (AI) models are subjective to hyperparameter tuning [⁴⁹]. The new era of computer aid application has advanced to the hybridization of AI models with various nature-inspired optimization algorithms, such as particle swarm optimization (PSO) [⁵⁰], ant colony (ACO) [⁵¹], differential evolution (DE) [⁵²], and genetic algorithm (GA) [⁵³], which can train AI methods and optimize their results, particularly when solving nonlinear and high-dimensional problems.

The main aim of hybridization is to achieve a stable and reliable learning process [⁴⁹,⁵⁴,⁵⁵]. The hybridization of the ANFIS model with nature-inspired optimization algorithms demonstrated a remarkable improvement in the prediction of several engineering applications [⁵⁶–⁵⁹]. To the best of the authors’ knowledge, hybrid ANFIS models with four different nature-inspired optimization algorithms (e.g., PSO, ACO, DE, and GA) have been developed to predict the compressive strength of foamed mixture concrete.

Despite being used in several fields, hybrid ANFIs models have not been used to predict foamed mixture concrete compressive strength. The modeling procedure is conducted using various related concrete characteristics, including cement content, oven dry density, water-to-binder ratio, and foamed volume, as predictive attributes for the prediction matrix. The proposed hybrid intelligence models comprising ANFIS and evolutionary algorithms are appropriate for nonlinear, discontinuous, implicitly defined models and multiobjective problems; furthermore, they can provide better predictability results.

Data regarding foamed concrete

The developed classical and hybrid ANFIS models were applied to a foamed concrete mixture comprising cement paste, mortar (a combination of sand, cement, and water), and a certain volume of foamed bubbles. Various additives such as fly ash and silica were used to improve the compressive strength of the foamed concrete mixture. The reduction in water percentage and the increase in the superplasticizer water/binder ratio increased the compressive strength. The concrete mixture was designed based on various foamed volumes, densities, and mixture constituents. Other additives, such as steel, class, carbon, and synthetic fibers, can be used to improve the tensile strength, flexural performance, and toughness of foamed concrete; however, they exert a less significant effect on the compressive strength.

This study disregards the magnitude of fibers as a parameter that affects the compressive strength. However, the effect of superplasticizers was implicitly accounted for via the amount of water used in the mixing process. Supplementary cementitious materials, such as fly ash and silica fume, were included as cementitious materials, and their effect on the compressive strength was considered. The experimental data sets pertaining to the concrete compressive strength and related concrete properties were obtained from various published open-access studies [⁶⁰–⁶⁴]. These studies focused on the simulation of foamed lightweight concrete with a specific density between 600 and 1800 kg/m³.

The test data set was obtained from 91 experiments, in which 70% of the total data set was abstracted for use in the training phase of the applied predictive models, whereas the remaining 30% was used to validate the constructed optimal predictive models (in the testing phase). The data partition was conducted based on a trial-and-error procedure. The input variables used to predict the compressive strength included the cement content (C), oven dry density (O), water-to-binder ratio (W), and foamed volume (F).

Methodological overview

Soft computing models are used to solve complex application problems that are highly stochastic, nonlinear, and not solvable using classical models. Soft computing is the most basic form of AI. Soft computing eliminates some of the disadvantages of hard computing. For instance, hard computing requires extremely precise models; however, because the procedure of soft computing is similar to that of a black box, the problem does not need to be modeled perfectly. Soft computing can consider both partial truth and approximation, and models that are based on soft computing methods can incorporate uncertainties.

The goal of the present study is to predict the strength of concrete using efficient hybrid algorithms that are based on the ANFIS. These algorithms include integrated ones such as ANFIS–ACO, ANFIS–PSO, ANFIS–DE, and ANFIS–GA. This section describes the concepts on which these algorithms are based as well as the corresponding flowcharts. The collected data were partitioned into training and test sets. All of the data were evaluated using basic ANFIS and hybrid algorithms.

ANFIS model

FL was conceived many decades ago as a method for processing data that can consider partial set membership [⁶⁵]. The main reason for its popularity is that it allows inputs that are not precise numerical inputs [⁶⁶]. The most important advantage of FL is that it easily generates conclusions from noisy or imprecise input data.

Relevant experience and knowledge are required to select the appropriate shapes of membership functions and fuzzy rules to yield the best results. In some cases, time-consuming methods, such as trial and error, must be used. ANNs can be used for model training. The synthesis of neural networks with fuzzy systems can provide a more powerful tool that incorporates the advantages of both methods [⁶⁷–⁶⁹].

