Improvement of ANFIS Model for Prediction of Compressive Strength of Manufactured Sand Concrete

Abstract

Use of manufactured sand to replace natural sand is increasing in the last several decades. This study is devoted to the assessment of using Principal Component Analysis (PCA) together with Teaching-Learning-Based Optimization (TLBO) for enhancing the prediction accuracy of individual Adaptive Neuro Fuzzy Inference System (ANFIS) in predicting the compressive strength of manufactured sand concrete (MSC). The PCA technique was applied for reducing the noise in the input space, whereas, TLBO was employed to increase the prediction performance of single ANFIS model in searching the optimal weights of input parameters. A number of 289 configurations of MSC were used for the simulation, especially including the sand characteristics and the MSC long-term compressive strength. Using various validation criteria such as Correlation Coefficient (R), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE), the proposed method was validated and compared with several models, including individual ANFIS, Artificial Neural Networks (ANN) and existing empirical equations. The results showed that the proposed model exhibited great prediction capability compared with other models. Thus, it appeared as a robust alternative computing tool or an efficient soft computing technique for quick and accurate prediction of the MSC compressive strength.

1. Introduction

In recent years, manufactured sand has been progressively used to replace natural sand in building materials [1]. This new type of material, produced by crushing rock depositions, exhibits some morphology features that are unique and cannot be found in natural sand, such as the angular and rougher texture [2,3]. Unlike manufactured sand, the common negative characteristic while using natural sand in concrete is the alkali-silica reactions between the reactive silica components and the alkali hydroxides derived from the cement. Such reactions lead to the expansion, deterioration and even failure of concrete structural elements [4]. On the contrary, using manufactured sand is advantageous with the possibility of preselection of low silica content rocks as to avoid such alkali-silica reactions. Besides, manufactured sand could positively influence the strength and durability of concrete, mainly due to better interlocking between highly irregular particles [2] or a better bonding between steel bar and concrete [5]. Indeed, as demonstrated by Li et al. [6] in an experimental investigation, machine-made sand particles exhibit usually a higher degree of angularity than river sand particles (i.e., in terms of morphology). The same conclusion was also achieved by Li et al. [6] in terms of surface texture. Logically, they found that the compressive and flexural strengths of pavement cement concrete, as well as its abrasion resistance, have been largely improved compared to conventional sand concrete. Various other comparative studies have been introduced in the literature in proving the performance of machine-made sand such as Gonçalves et al. [7], Yamei et al. [8] or Mundra et al. [9]. Therefore, manufactured sand has currently caught great attention from researchers, as well as the whole construction industry [10,11].

In the literature, many works have focused on the quantification of how the concrete workability and durability could be influenced by using manufactured sand [6,12], especially its compressive behavior [1,11,13,14,15]. In a series of papers, Shen et al. [3,13] have identified the influence of the shape of manufactured sand and roughness on the concrete compressive strength based on digital image analysis and experimental compression tests. The authors have also pointed out that the stone powder of manufactured sand exhibited a considerable effect on the performance of concrete. In another study, Li et al. [6] have prepared manufactured sand concrete (MSC) samples from different petrographic sands such as limestone, quartzite, granite, basalt and granite gneiss, respectively. The authors discovered that morphology and texture characteristics of manufactured sand hugely influenced the performance of concrete. As an example of results, the flexural strength and abrasion resistance of MSC were better than natural sand concrete if the crushing value of manufactured sand particles is less than 26.5%. It was also proven in many other experimental studies that MSC exposed better compressive behavior than that composed of natural river sand [16,17].

