# Predicting Marshall Flow and Marshall Stability of Asphalt Pavements Using Multi Expression Programming

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## Abstract

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## 1. Introduction

## 2. Experimental Database

#### 2.1. Data Division and Preprocessing

#### 2.2. Multi-Collinearity

#### 2.3. Data Statistical Information

## 3. Model Development and Evaluation Criteria

_{i}, x

_{i}, ${\overline{\mathrm{p}}}_{\mathrm{i}}$ and ${\overline{\mathrm{x}}}_{\mathrm{i}}$, denote the ith predicted, experimental, mean predicted and mean experimental values, respectively, and the symbol n denotes the total number of values in the dataset used for the development of the models. The training and testing sets are denoted by abbreviations T and TE, respectively. An accurate model has a high R value, while the statistical errors are low. R has been recommended by the researchers to assess the linear dependency among input and output parameters [75], with a value greater than 0.8 indicating a decent correlation between experimental and predicted values [41,76]. Due to the insensitivity of R with the division and multiplication of outputs with a constant, it could not be considered solely as a measure for overall model efficiency. The average magnitude of errors can be measured using MAE and RMSE. Both parameters, however, have their own implication. In RMSE, errors are squared before average is estimated, giving larger errors more weight. A high RMSE value indicates that the amount of high-error predictions is significantly more than the expected and should be excluded. MAE, however, gives large errors a low weight and is always smaller than RMSE. Likewise, Despotovic et al., (2016) recommended that a model is considered to be outstanding if RRMSE values are between 0 and 0.10 and fair if the value is between 0.11 and 0.20 [77]. The range of values for OF and ρ is 0–infinity. If the values of ρ and OF are less than 0.2, the model can be considered as good [66]. While using OF, it considers three factors at the same time, i.e., R and RRMSE with relative percentage of data in various datasets (training and testing). As a result, a low value of OF indicates that the model’s overall performance is superior. As stated previously, numerous trial runs were carried out, and the models with the lowest values of OF stated in this research study. Additionally, the validation of developed models was also carried out using criteria suggested by various researchers, which are described in Table 6.

## 4. Discussions

#### Formulation of Mechanical Properties

## 5. Results

#### 5.1. Performance Evaluation of MEP Models

#### 5.2. External Validation

#### 5.3. Parametric Analysis

#### 5.3.1. Marshall Stability Analysis

#### 5.3.2. Marshall Flow Analysis

## 6. Conclusions

- The developed models have produced results that are consistent with the experimental data and function equally well for unknown data.
- Various performance measures such as R, RRMSE, RSE, RMSE, MAE were used to assess the reliability and correction of the developed models. Furthermore, OF and ρ showed highly generalization capability of the developed models, with the issue of overfitting effectively addressed. The results of the statistical parameters validated the accuracy of the proposed MEP developed models.
- The value of R lies in between 0.90 and 0.98 for MS and MF of ABC and AWC. MAE ranges from 24.64 to 36.94 kg for MS of ABC and AWC, while it ranges from 0.31 (0.25 mm) to 0.71 (0.25 mm) for MF of ABC and AWC.
- The developed models also met a number of external validation criteria taken from the literature.
- The models developed have successfully incorporated input parameters and have the capability to predict the trends of MS and MF for flexible pavements, as revealed from the parametric study.
- It is convincing from the modeling approach being proposed, i.e., MEP in conjunction with validation parameters, that MEP can be utilized for predicting the Marshall parameters.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Ps (%) | Pb (%) | Gmb | Gsb | Gmm | Va (%) | VMA (%) | VFA (%) | |
---|---|---|---|---|---|---|---|---|

