## 3 Computational Examples and Analysis
### Case Study I: Bi-Objective Optimization
#### 3.1.3 Results of Asphalt Mixture Proportion Optimization Using GPR-AWMOPSO
Figure 5: The reliable results of the GPR model versus size of training dataset. Figure 6: Comparison of the predictive values by GPR versus the experimental values: (**a**) \(Y_{1^{2}}\); (**b**) \(Y_{2^{2}}\); (**c**) \(Y_{3^{2}}\); (**d**) \(Y_{4^{2}}\); and (**e**) \(Y_{5}\). (2) Proportion Optimization Results for Basalt Fibre-Modified Asphalt Mixture using GPR-AWMOPSO  
According to the above AWMOPSO algorithm, the optimal solution can be obtained for the proportion of asphalt mixture containing basalt fibre. The bi-objective modified asphalt mixture optimization results are shown in Figure 7a. GPR is used as the objective function for modelling proportion design of asphalt mixture. The Pareto front based on mechanical performance (Marshall stability) and cost is obtained. These points are distributed within a reasonable range, showing a good effectiveness of GPR-AWMOPSO. Meanwhile, it is observed that the asphalt mixture cost would increase with the increasing of mechanical performance. In total, several non-dominated solutions (the optimal mixture proportions) are selected. In Figure 7a, the solution with higher mechanical performance on the Pareto front is more expensive. In general, the optimal solution depends on the mixture design consideration. According to the closeness coefficient of TOPSIS on the Pareto front, the point A has the highest TOPSIS score. The solution B on the Pareto front is more expensive but has a higher mechanical performance, whereas the solution C has a lower mechanical performance with a correspondingly lower cost. It also shows that the solution A by TOPSIS has an intermediate cost and Marshall stability compared with the solutions B and C. With the mechanical performance as the primary goal, the point B marked by a red five-pointed star will be the best option to achieve the maximum mechanical performance. While minimizing cost is the main objective, the point C marked by a green five-pointed star will be the best option. To verify the proposed GPR-AWMOPSO algorithm, the optimized proportion of asphalt mixture is compared with the results of RSM, traditional MOPSO [43, 44, 45] and mesh-adaptive direct search (MADS) [46, 47, 48]. When the objective is to achieve the maximum Marshall stability, the three inputs and five outputs variables are summarized in Table 5. From Table 5, it can be found that the optimal results yield higher mechanical properties without a noticeable increase in the mixture proportions. Meanwhile, these results are well satisfied with the constraints shown in Table 3 following the Chinese specification JTG F40-2004. Moreover, Figure 7b shows the swarm maximum value of the objective function (at the swarm best position) versus iteration for traditional MOPSO, MADS and GPR-AWMOPSO. After several numbers of iteration, it can be found that a significant increase in the maximum objective function value occurs, indicating that both traditional MOPSO, MADS and GPR-AWMOPSO algorithms are efficient in the proportion optimization of composite modified asphalt mixture. The traditional MOPSO, MADS and GPR-AWMOPSO algorithms also possess good precision. As for calculation speed, the proposed GPR-AWMOPSO in this paper starts to converge rapidly after about 6 iterations, while the convergence rate of the traditional MOPSO is slightly slower. In contrast, the MADS method also has better results. The average iteration number of the MADS method is slightly larger than that of GPR-AWMOPSO, while the convergence rate of GPR-AWMOPSO in this paper is stable and fast. The optimal solution generally relies on the considerations of mixture design. Compared with the RSM design and traditional MOPSO, the proposed GPR-AWMOPSO