One of the most important parameters of MSC is the compressive strength, which is often obtained by experimental laboratory tests. However, these laboratory investigations were generally complexes, cost and time consuming. Moreover, the laboratory tests were often limited within 90–180 days periods [18]. Despite all the efforts, it was not often possible to perform too many variations of ingredients (i.e., proportion of cement, water and machine-made sand as well as stone powder content in sand). Recently, Artificial Intelligence (AI) based methods have been successfully employed in the field of materials [19], especially for predicting compressive strength of concrete [20,21,22,23,24]. Dao et al. [25,26] have proposed a comparative study between Adaptive Neuro Fuzzy Inference System (ANFIS), Artificial Neural Networks (ANN) and Support Vector Machine (SVM) in predicting the compressive strength of geo-polymer concrete based on mixture inputs. Similarly, Golafshani et al. [27] has optimized the mix design of silica fume concrete in order to obtain desired compressive strength using Biogeography-based programming. In another study, Bingöl et al. [28] has shown that the ANN model provided more accurate results than traditional regression in predicting compressive strength of lightweight concrete under heat treatment. Mishra et al. [29] has used ANN and ANFIS for investigating the compressive strength of brick–mortar masonry. In combining AI methods with global optimization techniques, Bui et al. [30] have implemented whale algorithm in order to better optimize the weight of a ANN model when predicting compressive strength of concrete. In another attempt, Behnood et al. [31] have also predicted compressive strength of silica fume concrete based on a hybrid model involving ANN and multi-objective grey wolves technique. Based on these mentioned studies, it can be stated that the AI based methods could be able to analyze the nonlinear relationship between ingredients and compressive strength of various types of concrete for better prediction and assessment [32,33,34,35,36].

In this study, the main objective is to improve the performance of individual ANFIS model, which is one of the effective AI models, by using two optimized techniques namely Principal Component Analysis (PCA) and Teaching-Learning-Based Optimization (TLBO) for better prediction of the compressive strength of MSC. Out of these optimized techniques, PCA technique was applied for reducing the noise in the input space whereas TLBO was employed to increase the prediction performance of single ANFIS model in searching the optimal weights of input parameters. A number of 289 compression tests of MSC were gathered from the available literature, including inputs parameters such as mixture proportions, cement’s compressive strength, water content, as well as manufactured sand’s characteristics. The target response of the present study is the long-term compressive strength of MSC, ranging from 3 to 388 days of curing age. Correlation Coefficient (R), Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) were employed to validate and test the effectiveness of the proposed model, as well as to compare with other AI models such as individual ANFIS and ANN.

2. Research Significance

As indicated in the introduction part, the estimation of the compressive strength of MSC is important in civil engineering applications. Although miscellaneous experimental investigations have reported this problem, it is difficult to deduce a generalized formulation, taking into account all the parameters that affect compressive strength of MSC, especially in terms of sand’s characteristics. The employment of AI approaches could help to look into the (nonlinear or not) relationships between the desired outputs and the corresponding inputs. That way, the influence on the MSC’s compressive strength of input variables, particularly sand’s characteristics, could be fully achieved and quantified. Moreover, pre-processing data technique such as PCA and optimization technique namely TLBO could be applied to improve the performance of a popular ANFIS model for better accurate prediction of the compressive strength of MSC. Thus, it can be a great approach which could help engineers and/or researchers to decrease cost and time in laboratory experiments for determination of the compressive strength of MSC.

3. Materials and Methods

3.1. Adaptive Neuro Fuzzy Inference System (ANFIS)

Early introduced in the 1990s by Jang [37], ANFIS is well-known as a hybrid AI model in merging ANN [38,39,40,41] and Fuzzy Logic (FL) [42]. The ANFIS architecture consists of five principal layers such as fuzzification, rule, normalization, defuzzification and aggregation [37,43,44,45,46]. Based on such construction, the neural network exhibits the ability to identify the parameters of FL algorithm [47,48,49]. In ANFIS, the Takagi–Sugeno if–then rules and appropriate membership function are employed for the fuzzy inference system [50,51]. Similar to ANN, hybrid ANFIS technique has also an ability to identify the nonlinear relationship between inputs and outputs [43,52,53]. It is worth noticed that ANFIS exhibits a better prediction efficiency than individual ANN or FL, as demonstrated in several investigations, such as in the works of Mukerji Aditya et al. [54] or Nayak et al. [55]. However, there are several limitations in ANFIS model such as ANFIS is not powerful in searching the best firing strength (i.e., weight) [56,57], which greatly impact the prediction effectiveness [47]. Various investigations have employed different optimization methods to find the weighs of parameters in a better way, for instance using Genetic Algorithm, Particle Swarm Optimization, Grey Wolf Optimizer or Invasive Weed Optimization [58,59,60,61,62,63]. In this paper, we proposed the Teaching-Learning-based Optimization (TLBO) to optimize the parameter’s weights in ANFIS.

