# Memory-Enhanced Dynamic Multi-Objective Evolutionary Algorithm Based on \(L_{p}\) Decomposition
## 3 Memory-Enhanced Dynamic Multi-Objective Evolutionary Algorithm Based on \(L_{p}\) Decomposition
### Detailed Description of dMOEA/D-\(L_{p}\) and Its Time Complexity Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**(\(\mathbf{\tau_{T}}\),\(\mathbf{n_{T}}\))**} & \multirow{2}{*}{**Statistic**} & \multicolumn{4}{c}{**rGD(\(t\)) Metrics**} \\ \cline{4-7}  & & & **dMOEA/D-\(\mathbf{L_{p}}\)** & **dMOEA/D-TCH** & **dMOEA/D-WS** & **dMOEA/D-PBI** \\ \hline \multirow{6}{*}{FDA1(2)} & \multirow{2}{*}{(\(10,10\))} & Min & \((6.69\times 10^{-1})\) & \((\mathbf{5.76\times 10^{-1}})\) & \((1.94\times 10^{0})\) & \((8.01\times 10^{-1})\) \\  & & Mean & \((\mathbf{1.05\times 10^{0}})\) & \((1.12\times 10^{0})\) & \((2.05\times 10^{0})\) & \((1.16\times 10^{0})\) \\  & & Std & \((\mathbf{2.39\times 10^{-2}})\) & \((3.20\times 10^{-1})\) & \((1.27\times 10^{-1})\) & \((5.13\times 10^{-1})\) \\ \cline{2-7}  & \multirow{2}{*}{(25,10)} & Min & \((\mathbf{6.75\times 10^{-1}})\) & \((7.33\times 10^{-1})\) & \((1.99\times 10^{0})\) & \((8.01\times 10^{-1})\) \\  & & Mean & \((\mathbf{7.27\times 10^{-1}})\) & \((1.96\times 10^{0})\) & \((2.01\times 10^{0})\) & \((8.74\times 10^{-1})\) \\  & & Std & \((\mathbf{2.03\times 10^{-2}})\) & \((2.32\times 10^{-1})\) & \((3.49\times 10^{-2})\) & \((2.11\times 10^{-1})\) \\ \cline{2-7}  & \multirow{2}{*}{(25,5)} & Min & \((6.75\times 10^{-1})\) & \((\mathbf{6.03\times 10^{-1}})\) & \((1.98\times 10^{0})\) & \((8.85\times 10^{-1})\) \\  & & Mean & \((\mathbf{1.06\times 10^{-1}})\) & \((9.81\times 10^{-1})\) & \((2.02\times 10^{0})\) & \((9.13\times 10^{-1})\) \\  & & Std & \((\mathbf{2.41\times 10^{-2}})\) & \((2.55\times 10^{-1})\) & \((8.11\times 10^{-2})\) & \((2.32\times 10^{-1})\) \\ \hline \multirow{6}{*}{FDA2(2)} & \multirow{2}{*}{(5,10)} & Min & \((\mathbf{2.25\times 10^{-1}})\) & \((1.79\times 10^{0})\) & \((1.05\times 10^{0})\) & \((2.33\times 10^{-1})\) \\  & & Mean & \((\mathbf{3.32\times 10^{-1}})\) & \((2.75\times 10^{0})\) & \((2.44\times 10^{0})\) & \((1.13\times 10^{0})\) \\  & & Std & \((\mathbf{5.78\times 10^{-2}})\) & \((6.91\times 10^{0})\) & \((6.01\times 10^{-1})\) & \((5.25\times 10^{-1})\) \\ \cline{2-7}  & \multirow{2}{*}{(15,10)} & Min & \((\mathbf{1.21\times 10^{-1}})\) & \((3.49\times 10^{-1})\) & \((1.77\times 