Ps (%) | 1.00 | −1.00 | −0.15 | 0.01 | 0.59 | 0.76 | −0.28 | −0.84 |

Pb (%) | −1.00 | 1.00 | 0.15 | −0.01 | −0.59 | −0.76 | 0.28 | 0.84 |

Gmb | −0.15 | 0.15 | 1.00 | 0.68 | 0.50 | −0.48 | −0.65 | 0.32 |

Gsb | 0.01 | −0.01 | 0.68 | 1.00 | 0.63 | −0.03 | −0.01 | 0.03 |

Gmm | 0.59 | −0.59 | 0.50 | 0.63 | 1.00 | 0.52 | −0.40 | −0.64 |

Va (%) | 0.76 | −0.76 | −0.48 | −0.03 | 0.52 | 1.00 | 0.24 | −0.96 |

VMA (%) | −0.28 | 0.28 | −0.65 | −0.01 | −0.40 | 0.24 | 1.00 | 0.02 |

VFA (%) | −0.84 | 0.84 | 0.32 | 0.03 | −0.64 | −0.96 | 0.02 | 1.00 |

Ps (%) | Pb (%) | Gmb | Gsb | Gmm | Va (%) | VMA (%) | VFA (%) | |
---|---|---|---|---|---|---|---|---|

Ps (%) | 1.00 | −1.00 | −0.37 | 0.09 | 0.68 | 0.93 | −0.09 | −0.95 |

Pb (%) | −1.00 | 1.00 | 0.37 | −0.09 | −0.68 | −0.93 | 0.09 | 0.95 |

Gmb | −0.37 | 0.37 | 1.00 | 0.68 | 0.35 | −0.50 | −0.30 | 0.46 |

Gsb | 0.09 | −0.09 | 0.68 | 1.00 | 0.70 | 0.08 | 0.31 | −0.02 |

Gmm | 0.68 | −0.68 | 0.35 | 0.70 | 1.00 | 0.64 | −0.09 | −0.65 |

Va (%) | 0.93 | −0.93 | −0.50 | 0.08 | 0.64 | 1.00 | 0.16 | −0.99 |

VMA (%) | −0.09 | 0.09 | −0.30 | 0.31 | −0.09 | 0.16 | 1.00 | 0.00 |

VFA (%) | −0.95 | 0.95 | 0.46 | −0.02 | −0.65 | −0.99 | 0.00 | 1.00 |

Parameters | MS | MF | Ps (%) | Pb (%) | Gmb | Gsb | Gmm | Va (%) | VMA (%) | VFA (%) |
---|---|---|---|---|---|---|---|---|---|---|

Unit | Kg | 0.25 mm | % | % | g/cm^{3} | g/cm^{3} | g/cm^{3} | % | % | % |

Mean | 2630 | 14.50 | 96.50 | 3.50 | 2.400 | 2.677 | 2.542 | 5.58 | 13.48 | 58.57 |

Standard Error | 11.07 | 0.22 | 0.04 | 0.04 | 0.003 | 0.002 | 0.003 | 0.10 | 0.07 | 0.73 |

Median | 2670 | 14.40 | 96.50 | 3.50 | 2.401 | 2.676 | 2.540 | 5.25 | 13.37 | 59.73 |

Mode | 2680 | 16.60 | 96.50 | 3.50 | 2.368 | 2.689 | 2.545 | 6.27 | 15.21 | 61.92 |

Standard Deviation | 176.15 | 3.51 | 0.62 | 0.62 | 0.042 | 0.031 | 0.045 | 1.64 | 1.12 | 11.66 |

Sample Variance | 31,027.69 | 12.31 | 0.39 | 0.39 | 0.002 | 0.001 | 0.002 | 2.70 | 1.26 | 135.97 |

Coefficient of Variation | 6.70 | 24.20 | 0.65 | 17.79 | 1.731 | 1.164 | 1.769 | 29.41 | 8.34 | 19.91 |

Kurtosis | −0.816 | −0.46 | −0.57 | −0.57 | −0.578 | −0.625 | 0.608 | 0.64 | 0.46 | 0.62 |

Skewness | −0.433 | −0.04 | −0.14 | 0.14 | 0.199 | 0.783 | 0.139 | 0.59 | 0.73 | −0.42 |

Range | 752 | 16.80 | 2.50 | 2.50 | 0.177 | 0.097 | 0.240 | 9.33 | 5.67 | 69.41 |

Minimum | 2220 | 6.00 | 95.00 | 2.50 | 2.306 | 2.641 | 2.418 | 1.27 | 11.23 | 19.89 |

