## 4 Proposed Algorithm
### Main Framework of the Proposed Algorithm
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **ZS-MMBSO\_2** & & **ZS-MMBSO** \\ \hline MMF1 & \(1.24\times 10^{2}\) (\(4.88\times 10^{0}\)) & + & **1.99 \(\times\) 10\({}^{2}\) (1.25 \(\times\) 10\({}^{1}\))** \\ MMF2 & \(1.69\times 10^{2}\) (\(9.98\times 10^{0}\)) & + & **1.22 \(\times\) 10\({}^{3}\) (2.21 \(\times\) 10\({}^{2}\))** \\ MMF3 & \(1.74\times 10^{2}\) (\(1.34\times 10^{1}\)) & + & 9.28 \(\times\) 10\({}^{2}\) (1.48 \(\times\) 10\({}^{2}\)) \\ MMF4 & \(2.64\times 10^{2}\) (\(9.27\times 10^{0}\)) & + & 3.64 \(\times\) 10\({}^{2}\) (1.10 \(\times\) 10\({}^{1}\)) \\ MMF5 & \(6.17\times 10^{1}\) (\(2.54\times 10^{0}\)) & + & 8.24 \(\times\) 10\({}^{1}\) (5.29 \(\times\) 10\({}^{0}\)) \\ MMF6 & \(7.52\times 10^{1}\) (\(3.69\times 10^{0}\)) & + & 8.04 \(\times\) 10\({}^{1}\) (4.25 \(\times\) 10\({}^{0}\)) \\ MMF7 & \(2.54\times 10^{2}\) (\(1.13\times 10^{1}\)) & + & 2.86 \(\times\) 10\({}^{2}\) (1.19 \(\times\) 10\({}^{0}\)) \\ MMF8 & \(1.21\times 10^{2}\) (\(6.61\times 10^{0}\)) & + & 1.67 \(\times\) 10\({}^{2}\) (2.03 \(\times\) 10\({}^{1}\)) \\ MMF9 & \(7.15\times 10^{2}\) (\(1.94\times 10^{1}\)) & + & 1.53 \(\times\) 10\({}^{3}\) (5.89 \(\times\) 10\({}^{1}\)) \\ MMF10 & \(2.30\times 10^{2}\) (\(2.85\times 10^{1}\)) & + & 3.16 \(\times\) 10\({}^{3}\) (2.91 \(\times\) 10\({}^{1}\)) \\ MMF11 & \(7.14\times 10^{2}\) (\(2.91\times 10^{1}\)) & + & **1.15 \(\times\) 10\({}^{3}\) (4.77 \(\times\) 10\({}^{1}\))** \\ MMF12 & \(7.08\times 10^{2}\) (\(4.92\times 10^{1}\)) & + & 1.83 \(\times\) 10\({}^{3}\) (6.82 \(\times\) 10\({}^{1}\)) \\ MMF13 & \(6.23\times 10^{1}\) (\(2.15\times 10^{0}\)) & + & 7.76 \(\times\) 10\({}^{1}\) (3.84 \(\times\) 10\({}^{0}\)) \\ MMF14 & \(5.77\times 10^{1}\) (\(6.05\times 10^{-1}\)) & = & 5.77 \(\times\) 10\({}^{1}\) (8.39 \(\times\) 10\({}^{-1}\)) \\ MMF15 & \(7.41\times 10^{1}\) (\(9.13\times 10^{-1}\)) & + & 7.76 \(\times\) 10\({}^{1}\) (1.27 \(\times\) 10\({}^{0}\)) \\ MMF1\_z & \(1.63\times 10^{2}\) (\(5.58\times 10^{0}\)) & + & 2.70 \(\times\) 10\({}^{2}\) (1.90 \(\times\) 10\({}^{1}\)) \\ MMF1\_e & \(6.76\times 10^{0}\) (\(7.15\times 10^{-1}\)) & + & **8.23 \(\times\) 10\({}^{0}\) (6.20 \(\times\) 10\({}^{-1}\))** \\ MMF14\_a & \(\textbf{5.11}\times\textbf{10}^{1}\) (\(\textbf{5.41}\times\textbf{10}^{-1}\)) & - & 4.94 \(\times\) 10\({}^{1}\) (9.01 \(\times\) 10\({}^{-1}\)) \\ MMF15\_a & \(6.68\times 10^{1}\) (\(8.58\times 10^{-1}\)) & + & **6.80 \(\times\) 10\({}^{1}\) (1.40 \(\times\) 10\({}^{0}\))** \\ SYM\_PART simple & \(\textbf{1.00}\times\textbf{10}^{2}\) (\(\textbf{4.37}\times\textbf{10}^{0}\)) & - & 9.22 \(\times\) 10\({}^{1}\) (9.45 \(\times\) 10\({}^{0}\)) \\ \hline SYM\_PART rotated & \(1.01\times 10^{2}\) (\(4.36\times 10^{0}\)) & + & \(\textbf{1.25}\times\textbf{10}^{2}\) (\(\textbf{1.45}\times\textbf{10}^{1}\)) \\ Omni\_test & \(1.99\times 10^{1}\) (\(1.62\times 10^{0}\)) & + & \(\textbf{3.04}\times\textbf{10}^{1}\) (\(\textbf{4.22}\times\textbf{10}^{0}\)) \\ \hline + & & 19 & \\ = & & 1 & \\ - & & 2 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean and standard deviation values of _PSP_ obtained by ZS-MMBSO and ZS-MMBSO_2. ## 6 Discussion

In this study, a zoning search-based multimodal multi-objective brain storm optimization (ZS-MMBSO) is proposed. In the ZS-MMBSO, the ZS is employed to reduce the complexity of MMOPs. A novel individual generation strategy is proposed to strengthen the global exploration ability in the early stage and improve local exploitation in the late stage. To demonstrate the effectiveness of the ZS-MMBSO, it is compared with five famous MMOEAs on 22 test functions. The results demonstrate ZS_MMBSO is competitive in both decision space and objective space. The effectiveness of the proposed strategy is analyzed in Section 5.3.1 and Section 5.3.2. The experimental analysis results indicate that these strategies can help the ZS-MMBSO solve MMOPs. Section 5.3.3 analyzed the sensitivity of the cluster number \(K\). Conceptualization, J.F. and Q.F. ; methodology, J.F., W.H., Q.J. and Q.F; investigation, J.F., W.H., Q.J.