10^{0})\) & \((2.33\times 10^{-1})\) \\  & & Mean & \((\mathbf{1.30\times 10^{-1}})\) & \((7.33\times 10^{-1})\) & \((2.74\times 10^{0})\) & \((1.13\times 10^{0})\) \\  & & Std & \((\mathbf{5.71\times 10^{-2}})\) & \((5.69\times 10^{-1})\) & \((6.92\times 10^{-1})\) & \((5.25\times 10^{-1})\) \\ \cline{2-7}  & \multirow{2}{*}{(15,5)} & Min & \((\mathbf{6.33\times 10^{-2}})\) & \((1.11\times 10^{-1})\) & \((2.10\times 10^{0})\) & \((1.00\times 10^{0})\) \\  & & Mean & \((\mathbf{7.34\times 10^{-2}})\) & \((1.83\times 10^{-1})\) & \((2.91\times 10^{0})\) & \((1.77\times 10^{0})\) \\  & & Std & \((\mathbf{5.04\times 10^{-3}})\) & \((7.47\times 10^{-2})\) & \((4.20\times 10^{-1})\) & \((7.14\times 10^{-1})\) \\ \hline \multirow{6}{*}{FDA3(2)} & \multirow{2}{*}{(25,10)} & Min & \((\mathbf{1.00\times 10^{-1}})\) & \((2.64\times 10^{-1})\) & \((5.74\times 10^{-1})\) & \((1.51\times 10^{-1})\) \\  & & Mean & \((\mathbf{1.03\times 10^{-1}})\) & \((4.03\times 10^{-1})\) & \((6.56\times 10^{-1})\) & \((5.15\times 10^{-1})\) \\  & & Std & \((\mathbf{2.05\times 10^{-3}})\) & \((2.38\times 10^{-3})\) & \((2.10\times 10^{-2})\) & \((2.54\times 10^{-1})\) \\ \cline{2-7}  & \multirow{2}{*}{(35,10)} & Min & \((4.04\times 10^{-1})\) & \((6.06\times 10^{-1})\) & \((6.13\times 10^{-2})\) & \((\mathbf{2.17\times 10^{-1}})\) \\  & & Mean & \((\mathbf{6.05\times 10^{-1}})\) & \((7.56\times 10^{-1})\) & \((7.23\times 10^{-2})\) & \((6.06\times 10^{-1})\) \\  & & Std & \((2.73\times 10^{-2})\) & \((\mathbf{2.05\times 10^{-2}})\) & \((1.34\times 10^{-1})\) & \((2.31\times 10^{-1})\) \\ \cline{2-7}  & \multirow{2}{*}{(35,5)} & Min & \((4.79\times 10^{-1})\) & \((4.13\times 10^{-1})\) & \((5.01\times 10^{-1})\) & \((\mathbf{1.35\times 10^{-1}})\) \\  & & Mean & \((\mathbf{5.16\times 10^{-1}})\) & \((\mathbf{4.60\times 10^{-1}})\) & \((6.23\times 10^{-1})\) & \((5.33\times 10^{-1})\) \\  & & Std & \((\mathbf{2.73\times 10^{-2}})\) & \((1.84\times 10^{-1})\) & \((8.71\times 10^{-2})\) & \((2.08\times 10^{-1})\) \\ \hline \multirow{6}{*}{FDA4(3)} & \multirow{2}{*}{(20,10)} & Min & \((\mathbf{1.29\times 10^{-2}})\) & \((7.01\times 10^{-1})\) & \((1.96\times 10^{0})\) & \((4.05\times 10^{-1})\) \\  & & Mean & \((\mathbf{2.43\times 10^{-2}})\) & \((8.00\times 10^{-1})\) & \((2.74\times 10^{0})\) & \((4.14\times 10^{-1})\) \\  & & Std & \((\mathbf{5.67\times 10^{-3}})\) & \((2.34\times 