# Prediction of Rapid Chloride Penetration Resistance to Assess the Influence of Affecting Variables on Metakaolin-Based Concrete Using Gene Expression Programming

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## Abstract

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_{c}), genes (N

_{g}) and, the head size (H

_{s}) of the gene expression programming (GEP) model were varied to study their influence on the predicted RCP values. The performance of all the GEP models was assessed using a variety of performance indices, i.e., R

^{2}, RMSE and comparison of regression slopes. The optimal GEP model (Model T3) was obtained when the N

_{c}= 100, H

_{s}= 8 and N

_{g}= 3. This model exhibits an R

^{2}of 0.89 and 0.92 in the training and testing phases, respectively. The regression slope analysis revealed that the predicted values are in good agreement with the experimental values, as evident from their higher R

^{2}values. Similarly, parametric analysis was also conducted for the best performing Model T3. The analysis showed that the amount of binder, compressive strength and age of the sample enhanced the RCP resistance of the concrete specimens. Among the different input variables, the RCP resistance sharply increased during initial stages of curing (28-d), thus validating the model results.

## 1. Introduction

^{2}= 0.98 and 0.96, and RMSE = 28.6 and 41.4 for the training and testing phases, respectively. Similarly, Yaman et al. [35] employed ANN for predicting the constituents of self-compacting concrete. For this purpose, two different methodologies viz., ANN model with multi input–multi output and ANN model with multi input–single output network were developed using 28 days of compressive strength and diameter of slump flow as the input parameters. It was found that the ANN model with multi input–single output methodology produced better results for the outputs, as evident from their higher R

^{2}values (0.63–1.0).

^{2}> 0.85) and, at the same time, derive a new empirical equation for the output in terms of input variables.

## 2. Methodology

#### 2.1. Database Compilation

^{3}), water-to-binder (w/b) ratio, MK (%) and the compressive strength value. Table 1 shows the descriptive statistics of the database that was used to develop the GEP models. Moreover, Figure 1 illustrates the frequency histograms of the input variables. It can be inferred from the histograms that except for the compressive strength, a majority of the input variables, such as age of the sample, b, Fag, Cag and w/b, were not normally distributed. Since the distribution of data depends upon the source used, it is not necessary that the data be normally distributed [23]. Similarly, it can be seen from Table 2 that the kurtosis value of binder was positive, while all other values of the input parameters were negative. Since, kurtosis represents the deviation of the distribution’s tail from the normal distribution tail, these values are also in accordance with the plots in Figure 1.

#### 2.2. GEP Modelling

_{c}) were varied from 30 to 200, numbers of genes (N

_{g}) from 3 to 5, and the H

_{s}from 8 to 12. Different linking functions (+, −, ×, /) between the N

_{g}were explored during the performance of trials, and it was found that the addition function provided the best performance. The flowchart of GEP modelling is shown in Figure 2, while the details of the undertaken trials are given in Table 2.

^{2}), RMSE and mean absolute error (MAE), have been employed for evaluating the performance of the proposed GEP model. Table 3 shows the ideal values of these indices.

_{c}, N

_{g}and H

_{s}(Table 2). Initially, the N

_{c}were changed from 30 to 200 while keeping the H

_{s}and N

_{g}constant (i.e., 8 and 3, respectively). Similarly, the H

_{s}was changed from 8 to 12, keeping the other two variables constant. A similar procedure was followed to find the optimal number of N

_{g}. Thus, the N

_{c}, H

_{s}and N

_{g}for an optimally performing GEP model (Model T3) were recorded to be 100, 8 and 3, respectively.

## 3. Results & Discussion

#### 3.1. Effect of Variable Genetic Parameters

_{c}, H

_{s}, and N

_{g}on the performance of the models (T1–T11) evaluated using different indices such as R

^{2}, RMSE and MAE, for both the TR and TS phases, respectively. The hyperparameter investigation was carried out in 11 distinct trials by varying the genetic parameters, i.e., N

_{c}, H

_{s}, and N

_{g}. The performance was observed in terms of the aforementioned statistical indices. It can be seen from Table 2 that initially the N

_{c}was varied from 30 to 200, maintaining the other two genetic parameters constant (H

_{s}= 8 and N

_{g}= 3). Subsequently, the optimum model (Model T3) obtained during this investigation was further subjected to an increase in the H

_{s}from 8 to 12. The N

_{g}was changed from 3 to 5 in Models T9, T10, and T11 in order to obtain the optimal N

_{g}value.