3.2. Teaching-Learning-Based Optimization (TLBO)

TLBO is an optimization technique that is recently introduced for solving scheduling problems [64,65,66]. As the name indicated, TLBO includes two main phases: the “Teacher Phase”, where the model learns something from a teacher and the “Learner phase”, where the model learns itself by interacting with other learners. The structure of TLBO can be divided in three main steps as the followings [67,68,69,70]:

3.2.1. Initialization of the Population

In this step, a population of size N is generated along with the initial parameters such as number of jobs or number of machines. Each element in the population is considered as a student in a class.

3.2.2. Teacher Phase

In this step, the main objective of the teacher is to increase the mean result of the class in a subject based on each student’s capability. Let us define “kbest” is the best learner in the class and X_kbest is the highest overall result in the entire class in all subjects. The mean result of the class can be improved using the difference between the current mean result and the result given by the teacher such as:

$$Diff=r_i\left(X_{kbest}-T_F{M}_{j,i}\right),$$

(1)

where r_i is a random number chosen between 0 and 1, T_F is the teaching factor and M_j,i is the mean of all learner’s results in subject j at iteration i.

The current solution is then updated using the following equation:

$$X_i^{\prime }=X_i+Diff,$$

(2)

where X_i is the current solution at iteration i and X_i^’ is the updated solution. If the updated solution gives a better function value, then accept it, otherwise keep the current solution.

3.2.3. Learner Phase

In this step, the learners increase their results by interacting between themselves. Considering two learners A and B, the solution is updated using the following relations:

$$X_{j,i,A}^{"}=X_{j,i,A}^{\prime }+r_i\left(X_{j,i,A}^{\prime }-X_{j,i,B}^{\prime }\right)\mathrm{if}{X}_{j,i,A,total}^{\prime }<X_{j,i,B,total}^{\prime },$$

(3)

$$X_{j,i,A}^{"}=X_{j,i,A}^{\prime }+r_i\left(X_{j,i,B}^{\prime }-X_{j,i,A}^{\prime }\right)\mathrm{if}{X}_{j,i,A,total}^{\prime }\ge X_{j,i,B,total}^{\prime }$$

(4)

If the updated solution gives a better function value then it is accepted. When the number of generations is completed, then the algorithm is stopped.

3.3. Principal Component Analysis (PCA)

In multi-variable problems, particularly in the case of applying AI models, PCA exhibits many advantages in the step of pre-processing data. PCA could help the training phase of AI models in: (i) assessing the feasibility of space reduction when the problem has too many inputs [71,72], (ii) preventing the overtraining, as recommended in the work of Defernez et al. [73], and (iii) reducing the noise in the input data [74,75,76]. Using PCA technique, important statistical information in the data has been conserved without dropping any typical characteristics [77,78]. The main steps of PCA technique are given below [71,79,80]:

• Preparation and normalization of inputs;
• Calculation of the covariance matrix;
• Calculation of the eigenvalues and eigenvectors;
• Estimation of the proportion of total variance of each principal component;
• Identification of the loading of principal components and contribution of inputs.

In this study, PCA was applied in order to reduce noise in the input space for training the models. Two datasets were prepared: one with raw inputs and the data with PCA pre-processed. These two datasets were used as inputs for the training and testing phases of the models as to demonstrate the effectiveness of PCA technique.

3.4. Collection of Data

In this study, the target output is the compressive strength (both cubic and cylinder) of MSC at different curing ages ranging from 3 to 388 days. The database was collected from experimental tests in the available literature [14,81,82,83]. A number of 289 MSC samples have been mixed by raw materials such as: (i) ordinary silicate cements; (ii) admixture consisted of fly ash, slag and silica fume; (iii) crushed stone and (iv) manufactured sand. The properties of the mentioned materials (i.e., compressive strength and tensile strength of cement at 28 days), the mix proportions as well as MSC’s compressive strength were detailed in the database, as summarized in

Table 1. Statistical analysis of inputs and output used in this study.