10^{-1})\) & \((6.97\times

\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**(\(\mathbf{\tau_{T}}\),\(\mathbf{n_{T}}\))**} & \multirow{2}{*}{**Statistic**} & \multicolumn{5}{c}{**GD(\(t\)) Metrics**} \\ \cline{4-7}  & & & **dMOEA/D-\(L_{P}\)** & **dMOEA/D-TCH** & **dMOEA/D-WS** & **dMOEA/D-PBI** \\ \hline \multirow{8}{*}{FDA1(2)} & \multirow{4}{*}{(10,10)} & Min & \(\mathbf{(5.55\times 10^{-4})}\) & \((5.96\times 10^{-4})\) & \((7.74\times 10^{-4})\) & \((9.72\times 10^{-4})\) \\  & & Mean & \(\mathbf{(9.14\times 10^{-4})}\) & \((6.45\times 10^{-3})\) & \((9.61\times 10^{-3})\) & \((1.01\times 10^{-2})\) \\  & & Std & \(\mathbf{(4.79\times 10^{-4})}\) & \((4.45\times 10^{-3})\) & \((1.77\times 10^{-2})\) & \((6.49\times 10^{-3})\) \\ \hline \multirow{8}{*}{FDA1(2)} & \multirow{4}{*}{(25,10)} & Min & \(\mathbf{(4.18\times 10^{-4})}\) & \((4.92\times 10^{-4})\) & \((1.20\times 10^{-3})\) & \((6.71\times 10^{-4})\) \\  & & Mean & \(\mathbf{(1.26\times 10^{-3})}\) & \((1.62\times 10^{-3})\) & \((1.66\times 10^{-3})\) & \((3.00\times 10^{-3})\) \\  & & Std & \(\mathbf{(5.18\times 10^{-4})}\) & \((6.00\times 10^{-4})\) & \((1.49\times 10^{-3})\) & \((1.24\times 10^{-3})\) \\ \hline \multirow{8}{*}{FDA2(2)} & \multirow{4}{*}{(25,5)} & Min & \(\mathbf{(5.97\times 10^{-4})}\) & \((6.26\times 10^{-4})\) & \((1.12\times 10^{-3})\) & \((7.64\times 10^{-4})\) \\  & & Mean & \(\mathbf{(9.13\times 10^{-4})}\) & \((2.26\times 10^{-3})\) & \((3.35\times 10^{-3})\) & \((4.12\times 10^{-3})\) \\  & & Std & \(\mathbf{(4.79\times 10^{-4})}\) & \((1.29\times 10^{-3})\) & \((6.37\times 10^{-3})\) & \((1.86\times 10^{-3})\) \\ \hline \multirow{8}{*}{FDA2(2)} & \multirow{4}{*}{(5,10)} & Min & \(\mathbf{(7.89\times 10^{-20})}\) & \((1.20\times 10^{-6})\) & \((3.16\times 10^{-18})\) & \((8.04\times 10^{-4})\) \\  & & Mean & \(\mathbf{(5.10\times 10^{-5})}\) & \((3.31\times 10^{-3})\) & \((1.55\times 10^{-4})\) & \((5.79\times 10^{-3})\) \\  & & Std & \(\mathbf{(4.92\times 10^{-3})}\) & \((2.54\times 10^{-3})\) & \((\mathbf{1.26\times 10^{-3})}\) & \((3.43\times 10^{-3})\) \\ \hline \multirow{8}{*}{FDA3(2)} & \multirow{4}{*}{(25,10)} & Min & \(\mathbf{(6.16\times 10^{-4})}\) & \((6.70\times 10^{-4})\) & \((1.02\times 10^{-3})\) & \((6.49\times 10^{-4})\) \\  & & Mean & \(\mathbf{(2.43\times 10^{-3})}\) & \((2.46\times 10^{-3})\) & \((**1.32\times 10^{-3})\) & \((2.57\times 