Maximum | 2972 | 22.80 | 97.50 | 5.00 | 2.483 | 2.738 | 2.658 | 10.60 | 16.89 | 89.30 |

Parameters | MS | MF | Ps (%) | Pb (%) | Gmb | Gsb | Gmm | Va (%) | VMA (%) | VFA (%) |
---|---|---|---|---|---|---|---|---|---|---|

Unit | Kg | 0.25 mm | % | % | g/cm^{3} | g/cm^{3} | g/cm^{3} | % | % | % |

Mean | 1358 | 10.97 | 95.94 | 4.06 | 2.363 | 2.660 | 2.501 | 5.50 | 14.79 | 62.81 |

Standard Error | 5.91 | 0.09 | 0.04 | 0.04 | 0.002 | 0.002 | 0.002 | 0.08 | 0.04 | 0.54 |

Median | 1372 | 10.90 | 95.90 | 4.10 | 2.355 | 2.655 | 2.495 | 5.25 | 14.68 | 63.89 |

Mode | 1410 | 10.40 | 96.00 | 4.00 | 2.340 | 2.625 | 2.462 | 5.18 | 14.31 | 63.81 |

Standard Deviation | 109.40 | 1.70 | 0.66 | 0.66 | 0.032 | 0.033 | 0.038 | 1.53 | 0.72 | 10.06 |

Sample Variance | 11,968.28 | 2.88 | 0.43 | 0.43 | 0.001 | 0.001 | 0.001 | 2.34 | 0.52 | 101.21 |

Coefficient of Variation | 8.06 | 15.46 | 0.68 | 16.16 | 1.344 | 1.238 | 1.507 | 27.82 | 4.87 | 16.02 |

Kurtosis | 0.838 | −0.48 | −0.47 | −0.47 | −0.474 | 1.744 | −0.212 | −0.15 | 1.19 | −0.36 |

Skewness | −0.129 | −0.06 | 0.08 | −0.08 | 0.413 | 1.486 | 0.497 | 0.65 | 0.69 | −0.50 |

Range | 656 | 8.70 | 3.00 | 3.00 | 0.141 | 0.126 | 0.172 | 7.65 | 4.14 | 48.84 |

Minimum | 1024 | 6.40 | 94.50 | 2.50 | 2.290 | 2.625 | 2.427 | 2.20 | 13.24 | 34.82 |

Maximum | 1680 | 15.10 | 97.50 | 5.50 | 2.431 | 2.751 | 2.599 | 9.85 | 17.39 | 83.65 |

Number of Subpopulations | 50 |
---|---|

Subpopulation Size | 100 |

Code Length | 50 |

Crossover Probability | 0.9 |

Crossover Type | Uniform |

Error Measure | MAE |

Mathematical Operators | $+,-,\times ,\xf7,\mathrm{Power},\mathrm{Sqrt},\mathrm{Exp},\mathrm{Sin},\mathrm{Cos},\mathrm{Tan}$ |

Mutation Probability | 0.01 |

Tournament Size | 2 |

Functions | 0.5 |

Variables | 0.5 |

Number of Generations | 1000 |

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

1 | $k=\frac{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{x}}_{\mathrm{i}}\times {\mathrm{p}}_{\mathrm{i}}\right)}{{{\displaystyle \sum}}_{\mathrm{i}}^{\mathrm{n}}{\mathrm{p}}_{\mathrm{i}}^{2}}$ | 0.85 < k < 1.15 | Golbraikh and Tropsha, 2002 |

2 | ${k}^{\prime}=\frac{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{x}}_{\mathrm{i}}\times {\mathrm{p}}_{\mathrm{i}}\right)}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{i}}^{2}}$ | 0.85 < k′ < 1.15 | Golbraikh and Tropsha, 2002 |

3 | ${R}_{m}={{R}^{\prime}}_{0}^{2}\times \left(1-\left|\sqrt{{{R}^{\prime}}_{0}^{2}-{R}_{0}^{2}}\right|\right)$ | R_{m} > 0.5 | Roy and Roy, 2008 |