Parameter	Compressive Strength of Cement	Tensile Strength of Cement	Curing Age	Dmax of Crushed Stone	Stone Powder Content in Sand	Fineness Modulus of Sand
Unit	MPa	MPa	Days	mm	%	[-]
Notation	I1	I2	I3	I4	I5	I6
Min	35.50	6.90	3.00	16.00	0.00	2.20
Average	47.95	8.25	80.93	28.31	7.54	3.06
Median	46.80	8.00	28.00	31.50	6.60	3.15
Max	63.40	10.20	388.00	31.50	20.00	3.50
Std	4.29	0.60	102.36	3.68	4.48	0.27
CV (%)	8.95	7.29	126.48	12.99	59.42	8.98
Parameter	Water to Binder Ratio	Water to Cement Ratio	Water	Sand Ratio	Slump	Cubic Compressive Strength of Concrete *
Unit	[-]	[-]	kg/m3	%	mm	MPa
Notation	I7	I8	I9	I10	I11	Y
Min	0.25	0.31	120.00	28.00	11.00	19.00
Average	0.43	0.46	175.49	37.23	98.34	55.80
Median	0.45	0.45	180.00	36.00	70.00	56.45
Max	0.69	0.69	291.00	44.00	260.00	96.30
Std	0.09	0.07	15.16	4.00	66.64	16.70
CV (%)	20.81	14.42	8.64	10.74	67.77	29.93

* Cylinder compressive strength = 0.82 × Cubic compressive strength.

. A factor of 0.82 was used to convert the cylinder and cubic compressive strength of MSC as suggested by Zhao et al. [83]. As indicated, the cubic compressive strength of MSC, denoted by Y, ranges from 19 to 96.3 MPa, with an average value of 55.80 MPa, a standard deviation of 1670 MPa (corresponding to a coefficient of variation CV of 29.93%). The characteristics of sand such as stone powder content and fineness modulus are also indicated. As reported by Shen et al. [13], stone powder content in sand was most influenced factor on the compressive performance of MSC.

Initial data analysis (histogram) was performed and showed that most inputs, except I₃, I₄ and I₉, are well distributed and suitable for the training of the models (i.e., the histogram is close to a Gaussian distribution). For that reason, PCA was performed in this study to reduce the noise in the 11-dimensional input space. As mentioned above, two datasets were prepared for training the models using the raw and PCA pre-processed inputs. The detailed analysis of PCA technique are indicated in the results and discussions section.

3.5. Quality Assessment Criteria

In this study, three common-used quality assessment criteria such as R, MAE and RMSE were calculated to evaluate the prediction performance of the models. Detailed information and formulations of these criteria could be found in the literature [84,85,86,87,88,89,90,91].

4. Results and Discussions

4.1. PCA’s Results

Table 2. Principal Component Analysis (PCA’s) results. Principal Component Analysis (PCA).

Input	PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8	PC9	PC10	PC11
I1	14.02	5.43	17.64	0.92	12.21	0.24	0.18	2.43	2.92	8.73	35.28
I2	11.81	4.96	26.94	0.00	5.63	1.38	0.00	0.05	2.70	12.69	33.85
I3	3.57	1.07	3.60	40.88	36.66	0.86	0.34	10.97	1.84	0.21	0.00
I4	18.43	1.64	0.81	1.73	5.55	2.68	1.27	61.74	2.57	0.05	3.53
I5	10.04	0.58	3.31	10.17	2.96	57.17	6.72	0.42	5.69	2.93	0.00
I6	0.09	1.55	10.78	42.35	33.39	3.42	4.56	0.22	2.40	1.05	0.19
I7	18.97	10.34	4.49	0.01	1.73	0.16	0.93	0.18	11.30	31.64	20.26
I8	6.48	31.74	0.04	0.06	0.35	0.91	1.91	0.07	15.77	37.60	5.07
I9	9.87	1.82	15.68	0.04	0.51	0.17	57.86	12.10	0.08	1.80	0.06
I10	1.70	34.07	2.63	0.85	0.02	0.96	0.28	2.83	51.88	3.23	1.54
I11	5.03	6.79	14.08	3.00	0.99	32.05	25.95	9.00	2.84	0.07	0.21
EV	29.71	19.71	14.32	9.80	7.57	6.60	5.25	3.69	2.30	0.90	0.16
CS	29.71	49.42	63.73	73.54	81.10	87.70	92.95	96.64	98.94	99.84	100.00

