# Hybrid Particle Swarm Optimization for High-Dimensional Latin Hypercube Design Problem
## 4 Numerical Results and Discussion
### Comparison Results  
Several state-of-the-art methods are compared with our proposed method in this section, including Comprehensive Learning PSO (CLPSO) [52; 53], Adaptive Memetic PSO (AMPSO) [36], traditional PSO [26; 28], Enhanced Stochastic Evolutionary Algorithm (ESE) [16], Genetic Algorithm (GA) [12], and LaPSO [35]; other algorithms perform performance comparison, after scaling LHD to \([0,1]^{k}\), and calculate the value of \(\phi p\). Tables 1 and 2 show 20 sets of the value of \(\phi p\) with dimensions \(m=2\), \(6\), \(10\), \(14\), \(18\) and \(n=20,40,60,80\). The best results are shown in bold. Meanwhile, we selected five dimensions of \(2\)\(\times\)\(60\), \(6\)\(\times\)\(60\), \(10\)\(\times\)\(60\), \(14\)\(\times\)\(60\), and \(18\)\(\times\)\(60\) to draw the population evolution diagram, shown in Figure 9. Furthermore, a box plot of \(18\)\(\times\)\(60\) between the proposed algorithm and the compared algorithm was drawn, as shown in Figure 10.  
Figure 8: Numerical results of different refresh intervals. Figure 7: The variation of \(\phi p\) with MPD based evolutionary strategy. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Dimension** & **Algorithm** & **Measure** & \(n=20\) & \(n=40\) & \(n=60\) & \(n=80\) \\ \hline \multirow{9}{*}{\(m=2\)} & CLPSO & Mean & 4.27695 & 8.03165 & 11.7084 & 15.1155 \\  & & Std & 0.268134 & 0.314826 & 0.694002 & 1.082523 \\  & & Mean & 4.082846 & 6.430663 & 8.770171 & 11.02683 \\  & & Std & 0.082087 & 0.338895 & 0.419517 & 0.576738 \\  & & Mean & 4.055958 & 6.051447 & 8.306216 & 17.43493 \\  & & Std & 8.88 \(\times 10^{-16}\) & 0.315769 & 1.198285 & 1.608196 \\  & & Mean & 5.070727 & 9.852305 & 14.73661 & 19.39307 \\  & & Std & 0.042239 & 0.558617 & 0.912121 & 1.359123 \\  & & Mean & 4.062401 & 5.852419 & 7.473188 & 8.786267 \\  & & Std & 0.112058 & 0.149028 & 0.340156 & 0.418199 \\  & & Mean & 4.118622 & 6.628554 & 8.190774 & 11.61435 \\  & & Std & 0.087171 & 0.339539 & 0.39093 & 0.261372 \\  & & Mean & **3.981064** & **5.825657** & **7.370558** & **8.743883** \\  & & Std & **0.188432** & **0.123855** & **0.27801** & **0.316564** \\ \hline \multirow{9}{*}{\(m=6\)} & CLPSO & Mean & 0.755957 & 0.999181 & 1.182143 & 1.325422 \\  & & Std & 0.013096 & 0.014428 & 0.018933 & 0.020479 \\  & & Mean & 0.690927 & 0.882016 & 1.015902 & 1.118163 \\  & & Std & 0.009545 & 0.041475 & 0.010908 & 0.018196 \\  & & Mean & 0.695391 & 0.964931 & 1.154155 & 1.300088 \\  & & Std & 0.011854 & 0.019534 & 0.023778 & 0.034823 \\  & & Mean & 0.843597 & 1.130684 & 1.338438 & 1.505415 \\  & & Std & 0.010928 & 0.02177 & 0.021391 & 0.02172 \\  & & Mean & 0.659701 & 0.798081 & 0.895581 & 0.971953 \\  & & Std & 0.008956 & 0.007465 & 0.007773 & 0.005263 \\  & & Mean & 0.667212 & 0.828105 & 0.936295 & 1.027103 \\  & & Std & 0.009503 & 0.009213 & 0.010532 & 0.010696 \\  & & Mean & **0.666972** & **0.806242** & **0.906449** & **0.984813** \\  & & Std & **0.009571** & **0.009701** & **0.014546** & **0.020373** \\ \hline \multirow{9}{*}{\(m=10\)} & CLPSO & Mean & 0.400005 & 0.490980 & 0.550981 & 0.595821 \\  & & Std & 0.004002 & 0.004836 & 0.004982 & 0.005118 \\  & & Mean & 0.372430 & 0.444516 & 0.492998 & 0.528137 \\  & & Std & 0.003088 & 0.00411 & 0.00387 & 0.004777 \\  & & Mean & 0. Figure 9: Performance trends of different LHD algorithms: **(a)** 2 \(\times\) 60, **(b)** 6 \(\times\) 60, **(c)** 10 \(\times\) 60, **(d)** 14 \(\times\) 60, and **(e)** 18 \(\times\) 60. As shown in Tables 1 and 2, the proposed algorithm outperforms all the compared algorithms in most cases of numerical tests. The numerical results in six dimensions are also close to the optimal data in the compared algorithms. The proposed algorithm is superior to the compared algorithms in high dimensions after parameter adjustment and has greater stability. Experiments find that the initial values of \(c_{1}\), \(c_{2}\), and \(\omega\) in Equation (2) are 0.5, 2.7, and 0.25, respectively. When the global optimal particle _gBest_ is updated, the parameters in Equation (2) change into the initial value.