## 4 MSIMP Based on EMOEAs
### Surrogate Models
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Types** & **Surrogate Model** & **Main Ideas** & **Advantages and Disadvantages** \\ \hline \multirow{8}{*}{**Statistical Model**} & \multirow{8}{*}{Kriging} & Assuming that the value of & **Advantages:** it can provide the prediction \\  & & the objective function follows & and variance of the objective function value, \\  & & the multivariable normal & and is suitable for dealing with highly \\  & & distribution, the candidate & nonlinear problems. \\  & & solution is predicted by & **Disadvantages:** it has high computational \\  & & historical data. & complexity for large-scale data sets. \\ \cline{2-4}  & \multirow{4}{*}{Support Vector Regression} & To minimize the error, a & **Advantages:** it performs well in \\  & & hyperplane is found in the & high-dimensional space and has good \\  & & higher-dimensional space by & generalization ability. \\  & & classification and analysis. & **Disadvantages:** the selection kernel function \\  & & Bayesian statistics and & and parameter tuning are complex. \\  & & surrogate models are used to & **Advantages:** it effectively handles expensive \\  & & approximate the objective & black-box function optimization problems, \\  & & function, and the acquisition & balancing exploration and development. \\  & & function is used to select new & **Disadvantages:** it is very sensitive to the \\  & & evaluation points. & quality and quantity of initial data, and \\  & & & improper selection of initial samples may \\  & & & lead to a poor performance of proxy models. \\ \hline \multirow{8}{*}{**Function approximation model**} & \multirow{8}{*}{Radial Basis Function} & Lower-order polynomial & **Advantages:** the calculation is simple and \\  & & functions, such as linear & suitable for linear or low-order nonlinear \\  & & polynomials, are often used to & problems. \\ \cline{1-1}  & & approximate objective & **Disadvantages:** for complex systems, the \\ \cline{1-1}  & & functions and optimize & model is overly sensitive to training data, \\ \cline{1-1}  & & stochastic processes. & which can lead to overfitting and affect \\ \cline{1-1} \cline{2-4}  & \multirow{4}{*}{Radial Basis Function} & The influence of each sample & **Advantages:** it is suitable for processing \\  & & point is expressed as a & multidimensional nonlinear problems with \\ \cline{1-1}  & & function of distance, and the & high flexibility. \\ \cline{1-1}  & & objective function is & **Disadvantages:** it has low computational \\ \cline{1-1}  & & approximated and fitted. & efficiency for processing large-scale data sets. \\ \hline \multirow{8}{*}{Polynomial Regression} & \multirow{8}{*}{Polynomial Regression} & **Advantages:** it is suitable for \\ \cline{1-1}  & & the objective function is fitted & low-dimensional problems and the \\ \cline{1-1}  & & using polynomial functions. & calculation is simple. \\ \cline{1-1} \cline{2-4}  & & & **Disadvantages:** it is not effective for \\ \cline{1-1} \cline{2-4}  & & high-dimensional nonlinear problems. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The main ideas, advantages, and disadvantages of the three types of surrogate models. According to Table 3, the three types of surrogate models differ in their scope of application and ability to handle data. Among them, statistical models require a medium-sized dataset to train the surrogate model and are suitable for complex, high-dimensional, and highly nonlinear problems. Function approximation models, on the other hand, require fewer data to train and are suitable for low-dimensional, linear, and low-order nonlinear problems. Machine learning models are more data-demanding, especially for big data and complex tasks requiring large amounts of data for training, and are suitable for complex and high-dimensional problems. The MSIMP problem can be solved in practice through choosing an appropriate surrogate model according to the specific user requirements in order to improve the efficiency and quality of problem solving.