EV: explained variance (%); CS: cumulative sum (%).

indicates the results of PCA, including: (i) the contribution of 11 inputs to 11 principal components (denoted by PC_k with k = 1:11); (ii) the explained variance (EV) of each PC_k and (iii) the cumulative sum (CS) of explained variance, respectively. It can be seen that PC₁ exhibits 29.71% and the first 9 PC_s expose 98.94% of total variance. The higher explained variance, the more statistical information of the input space is propagated. The PCA pre-processed dataset was then created for training the models together with raw inputs.

4.2. Optimization Procedure: Determination of Optimal Population Size

In this section, a parametric study for optimizing the number of populations of TLBO is presented. Simulations were performed using the number of populations ranging from 10 to 200 with a step of 10. Quality criteria, i.e., Correlation Coefficient (R), RMSE and MAE, were employed to identify the optimal number of population size, as shown in Figure 1a,b for R and RMSE, respectively. It should be noticed that a maximum number of iterations of 1000 was applied in the parametric study. As can be seen, an optimal number of population size of 40 was obtained, based on two assessments: (i) the values of R and RMSE at population size of 40 and (ii) the time consuming. Indeed, the higher population size used, the more computer processing time needed. Figure 2 shows the convergence of optimization cost with respect to R and RMSE using the optimal population size of 40. It is shown that 1000 iterations were relevant as the stopping criterion in order to obtain optimized results with respect to both R and RMSE.

4.3. Prediction Capability: Improvement of Single ANFIS

In this section, the prediction capability of the proposed models is analyzed thanks to statistical error criteria such as R, RMSE and MAE. A linear fitting line, defined as “Predicted = Slope *Actual + C” is also used for the prediction capability analysis of these models. The statistical results of the ANFIS model optimized by TLBO and using PCA, denoted as ANFIS-TLBO/P, including the values of R, RMSE, MAE, error mean, error standard deviation and slope are summarized in

Table 3. Summary information of ANFIS-TLBO/P model after using PCA. Adaptive Neuro Fuzzy Inference System (ANFIS). Teaching-Learning-Based Optimization (TLBO). Principal Component Analysis (PCA). Mean Absolute Error (MAE).

Criteria	Training Dataset	Testing Dataset
R	0.92	0.96
RMSE	6.62	4.93
MAE	4.77	4.09
Error mean	−0.08	0.26
Error Std	6.64	4.95
Slope	0.86	0.94

, for the training and testing data sets. The correlation between actual and predicted compressive strength of ANFIS-TLBO/P was also plotted in Figure 3. For the training part, it can be noted that the correlation coefficient of ANFIS-TLBO/P was R = 0.92 with a slope of 0.86. For the testing part, the correlation coefficient was R = 0.96 with a slope of 0.94. In addition, the comparison of these criteria between ANFIS-TLBO/P and other models is also shown in

Table 4. Comparison of prediction capability of different models using raw and pre-processed data, respectively. Correlation Coefficient (R), Root Mean Squared Error (RMSE).

Model	Data Used	Designation	R	RMSE	MAE	Error Std	Slope
Individual ANFIS	raw	ANFIS/R	0.86	8.58	6.24	8.63	0.78
Individual ANFIS	pre-processed	ANFIS/P	0.93	6.46	5.28	6.49	0.89
Individual ANN	pre-processed	ANN/P	0.90	7.67	5.06	7.67	0.77
ANFIS+TLBO	raw	ANFIS-TLBO/R	0.88	7.65	5.02	7.59	0.76
ANFIS+TLBO	pre-processed	ANFIS-TLBO/P	0.96	4.93	4.09	4.95	0.94

. Let ANFIS/R and ANFIS/P are defined as the single ANFIS model using raw data and the processed data using PCA technique, respectively. Besides, the ANN model using PCA technique is designed as ANN/P, whereas ANFIS-TLBO/R is the ANFIS optimized with TLBO without the PCA treatment. In comparison with other techniques for the testing dataset, it can be seen that ANFIS-TLBO/P had the highest values of R and slope. Indeed, the value of R for ANFIS/R, ANFIS/P and ANFIS-TLBO/R was 0.86, 0.88 and 0.93, respectively. On the other hand, the value of slope for these three models was 0.78, 0.76 and 0.89, respectively.