A neuro-fuzzy system can be considered as a hybrid algorithm for making decisions from fuzzy modern soft computing-based modeling using an ANN. The ANFIS was introduced in 1993 by Jang [³⁵]. The neuro-fuzzy system was developed with the learning capability of the ANN. The most important components of fuzzy systems are rules, which are also the basic parts of an entire algorithm. The ANN is used to optimize these rules [⁷⁰].

The first proposed ANFIS model had five layers. Figure 1 schematically depicts the structure of the ANFIS. The rules are as follows [³⁵]:

(1)\mathrm{Rule1:If}X\mathrm{is}{A}_1\mathrm{and}Y\mathrm{is}{B}_1,\mathrm{then}{f}_1=p_{1}x+q_{1}y+r_1,

(2)\mathrm{Rule\; 2:If}X\mathrm{is}{A}_2\mathrm{and}Y\mathrm{is}{B}_2,\mathrm{then}{f}_2=p_{2}x+q_{2}y+r_2,

, where A₁-A₂ and B₁-B₂ are membership functions for inputs x and y, respectively.

In layer 1, each node is a square node for creating the membership grades. Utilizing the membership function, inputs x and y are translated into linguistic terms.

(3)O_{\mathrm{1},i}={\mu }_{A_i}(x),\hspace{1em}i=\mathrm{1},\mathrm{2},

where x is the input value to node i, A_i is the linguistic term, and O_1,i is the membership function of A_i. The types of membership functions available are Gaussian, triangular, and trapezoidal. The Gaussian function is expressed as follows:

(4){\mu }_{A_i}(x)=\exp \left(-{\left({\displaystyle \frac{x-a_i}{b_i}}\right)}^{\mathrm{2}}\right),

where a_i and b_i are the distribution parameters (antecedent parameters).

Similarly, in the second layer, each node is circular, and the output is obtained using the following function:

(5)O_{\mathrm{2},i}=w_i={\mu }_{A_i}(x)\ast {\mu }_{B_i}(x),\hspace{1em}i=\mathrm{1},\mathrm{2},

where w_i is the weight of the rule.

In the third layer, the nodes calculate the ratio of the weight of rules and then divides it by the sum of all weights, as follows:

(6)O_{\mathrm{3},i}={\bar{w}}_l={\displaystyle \frac{w_i}{w_{\mathrm{1}}+w_{\mathrm{2}}}},\hspace{1em}i=\mathrm{1},\mathrm{2.}

All the nodes in the fourth layer are square and expressed as follows:

(7)O_{\mathrm{4},i}={\bar{w}}_l{f}_i={\bar{w}}_l(p_{i}x+q_{i}y+r_i),\hspace{1em}i=\mathrm{1},\mathrm{2},

where {\bar{w}}_l is the output of the third layer; p_i, q_i, and r_i are consequent parameters that are updated during the training phase.

Finally, the fifth layer comprises the summation of the layers in one circle node (∑), as follows:

(8)O_{\mathrm{5},i}={\displaystyle \sum _i{\bar{w}}_l}{f}_i={\displaystyle \frac{{\displaystyle \sum _i{w}_i}{f}_i}{{\displaystyle \sum _i{w}_i}}},\hspace{1em}i=\mathrm{1},\mathrm{2.}

PSO Algorithm

PSO, which was developed by Eberhart and Kennedy a few decades ago, is based on the behavior and movements of a community of birds, fish, and insects [⁷¹]. Three operators are considered in PSO: alignment, separation, and cohesion. The algorithm utilizes a group of particles that flow in the search space to search for the optimum point. The positions of all particles in the space are adjusted based on one’s own experience as well as those of others [⁷²]. Their speeds are similarly adjusted. The positions of the particles change according to their current position/velocity/distance to the best particle.

The particle update rule is as follows:

(9)p=p+v,

with

(10)v=v+c_1·\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}·(p_{\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{t}}-p)+c_2·\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}·(g_{\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{t}}-p),

where p, v, c₁, c₂, p_Best, g_Best, and rand are the position, direction, local weight, global weight, best position of the particles, best position of the swarm, and a random value, respectively.

The velocities of the particles are updated using the following equation:

(11)V_i^{t+1}=v_i^t+c_1{U}_1^t(p{b}_i^t-p_i^t)+c_2{U}_2^t(g{b}_t-p_i^t)\mathrm{.}

The three terms from left to right in the equation above are the inertia, personal influence, and social influence, respectively.

