# Kriging-Based Variable Screening Method for Aircraft Optimization Problems with Expensive Functions
## 3 Proposed KRG-VSM
### Test on Benchmark Problem  
#### Benchmark Description and Screening Results  
To verify the effectiveness of the proposed algorithm, we test it on five well-known variable screening problems. The screening results are presented and discussed. To further demonstrate the effectiveness of the proposed algorithm relative to the original KRG-VSM, this subsection employs a benchmark numerical case, i.e., the Welch function, from the original KRG-VSM to conduct comparative simulations with our enhanced method, and the simulation results demonstrate the advantages of our proposed approach. \(\theta_{\text{common}}\) means the value of theta for the inputs left in the set \(S\). \(\theta_{\text{end}}\) means the values of theta for the unrestrained inputs at the end of the algorithm. We obtain the samples utilizing the LHS approach.  
Throughout all test cases, sample points were generated using LHS with randomized initialization, ensuring statistically equivalent comparison conditions across all simulations.  
Welch function  
\[\begin{split} f(x)=&\frac{5x_{12}}{(1+x_{1})}+5{(x_{ 4}-x_{20})}^{2}+x_{5}+40x_{19}^{3}-5x_{19}+0.05x_{2}+0.08x_{3}\\ &-0.03x_{6}+0.03x_{7}-0.09x_{9}-0.01x_{10}-0.07x_{11}+0.25x_{13} ^{2}\\ &-0.04x_{14}+0.06x_{15}-0.01x_{17}-0.03  
Linkletter sinusoidal function  
\[f(x)=\sin(x_{1})+\sin(5x_{2}) \tag{11}\]  
Equation (11) [15] is evaluated on the scaled input domain \(x_{i}\in[0,1]\), for all \(i=1,\dots,10\). It is shown above that there are only two active variables in this function. The last eight inputs do not appear in the expression. The input variable ranges for most numerical examples are defined as symmetric distributions centered around zero. The choice of min-max scaling to [0, 1] for the input variables in this function is made to maintain consistency with the benchmark problem we adopted from the well-established Virtual Library of Simulation Experiments (VASE) at Simon Fraser University ([[https://www.sfu.ca/~ssurjano/linketal0](https://www.sfu.ca/~ssurjano/linketal0)]([https://www.sfu.ca/~ssurjano/linketal0](https://www.sfu.ca/~ssurjano/linketal0)) 6 sin.html, accessed on 17 March 2025).  
Eighty samples are used for the variable screening. The results are shown in Table 8. The first two factors to be selected are \(x_{2}\) and \(x_{1}\). Then, the procedure terminated as the value of the common theta was less than \(10\times 10^{-6}\).  
Loeppy function  
\[\begin{split} f(x)=&\ 6x_{1}+4x_{2}+5.5x_{3}+3x_{1}x_{ 2}+2.2x_{1}x_{3}\\ &+1.4x_{2}x_{3}+x_{4}+0.5x_{5}+0.2x_{6}+0.1x_{7}\end{split} \tag{12}\]  
For Equation (12), the first three inputs are active, and the last three are inactive, while the rest are moderately active. The function is evaluated on the hypercube \(x_{i}\in[0,1]\)  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Selected** & **\(\boldsymbol{\theta}_{\text{common}}\)** & \
\begin{tabular}{c} **Log-** \\ **Likelihood** \\ \end{tabular} & \(\boldsymbol{\theta}_{\text{end}}\) & **Flag** &
\begin{tabular}{c} **Active** \\ **Input** \\ \end{tabular} \\ \hline \(x_{18}\) & 0.0161 & 24.68 & 0.0048 & 0 & Y \\ \(x_{1}\) & 0.0123 & 25.48 & 0.0517 & 0 & Y \\ \(x_{1  
The last three input variables do not appear in the expression. The screening procedure terminated as the value of the common theta was \(7.7\times 10^{-6}\), which is less than \(10\times 10^{-6}\). No variable is selected from the set \(S\).  
Linkletter simple function  
\[f(x)=0.2x_{1}+0.2x_{2}+0.2x_{3}+0.2x_{4} \tag{13}\]  
Equation (13) is used for input variable screening because only four of its input variables have a nonzero effect on the response. The function is evaluated on the scaled input domain \(x_{i}\in[0,1]\), for all \(i=1,\ldots,10\).  
The screening procedure terminated as the value of common theta is \(2.8\times 10^{-8}\), which is less than \(10\times 10^{-6}\). No variable is selected from the set \(S\).