10^{-3})\) \\  & & Std & \(\mathbf{(4.92\times 10^{-4})}\) & \((1.22\times 10^{-3})\) & \((1.31\times 10^{-3})\) & \((1.54\times 10^{-3})\) \\ \hline \multirow{8}{*}{FDA3(2)} & \multirow{4}{*}{(35,10)} & Min & \(\mathbf{(7.56\times 10^{-4})}\) & \((5.99\times 10^{-4})\) & \((\mathbf{6.02\times 10^{-9})}\) & \((6.60\times 10^{-4})\) \\  & & Mean & \(\mathbf{(4.75\times 10^{-3})}\) & \((1.91\times 10^{-2})\) & \((7.66\times 10^{-3})\) & \((1.98\times 10^{-2})\) \\  & & Std & \(\mathbf{(2.53\times 10^{-3})}\) & \((1.40\times 10^{-2})\) & \((1.28\times 10^{-2})\) & \((1.36\times 10^{-2})\) \\ \hline \multirow{8}{*}{FDA3(2)} & \multirow{4}{*}{(25,10)} & Min & \(\mathbf{(1.00\times 10^{-3})}\) & \((8.07\times 10^{-3})\) & \((1.00\times 10^{-2})\) & \((3.37\times 10^{-2})\) \\  & & Mean & \(\mathbf{(3.48\times 10^{-2})}\) & \((4.92\times 10^{-2})\) & \((2.20\times 10^{-1})\) & \((3.71\times 10^{-2})\) \\  & & Std & \(\mathbf{(3.91\times 10^{-2})}\) & \((7.34\times 10^{-2})\) & \((1.08\times 10^{-1})\) & \((4.17\times 10^{-2})\) \\ \hline \multirow{8}{*}{FDA4(2)} & \multirow{4}{*}{(35,10)} & Min & \(\mathbf{(1.02\times 10^{-3})}\) & \((7.56\times 10^{-1})\) & \((5.90\times 10^{-3})\) & \((3.23\times 10^{-3})\) \\  & & Mean & \(\mathbf{(3.15\times 10^{-2})}\) & \((2.16\times 10^{0})\) & \((6.89\times 10^{-2})\) & \((3.36\times 10^{-2})\) \\  & & Std & \(\mathbf{(3.48\times 10^{-2})}\) & \((1.06\times 10^{0})\) & \((5.35\times 10^{-2})\) & \((3.87\times 10^{-2})\) \\ \hline \multirow{8}{*}{FDA3(2)} & \multirow{4}{*}{(35,5)} & Min & \(\mathbf{(1.06\times 10^{-3})}\) & \((9.18\times 10^{-3})\) & \((1.11\times 10^{-2})\) & \((7.72\times 10^{-3})\) \\  & & Mean & \(\mathbf{(3.64\times 10^{-2})}\) & \((3.90\times 10^{-2})\) & \((6.67\times 10^{-2})\) & \((3.63\times 10^{-2})\) \\  & & Std & \(\mathbf{(3.39\times 10^{-2})}\) & \((3.89\times 10^{-2})\) & \((5.53\times 10^{-2})\) & \((3.48\times 10^{-2})\) \\ \hline \multirow{8}{*}{FDA3(2)} & \multirow{4}{*}{(20,10)} & Min & \(\mathbf{(2.13\times 10^{-2})}\) & \((2.52\times 10^{-2})\) & \((3.24\times 10^{-2})\) & \((2.31\times 10^{-2})\) \\  & & Mean & \(\mathbf{(2.16\times 10^{-2})}\) & \((2.56\times 10^{-2})\) & \((4.64\times 10^{-2})\) & \((2

