## 3 Computational Examples and Analysis
### Case Study III: Optimization with Four Objectives
#### 3.3.3 Results of Concrete Mix Proportion Optimization using GPR-AWMOPSO  
(1) System Models Based on GPR  
For the GPR system model, 12 groups are selected randomly from 17 groups of RSM experimental design and also divided into training and testing dataset. Then we can predict the performance properties of rubber and basalt fibre composite modified concrete from 25% of the selected dataset by using the training dataset from 75% of the selected dataset randomly.  
Figure 12 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 with lower RMSE values and higher R\({}^{2}\) values was successful in learning the relationship between the input and output variables. It can be indicated that the GPR model can accurately model the relationship between input and output variables, which could be used for the performance prediction of rubber and basalt fibre modified concrete.  
(2) Proportion Optimization Results Based on GPR-AWMOPSO  
According to the above AWMOPSO algorithm, the optimal solution can be obtained for the proportion of concretes modified by rubber and basalt fibre. The modified concrete optimization results are shown in Figure 13. GPR is used as the objective function for modelling proportion design of asphalt mixture. The Pareto front based on response variables and cost is obtained. These points are distributed within a reasonable range, showing a good effectiveness of GPR-AWMOPSO. In total, several non-dominated solutions (the optimal mixture proportions) are selected. In Figure 13, 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. Solution B on the Pareto front is more expensive and has higher compression strength and flexural strength but lower slump. However, solution C has lower compression strength and flexural strength with a correspondingly lower cost. It also shows that solution A by TOPSIS has an intermediate cost and response variables compared with solutions B and C. With the mechanical performance including compression strength and flexural strength as the primary goal, point B will be the best option to achieve the maximum mechanical performance. While when minimizing cost is the main objective, point C will be the best option. Moreover, Table 11 also summarizes the comparison results of three inputs and three outputs variables. From Table 10, compared with the results of the orthogonal de  
Figure 12: Comparison of the predictive values by GPR versus the experimental values: (**a**) Y\({}_{1}\); (**b**) Y\({}_{2}\); and (**c**) Y\({}_{3}\).  
sign, traditional MOPSO and MADS, it can be found that the optimal results yield higher mechanical performance including compression strength and flexural strength.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Optimization**} & \multicolumn{3}{c}{**Input Variables**} & \multicolumn{3}{c}{**Output Variables**} \\ \cline{2-7}  & **X1** & **X2 (kg/m3)** & **X3 (\%)** & **Y1 (mm)** & **Y2 (MPa)** & **Y3 (MPa)** \\ \hline Orthogonal & 0.39 & 4.56 & 10 & 64 & 39.96 & 3.8 \\ Traditional MOPSO & 0.3978 & 4.5762 & 10.6711 & 65.1515 & 38.7927 & 3.8367 \\ MADS & 0.4012 & 4.5886 & 10.7919 & 65.5083 & 40.0506 & 3.9598 \\ GPR-AWMOPSO & 0.4049 & 4.5802 & 10.6861 & 65.3504 & 40.2792 & 3.9766 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Comparison of optimization results among the traditional orthogonal design [53], traditional MOPSO, MADS and GPR-AWMOPSO algorithm.  
Figure 13: The proportion optimization results for rubber and basalt fibre modified concrete using GPR-AWMOPSO: (**a**) Pareto front showing three objectives (F1, F2, and F3); (**b**) Pareto front showing three objectives (F1, F2, and F4) and (**c**) Pareto front showing three objectives (F2, F3 and F4).