^{2}with changing N

_{c}for TR phase, TS phase, and in case of the overall dataset. Figure 3 shows that when the N

_{c}was increased from 30 to 200, the optimal N

_{c}was revealed to be 100. Moreover, the detailed illustration of the N

_{c}variation in Figure 3a–c shows that the overall values of MAE, RMSE, and R

^{2}(i.e., 375.05, 489, and 0.905, respectively) indicated improved accuracy for N

_{c}equaling 100. Afterwards, the performance of the other models plummeted. Similarly, the optimal values of H

_{s}and N

_{g}were observed as 8 and 3, as shown in Figure 4 and Figure 5, respectively. This strongly suggests that the hyperparameter tuning in the GEP modelling is solely a trial and error process, and there are no concrete recommendations regarding the effect of changing the genetic parameters. Furthermore, it is evident from the previous studies that increasing N

_{g}and employing complex linking functions may increase the robustness of the models, but perplexes the traceable output mathematical equation [38,40].

^{2}= 0.88 in the TR phase and R

^{2}= 0.92 in TS phase, as evident from Table 2. Lower values of R

^{2}and higher values of RMSE and MAE were observed when the number of genes was 5. The effect of number of genes on the performance of the models can also be seen in Figure 4. It is evident that the models had lower R

^{2}(Figure 4a) and higher RMSE (Figure 4b) when the number of genes was equal to 4.

_{s}changed from 8 to 12 in Model T6 to Model T9. It can be inferred that when the H

_{s}is increased while keeping the other two parameters constant, the R

^{2}value increases for both the TR and TS phases (Figure 5b). The highest value of R

^{2}= 0.87 and 0.92, can be observed for both TR and TS phase, respectively, in Model T9. Similarly, smaller values of RMSE = 564.8 and 478.9 and MAE = 447 and 399.3 were observed for both TR and TS phase, respectively, in Model T9. Looking at the performance of the model in different trials, it can be concluded from Table 2 that the trial T3 was the most optimal compared to the others since it had the highest R

^{2}value and a smaller RMSE and MAE value.

#### 3.2. Performance of Models

#### 3.2.1. Statistical Evaluation

^{2}= 0.89 for TR phase, and 0.92 for TS phase), followed by Model T11 (R

^{2}= 0.88 for TR phase, and 0.91 for TS phase). The observed values of R

^{2}indicate a good agreement between the predicted and actual values. However, deciding about the performance of a model based on “R

^{2}” alone is not sufficient, and other statistical error indices must also be considered. In this regard, the values of RMSE and MAE were studied to evaluate the performance of the different models, in addition to the R

^{2}value (Table 2). It is evident from Table 2 that besides having a higher R

^{2}value, Model T3 exhibited the lowest RMSE (513.9 in TR phase, and 464.1 in TS phase) and MAE (385.2 in TR phase and 364.9 in TS phase). Similarly, Model T11 performed as the second-best model with R

^{2}= 0.88, RMSE = 525.3 and MAE = 394.8 in the TR phase, while Model T1 performed as second-best model with R

^{2}= 0.92, RMSE = 478.2, and MAE = 386.6 in the TS phase, respectively. The ranking of the models based on the different statistical indices is shown in Table 4.

#### 3.2.2. Comparison of Regression Slopes

^{2}and regression slopes for both the TR and TS phases, respectively. It is evident that the value of R

^{2}exceeded 0.8 for most of the models. In the TR phase, Model T3 had the finest fit with an R

^{2}value of 0.89, whereas Model T6 and Model T7 had lower values of R

^{2}= 0.78, each. Similarly, the performance of the models seemed to be better in the TS phase, as evident from their higher R

^{2}values, and all the models had R

^{2}≥ 0.85. Similarly, it can also be observed from Figure 6 and Figure 7 that the highest values of slope “m” = 0.89 and 0.92 were obtained for the optimal model “T3” in both the TR and TS phases, respectively. It is important to mention here that for m = 1, the slope of the regression line will be exactly 45

^{°}. The values of “m” observed for the model “T3” were closer to one as compared to other models; therefore, it can be concluded from the higher R

^{2}and m values that this was the optimal performing model compared to the others.

#### 3.2.3. Model Predicted to Experimental (P/E) Ratio

#### 3.2.4. Visual Interpretation of Results via Taylor Diagram

#### 3.3. GEP Formulations

#### 3.4. Parametric and Sensitivity Analyses

^{2}above 0.97. Figure 11 shows the sensitivity of each variable in resisting chloride penetration. Concrete age is the most significant parameter that influences RCP values, followed by Fag, Cag, w/b ratio, MK content, and finally, the compressive strength of concrete.

