## 3 Computational Examples and Analysis
### Case Study II: Tri-Objective Optimization
#### 3.2.3 Results of Asphalt Mixture Proportion Optimization using GPR-AWMOPSO  
(1) System Models Based on GPR  
For the GPR system model, 12 groups are selected randomly from 16 orthogonal groups and also divided into training and testing dataset. Then we can predict the volumetric and physical properties of diatomite and basalt fibre composite-modified asphalt mixture from 25% of the selected dataset by using the training dataset from 75% of the selected dataset randomly. The RMSE and R\({}^{2}\) can be calculated using Equations (21) and (22), and the prediction results for the volumetric and physical properties of asphalt mixture containing diatomite and basalt fibre are shown in Table 9. Figure 9 shows the predictive values by the GPR model versus the experimental values for training and testing sets. The regression analysis indicates that the optimal GPR model was successful in learning the relationship between the input and output variables. There are lower RMSE values and higher R\({}^{2}\) values for both training and testing datasets. It can be indicated that the GPR model can accurately model the relationship between input and output variables. Thus, it could be confirmed that the optimal GPR model has been well trained based on the comparison between the volumetric and physical properties of training and testing datasets  
(2) Proportion Optimization Results for Modified Asphalt Mixture Based on GPR-AWMOPSO  
According to the above AWMOPSO algorithm, the optimal solution can be obtained for the proportion of asphalt mixture containing diatomite and basalt fibre. The tri-objective modified asphalt mixture optimization results are shown in Figure 10a. GPR is also used as the objective function for modelling proportion design of asphalt mixture, The Pareto front based on mechanical performances (i.e., Marshall stability and splitting strength at \(-10\)\({}^{\circ}\)C) and cost could be obtained. It is observed that these points are distributed within a reasonable range, showing a good effectiveness of GPR-AWMOPSO. Meanwhile, it is seen that the modified asphalt mixture cost would increase with increasing mechanical performance. In total, several non-dominated solutions (the optimal mixture proportions) are selected. In Figure 10a, the solution with higher mechanical performance on the Pareto front is more expensive. In general, the optimal solution depends on the mixture design consideration. Solution B on the Pareto front is more expensive but has higher mechanical performances, whereas solution C has lower mechanical performances with a correspondingly lower cost. Point A on the Pareto front has the highest TOPSIS score. It also shows that the solution A by TOPSIS has an intermediate cost and performance responses compared with solutions B and C. With the mechanical performances as the primary goal, point C marked by a red cube will be the best option to achieve the maximum mechanical performances. Figure 9: Comparison of the predictive values by GPR versus the experimental values: (**a**) Y\({}_{1}\); (**b**) Y\({}_{2}\); (**c**) Y\({}_{3}\); (**d**) Y\({}_{4}\); and (**e**) Y\({}_{5}\). To verify the proposed GPR-AWMOPSO algorithm, the optimized proportion of asphalt mixture is compared with the results of the orthogonal design, traditional MOPSO and MADS. The three inputs and five outputs variables are summarized in Table 10. From Table 10, it can be found that the optimal results yield higher high-temperature and low-temperature physical performances without a noticeable increase in the mixture proportions. Meanwhile, these results are satisfied well with the constraints following JTG F40-2004. The optimal solution generally relies on the considerations of mixture design. Compared with the traditional orthogonal design, traditional MOPSO and MADS, the proposed GPR-AWMOPSO algorithm has higher physical performances. When maximizing performances is the main objective, the proposed GPR-AWMOPSO algorithm is the best option.