Ant colony algorithm

The ACO algorithm was introduced by Dorigo approximately 30 years ago [⁷³]. Subsequently, many researchers have extended the system. Because ACO algorithms can solve both static and dynamic problems, they are applicable to various optimization problems.

Activities such as foraging (food searching), division of labor, brood sorting, and cooperative transport are coordinated by stigmergy, which enables self-organization. Although a colony of ants comprises simple individuals, it is considered to be one of the most complicated and well-organized structures in nature, whose activities are coordinated via stigmergy.

The pheromone trails of ants are sensed by other ants, which seek the shortest path to food. A similar procedure is used in the ACO algorithm to obtain the optimal point in the search space. The movement modes of the ants are the forward and backward modes. The ants implement a step-by-step decision procedure to obtain the best solution to a problem [⁷⁴,⁷⁵].

DE algorithm

In engineering problems, objective functions may be discrete, nonlinear, or multidimensional. Some may have local minima. In such cases, a population-based algorithm with stochastic features is required to obtain the solution. The DE algorithm, which was introduced by Storn and Price in 1996, exhibits the abovementioned features [⁷⁶,⁷⁷].

To optimize a function with n real parameters, the vectors used are of the following form:

,

(12),xi,G=[x1,i,G,x2,i,G,\mathrm{...}xn,i,G]i=1,2,\mathrm{...},k,

where G is the generation number. Defining the upper and lower bounds for each parameter yields

(13)xLj\le xj,i,1\le xUj\mathrm{.}

Therefore, the initial values of the parameters are set with identical probabilities.

Genetic algorithm

The GA is an evolutionary search algorithm that can be utilized to solve optimization problems [⁷⁸,⁷⁹]. It is based on the Darwinian principle of natural selection. The algorithm starts by randomly generating the initial population. Next, the fitness of the individuals is evaluated using the fitness function. Subsequently, in the selection stage, methods such as the Roulette wheel method are used. Crossover and mutation are two operators used to generate offspring as new solutions in the search space [⁸⁰].

ANFIS training procedure

Several studies have reported standalone ANFIS being trapped in a local optimum solution with a low convergence speed when their parameters were tuned [⁸¹,⁸²]. In addition, it was highlighted that the combination of an ANFIS with nature-inspired optimization algorithms can substantially enhance the convergence speed, thereby reducing the computational expenses of hybrid models not trapped in the local minima. As mentioned earlier, an ANFIS comprises two parameter sets, i.e., the antecedent (a_i and b_i in Eq. (4)) and consequent parameters (p_i, q_i, and r_i in Eq. (7)). A standalone ANFIS uses gradient-based approaches to obtain its parameters. However, obtaining the local optimum solution is the major weakness of these methods [⁸³].

In this study, the optimal values of the ANFIS parameters (antecedent and consequent parameters) were obtained using several nature-inspired optimization algorithms (e.g., PSO, ACO, DE, and GA). The implementation of evolutionary algorithms through ANFIS training is as follows.

1) The data are loaded to model.

2) An ANFIS model structure is created.

3) The primary values of the training parameters (antecedent and consequent parameters) are considered.

4) In the iterative process, the ANFIS parameters are tuned by the evolutionary algorithms. The evolutionary algorithms attempt to minimize the fitness function by obtaining the best combinations of the parameters mentioned in the previous section. This fitness function is based on the root mean square error (RMSE) index.

5) In each iteration, the value of the fitness function is evaluated.

6) If the termination criteria are satisfied, the best parameters of the fuzzy system and corresponding prediction are saved; otherwise, the processes beginning from step iv are repeated.

The training procedure of the ANFIS using nature-inspired algorithms is presented in Fig. 2.

The computational costs of the hybrid ANFIS models depend on several factors, including the parameters of the fuzzy c-means clustering (Table 1) and evolutionary algorithms (Table 2). They are defined by a trial-and-error procedure to prevent early convergences, generating precise results with an acceptable central processing unit (CPU) time.

Description of performance indices

To assess the accuracy of the proposed predictive models, performance indices such as the RMSE, mean absolute error (MAE), Legate and McCabe’s index (LMI), correlation coefficient (CC), PBIAS (percent bias), Willmott’s index (WI), and relative root mean square error (RRMSE) were employed. They are expressed as follows [⁸⁴–⁸⁹]:

(14)RMSE=\sqrt{\frac{1}{N_{\mathrm{s}}}{\displaystyle \sum _{j=1}^{N_{\mathrm{s}}}{\left({\left(f_{\mathrm{cc}}\right)}_{\exp }-{\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}\right)}^2}},