## 4. Conclusions

_{c}), head size (H

_{s}), and number of genes (N

_{g}) of the GEP model were varied to study their influence on the predicted values of the RCP. Following are the main conclusions drawn from this study:

- The tuning of the hyperparameter settings for the GEP model revealed that the model with N
_{c}= 100, H_{s}= 8 and N_{g}= 3 (Model T3) resulted in an optimal GEP model, as evident from its high R^{2}values (i.e., 0.89 in the TR phase and 0.92 in the TS phase, respectively). Similarly, the values of RMSE = 513.9 and 464.1, and of MAE = 385.2 and 364.9, were also comparatively smaller than in all the other models in the TR and TS phases, respectively. - The regression slope analysis showed that the predicted values were in good agreement with the experimental values, as indicated from the higher R
^{2}values. It was also observed that the performance of the models improved in the TS phase, which was reflected in their higher R^{2}values, with the majority of developed models having R^{2}> 0.8. In addition, the P/E ratio analysis revealed that Model T3 was the best performing model, because a larger frequency was observed for the P/E ratio proximal to one. - Similarly, the parametric analysis for the best performing Model T3 revealed that the amount of binder, compressive strength and age of the sample enhanced the RCP resistance of concrete specimens. However, among the different input variables, the RCP resistance sharply increased within the first 28 days of age of the concrete specimen.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Rapid chloride ion penetration | RCP |

Number of chromosomes | N_{c} |

Number of genes | N_{g} |

Head size | H_{s} |

Gene expression programming | GEP |

Amount of binder | b |

Fine aggregate | Fag |

Coarse aggregate | Cag |

Water to binder ratio | w/b |

Metakaolin | MK |

Model predicted to experimental | P/E |

## Appendix A

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**Figure 1.**Frequency histograms of input and output variables; Training dataset (dark blue histograms) and test data (pink colored histograms) (

**a**–

**h**) for age, binder, fine aggregate, coarse aggregate, w/b ratio, metakaolin, compression strength and RCPT, respectively.

**Figure 3.**Effect of number of chromosomes on the performance of models. (

**a**) MAE, (

**b**) RMSE, (

**c**) R

^{2}.

**Figure 10.**Parametric analysis of input variables. (

**a**) age, (

**b**) binder, (

**c**) Fag, (

**d**) Cag, (

**e**) w/b ratio, (

**f**) MK, and (

**g**) compressive strength.

Descriptive Statistics | Age | b | Fag | Cag | w/b | MK | Compressive Strength | RCPT |
---|---|---|---|---|---|---|---|---|

Average | 63.53 | 389.92 | 0.42 | 10.76 | 878.46 | 874.97 | 55.11 | 2309.98 |

Standard Error | 4.35 | 4.88 | 0.004 | 0.48 | 7.70 | 6.18 | 1.19 | 111.36 |

Median | 28 | 360 | 0.45 | 12.5 | 881.30 | 832.5 | 52.7 | 1973 |

Standard Deviation | 61.63 | 69.21 | 0.058 | 6.74 | 109.21 | 87.69 | 17.01 | 1578.79 |

Sample Variance | 3798.46 | 4790.43 | 0.003 | 45.41 | 11,925.88 | 7689.79 | 289.30 | 2492.57 × 10^{3} |

Kurtosis | −0.42 | 2.19 | −0.90 | −0.49 | −0.99 | −0.82 | −0.31 | 0.11 |

Skewness | 1.04 | 1.69 | −0.24 | −0.10 | −0.25 | 0.34 | 0.25 | 0.93 |

Minimum | 7 | 320 | 0.3 | 0 | 589.2 | 707 | 19 | 203 |

Maximum | 180 | 600 | 0.5 | 25 | 1017.5 | 1111.7 | 108 | 6982 |

Trial/Model | No. of Variables | No. of Chromosomes | Head Size | No. of Genes | TR Phase | TS Phase | ||||
---|---|---|---|---|---|---|---|---|---|---|

