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

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{*}

## Abstract

**:**

## 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

- Asphalt Institute. MS-2 Asphalt Mix Design Methods; Asphalt Institute: Lexington, KY, USA, 2014. [Google Scholar]
- Zumrawi, M.M.; Edrees, S.A.S. Comparison of Marshall and Superpave asphalt design methods for Sudan pavement mixes. Int. J. Sci. Tech. Adv.
**2016**, 2, 29–35. [Google Scholar] - Ministry of Communications. National Highway Authority, Government of Pakistan; Ministry of Communications: Islamabad, Pakistan, 2009; pp. 301-1–301-3d.
- Kuloǧlu, N. Effect of astragalus on characteristics of asphalt concrete. J. Mater. Civ. Eng.
**1999**, 11, 283–286. [Google Scholar] [CrossRef] - Hınıslıoğlu, S.; Ağar, E. Use of waste high density polyethylene as bitumen modifier in asphalt concrete mix. Mater. Lett.
**2004**, 58, 267–271. [Google Scholar] [CrossRef] - Azarhoosh, A.; Pouresmaeil, S. Prediction of Marshall Mix Design Parameters in Flexible Pavements Using Genetic Programming. Arab. J. Sci. Eng.
**2020**, 45, 8427–8441. [Google Scholar] [CrossRef] - Alsugair, A.M.; Al-Qudrah, A.A. Artificial neural network approach for pavement maintenance. J. Comput. Civ. Eng.
**1998**, 12, 249–255. [Google Scholar] [CrossRef] - Tapkın, S.; Çevik, A.; Uşar, Ü. Prediction of Marshall test results for polypropylene modified dense bituminous mixtures using neural networks. Expert Syst. Appl.
**2010**, 37, 4660–4670. [Google Scholar] [CrossRef] - Milad, A.A. Development of a Hybrid Machine Learning Model for Asphalt Pavement Temperature Prediction. IEEE Access
**2021**, 9, 158041–158056. [Google Scholar] [CrossRef] - Aslam, F. Compressive strength prediction of rice husk ash using multiphysics genetic expression programming. Ain Shams Eng. J.
**2021**, 13, 101593. [Google Scholar] [CrossRef] - Zhao, T.H.; Khan, M.I.; Chu, Y.M. Artificial neural networking (ANN) analysis for heat and entropy generation in flow of non-Newtonian fluid between two rotating disks. Math. Methods Appl. Sci.
**2021**. [Google Scholar] [CrossRef] - Zha, T.-H. A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak. Appl. Comput. Math.
**2021**, 20, 160–176. [Google Scholar] - Chu, H.-H.; Zhao, T.-H.; Chu, Y.-M. Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means. Math. Slovaca
**2020**, 70, 1097–1112. [Google Scholar] [CrossRef] - Zhao, T.-H.; He, Z.-Y.; Chu, Y.-M. On some refinements for inequalities involving zero-balanced hypergeometric function. AIMS Math
**2020**, 5, 6479–6495. [Google Scholar] [CrossRef] - Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math
**2020**, 5, 4512–4528. [Google Scholar] [CrossRef] - Zhao, T.-H. Quadratic transformation inequalities for Gaussian hypergeometric function. J. Inequalities Appl.
**2018**, 2018, 251. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhao, T.-H. On approximating the quasi-arithmetic mean. J. Inequalities Appl.
**2019**, 2019, 42. [Google Scholar] [CrossRef] - Ozgan, E. Artificial neural network based modelling of the Marshall Stability of asphalt concrete. Expert Syst. Appl.
**2011**, 38, 6025–6030. [Google Scholar] [CrossRef] - Baldo, N.; Manthos, E.; Miani, M. Stiffness modulus and marshall parameters of hot mix asphalts: Laboratory data modeling by artificial neural networks characterized by cross-validation. Appl. Sci.
**2019**, 9, 3502. [Google Scholar] [CrossRef][Green Version] - Shah, S.A.R. Marshall stability and flow analysis of asphalt concrete under progressive temperature conditions: An application of advance decision-making approach. Constr. Build. Mater.
**2020**, 262, 120756. [Google Scholar] [CrossRef] - Saffarzadeh, M.; Heidaripanah, A. Effect of asphalt content on the marshall stability of asphalt concrete using artificial neural networks. Sci. Iran.