\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**(\(\mathbf{\tau_{T}}\),\(\mathbf{n_{T}}\))**} & \multirow{2}{*}{**Statistic**} & \multicolumn{4}{c}{**rGD(\(\mathbf{t}\)) Metrics**} \\ \cline{4-6}  & & & **dMOEA/D-\(\mathbf{L_{p}}\)** & **DNSGA-II** & **QICCA** \\ \hline \multirow{6}{*}{FDA1(2)} & \multirow{3}{*}{(10,10)} & Min & **(6.69 \(\times\mathbf{10^{-1}}\))** & (1.115 \(\times\mathbf{10^{0}}\)) & (1.05 \(\times\mathbf{10^{0}}\)) \\  & & Mean & **(1.05 \(\times\mathbf{10^{0}}\))** & (1.21 \(\times\mathbf{10^{0}}\)) & (1.08 \(\times\mathbf{10^{0}}\)) \\  & & Std & **(2.39 \(\times\mathbf{10^{-2}}\))** & (3.11 \(\times\mathbf{10^{-1}}\)) & (2.45 \(\times\mathbf{10^{-1}}\)) \\ \cline{2-6}  & \multirow{3}{*}{(25,10)} & Min & **(6.75 \(\times\mathbf{10^{-1}}\))** & (1.07 \(\times\mathbf{10^{0}}\)) & (1.05 \(\times\mathbf{10^{0}}\)) \\  & & Mean & **(7.27 \(\times\mathbf{10^{-1}}\))** & (1.08 \(\times\mathbf{10^{0}}\)) & (1.06 \(\times\mathbf{10^{0}}\)) \\  & & Std & **(2.03 \(\times\mathbf{10^{-2}}\))** & (5.48 \(\times\mathbf{10^{-2}}\)) & (4.43 \(\times\mathbf{10^{-2}}\)) \\ \cline{2-6}  & \multirow{3}{*}{(25,5)} & Min & **(6.75 \(\times\mathbf{10^{-1}}\))** & (1.06 \(\times\mathbf{10^{0}}\)) & (1.06 \(\times\mathbf{10^{0}}\)) \\  & & Mean & **(1.06 \(\times\mathbf{10^{0}}\))** & (1.09 \(\times\mathbf{10^{0}}\)) & (1.07 \(\times\mathbf{10^{0}}\)) \\  & & Std & **(2.41 \(\times\mathbf{10^{-2}}\))** & (1.36 \(\times\mathbf{10^{-1}}\)) & (7.44 \(\times\mathbf{10^{-2}}\)) \\ \hline \multirow{6}{*}{FDA2(2)} & \multirow{3}{*}{(5,10)} & Min & **(2.25 \(\times\mathbf{10^{-1}}\))** & (1.50 \(\times\mathbf{10^{0}}\)) & (1.07 \(\times\mathbf{10^{0}}\)) \\  & & Mean & **(3.32 \(\times\mathbf{10^{-1}}\))** & (3.65 \(\times\mathbf{10^{0}}\)) & (3.32 \(\times\mathbf{10^{0}}\)) \\  & & Std & **(5.78 \(\times\mathbf{10^{-2}}\))** & (7.53 \(\times\mathbf{10^{-1}}\)) & (7.11 \(\times\mathbf{10^{-1}}\)) \\ \cline{2-6}  & \multirow{3}{*}{(15,10)} & Min & **(1.21 \(\times\mathbf{10^{-1}}\))** & (1.34 \(\times\mathbf{10^{0}}\)) & (1.47 \(\times\mathbf{10^{0}}\)) \\  & & Mean & **(1.30 \(\times\mathbf{10^{-1}}\))** & (4.30 \(\times\mathbf{10^{0}}\)) & (3.95 \(\times\mathbf{10^{0}}\)) \\  & & Std & **(5.71 \(\times\mathbf{10^{-2}}\))** & (5.95 \(\times\mathbf{10^{-1}}\)) & (6.36 \(\times\mathbf{10^{-1}}\)) \\ \cline{2-6}  & \multirow{3}{*}{(15,5)} & Min & **(6.33 \(\times\mathbf{10^{-2}}\))** & (1.88 \(\times\mathbf{10^{0}}\)) & (1.40 \(\times\mathbf{10^{0}}\)) \\  & & Mean & **(7.34 \(\times\mathbf{10^{-2}}\))** & (3.75 \(\times\mathbf{10^{0}}\)) & (3.63 \(\times\mathbf{10^{0}}\)) \\  & & Std & **(5.04 \(\times\mathbf{10^{-3}}\))** & (6.33 \(\times\mathbf{10^{-1}}\)) & (6.95 \(\times\mathbf{10^{-1}}\)) \\ \hline \multirow{6}{*}{FDA3(2)} & \multirow{3}{*}{(25,10)} & Min & **(1.00 \(\times\mathbf{10^{-1}}\))** & (1.17 \(\times\mathbf{10^{-1}}\)) & (1.40 \(\times\mathbf{10^{-1}}\)) \\  & & Mean & **(1.03 \(\times\mathbf{10^{-1}}\))** & (6.36 \(\times\mathbf{10^{-1}}\)) & (6.40 \(\times\mathbf{10^{-1}}\)) \\  & & Std & **(2.05 \(\times\mathbf{10^{-3}}\))** & (2.61 \(\times\mathbf{10^{-2}}\)) & (2.63 \(\times\mathbf{10^{-2}}\)) \\ \cline{2-6}  & \multirow{3}{*}{(35,10)} & Min & **(4.04 \(\times\mathbf{10^{-1}}\))** & (6.30 \(\times\mathbf{10^{-1}}\)) & (6.35 \(\times\mathbf{10^{-1}}\)) \\  & & Mean & **(6.05 \(\times\mathbf{10^{-1}}\))** & (6.35 \(\times\mathbf{10^{-1}}\)) & (6.38 \(\times\mathbf{10^{-1}}\)) \\  & & Std & (2.73 \(\times\mathbf{10^{-2}}\)) & **(2.67 \(\times\mathbf{10^{-2}}\))** & (2.68 \(\times\mathbf{10^{-2}}\)) \\ \cline{2-6}  & \multirow{3}{*}{(35,5)} & Min & **(4.79 \(\times\mathbf{10^{-1}}\))** & (6.06 \(\times\mathbf{10^{-1}}\)) & (6.09 \(\times\mathbf{10^{-1}}\)) \\  & & Mean & **(5.16 \(\times\mathbf{10^{-1}}\))** & (6.10 \(\times\mathbf{10^{-1}}\)) & (6.11 \(\times\mathbf{10^{-1}}\)) \\  & & Std & (2.73 \(\times\mathbf{10^{-2}}\)) & (2.70 \(\times\mathbf{10^{-2}}\)) & **(2.52 \(\times\mathbf{10^{-2}}\))** \\ \hline \multirow{6}{*}{FDA4(3)} & \multirow{3}{*}{(20,10)} & Min & **(2.43 \(\times\mathbf{10^{-2}}\))** & (1.39 \(\times\mathbf{10^{-1}}\)) & (1.41 \(\times\mathbf{10^{0}}\)) \\  & & Mean & **(1.29 \(\times\mathbf{10^{-2}}\))** & (1.40 \(\times\mathbf{10^{-1}}\)) & (1.77 \(\times\mathbf{10^{0}}\)) \\  & & Std & **(5.67 \(\times\mathbf{10^{-3}}\))** & (5.19 \(\times\mathbf{10^{-1}}\)) & (3.13 \(\times\mathbf{10^{-1}}\)) \\ \cline{2-6}  & \multirow{3}{*}{(25,10)} & Min & **(1.26 \(\times\mathbf{10^{-1}}\))** & (1.27 \(\times\mathbf{10^{-1}}\)) & (1.38 \(\times\mathbf{10^{-1}}\)) \\  & & Mean & **(1.40 \(\times\mathbf{10^{-1}}\))** & (1.42 \(\times\mathbf{10^{-1}}\)) & (1.45 \(\times\mathbf{10^{-1}}\)) \\  & & Std & **(3.36 \(\times\mathbf{10^{-2}}\))** & (4.38 \(\times\mathbf{10^{-1}}\)) & (3.83 \(\times\mathbf{10^{-2}}\)) \\ \cline{2-6}  & \multirow{3}{*}{(25,5)} & Min & **(1.30 \(\times\mathbf{10^{-1}}\))** & (1.32 \(\times\mathbf{10^{-1}