(15)RRMSE=\frac{\sqrt{\frac{1}{N_{\mathrm{s}}}{\displaystyle \sum _{j=1}^{N_{\mathrm{s}}}{\left({\left(f_{\mathrm{cc}}\right)}_{\exp }-{\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}\right)}^2}}}{\overline{{\left(f_{\mathrm{cc}}\right)}_{\exp }}},

(16)MAE=\frac{1}{N_{\mathrm{s}}}{\displaystyle \sum _{j=1}^{N_{\mathrm{s}}}\left|{\left(f_{\mathrm{cc}}\right)}_{\exp }-{\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}\right|},

(17)CC=\frac{{\displaystyle \sum _{j=1}^{N_{\mathrm{s}}}\left({\left(f_{\mathrm{cc}}\right)}_{\exp }-\overline{{\left(f_{\mathrm{cc}}\right)}_{\exp }}\right)\left({\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}-\overline{{\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}}\right)}}{\sqrt{{\displaystyle \sum _{j=1}^{N_{\mathrm{s}}}{\left({\left(f_{\mathrm{cc}}\right)}_{\exp }-\overline{{\left(f_{\mathrm{cc}}\right)}_{\exp }}\right)}^2}{\displaystyle \sum _{j=1}^{N_{\mathrm{T}}}{\left({\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}-\overline{{\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}}\right)}^2}}},

(18)WI=\mathrm{1}-\left[{\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{s}}}{({(f_{\mathrm{cc}})}_{\exp }-{(f_{\mathrm{cc}})}_{\mathrm{s}\mathrm{i}\mathrm{m}})}^{\mathrm{2}}}}{{{\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{s}}}\left(\left|{(f_{\mathrm{cc}})}_{\mathrm{s}\mathrm{i}\mathrm{m}}-\overline{{(f_{\mathrm{s}\mathrm{i}\mathrm{m}})}_{\exp }}\right|+\left|{(f_{\mathrm{cc}})}_{\exp }-\overline{{(f_{\mathrm{cc}})}_{\exp }}\right|\right)}}^{\mathrm{2}}}}\right],

(19)LMI=1-\left[\frac{{\displaystyle \sum _{i=1}^{N_{\mathrm{s}}}\left|{\left(f_{\mathrm{cc}}\right)}_{\exp }-{\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}\right|}}{{\displaystyle \sum _{i=1}^{N_{\mathrm{s}}}\left|{\left(f_{\mathrm{cc}}\right)}_{\exp }-\overline{{\left(f_{\mathrm{cc}}\right)}_{\exp }}\right|}}\right],

(20)PBIAS=\left[\frac{{\displaystyle \sum _{i=1}^{N_{\mathrm{s}}}\left({\left(f_{\mathrm{cc}}\right)}_{\exp }-{\left(f_{\mathrm{cc}}\right)}_{\mathrm{sim}}\right)}}{{\displaystyle \sum _{i=1}^{N_{\mathrm{S}}}{\left(f_{\mathrm{cc}}\right)}_{\exp }}}\right]\times 100,

where {(f_{\mathrm{cc}})}_{\exp }and {(f_{\mathrm{cc}})}_{\mathrm{sim}}are the experimental and simulated compressive strength of foamed concrete, respectively\overline{{(f_{\mathrm{cc}})}_{\exp }}. and \overline{{(f_{\mathrm{cc}})}_{\mathrm{sim}}} are their mean values, and N_S is the sample size.

Among the proposed predictive models, the superior model may be different in terms of different performance indices. This weakness can be solved using a new index called mean performance (MP), which integrates all employed indices. To compute the MP value of each predictive model, it is necessary to convert the indices to a standardized form in the range of [0 1] using the following Eqs. [⁹⁰]:

(21){\widehat{RMSE}}_{\mathrm{Model}(i)}={\displaystyle \frac{(RMS{E}_{\mathrm{All}}^{\max }-RMS{E}_{\mathrm{Model}(i)})}{(RMS{E}_{\mathrm{All}}^{\max }-RMS{E}_{\mathrm{All}}^{\min })}},

(22) {\widehat{MAE}}_{\mathrm{Model}(i)}={\displaystyle \frac{(MA{E}_{\mathrm{All}}^{\max }-MA{E}_{\mathrm{Model}(i)})}{(MA{E}_{\mathrm{All}}^{\max }-MA{E}_{\mathrm{All}}^{\min })}},

(23){\widehat{LMI}}_{\mathrm{Model}(i)}={\displaystyle \frac{(LM{I}_{\mathrm{Model}(i)}-LM{I}_{\mathrm{All}}^{\min })}{(LM{I}_{\mathrm{All}}^{\max }-LM{I}_{\mathrm{All}}^{\min })}},