**2009**, 16, 98–105. [Google Scholar] - Morova, N. Modeling Marshall Stability of light asphalt concretes fabricated using expanded clay aggregate with Artificial Neural Networks. In Proceedings of the 2012 International Symposium on Innovations in Intelligent Systems and Applications, Trabzon, Turkey, 2–4 July 2012. [Google Scholar]
- Morova, N. Modelling Marshall Stability of fiber reinforced asphalt mixtures with ANFIS. In Proceedings of the 2017 IEEE International Conference on Innovations in Intelligent Systems and Applications (INISTA), Gdynia, Poland, 3–5 July 2017. [Google Scholar]
- SERİN, S. The Fuzzy Logic Model for the Prediction of Marshall Stability of Lightweight Asphalt Concretes Fabricated using Expanded Clay Aggregate. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Derg.
**2013**, 17, 163–172. [Google Scholar] - Ozgan, E. Fuzzy logic and statistical-based modelling of the Marshall Stability of asphalt concrete under varying temperatures and exposure times. Adv. Eng. Softw.
**2009**, 40, 527–534. [Google Scholar] [CrossRef] - Tsompanakis, Y. Stability Prediction of Asphaltic Concrete Mixes Using Multiple Additive Regression Trees. In Proceedings of the Fourth International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering; Civil-Comp Press: Prague, Czech Republic, 2015. [Google Scholar]
- Nguyen, H.-L. Development of hybrid artificial intelligence approaches and a support vector machine algorithm for predicting the marshall parameters of stone matrix asphalt. Appl. Sci.
**2019**, 9, 3172. [Google Scholar] [CrossRef][Green Version] - Khuntia, S. Prediction of Marshall parameters of modified bituminous mixtures using artificial intelligence techniques. Int. J. Transp. Sci. Technol.
**2014**, 3, 211–227. [Google Scholar] [CrossRef][Green Version] - Ghanizadeh, A.R. Predicting flow number of asphalt mixtures based on the marshall mix design parameters using multivariate adaptive regression spline (MARS). Int. J. Transp. Eng.
**2020**, 7, 433–448. [Google Scholar] - Yan, K.-Z.; Ge, D.-D.; Zhang, Z. Support vector machine models for prediction of flow number of asphalt mixtures. Int. J. Pavement Res. Technol.
**2014**, 7, 31. [Google Scholar] - Sharifi, S.; Abrishami, S.; Gandomi, A.H. Consolidation assessment using multi expression programming. Appl. Soft Comput.
**2020**, 86, 105842. [Google Scholar] [CrossRef] - Alavi, A.H. Multi expression programming: A new approach to formulation of soil classification. Eng. Comput.
**2010**, 26, 111–118. [Google Scholar] [CrossRef] - Chu, H.-H. Sustainable use of fly-ash: Use of gene-expression programming (GEP) and multi-expression programming (MEP) for forecasting the compressive strength geopolymer concrete. Ain Shams Eng. J.
**2021**, 12, 3603–3617. [Google Scholar] [CrossRef] - Gandomi, A.H. New design equations for elastic modulus of concrete using multi expression programming. J. Civ. Eng. Manag.
**2015**, 21, 761–774. [Google Scholar] [CrossRef][Green Version] - Iqbal, M.F. Sustainable utilization of foundry waste: Forecasting mechanical properties of foundry sand based concrete using multi-expression programming. Sci. Total Environ.
**2021**, 780, 146524. [Google Scholar] [CrossRef] - Zhang, Q. Predicting cement compressive strength using double-layer multi-expression Programming. In Proceedings of the 2012 Fourth International Conference on Computational and Information Sciences, Chongqing, China, 17–19 August 2012. [Google Scholar]
- Abdalla, A.; Salih, A. Implementation of multi-expression programming (MEP), artificial neural network (ANN), and M5P-tree to forecast the compression strength cement-based mortar modified by calcium hydroxide at different mix proportions and curing ages. Innov. Infrastruct. Solut.
**2022**, 7, 153. [Google Scholar] [CrossRef] - Heshmati, A. A multi expression programming application to high performance concrete. World Appl. Sci. J.
**2008**, 5, 215–223. [Google Scholar] - Amin, M.N. Multigene Expression Programming Based Forecasting the Hardened Properties of Sustainable Bagasse Ash Concrete. Materials
**2021**, 14, 5659. [Google Scholar] [CrossRef] [PubMed] - Wang, H.-L.; Yin, Z.-Y. High performance prediction of soil compaction parameters using multi expression programming. Eng. Geol.
