# Forecasting Compressive Strength of RHA Based Concrete Using Multi-Expression Programming

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## Abstract

**:**

## 1. Introduction

## 2. Multi-Expression Programming (MEP)

## 3. Data Collection

## 4. Model Development

#### 4.1. Shapley Additive Explanations (SHAP)

^{2}), relative root mean square error (RRMSE), relative squared error (RE), and performance index $\mathsf{\rho}$ (Equations (2)–(8), respectively) have been used in this study.

^{2}, as well as the pre-selected significance value, i.e., alpha (usually 0.05) from F and t-tests, indicate that the predictive model performs well and has a better accuracy. Additionally, it implies that the experimental and anticipated values are highly connected. Additionally, it is worth noting that a R value larger than 0.8, an R

^{2}value nearer to 1, an RMSE value nearer to or equal to zero, and ρ value (0 to infinity) approaching zero all contribute to improved model calibration. Unlike the RMSE, MAE is a positive evolution metric when the original data is relatively smooth [60]. On the other hand, the normalized mean square error (NSE) runs between 0 and 1.0 (1 inclusive), with 1 regarded as the best number. Additionally, a significant issue linked with AI systems is overfitting, which occurs because of extensive training and results in higher mistakes in the testing set. As demonstrated in Equation (9), the objective function (OBF) is assessed and decreased prior to selecting the best predictive mode [61]. The OBF is used to evaluate the trained model’s performance by including changes in the error function (RRMSE) and correlation coefficient (R). A low OBF value aids in overcoming the issue of overfitting.

#### 4.2. Cross-Validation Using 10 K-Fold Method

## 5. Results and Discussion

#### 5.1. MEP Analysis of CRHA

#### 5.2. Performance Evaluation of MEP Model

#### 5.3. External Validation

_{m}> 0.5, is met [63,64,65]. Additionally, the squared correlation coefficient (${{R}^{\prime}}_{o}^{2}$) between the estimated and experimental datasets, as well as the correlation coefficient (${R}_{o}^{2}$) between the experimental and estimated values, must approach one [66,67,68]. As seen in Table 6, the suggested MEP model meets nearly all the stated requirements, which is consistent with the findings of existing literature and recommendations [69,70,71,72].

#### 5.4. 10-K Fold Cross Validation

^{2}(regression tool) along with MAE and RMSE (statistical error tools) as shown in Table 7. In Figure 11, fluctuation in the value R

^{2}is observed for the 10 K-fold validation of different ML techniques, but still, a high level of accuracy is maintained in each fold. The accuracy of the cross-validation was also assessed in terms of MAE and RMSE and is given in Figure 11, respectively. The average value of MAE for is 4.2 MPa, respectively, as shown in Figure 11.