(24){\widehat{CC}}_{\mathrm{Model}(i)}={\displaystyle \frac{(C{C}_{\mathrm{Model}(i)}-C{C}_{\mathrm{All}}^{\min })}{(C{C}_{\mathrm{All}}^{\max }-C{C}_{\mathrm{All}}^{\min })}},

(25){\widehat{PBIAS}}_{\mathrm{Model}(i)}={\displaystyle \frac{(PBIA{S}_{\mathrm{All}}^{\max }-PBIA{S}_{\mathrm{Model}(i)})}{(PBIA{S}_{\mathrm{All}}^{\max }-PBIA{S}_{\mathrm{All}}^{\min })}},

(26){\widehat{WI}}_{\mathrm{Model}(i)}={\displaystyle \frac{(W{I}_{\mathrm{Model}(i)}-W{I}_{\mathrm{All}}^{\min })}{(W{I}_{\mathrm{All}}^{\max }-W{I}_{\mathrm{All}}^{\min })}},

(27){\widehat{RRMSE}}_{\mathrm{Model}(i)}={\displaystyle \frac{(RRMS{E}_{\mathrm{All}}^{\max }-RRMS{E}_{\mathrm{Model}(i)})}{(RRMS{E}_{\mathrm{All}}^{\max }-RRMS{E}_{\mathrm{All}}^{\min })}},

where{\widehat{RMSE}}_{\mathrm{Model}(i)}, {\widehat{MAE}}_{\mathrm{Model}(i)}, {\widehat{LMI}}_{\mathrm{Model}(i)}, {\widehat{CC}}_{\mathrm{Model}(i)}, {\widehat{PBIAS}}_{\mathrm{Model}(i)}, {\widehat{WI}}_{\mathrm{Model}(i)}, {\widehat{RRMSE}}_{\mathrm{Model}(i)} and are the standardized values of the employed performance indices of the i^th model (hybrid ANFIS model); RMS{E}_{\mathrm{All}}^{\max }, RRMS{E}_{\mathrm{All}}^{\max }, LM{I}_{\mathrm{All}}^{\max }, C{C}_{\mathrm{All}}^{\max }, PBIA{S}_{\mathrm{All}}^{\max }, W{I}_{\mathrm{All}}^{\max }, and RRMS{E}_{\mathrm{All}}^{\max } are the maximum values of the indices among all predictive models; RMS{E}_{\mathrm{All}}^{\min }, RRMS{E}_{\mathrm{All}}^{\min }, LM{I}_{\mathrm{All}}^{\min }, C{C}_{\mathrm{All}}^{\min }, PBIA{S}_{\mathrm{All}}^{\min }, W{I}_{\mathrm{All}}^{\min }, and RRMS{E}_{\mathrm{All}}^{\min } are the minimum values.

Furthermore, the MP of the i^th model (hybrid ANFIS) can be expressed mathematically as follows [⁹¹]:

(28)M{P}_{\mathrm{Model}(i)}={\displaystyle \frac{\left(\begin{array}{l}{\widehat{RMSE}}_{\mathrm{Model}(i)}+{\widehat{MAE}}_{\mathrm{Model}(i)}+{\widehat{LMI}}_{\mathrm{Model}(i)}\\ +{\widehat{CC}}_{\mathrm{Model}(i)}+{\widehat{PBIAS}}_{\mathrm{Model}(i)}+{\widehat{WI}}_{\mathrm{Model}(i)}+\widehat{RRMS}{E}_{\mathrm{Model}(i)}\end{array}\right)}{\mathrm{7}}}.

From Eq. (28), it is clear that the MP value of each model is the mean value of its standardized performance indices, which is in the range of [0 1]. However, the superior model has the highest MP value compared with the other models.

Application results and analysis

The main purpose of the current study is to investigate the viability of using different versions of hybrid ANFIS models for predicting the compressive strength of foamed concrete. A concrete mixture combines several constituents, such as fine coarse aggregates, cement, mixed chemicals, water, additive cementitious materials, and sometimes fibers. The goal of concrete design is to achieve a high compressive strength.

However, the main concern in concrete design is that a high compressive strength is highly dynamic, nonstationary, varies nonlinearly with the constituents and their properties. Limitations regarding the use of empirical formulas to simulate the exact relationships between the compressive strength of concrete and other properties have motivated the use of robust and reliable predictive data-intelligence models to determine the effects of independent variables on dependent variables. In fact, such a reliable data-intelligence predictive model can contribute to the design of materials with reliable properties from various structural engineering perspectives.