**2020**, 276, 105758. [Google Scholar] [CrossRef] - Gandomi, A.H.; Alavi, A.H.; Yun, G.J. Formulation of uplift capacity of suction caissons using multi expression programming. KSCE J. Civ. Eng.
**2011**, 15, 363–373. [Google Scholar] [CrossRef] - Koza, J.R. Genetic Programming: On the Programming of Computers by Means of Natural Selection; MIT Press: Cambridge, MA, USA, 1992; Volume 1. [Google Scholar]
- Banzhaf, W. Genetic Programming: An Introduction: On the Automatic Evolution of Computer Programs and Its Applications; Morgan Kaufmann Publishers Inc.: Burlington, MA, USA, 1998. [Google Scholar]
- Back, T. Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms; Oxford University Press: Oxford, UK, 1996. [Google Scholar]
- Oltean, M.; Dumitrescu, D. Multi expression programming. J. Genet. Program. Evolvable Mach. Kluwer Second. Tour Rev.
**2002**. [Google Scholar] [CrossRef] - Oltean, M.; Grosan, C. A comparison of several linear genetic programming techniques. Complex Syst.
**2003**, 14, 285–314. [Google Scholar] - Khan, M.A. Simulation of Depth of Wear of Eco-Friendly Concrete Using Machine Learning Based Computational Approaches. Materials
**2022**, 15, 58. [Google Scholar] [CrossRef] - Ceylan, H. Backcalculation of full-depth asphalt pavement layer moduli considering nonlinear stress-dependent subgrade behavior. Int. J. Pavement Eng.
**2005**, 6, 171–182. [Google Scholar] [CrossRef] - Ceylan, H. Accuracy of predictive models for dynamic modulus of hot-mix asphalt. J. Mater. Civ. Eng.
**2009**, 21, 286–293. [Google Scholar] [CrossRef][Green Version] - Vapnik, V. The Nature of Statistical Learning Theory; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Alavi, A.H. Formulation of flow number of asphalt mixes using a hybrid computational method. Constr. Build. Mater.
**2011**, 25, 1338–1355. [Google Scholar] [CrossRef] - Alawi, M.; Rajab, M. Determination of optimum bitumen content and Marshall stability using neural networks for asphaltic concrete mixtures. In Proceedings of the 9th WSEAS International Conference on Computers, World Scientific and Engineering Academy and Society (WSEAS), Athens, Greece, 11–13 July 2005. [Google Scholar]
- Serin, S. Determining amount of bituminous effects on asphalt concrete strength with artificial intelligence and statistical analysis methods. In Proceedings of the 2011 International Symposium on Innovations in Intelligent Systems and Applications, Istanbul, Turkey, 15–18 June 2011. [Google Scholar]
- Bagheri, M. Investigating plant uptake of organic contaminants through transpiration stream concentration factor and neural network models. Sci. Total Environ.
**2021**, 751, 141418. [Google Scholar] [CrossRef] [PubMed] - Abdolrasol, M.G. Artificial Neural Networks Based Optimization Techniques: A Review. Electronics
**2021**, 10, 2689. [Google Scholar] [CrossRef] - Nazeer, M. Theoretical study of MHD electro-osmotically flow of third-grade fluid in micro channel. Appl. Math. Comput.
**2022**, 420, 126868. [Google Scholar] [CrossRef] - Oltean, M.; Groşan, C. Evolving evolutionary algorithms using multi expression programming. In Proceedings of the European Conference on Artificial Life, Dortmund, Germany, 14–17 September 2003. [Google Scholar]
- Khan, M.A. Geopolymer Concrete Compressive Strength via Artificial Neural Network, Adaptive Neuro Fuzzy Interface System, and Gene Expression Programming with K-Fold Cross Validation. Front. Mater.
**2021**, 8, 621163. [Google Scholar] [CrossRef] - Khan, S. Predicting the Ultimate Axial Capacity of Uniaxially Loaded CFST Columns Using Multiphysics Artificial Intelligence. Materials
**2022**, 15, 39. [Google Scholar] [CrossRef] - Faris, H.; Mirjalili, S.; Aljarah, I. Automatic selection of hidden neurons and weights in neural networks using grey wolf optimizer based on a hybrid encoding scheme. Int. J. Mach. Learn. Cybern.
