# 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
Except for the decomposition method, all the other operators were the same for all four compared algorithms. The final rGD(\(t\))-metric statistical results are shown in Table 4, and the final GD(\(t\))-metric statistical results are shown in Table 5. Each table includes the best, mean, and standard deviation (std) of the performance metric values, and the best performance of the metric value is highlighted in bold. As shown in Table 4, in terms of the rGD(\(t\))-metric, we can see that dMOEA/D-\(L_{p}\) achieved the best \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Population size (\(N\)) & Two objectives: \(N=100\); Three objectives: \(N=300\) \\ Crossover probability & 0.9 \\ Mutation probability & \(1/N\) \\ Frequency of change (\(\tau_{T}\)) & 5, 10, 15, 20, 25, 35 \\ Severity of change (\(n_{T}\)) & 5, 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Public parameter settings in the experiments. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**(\(\tau_{T}\),\(n_{T}\))**} & \multirow{2}{*}{**Statistic**} & \multicolumn{5}{c}{**rGD(\(t\))**} \\ \cline{3-8}  & & & \(\mathbf{p=\infty}\) & \(\mathbf{p=1/2}\) & \(\mathbf{p=1/3}\) & \(\mathbf{p=1}\) & \(\mathbf{p=2}\) \\ \hline \multirow{3}{*}{FDA1(2)} & \multirow{3}{*}{(25,5)} & Min & \(\mathbf{(6.03\times 10^{-1})}\) & \((8.30\times 10^{-1})\) & \((8.30\times 10^{-1})\) & \((1.98\times 10^{0})\) & \((6.75\times 10^{-1})\) \\  & & Mean & \((9.81\times 10^{-1})\) & \((1.07\times 10^{0})\) & \((1.07\times 10^{0})\) & \((2.02\times 10^{0})\) & \(\mathbf{(1.06\times 10^{-1})}\) \\  & & Std & \((2.55\times 10^{-1})\) & \((8.98\times 10^{-2})\) & \((8.98\times 10^{-2})\) & \((8.11\times 10^{-2})\) & \(\mathbf{(2.41\times 10^{-2})}\) \\ \hline \multirow{3}{*}{FDA2(2)} & \multirow{3}{*}{(15,5)} & Min & \((1.11\times 10^{-1})\) & \((9.33\times 10^{-1})\) & \((9.33\times 10^{-1})\) & \((2.10\times 10^{0})\) & \(\mathbf{(6.33\times 10^{-2})}\) \\  & & Mean & \((1.83\times 10^{-1})\) & \((1.56\times 10^{0})\) & \((1.56\times 10^{0})\) & \((2.91\times 10^{0})\) & \(\mathbf{(7.34\times 10^{-2})}\) \\  & & Std & \((7.47\times 10^{-2})\) & \((5.30\times 10^{-1})\) & \((5.30\times 10^{-1})\) & \((4.20\times 10^{-1})\) & \(\mathbf{(5.04\times 10^{-3})}\) \\ \hline \multirow{3}{*}{FDA3(2)} & \multirow{3}{*}{(35,5)} & Min & \(\mathbf{(4.13\times 10^{-1})}\) & \((6.94\times 10^{-1})\) & \((6.94\times 10^{-1})\) & \((5.01\times 10^{-1})\) & \((4.79\times 10^{-1})\) \\  & & Mean & \(\mathbf{(4.60\times 10^{-1})}\) & \((3.33\times 10^{0})\) & \((3.33\times 10^{0})\) & \((6.23\times 10^{-1})\) & \((5.16\times 10^{-1})\) \\  & & Std & \((1.84\times 10^{-1})\) & \((1.34\times 10^{0})\) & \((1.34\times 10^{0})\) & \((8.71\times 10^{-2})\) & \(\mathbf{(2.73\times 10^{-2})}\) \\ \hline \multirow{3}{*}{FDA4(3)} & \multirow{3}{*}{(25,5)} & Min & \((2.02\times 10^{-1})\) & \((8.31\times 10^{0})\) & \((8.31\times 10^{0})\) & \((8.91\times 10^{-1})\) & \(\mathbf{(1.30\times 10^{-1})}\) \\  & & Mean & \((4.64\times 10^{-1})\) & \((8.36\times 10^{0})\) & \((8.36\times 10^{0})\) & \((1.03\times 10^{0})\) & \(\mathbf{(1.35\times 10^{-1})}\) \\  & & Std & \((2.64\times 10^{-2})\) & \((2.71\times 10^{-2})\) & \((2.71\times 10^{-2})\) & \((6.38\times 10^{-2})\) & \(\mathbf{(2.51\times 10^{-2})}\) \\ \hline \multirow{3}{*}{FDA5(3)} & \multirow{3}{*}{(25,5)} & Min & \((8.43\times 10^{-2})\) & \((8.32\times 10^{0})\) & \((8.32\times 10^{0})\) & \((1.01\times 10^{-1})\) & \(\mathbf{(5.01\times 10^{-2})}\) \\  & & Mean & \((1.02\times 10^{-1})\) & \((1.26\times 10^{+1})\) & \((1.26\times 10^{+1})\) & \((1.69\times 10^{-1})\) & \(\mathbf{(5.69\times 10^{-2})}\) \\ \cline{1-1}  & & Std & \((8.43\times 10^{-4})\) & \((2.37\times 10^{0})\) & \((2.37\times 10^{0})\) & \((1.86\times 10^{-2})\) & \(\mathbf{(3.05\times 10^{-4})}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Verification of \(p\) value in the \(L_{p}\) decomposition method for the FDA series problem.