${R}_{0}^{2}=1-\frac{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{p}}_{\mathrm{i}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{r}0}\right)}^{2}}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{p}}_{\mathrm{i}}-\overline{{\mathrm{p}}_{\mathrm{i}}}\right)}^{2}}$ ${{R}^{\prime}}_{0}^{2}=1-\frac{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{i}}^{\mathrm{r}0}\right)}^{2}}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}}-\overline{{\mathrm{x}}_{\mathrm{i}}}\right)}^{2}}$ ${\mathrm{x}}_{\mathrm{i}}^{\mathrm{r}0}=\mathrm{k}\times {\mathrm{p}}_{\mathrm{i}}$ ${\mathrm{p}}_{\mathrm{i}}^{\mathrm{r}0}={\mathrm{k}}^{\prime}\times {\mathrm{e}}_{\mathrm{i}}$ | ${R}_{0}^{2}\cong 1$ ${{R}^{\prime}}_{0}^{2}\cong 1$ |

Model | Dataset | R | MAE | RMSE | RSE | RRMSE | ρ | OF |
---|---|---|---|---|---|---|---|---|

ABC–MS | Training | 0.96 | 36.30 | 46.62 | 0.07 | 0.01 | 0.004 | 0.033 |

Validation | 0.96 | 33.51 | 41.39 | 0.08 | 0.03 | 0.017 | ||

Testing | 0.97 | 36.94 | 46.71 | 0.06 | 0.03 | 0.017 | ||

ABC–MF | Training | 0.97 | 0.62 | 0.80 | 0.05 | 0.01 | 0.004 | |

Validation | 0.98 | 0.53 | 0.73 | 0.05 | 0.04 | 0.018 | ||

Testing | 0.96 | 0.71 | 0.90 | 0.09 | 0.03 | 0.017 | ||

AWC–MS | Training | 0.95 | 26.65 | 33.72 | 0.13 | 0.01 | 0.003 | 0.046 |

Validation | 0.97 | 24.64 | 30.55 | 0.07 | 0.02 | 0.012 | ||

Testing | 0.90 | 29.59 | 43.32 | 0.29 | 0.03 | 0.015 | ||

AWC–MF | Training | 0.96 | 0.37 | 0.47 | 0.09 | 0.01 | 0.003 | |

Validation | 0.98 | 0.34 | 0.40 | 0.05 | 0.02 | 0.012 | ||

Testing | 0.97 | 0.31 | 0.41 | 0.06 | 0.03 | 0.013 |

Model | k | k′ | Rm | R20 | R’20 |
---|---|---|---|---|---|

ABC–MS | 1.0008 | 0.9989 | 0.9887 | 0.9999 | 0.9997 |

ABC–MF | 0.9996 | 0.9975 | 0.9894 | 1.0000 | 0.9999 |

AWC–MS | 0.9994 | 1.0000 | 0.9913 | 0.9999 | 1.0000 |

AWC–MF | 0.9985 | 1.00 | 1.0000 | 1.0000 | 1.0000 |

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## Share and Cite

**MDPI and ACS Style**

Awan, H.H.; Hussain, A.; Javed, M.F.; Qiu, Y.; Alrowais, R.; Mohamed, A.M.; Fathi, D.; Alzahrani, A.M.
Predicting Marshall Flow and Marshall Stability of Asphalt Pavements Using Multi Expression Programming. *Buildings* **2022**, *12*, 314.
https://doi.org/10.3390/buildings12030314

**AMA Style**

Awan HH, Hussain A, Javed MF, Qiu Y, Alrowais R, Mohamed AM, Fathi D, Alzahrani AM.
Predicting Marshall Flow and Marshall Stability of Asphalt Pavements Using Multi Expression Programming. *Buildings*. 2022; 12(3):314.
https://doi.org/10.3390/buildings12030314

**Chicago/Turabian Style**

Awan, Hamad Hassan, Arshad Hussain, Muhammad Faisal Javed, Yanjun Qiu, Raid Alrowais, Abdeliazim Mustafa Mohamed, Dina Fathi, and Abdullah Mossa Alzahrani.
2022. "Predicting Marshall Flow and Marshall Stability of Asphalt Pavements Using Multi Expression Programming" *Buildings* 12, no. 3: 314.
https://doi.org/10.3390/buildings12030314