Considering now RMSE and MAE, it can be seen that the values of RMSE, MAE, error mean and error standard deviation of ANFIS-TLBO/P were 6.62, 4.77, −0.08 and 6.64, respectively for the training part; and 4.93, 4.09, 0.26 and 4.95, respectively, for the testing part. In comparison with other models with respect to the testing part, ANFIS-TLBO/P produced the highest performance, meaning the lowest values of RMSE, MAE and error standard deviation. In particular, the values of these criteria were 8.58, 6.24 and 8.63 for ANFIS/R; 7.65, 5.02 and 7.59 for ANFIS-TLBO/R; and 6.46, 5.28 and 6.49 for ANFIS/P, respectively. In addition, ANN/P was also applied to check the performance of the ANFIS-TLBO/P model and the results show that ANFIS-TLBO/P outperforms ANN/P (Table 4).

For conclusion, firstly, PCA technique helped increasing the prediction performance of the AI models by reducing noise in the input space. Secondly, TLBO technique well optimized the parameters of individual the models. Finally, out of all tested models, ANFIS-TLBO/P was the best model for the prediction of MSC compressive strength, in taking into account sand’s characteristics.

4.4. Sensitivity Analysis

In this section, the influence of Principal Components (PCs) as well as inputs on the prediction of compressive strength of MSC is analyzed. The influence is characterized by the sensitivity index (in %), which was calculated based on various levels of variables from quantiles 0 to 100%. Details of the calculation could be found in the literature [61,92]. Figure 4a, 4b show the sensitivity index of 11 PCs and inputs, respectively, in a descending order. All explicit values are indicated in

Table 5. Sensitivity index of Principal Components (PCs) and Inputs, respectively.

PCs	PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8	PC9	PC10	PC11
Sensitivity index (%)	12.48	17.04	10.53	17.41	15.16	1.80	0.72	11.68	4.37	3.85	4.95
Inputs	I1	I2	I3	I4	I5	I6	I7	I8	I9	I10	I11
Sensitivity index (%)	9.04	8.32	15.07	11.36	5.41	14.12	7.61	8.71	5.19	9.27	5.89

. It should be noticed that the contribution of each inputs on PCs was indicated in Table 2.

Figure 4b shows that the most three distinguished important input are curing age (I₃), following by fineness modulus of sand (I₆) and maximum diameter of crushed stone (I₄). The sand ratio (I₁₀), compressive strength of cement at 28 days (I₁), water to cement ratio (I₈), tensile strength of cement at 28 days (I₂) and water to binder ratio (I₇) form the second group of degree of importance. The last three inputs, such as slump (I₁₁), stone powder content in sand (I₅) and water (I₉), each contributes at least 5% for the compressive strength of MSC.

4.5. Comparison with Existing Models in the Literature

In this section, the prediction performance of ANFIS-TLBO/P model is compared with existing formula in literature. In the work of Ding et al. [14], an empirical equation has been proposed in order to estimate the MSC compressive strength based on the cement compressive strength at 28 days, the cement density and the water to cement ratio.

Table 6. Comparison between ANFIS-TLBO/P and existing formula in the literature.

Curing Age	Model	R	RMSE	MAE	Error Std	Slope
7 days	Ding et al. [14] (Equation (6) in [14] *)	0.61	14.42	10.25	10.41	1.77
	Our model	0.90	7.43	5.22	6.06	0.99
	Δ (%)	+29.00	+48.49	+49.03	+41.78	+76.00
28 days	Ding et al. [14] (Equation (5))	0.92	5.31	4.44	5.32	0.94
	Our model	0.95	5.20	4.08	5.18	0.87
	Δ (%)	+3.00	+2.07	+8.11	+2.63	−7.00
56 days	Ding et al. [14] (Equation (6) in [14] *)	0.91	6.36	3.90	5.38	0.68
	Our model	0.87	5.94	5.43	4.46	0.83
	Δ (%)	−4.00	+6.53	−39.23	+17.01	+15.00
3–388 days	Ding et al. [14] (Equation (6) in [14] *)	0.77	12.82	9.96	12.54	0.66
	Our model	0.93	6.16	4.57	6.17	0.88
	Δ (%)	+16.00	+51.92	+54.11	+50.76	+22.00