#### 5.5. Explanation Using MEP Model

#### 5.6. Sensitivity Analysis

## 6. Conclusions

#### Future Recommendation

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RHA | Rice husk ash |

MEP | Multi-expression programming |

CRHA | Concrete made with rice husk ash |

SHAP | SHapley Additive exPlanations |

OPC | Ordinary Portland cement |

AIA | Artificial intelligence algorithms |

BBA | Black-box algorithms |

EA | Evolutionary algorithms |

GA | Genetic algorithm |

NN | Neural network |

GEP | Gene expression programming |

SP | Superplasticizer |

C | Amount of cement |

W | Amount of water |

A | Amount of aggregate |

AC | Age of concrete |

CS | Compressive strength |

RMSE | Root mean square error |

R | Coefficient of correlation |

MAE | Mean absolute error |

R^{2} | Coefficient of regression |

RRMSE | Relative root mean square error |

RE | Relative squared error |

$\mathsf{\rho}$ | Performance index |

OBF | Objective function |

NSE | Normalized mean square error |

## Appendix A

pg[0] = x[0]; |

pg[1] = sqrt(pg[0]); |

pg[2] = x[1]; |

pg[3] = x[3]; |

pg[4] = exp(pg[1]); |

pg[5] = pg[1] × pg[2]; |

pg[6] = pg[4] / pg[3]; |

pg[7] = pg[0] + pg[3]; |

pg[8] = pg[2] − pg[7]; |

pg[9] = x[4]; |

pg[10] = pg[8] × pg[9]; |

pg[11] = pg[10] + pg[5]; |

pg[12] = pg[1] × pg[0]; |

pg[13] = pg[8] + pg[11]; |

pg[14] = pow(pg[1], pg[9]); |

pg[15] = pg[13] − pg[12]; |

pg[16] = sqrt(pg[15]); |

pg[17] = x[2]; |

pg[18] = pg[6] + pg[8]; |

pg[19] = pg[9] × pg[17]; |

pg[20] = pg[12] / pg[18]; |

pg[21] = pg[19] + pg[18]; |

pg[22] = sqrt(pg[11]); |

pg[23] = pg[5] / pg[21]; |

pg[24] = sqrt(pg[14]); |

pg[25] = pg[24] × pg[4]; |

pg[26] = pg[6] − pg[20]; |

pg[27] = pg[22] / pg[25]; |

pg[28] = pg[26] / pg[23]; |

pg[29] = pg[16] − pg[27]; |

pg[30] = pg[29] + pg[28]; |

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**Figure 4.**(

**a**) Procedures involved in implementing MEP, (

**b**) Flowchart for expressions encoded by an MEP chromosome.

**Figure 6.**K-fold cross-validation algorithm [61].

AS * (Day) | C * (kg/m ^{3}) | RHA * (kg/m^{3}) | W * (kg/m ^{3}) | SP * (kg/m^{3}) | A * (kg/m ^{3}) | CS (MPa) | |
---|---|---|---|---|---|---|---|

AS (day) | 1.00 | ||||||

C30 (kg/m^{3}) | −0.11 | 1.00 | |||||

RHA (kg/m^{3}) | −0.03 | −0.22 | 1.00 | ||||

W (kg/m^{3}) | 0.01 | 0.08 | 0.14 | 1.00 | |||

SP (kg/m^{3}) | 0.00 | 0.25 | −0.02 | 0.27 | 1.00 | ||

A (kg/m^{3}) | −0.06 | −0.24 | −0.14 | −0.55 | −0.21 | 1.00 | |

CS (MPa) | 0.49 | 0.37 | −0.02 | −0.24 | 0.30 | 0.15 | 1.00 |

Description of Variables | AS (Day) | C (kg/m^{3}) | RHA (kg/m^{3}) | W (kg/m^{3}) | SP (kg/m^{3}) | A (kg/m ^{3}) | CS (MPa) |
---|---|---|---|---|---|---|---|

Mean | 34.57 | 409.02 | 62.33 | 193.54 | 3.34 | 1621.51 | 48.14 |

Median | 28.00 | 400.00 | 57.00 | 203.00 | 1.85 | 1725.00 | 45.95 |

Mode | 28.00 | 400.00 | 0.00 | 203.00 | 0.00 | 1725.00 | 47.00 |

Standard Deviation | 33.52 | 105.47 | 41.55 | 31.93 | 3.52 | 267.77 | 17.54 |

Sample Variance | 1123.61 | 11,124.88 | 1726.77 | 1019.71 | 12.37 | 71,702.44 | 307.70 |

Skewness | 0.75 | 1.55 | 0.44 | −0.42 | 0.69 | −0.74 | 0.83 |

Range | 89.00 | 534.00 | 171.00 | 118.00 | 11.25 | 930.00 | 88.10 |

Minimum | 1.00 | 249.00 | 0.00 | 120.00 | 0.00 | 1040.00 | 16.00 |

Maximum | 90.00 | 783.00 | 171.00 | 238.00 | 11.25 | 1970.00 | 104.10 |

Sum | 6638.00 | 78,531.00 | 11,967.10 | 37,158.91 | 640.35 | 311,330.00 | 9243.10 |

Count | 192.00 | 192.00 | 192.00 | 192.00 | 192.00 | 192.00 | 192.00 |

Parameters | MEP |
---|---|

Num of subpopulation | 20 |

Subpopulation size | 1000 |

Code length | 50 |

Crossover probability | 0.9 |

Crossover type | Uniform |

Mutation probability | 0.001 |

Tournament size | 2 |

Operators | 0.5 |

Variables | 0.5 |

Number of generations | 1000 |

Function set | +, −, ×, / |

Terminal set | Problem input |

Replication number | 10 |

Error measure | Mean squared error |

Problem type | Regression |

Simplified | Yes |

Random seed | 0 |

Number of runs | 10 |

Number of threads | 1 |

Trial No. | No. of Subpopulation | Subpopulation Size | Code Length | No. of Generation | Functions Used | R^{2} | RMSE | MAE | RRSE | Time (Min) |
---|---|---|---|---|---|---|---|---|---|---|

