## 4 Conclusions  
In this study, a novel AWMOPSO algorithm based on GPR is proposed for the proportion design and optimization of asphalt mixture, which could be adopted to deal with multi-variable, multi-level problems with multiple constraints. The objective function based on GPR was established and evaluated in terms of RMSE and R\({}^{2}\). The results showed that the optimal GPR can accurately model the relationship between input and output variables. The well trained GPR model was then applied to the AWMOPSO design of asphalt mixture (SMA-13) containing basalt fibre as well as asphalt mixture (AC-13) containing diatomite and basalt fibre. The proportions of modified asphalt mixtures were considered as input variables and the volumetric and physical properties were output variables. After the optimization, the comprehensive pavement performances were enhanced in terms of the high-temperature permanent deformation resistance, low-temperature crack resistance as well as moisture stability. Therefore, the proposed GPR-AWMOPSO algorithm is regarded as the best option and efficient for maximizing the performances of modified asphalt mixture. The main objective of this study is the verification of the GPR-AWMOPSO algorithm for the prediction of volumetric and physical properties and mixture proportion optimization. The GPR-AWMOPSO algorithm has several advantages over traditional laboratory-based experimental methods:  
1. The GPR-AWMOPSO algorithm can be used to develop the objective functions without knowing the explicit relationships between variables and objectives. 2. The selected metaheuristic algorithm could solve the problem and find the global optimal solution for multiple competing objectives. 3. The GPR-AWMOPSO algorithm has higher accuracy to balance the competing objectives with the ML model as objective functions, which can serve as guidance for the proportion optimization design of asphalt mixture. 4. The GPR-AWMOPSO algorithm has advantages of less computational time and fewer samples to avoid the limitation that the traditional experimental design method cannot effectively deal with lots of variables and levels. In addition, the proposed GPR-AWMOPSO algorithm can be extended to other practical design problems. Conceptualization, C.L., H.C., W.W. and Z.G. ; Methodology, H.C., W.W. and G.T. ; Validation, Z.G. and G.T. ; Formal Analysis, W.W., K.Z. and H.C.; Investigation, X.X., W.W., H.C. and H.Z. ; Writing--Original Draft Preparation, W.W.; Writing--Review and Editing, C.L., H.C. and Z.G. ; Project Administration, C.L., W.W. and G.T. ; Funding Acquisition, C.L., W.W., K.Z. and G.T. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (grant number 51978309), Scientific and Technological project of Science and Technology Department of Jilin Province (grant number 20190303052 SF), Jilin Province Development and Reform Commission Project (grant number 2019C041-5) and Science and Technology Project of Education Department of Jilin Province (grant number JJKH20190150 KJ), Jilin Transportation Science and Technology Popularized Project (grant number 2020-1-3). This work was also supported in part by the Innovation special project of Zhongshan Science and Technology Bureau (grant 2019 AG001). Not applicable. The data presented in this study are available on request from the corresponding author. The authors would like to appreciate anonymous reviewers for their constructive suggestions and comments to improve the quality of the paper. The authors declare no conflict of interest. * _[PERSON] et al. (2004)_ [PERSON]; [PERSON]; [PERSON] Handling multiple objectives with particle swarm optimization. _IEEE Trans. Evol. Comput._**2004**, \(8\), 256-279. [CrossRef]
* _[PERSON] et al. (2005)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] A comparative study of metamodeling methods for multiobjective crashworthiness optimization. _Comput. Struct._**2005**, _83_, 2121-2136. [CrossRef]
* _[PERSON] et al.