R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | |||||

T1 | 7 | 30 | 8 | 3 | 0.84 | 602.1 | 475.3 | 0.92 | 478.2 | 386.6 |

T2 | 6 | 50 | 8 | 3 | 0.81 | 681.3 | 483.3 | 0.90 | 526.6 | 424 |

T3 | 7 | 100 | 8 | 3 | 0.89 | 513.9 | 385.2 | 0.92 | 464.1 | 364.9 |

T4 | 7 | 150 | 8 | 3 | 0.83 | 641.3 | 483.3 | 0.92 | 482.7 | 383.1 |

T5 | 6 | 200 | 8 | 3 | 0.79 | 641.3 | 568.3 | 0.89 | 562.2 | 477.3 |

T6 | 7 | 100 | 9 | 3 | 0.78 | 723.3 | 538.0 | 0.89 | 561.9 | 433.1 |

T7 | 7 | 100 | 10 | 3 | 0.78 | 728.7 | 593.5 | 0.88 | 625.6 | 521.3 |

T8 | 7 | 100 | 11 | 3 | 0.83 | 641.2 | 469.6 | 0.89 | 527.2 | 423.5 |

T9 | 7 | 100 | 12 | 3 | 0.87 | 564.8 | 447 | 0.92 | 478.9 | 399.3 |

T10 | 6 | 100 | 8 | 4 | 0.83 | 634.2 | 477.1 | 0.89 | 567.7 | 451.5 |

T11 | 7 | 100 | 8 | 5 | 0.88 | 525.3 | 394.8 | 0.91 | 494.6 | 387.7 |

Index | Range/Ideal Value |
---|---|

R^{2} | (0–1)/1 |

RMSE | $(0\text{\u2013}\infty $)/0 |

MAE | $(0\text{\u2013}\infty $)/0 |

Statistic | R^{2} | RMSE | MAE | |||
---|---|---|---|---|---|---|

Rank | 1st | 2nd | 1st | 2nd | 1st | 2nd |

TR Phase | T3 | T11 | T3 | T11 | T3 | T11 |

TS Phase | T3, T1 | - | T3 | T1 | T3 | T4 |

Input Variables | Constant Input Parameters | No. of Datapoints | |
---|---|---|---|

Parameter | Range | ||

Age | 7–180 | B = 389.93, w/b = 0.42, MK = 10.76, Fag = 878.46, Cag = 874.97, compressive strength = 55.12 | 9 |

b | 320–600 | Age = 63.53, w/b = 0.42, MK = 10.76, Fag = 878.46, Cag = 874.97, compressive strength = 55.12 | |

Fag | 589.2–1017.5 | Age = 63.53, B = 389.93, w/b = 0.42, MK = 10.76, Cag = 874.97, compressive strength = 55.12 | |

Cag | 707–1111.7 | Age = 63.53, B = 389.93, w/b = 0.42, MK = 10.76, Fag = 878.46, compressive strength = 55.12 | |

w/b | 0.3–0.5 | Age = 63.53, B = 389.93, MK = 10.76, Fag = 878.46, Cag = 874.97, compressive strength = 55.12 | |

MK | 0–25 | Age = 63.53, B = 389.93, w/b = 0.42, Fag = 878.46, Cag = 874.97, compressive strength = 55.12 | |

compressive strength | 19–108 | Age = 63.53, B = 389.93, w/b = 0.42, MK = 10.76, Fag = 878.46, Cag = 874.97 |

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**MDPI and ACS Style**

Amin, M.N.; Raheel, M.; Iqbal, M.; Khan, K.; Qadir, M.G.; Jalal, F.E.; Alabdullah, A.A.; Ajwad, A.; Al-Faiad, M.A.; Abu-Arab, A.M.
Prediction of Rapid Chloride Penetration Resistance to Assess the Influence of Affecting Variables on Metakaolin-Based Concrete Using Gene Expression Programming. *Materials* **2022**, *15*, 6959.
https://doi.org/10.3390/ma15196959

**AMA Style**

Amin MN, Raheel M, Iqbal M, Khan K, Qadir MG, Jalal FE, Alabdullah AA, Ajwad A, Al-Faiad MA, Abu-Arab AM.
Prediction of Rapid Chloride Penetration Resistance to Assess the Influence of Affecting Variables on Metakaolin-Based Concrete Using Gene Expression Programming. *Materials*. 2022; 15(19):6959.
https://doi.org/10.3390/ma15196959

**Chicago/Turabian Style**

Amin, Muhammad Nasir, Muhammad Raheel, Mudassir Iqbal, Kaffayatullah Khan, Muhammad Ghulam Qadir, Fazal E. Jalal, Anas Abdulalim Alabdullah, Ali Ajwad, Majdi Adel Al-Faiad, and Abdullah Mohammad Abu-Arab.
2022. "Prediction of Rapid Chloride Penetration Resistance to Assess the Influence of Affecting Variables on Metakaolin-Based Concrete Using Gene Expression Programming" *Materials* 15, no. 19: 6959.
https://doi.org/10.3390/ma15196959