* a coefficient of c = 0.3 was used according to [82].

indicates the value of performance indicators such as R, RMSE, MAE, Error standard deviation and slope, using ANFIS-TLBO/P and the formula proposed by Ding et al. [14]. The comparison is presented for 7, 28, 56 and all curing age, respectively. For other curing ages such as 14, 60, 70, 84, 90 and 180, the number of samples is lower than 10 and thus not be considered in the comparison. In Table 6, ∆ is the deviation (in %) of performance indicators between the present work ANFIS-TLBO/P model and Ding et al. [14]. Noting that the values of ∆ with respect to R and slope were calculated based on the deviation around 1, whereas the values of ∆ corresponding to RMSE, MAE and Error Std were computed based on changes regarding zero, as indicated by the following equation:

$$\Delta =\{\begin{array}{l}\left(({\lambda }^{\mathrm{our-model}}-1)-({\lambda }^{\mathrm{original-model}}-1)\right)\times 100\mathrm{in}\mathrm{case}\mathrm{of}:\mathrm{R},\mathrm{Slope}\\ \left(({\lambda }^{\mathrm{our-model}}-{\lambda }^{\mathrm{other-model}})/{\lambda }^{\mathrm{original-model}}\right)\times 100\mathrm{in}\mathrm{case}\mathrm{of}:\mathrm{RMSE},\mathrm{MAE}\mathrm{and}{\mathrm{Error}}_{\mathrm{std}}\end{array}$$

(5)

In cases of curing age of 7 and 28 days, the developed ANFIS-TLBO/P model provided better performance indicators than the proposed exponential functions in Ding et al. [14]. In case of curing age of 56 days, negative effects were observed regarding to R and MAE. However, in terms of RMSE, error Std and slope, the proposed model gave the better prediction performance. Overall, from 3–388 days of curing age, the proposed ANFIS-TLBO/P model outperforms the empirical formulas in Ding et al. [14], with respect to R, RMSE, MAE, error Std and slope. It is worth noticed that Ding et al. [14] have optimized their equations based on only three variables, such as cement compressive strength at 28 days, cement density and water to cement ratio. Other factors have not been considered in the exponential equation form. Consequently, the influence of sand’s characteristic, especially stone powder content, have not been investigated. In addition, it is worth mentioned that as reported in Shen et al. [13], stone powder content in sand had a major effect on the compressive strength of MSC. Moreover, the impact of other sand’s characteristics on the compressive behavior of MSC is also discussed in the works of Nanthagopalan et al. [1] and Ji et al. [2]. In short, the ANFIS-TLBO/P model developed in this study provided better prediction capability of compressive strength of MSC than other approaches and models in taking into account the variation of sand’s characteristics in the simulation.

5. Conclusions

In this study, two optimized soft computing techniques such as PCA and TLBO were applied to improve the performance of individual ANFIS model for predicting the compressive strength of MSC. A number of 289 experimental tests of MSC were used as dataset for the simulation including inputs parameters such as mixture proportions, cement’s compressive strength, water content as well as manufactured sand’s characteristics. Various validation criteria such as R, RMSE and MAE were employed to validate and test the performance of the proposed model, as well as to compare with other AI models such as individual ANFIS and ANN with or without PCA treatment. Based on the results of this study, several conclusions could be deduced: (i) data pre-processing technique such as PCA is indispensable in order to improve the prediction capability of the AI models; (ii) TLBO algorithm exhibited a strong efficiency in finding the optimal parameters of individual ANFIS model; (iii) hybrid ANFIS-TLBO/P model is the best model for predicting compressive strength of MSC in compared to other models; and (iv) sand’s characteristics should be integrated in the input space for the simulation of the AI models for predicting the MSC compressive strength.

In terms of perspectives, uncertainty quantification should also be addressed for quantifying the confidence interval of MSC’s compressive behavior in the presence of input variability.