Statistical performance of the proposed hybrid models

The developed hybrid ANFIS models were appraised with several input combinations using statistical metrics, diagnostic plots, and error distributions between the laboratory measures and compressive strength values predicted over the testing phase. Table 3 lists all possible input combinations based on four main correlated concrete properties, with emphasis on developing the predictive model.

Tables 4 to 8 present the statistical performance metrics (RMSE, MSE, LMI, CC, PIAS, WI, and RRMSE) for the classical ANFIS, ANFIS–PSO, ANFIS–ACO, ANFIS–DE, and ANFIS–GA models and all the proposed input combinations. In general, the variance prediction results achieved based on the developed classical and hybrid ANFIS models can be explained using different learning processes, whereas AI models can be achieved during the training and test modeling phases.

The hybridized ANFIS yielded the highest predictive accuracy owing to the tuning of the membership function in nature-inspired optimization. For a quantitative analysis and the best input combination, these tables (Tables 4–8) show the optimal prediction possibility using all the applied predictive models. Based on the results in Table 9, the minimum RMSE was accomplished using ANFIS–PSO (RMSE≈6.99), ANFIS–DE (RMSE≈7.20), ANFIS–GA (RMSE≈8.73), classical ANFIS (RMSE≈9.36), and ANFIS-ACO (RMSE≈9.72).

Graphical evaluation of the proposed hybrid models

Figure 3 shows a plot of the mean of all computed performance metrics for all predictive models over the test modeling phase based on several input combinations [⁵⁵]. These models included univariate, bivariate, trivariate, and four-variate models. Figure 3a demonstrates that the oven dry density as a univariate input was the most important attribute in determining the compressive strength using the hybrid ANFIS–DE predictive model (MP≈0.96).

However, the oven dry density as a univariate input was the best predictor for almost all the predictive models; therefore, this factor should be considered in the design of concrete mixtures. The bivariate input combination yielded poorer lower modeling results, with MP≈0.88 using ANFIS–PSO (Fig. 3b). The trivariate input combination yielded the second-best predictions using the same hybrid predictive model as for univariate modeling (ANFIS–DE & MP≈0.94) (Fig. 3c). The incorporation of the four predictors yielded an optimal MP≈0.89 when using the ANFIS–PSO hybrid predictive model (Fig. 3d).

Owing to the diverse prediction performances illustrated in Fig. 3, the best models were abstracted and validated using the input combinations of the other models (see Fig. 4). As shown in this figure, various variations in modeling predictability were avoided. Figure 4 shows that incorporating the cement content and oven dry density optimized the compressive strength prediction of foamed concrete when using the hybrid ANFIS–PSO, where the highest MP = 0.96. This clearly demonstrates the potential of the PSO tuning algorithm integrated with the standalone ANFIS in providing a robust and reliable predictive model for compressive strength prediction.

Figure 4 shows that the four-variate (C–O–W–F) input combination yielded the worst prediction possibility. This can be explained by the vague attributes supplied to the prediction matrix. Hence, it is not always effective to provide more information attributes to fit data with higher accuracy. On the contrary, building a hybrid predictive model based on related information can offer more possibilities for a positive learning process, which results in to an accurate prediction result.

The behavior of the model was visualized as a scatter plot (which shows the variance between the measured compressive strength and those computed using the models) (Fig. 5). The best predictive model was identified from the variance around the fitted line and the CC magnitude. Figure 5 shows the best predictive models and their input combinations. The ANFIS–PSO model (the bivariate case) yielded the best variation from the best fit line, with the highest determination coefficient R² = 0.89.

In Fig. 5, the experimental and forecast values are plotted on the x- and y-axes, respectively. The hollow circles represent the ANFIS–PSO model in the four-variate case, the solid circles represent the ANFIS–PSO model in the bivariate case, the triangles represent the ANFIS-DE model in the univariate case, and the blue crosses represent the ANFIS–DE model in the trivariate case. As shown in Fig. 5, the ANFIS–PSO model in the bivariate case indicated the highest CC (R² = 0.943). However, although the ANFIS–PSO model yielded an acceptable variation around the ideal line at 45°, the capturing of the compressive strength magnitude from 20 to 40 MPa was limited. This is attributable to the insufficient informative attributes for this compressive strength range.