**2019**, 10, 2901–2920. [Google Scholar] [CrossRef] - Ferreira, C. Gene Expression Programming: Mathematical Modeling by an Artificial Intelligence; Springer: Berlin/Heidelberg, Germany, 2006; Volume 21. [Google Scholar]
- Azim, I. Semi-analytical model for compressive arch action capacity of RC frame structures. In Structures; Elsevier: Amsterdam, The Netherlands, 2020. [Google Scholar]
- Azim, I. Prediction of catenary action capacity of RC beam-column substructures under a missing column scenario using evolutionary algorithm. KSCE J. Civ. Eng.
**2021**, 25, 891–905. [Google Scholar] [CrossRef] - Chu, Y.-M. Combined impact of Cattaneo-Christov double diffusion and radiative heat flux on bio-convective flow of Maxwell liquid configured by a stretched nano-material surface. Appl. Math. Comput.
**2022**, 419, 126883. [Google Scholar] [CrossRef] - Khan, M.A. Compressive strength of fly-ash-based geopolymer concrete by gene expression programming and random forest. Adv. Civ. Eng.
**2021**, 2021, 6618407. [Google Scholar] [CrossRef] - Gandomi, A.H.; Roke, D.A. Assessment of artificial neural network and genetic programming as predictive tools. Adv. Eng. Softw.
**2015**, 88, 63–72. [Google Scholar] [CrossRef] - Maeda, T. How to rationally compare the performances of different machine learning models? PeerJ Prepr.
**2018**. [Google Scholar] [CrossRef] - Abunama, T. Leachate generation rate modeling using artificial intelligence algorithms aided by input optimization method for an MSW landfill. Environ. Sci. Pollut. Res.
**2019**, 26, 3368–3381. [Google Scholar] [CrossRef] [PubMed] - Azim, I. Prediction Model for Compressive Arch Action Capacity of RC Frame Structures under Column Removal Scenario Using Gene Expression Programming. In Structures; Elsevier: Amsterdam, The Netherlands, 2020. [Google Scholar]
- Li, P. Sustainable Use of Chemically modified Tyre Rubber in Concrete: Machine Learning based Novel Predictive Model. Chem. Phys. Lett.
**2022**, 793, 139478. [Google Scholar] [CrossRef] - Smith, G.N. Probability and statistics in civil engineering. In Collins Professional and Technical Books; Collins: London, UK, 1986; Volume 244. [Google Scholar]
- Mousavi, S. A data mining approach to compressive strength of CFRP-confined concrete cylinders. Struct. Eng. Mech.
**2010**, 36, 759. [Google Scholar] [CrossRef] - Pyo, J. Estimation of heavy metals using deep neural network with visible and infrared spectroscopy of soil. Sci. Total Environ.
**2020**, 741, 140162. [Google Scholar] [CrossRef] - Qiu, R. Water temperature forecasting based on modified artificial neural network methods: Two cases of the Yangtze River. Sci. Total Environ.
**2020**, 737, 139729. [Google Scholar] [CrossRef] - Nguyen, T. Deep neural network with high-order neuron for the prediction of foamed concrete strength. Comput. Aided Civ. Infrastruct. Eng.
**2019**, 34, 316–332. [Google Scholar] [CrossRef] - Gandomi, A.H. Nonlinear genetic-based models for prediction of flow number of asphalt mixtures. J. Mater. Civ. Eng.
**2011**, 23, 248–263. [Google Scholar] [CrossRef] - Despotovic, M. Evaluation of empirical models for predicting monthly mean horizontal diffuse solar radiation. Renew. Sustain. Energy Rev.
**2016**, 56, 246–260. [Google Scholar] [CrossRef] - Gholampour, A.; Gandomi, A.H.; Ozbakkaloglu, T. New formulations for mechanical properties of recycled aggregate concrete using gene expression programming. Constr. Build. Mater.
**2017**, 130, 122–145. [Google Scholar] [CrossRef] - Ali Khan, M. Application of Gene Expression Programming (GEP) for the Prediction of Compressive Strength of Geopolymer Concrete. Materials
**2021**, 14, 1106. [Google Scholar] [CrossRef] [PubMed] - Khan, M.A. Application of random forest for modelling of surface water salinity. Ain Shams Eng. J.
**2021**, 13, 101635. [Google Scholar] [CrossRef] - Golbraikh, A.; Tropsha, A. Beware of q2! J. Mol. Graph. Model.
**2002**, 20, 269–276. [Google Scholar] [CrossRef] - Roy, P.P.; Roy, K. On some aspects of variable selection for partial least squares regression models. QSAR Comb. Sci.
**2008**, 27, 302–313. [Google Scholar] [CrossRef]

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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 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