# Prediction and Optimization of Pile Bearing Capacity Considering Effects of Time

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## Abstract

**:**

## 1. Introduction

## 2. Methodology Background

#### 2.1. Genetic Programming

#### 2.2. Gray Wolf Optimization

#### 2.3. Artificial Bee Colony

## 3. Database Establishment

#### 3.1. Case Study and Input Parameters

#### 3.2. Statistical Information on the Data

## 4. Prediction of Pile Capacity

#### 4.1. GP Modeling Procedure

- (1)
- A training set and a testing set were created by randomly dividing the database. Then, 80 percent of the database (204 datasets) was dedicated to the training set, while the remaining 20 percent was devoted to testing (52 datasets). The initial population is randomly generated from the database and function sets. The function sets include +, −, ×, ÷, $\sqrt{\hspace{0.33em}}$, sin, cos, and tan.
- (2)
- The testing set is adapted to fit the prediction equation. After the genetic operation, i.e., selection, crossover, and variation, the preliminary prediction formula is obtained [46].
- (3)
- The fitness function of the population is defined, and it is employed to evaluate the fitness of each formula in the population. Root mean square error (RMSE) as the fitness function was used in this study. The fitness value is calculated according to Equation (2), where M means the number of training or testing sets, and ${\mathrm{UC}}^{\prime}$ represents the predicted value of the formula generated by GP.$$\mathrm{RMSE}=\sqrt{\frac{1}{M}\sum _{i=1}^{M}{\left(\mathrm{UC}-{\mathrm{UC}}^{\prime}\right)}^{2}}$$
- (4)
- Repeat steps (2–3) until the training time reaches the termination rule.
- (5)
- At the end of GP, the final optimal formula is evaluated from the goodness of fit coefficient ${R}^{2}$ between the predicted UC obtained by the formula and the real UC. ${R}^{2}$ is calculated according to Equation (3).$${R}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{M}{\left(\mathrm{UC}-\widehat{\mathrm{UC}}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{M}{\left(\mathrm{UC}-{\mathrm{UC}}^{\xb4}\right)}^{2}}$$

#### 4.2. Results

## 5. Optimizing Pile Capacity Using Metaheuristic Algorithms

#### 5.1. Gray Wolf Optimization

#### 5.2. Artificial Bee Colony Algorithm

## 6. Discussion

## 7. Limitations and Future Works

## 8. Conclusions

- The proposed GP equation is easy to implement and is of interest to civil and geotechnical engineers. An intelligent equation proposed by GP showed an acceptable level of accuracy in predicting pile capacity. Results with ${R}^{2}$ values of 0.897 in the training stage and 0.844 in the testing stage indicate that this GP model is capable enough to be implemented for predicting pile capacity.
- In the optimization phase, two powerful algorithms, namely GWO and ABC, were applied to maximize pile capacity. Obtaining the highest capacity of the pile is considered the ultimate objective of such projects. Although both algorithms are powerful in maximizing pile capacity, GWO performed better. Increase percentages of 52.6 and 54 were obtained by ABC and GWO, respectively, in their pile capacity results.
- For the best optimization algorithm (i.e., GWO), values of 38.59 m, 0.247 m, 2273 kPa, 157.46 kPa, 153.18 days, and 6098.488 kPa were obtained for LOP, PD, IC, Su, T, and UC, respectively. The proposed models and obtained results of this study can be used in designing pile capacity before implementing relevant projects.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Tree structure representation of optimal results (X0 = LOP; X1 = PD; X2 = IC; X3 = Su; X4 = T).

**Figure 11.**Taylor diagram of training set (RF: random forest, ANN: artificial neural network, DT: decision tree, SVM: support vector machine, KNN: k-nearest neighbors, GBM: gradient boosting machine, AdaBoost: adaptive boosting machine).

**Figure 12.**Taylor diagram of the test set (RF: random forest, ANN: artificial neural network, DT: decision tree, SVM: support vector machine, KNN: k-nearest neighbors, GBM: gradient boosting machine, AdaBoost: adaptive boosting machine).

Parameters | Range |
---|---|

LOP | 12.009–64.008 |

PD | 0.236–1.067 |

IC | 57.3–2276 |

Su | 26.97–191.52 |

T | 0.008–154 |

Parameter | Actual Value | Optimized Value | |
---|---|---|---|

GWO | ABC | ||

LOP | 24.0 | 38.59 | 38.47 |

PD | 0.457 | 0.247 | 0.240 |

IC | 1642.7 | 2273 | 2276 |

Su | 172.0 | 157.46 | 157.46 |

T | 6.0 | 153.18 | 170.16 |

Maximum UC | 3960.0 | 6098.488 | 6043.64 |

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**MDPI and ACS Style**

Khanmohammadi, M.; Armaghani, D.J.; Sabri Sabri, M.M.
Prediction and Optimization of Pile Bearing Capacity Considering Effects of Time. *Mathematics* **2022**, *10*, 3563.
https://doi.org/10.3390/math10193563

**AMA Style**

Khanmohammadi M, Armaghani DJ, Sabri Sabri MM.
Prediction and Optimization of Pile Bearing Capacity Considering Effects of Time. *Mathematics*. 2022; 10(19):3563.
https://doi.org/10.3390/math10193563

**Chicago/Turabian Style**

Khanmohammadi, Mohammadreza, Danial Jahed Armaghani, and Mohanad Muayad Sabri Sabri.
2022. "Prediction and Optimization of Pile Bearing Capacity Considering Effects of Time" *Mathematics* 10, no. 19: 3563.
https://doi.org/10.3390/math10193563