Tables 4 to 8 present the performance statistics of the different combinations of input variables for the ANFIS, ANFIS–PSO, ANFIS–ACO, ANFIS–DE, and ANFIS–GA models, respectively. The values in these tables reveal the accuracy of the model. A model that yields a higher R² value and a lower residual error-based index demonstrates a better performance. The results indicate that the ANFIS–PSO with two input variables (C–O) yielded better predictions than the others (RMSE = 6.99958). Table 9 presents the performance statistics for the best input combinations for each developed predictive model over the testing phase.

Figure 6 presents the computed Taylor diagrams for the applied ANFIS, ANFIS–PSO, ANFIS–ACO, ANFIS–DE, and ANFIS–GA predictive models based on the selected optimal results shown in Fig. 4. It presents a statistical summary of the predicted and observed compressive strengths of foamed concrete, including the CCs, standard deviations, and RMSEs. The Taylor curve provides an excellent graphical representation of the similarity between the predicted and observed values. The results demonstrate that ANFIS–PSO with two input variables (C–O) yielded predictions that were the most similar to the measured benchmark.

The computed boxplot for the predictive models (Fig. 7) were analyzed to predict the compressive strength of foamed concrete, degree of spread in the predicted data, and quartiles (25, 50, 75, and the interquartile range). Based on the magnitudes of the lower (Q25%), median (Q50%), and upper (Q75%) quartiles, it was discovered that the ANFIS–PSO model outperformed the other classical and hybrid ANFIS models. Based on various statistical metrics and graphical representations, the proposed hybrid data-intelligence model is highly effective for predicting the compressive strength of lightweight foamed concrete.

To assess the tradeoff between the modeling accuracy and efficiency of the employed hybrid ANFIS models, their CPU times were compared, as shown in Fig. 8. Figure 8 indicates that ANFIS–PSO (442.82 s), and ANFIS–ACO (909.93 s) incurred the lowest and highest CPU times, respectively. It is evident that the modeling convergence capacity of the ANFIS–PSO model is better than those of the other hybrid models. This confirms the superiority of the ANFIS–PSO model in predicting the foamed concrete compressive strength accurately and efficiently compared with the other models.

AI models have been widely used in several applications in material science and engineering. Hence, material engineers are currently developing new and reliable AI models to obtain more reliable material mixture designs. In this context, the proposed hybrid ANFIS model provides an efficient and flexible mixture design compared with traditional models, which are conservative and costly.

Furthermore, the proposed model is highly compatible with new data sets and material mixtures for accurately predicting the compressive strength of foamed concrete. Moreover, a new fusion technology to eliminate boundaries between digital and engineering spheres was introduced in this study. The breakthroughs of novel AI models such as the proposed model are changing the paradigm for solving engineering problems. Hence, intelligent models such as the proposed hybrid ANFIS tool are applicable to future automated and semi-automated design platforms.

Validation of proposed models against literature models

For a fair evaluation of the proposed hybrid intelligence models, their results were compared with those of previous models. Three studies were selected to assess the accuracy of the proposed models in predicting the compressive concrete strength [⁹²–⁹⁴] (Table 10). Zarandi et al. [⁹²] proposed a fuzzy polynomial neural network (FPNN) to simulate the concrete compressive strength. To achieve a realistic model, six different FPNN models were developed with different structures. The best performance obtained using those models within the testing phase indicated CC values in the range of [0.35–0.82].

Chou et al. [⁹³] developed a few AI models including ANNs, multiple regression (MR), SVM, multiple additive regression trees (MART), and bagging regression trees (BRT) to predict the compressive strength of high-performance concrete using 17 concrete strength data sets. Based on the values of the performance indices obtained, it was revealed that the R² metric of the proposed models was within the range of [0.61–0.91], and the highest prediction power was obtained using the MART model (R² = 0.91).

Nikoo et al. assessed the potential of evolutionary ANNs to predict the compressive strength of concrete [⁹⁴]. The authors developed four different predictive models with different numbers of neurons and hidden layers. The obtained results showed optimistic results for the applied ANNs with CC = 0.89 over the testing phase. Compared with previous studies, the results of the current study indicate the better performance of the proposed model in predicting the foamed concrete compressive strength.

Overall modeling assessment and future studies

The current research was established based on the proposition of several hybrid ANFIS models for foamed concrete compressive strength prediction. The models were built based on multiple possible input combinations incorporating related concrete characteristics such as cement content, oven dry density, water-to-binder ratio, and foamed volume.

Based on the predictability performance assessed using statistical metrics and graphical visualization, the modeling results were varied from one model to another (and from one input combination to another). This demonstrated the existence of modeling uncertainty, which is one of the major limitations of the current study.