MP1 | 10 | 200 | 20 | 200 | +, −, ×, / | 0.9275 | 71.1 | 48.03 | 0.2693 | 0–2 |

MP2 | 20 | 20 | +, −, ×, / | 0.9448 | 62.17 | 41.82 | 0.2355 | |||

MP3 | 50 | 25 | +, −, ×, / | 0.9454 | 61.94 | 45.67 | 0.2346 | |||

MP4 | 70 | 25 | +, −, ×, / | 0.9233 | 74.09 | 47.03 | 0.2806 | |||

MP5 | 100 | 35 | +, −, ×, / | 0.9221 | 74.33 | 46.89 | 0.2815 | |||

MP6 | 20 | 400 | 35 | +, −, ×, / | 0.9156 | 88.17 | 60.35 | 0.334 | ||

MP7 | 600 | 35 | +, −, ×, / | 0.9496 | 59.68 | 41.9 | 0.226 | |||

MP10 | 40 | 400 | +, −, ×, / | 0.9614 | 53.41 | 38.12 | 0.2023 | 15 | ||

MP11 | 40 | 600 | +, −, ×, / | 0.9376 | 66.01 | 42.78 | 0.25 | 25 | ||

MP12 | 1000 | 50 | +, −, ×, / | 0.9298 | 70.13 | 43.56 | 0.2656 | |||

MP13 | 50 | 1000 | +, −, ×, / | 0.9362 | 66.97 | 45.06 | 0.2536 | 45 |

Indicators | Training | Validation |
---|---|---|

R^{2} | 0.976419 | 0.971378 |

R | 0.988139 | 0.985585 |

RMSE | 3.843116 | 3.406354 |

MAE | 3.067433 | 2.317413 |

RRMSE | 0.079188 | 0.072075 |

RE | 0.047253 | 0.048581 |

$\mathsf{\rho}$ | 0.03983 | 0.0363 |

OBF | 0.04 |

S. No. | Equation | Condition | MP | Suggested by |
---|---|---|---|---|

1 | $\mathrm{R}=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({x}_{i}-{\overline{x}}_{i}\right)\left({y}_{i}-{\overline{y}}_{i}\right)}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-{\overline{x}}_{i}\right)}^{2}{{\displaystyle \sum}}_{i=1}^{n}{\left({y}_{i}-{\overline{y}}_{i}\right)}^{2}}}$ | R > 0.8 | 0.98 | [63,64,65] |

2 | $\mathrm{k}=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({x}_{i}\times {y}_{i}\right)}{{x}_{i}^{2}}$ | 0.85 < k < 1.15 | 0.975 | [62] |

3 | ${\mathrm{k}}^{\prime}=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({x}_{i}\times {y}_{i}\right)}{{y}_{i}^{2}}$ | 0.85 < k′ < 1.15 | 0.976 | |

4 | ${\mathrm{R}}_{\mathrm{m}}={R}^{2}\times (1-\sqrt{\left|{R}^{2}-{R}_{0}^{2}\right|}$ where ${R}_{o}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({y}_{i}-{x}_{i}^{o}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({y}_{i}-\overline{{y}_{i}^{o}}\right)}^{2}},\text{}{x}_{i}^{o}=k\times {y}_{i}$ $\text{}{{R}^{\prime}}_{o}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-{y}_{i}^{o}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{{x}_{i}^{o}}\right)}^{2}},\text{}{y}_{i}^{o}={k}^{\prime}\times {x}_{i}$ | R_{m} > 0.5 | 0.856 | [66,67,68] |

${R}_{o}^{2}\text{}\cong 1$ | 0.989 | [69,70,71,72] | ||

${{R}^{\prime}}_{o}^{2}\text{}\cong 1$ | 1.000 |

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

4.47 | 5.4 | 0.919 |

4.209 | 7.68 | 0.91 |

4.71 | 5.82 | 0.86 |

2.97 | 4.14 | 0.91 |

1.60 | 2.71 | 0.95 |

11.1 | 15. | 0.89 |

2.99 | 3.45 | 0.90 |

4.04 | 5.21 | 0.87 |

3.30 | 4.22 | 0.89 |

2.97 | 2.73 | 0.93 |

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## Share and Cite

**MDPI and ACS Style**

Amin, M.N.; Khan, K.; Javed, M.F.; Ewais, D.Y.Z.; Qadir, M.G.; Faraz, M.I.; Alam, M.W.; Alabdullah, A.A.; Imran, M.
Forecasting Compressive Strength of RHA Based Concrete Using Multi-Expression Programming. *Materials* **2022**, *15*, 3808.
https://doi.org/10.3390/ma15113808

**AMA Style**

Amin MN, Khan K, Javed MF, Ewais DYZ, Qadir MG, Faraz MI, Alam MW, Alabdullah AA, Imran M.
Forecasting Compressive Strength of RHA Based Concrete Using Multi-Expression Programming. *Materials*. 2022; 15(11):3808.
https://doi.org/10.3390/ma15113808

**Chicago/Turabian Style**

Amin, Muhammad Nasir, Kaffayatullah Khan, Muhammad Faisal Javed, Dina Yehia Zakaria Ewais, Muhammad Ghulam Qadir, Muhammad Iftikhar Faraz, Mir Waqas Alam, Anas Abdulalim Alabdullah, and Muhammad Imran.
2022. "Forecasting Compressive Strength of RHA Based Concrete Using Multi-Expression Programming" *Materials* 15, no. 11: 3808.
https://doi.org/10.3390/ma15113808