\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**(\(\mathbf{\tau_{T}}\),\(\mathbf{n_{T}}\))**} & \multirow{2}{*}{**Statistic**} & \multicolumn{3}{c}{**GD(\(\mathbf{t}\)) Metrics**} \\ \cline{4-6}  & & & **dMOEA/D-\(\mathbf{L_{p}}\)** & **DNSGA-II** & **QICCA** \\ \hline \multirow{6}{*}{FDA1(2)} & \multirow{3}{*}{(10,10)} & Min & \(\mathbf{(5.55\times 10^{-4})}\) & (6.91 \(\times 10^{-3}\)) & (5.76 \(\times 10^{-3}\)) \\  & & Mean & \(\mathbf{(9.14\times 10^{-4})}\) & (8.08 \(\times 10^{-3}\)) & (6.49 \(\times 10^{-3}\)) \\  & & Std & \(\mathbf{(4.79\times 10^{-4})}\) & (7.00 \(\times 10^{-3}\)) & (6.08 \(\times 10^{-2}\)) \\ \cline{2-6}  & \multirow{3}{*}{(25,10)} & Min & \(\mathbf{(4.18\times 10^{-4})}\) & (1.81 \(\times 10^{-3}\)) & (1.93 \(\times 10^{-3}\)) \\  & & Mean & \(\mathbf{(1.26\times 10^{-3})}\) & (2.02 \(\times 10^{-3}\)) & (2.05 \(\times 10^{-3}\)) \\  & & Std & \(\mathbf{(5.18\times 10^{-4})}\) & (6.42 \(\times 10^{-4}\)) & (1.37 \(\times 10^{-3}\)) \\ \cline{2-6}  & \multirow{3}{*}{(25,5)} & Min & \(\mathbf{(5.97\times 10^{-4})}\) & (2.56 \(\times 10^{-3}\)) & (2.76 \(\times 10^{-3}\)) \\  & & Mean & \(\mathbf{(9.13\times 10^{-4})}\) & (2.97 \(\times 10^{-3}\)) & (2.92 \(\times 10^{-3}\)) \\  & & Std & \(\mathbf{(4.79\times 10^{-4})}\) & (3.75 \(\times 10^{-3}\)) & (2.56 \(\times 10^{-3}\)) \\ \hline \multirow{6}{*}{FDA2(2)} & \multirow{3}{*}{(5,10)} & Min & (7.89 \(\times 10^{-20}\)) & **(1.52 \(\times 10^{-20}\))** & (5.10 \(\times 10^{-5}\)) \\  & & Mean & \(\mathbf{(5.10\times 10^{-5})}\) & (1.50 \(\times 10^{-3}\)) & (2.51 \(\times 10^{-3}\)) \\  & & Std & \(\mathbf{(4.92\times 10^{-3})}\) & **(2.33 \(\times 10^{-3}\))** & (2.97 \(\times 10^{-3}\)) \\ \cline{2-6}  & \multirow{3}{*}{(15,10)} & Min & (6.16 \(\times 10^{-4}\)) & **(8.80 \(\times 10^{-5}\))** & (9.11 \(\times 10^{-5}\)) \\  & & Mean & \(\mathbf{(2.43\times 10^{-3})}\) & (4.37 \(\times 10^{-3}\)) & (7.08 \(\times 10^{-3}\)) \\  & & Std & \(\mathbf{(4.92\times 10^{-4})}\) & (3.82 \(\times 10^{-3}\)) & (5.44 \(\times 10^{-3}\)) \\ \cline{2-6}  & \multirow{3}{*}{(15,5)} & Min & (7.56 \(\times 10^{-4}\)) & **(1.02 \(\times 10^{-7}\))** & (2.43 \(\times 10^{-7}\)) \\  & & Mean & \(\mathbf{(4.75\times 10^{-3})}\) & (7.55 \(\times 10^{-3}\)) & (1.80 \(\times 10^{-2}\)) \\  & & Std & \(\mathbf{(2.53\times 10^{-3})}\) & (5.70 \(\times 10^{-3}\)) & (1.29 \(\times 10^{-2}\)) \\ \hline \multirow{6}{*}{FDA3(2)} & \multirow{3}{*}{(25,10)} & Min & \(\mathbf{(1.00\times 10^{-3})}\) & (8.07 \(\times 10^{-3}\)) & (1.00 \(\times 10^{-2}\)) \\  & & Mean & \(\mathbf{(3.48\times 10^{-2})}\) & (4.92 \(\times 10^{-2}\)) & (2.20 \(\times 10^{-1}\)) \\  & & Std & \(\mathbf{(3.91\times 10^{-2})}\) & (7.34 \(\times 10^{-2}\)) & (1.08 \(\times 10^{-1}\)) \\ \cline{2-6}  & \multirow{3}{*}{(35,10)} & Min & \(\mathbf{(1.02\times 10^{-3})}\) & (3.02 \(\times 10^{-2}\)) & (3.40 \(\times 10^{-2}\)) \\  & & Mean & \(\mathbf{(3.15\times 10^{-2})}\) & (3.23 \(\times 10^{-2}\)) & (3.68 \(\times 10^{-2}\)) \\  & & Std & \(\mathbf{(3.48\times 10^{-2})}\) & (3.50 \(\times 10^{-2}\)) & (4.32 \(\times 10^{-2}\)) \\ \cline{2-6}  & \multirow{3}{*}{(35,5)} & Min & \(\mathbf{(1.06\times 10^{-3})}\) & (3.79 \(\times 10^{-2}\)) & (4.05 \(\times 10^{-2}\)) \\  & & Mean & \(\mathbf{(3.64\times 10^{-2})}\) & (3.92 \(\times 10^{-2}\)) & (4.27 \(\times 10^{-2}\)) \\  & & Std & \(\mathbf{(3.39\times 10^{-2})}\) & (3.42 \(\times 10^{-2}\)) & (4.55 \(\times 10^{-2}\)) \\ \hline \multirow{6}{*}{FDA4(3)} & \multirow{3}{*}{(20,10)} & Min & (2.13 \(\times 10^{-2}\)) & (3.57 \(\times 10^{-2}\)) & (2.54 \(\times 10^{-2}\)) \\  & & Mean & \(\mathbf{(2.16\times 10^{-2})}\) & (3.88 \(\times 10^{-2}\)) & (2.66 \(\times 10^{-2}\)) \\  & & Std & \(\mathbf{(1.68\times 10^{-4})}\) & (2.03 \(\times 10^{-4}\)) & (1.83 \(\times 10^{-4}\)) \\ \cline{2-6}  & \multirow{3}{*}{(25,10)} & Min & (2.51 \(\times 10^{-2}\)) & (2.56 \(\times 10^{-2}\)) & **(2.43 \(\times 10^{-2}\))** \\  & & Mean & \(\mathbf{(2.55\times 10^{-2})}\) & (3.11 \(\times 10^{-2}\)) & (3.66 \(\times 10^{-2}\)) \\  & & Std & \(\mathbf{(1.48\times 10^{-4})}\) & (1.75 \(\times 10^{-3}\)) & (2.02 \(\times 10^{-4}\)) \\ \cline{2-6}  & \multirow{3}{*}{(25,5)} & Min & \(\mathbf{(1.12\times 10^{-2})}\) & (1.53 \(\times 10^{-2}\)) & (1.99 \(\times 10^{-2}\)) \\  & & Mean & \(\mathbf{(1.47\times 10^{-2})}\) & (1.67 \(\times 10^{-2}\)) & (2.47 \(\times 10^{-2}\)) \\  & & Std & \(\mathbf{(6.04\times 10^{-5})}\) & (1.44 \(\times 10^{-4}\)) & (1.68 \(\times 10^{-4}\)) \\ \hline \multirow{6}{*}{FDA5(3)} & \multirow{3}{*}{(20,10)} & Min & (2.69 It can be seen from Table 1 that some components (\(x_{II}\)) of the Pareto optimal solutions in FDA1 varied sinusoidally with the function \(G(t)\), so the \(POS(t)\) of FDA1 changed with time.