Another critical observation was the span of the investigated data set. The experimental data obtained from open-access published studies appeared to be insufficient for constructing the prediction matrix. Hence, based on these modeling limitations, the following can be performed in future studies.

1) Investigate the actual associated uncertainties, either by modeling the uncertainty or the input variable uncertainty [⁹⁵,⁹⁶].

2) Extend the span of the modeled data set based on related studies to provide more informative details regarding the input attributes.

3) Data modeling partition and various data partition percentages can be analyzed to achieve better learning processes of the established models.

4) Other crucial variables can be considered as input attributes for the prediction matrix. In reference to a recent study [⁹⁷], the authors investigated the effect of ash content on the strength of foamed concrete. The addition of fly ash reduced the compression strength, whereas increasing the cement content increased the compressive strength. Hence, fly ash must be included as an additional variable and its effect on the compressive strength investigated. Furthermore, it is noteworthy that the dependence of the compressive strength on the fly ash content varies by cement content. Furthermore, the types of cement/ash/foaming agent (synthetic or organic) can affect the magnitude of the compressive strength.

5) Input parameter sensitivity analysis should be performed as it is an essential modeling process for recognizing the effect of each parameter on the targeted compressive strength of foamed concrete [⁹⁸,⁹⁹].

6) The optimization algorithms were implemented in this study and variance modeling results were obtained. Hence, other optimization algorithms (e.g., inverse analysis method, nomadic people optimizer, and firefly optimizer) can be investigated to enhance the predictability performance of the ANFIS model [¹⁰⁰–¹⁰³].

vii. As an advanced AI model, the deep learning (DL) neural network is one of the well-established networks established recently for material science engineering [¹⁰⁴,¹⁰⁵]. Hence, the feasibility of the DL model can be investigated for predicting the compressive strength of foamed concrete.

Conclusions

Foamed concrete is typically used in construction; it is used in the form of precast elements (for blocks and walls) and insulation screeds (for floors and roofs) and provides excellent thermal and acoustic insulation. An effective estimation of the concrete compressive strength of foamed concrete from the characteristics of its constituents, such as cement content (C), oven dry density (O), water-to-binder ratio (W), and foamed volume (F) is challenging for structural engineers. Hence, the feasibility of using some optimization methods (PSO, ACO, DE, and GA) was evaluated for developing hybrid adaptive neuro-fuzzy inference system models (ANFIS–PSO, ANFIS–ACO, ANFIS–DE, and ANFIS–GA) to accurately predict the compressive strength of foamed concrete.

To obtain the best predictive model and input for predicting the compressive strength of foamed concrete, various combinations of input variables (univariate (C, O, W and F), bivariate (C–O, C–W, C–F, O–W, O–F, and W–F), trivariate (C–O–W, C–W–F, O–W–F), and four-variate (C–O–W–F)) were considered for each hybrid model. To measure the performance of the predictive models, a new error index, i.e., the mean performance (MP) was used; it incorporates other performance indices–RMSE, RRMSE, MAE, LMI, CC, PIAS, and WI. The results indicated that the oven dry density exerted the most significant effect on the compressive strength of foamed concrete.

The oven dry density was used as a predictive input variable in all of the best predictive models with different input variables (ANFIS–DE–(O), ANFIS–PSO–(C–O), ANFIS–DE–(O–W–F), and ANFIS–PSO–(C–O–W–F)). The ANFIS–PSO models performed the best among the bivariate (MP = 0.88) and four-variate (MP = 0.89) models. The ANFIS–PSO–(C–O) (MP = 0.96) was the most accurate model among all the considered models. Therefore, hybrid ANFIS–PSO model is a reliable model for predicting the compressive strength of foamed concrete.

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[101] Vu-Bac N, Duong T X, Lahmer T, Areias P, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis of thermal expansion induced morphing of thin shells. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 480–510

[102] Salih S Q, Alsewari A A. A new algorithm for normal and large-scale optimization problems: Nomadic People Optimizer. Neural Computing & Applications, 2019, 32: 1–28

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[104] Hamdia K M, Ghasemi H, Bazi Y, AlHichri H, Alajlan N, Rabczuk T. A novel deep learning based method for the computational material design of flexoelectric nanostructures with topology optimization. Finite Elements in Analysis and Design, 2019, 165: 21–30

[105] Guo H, Zhuang X, Rabczuk T. A deep collocation method for the bending analysis of Kirchhoff plate. Computers, Materials and Continua, 2019, 59(2): 433–456

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