# 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
In a sense, if the \(|POF^{*}(t)|\) is sufficiently large to represent the \(POF^{*}(t)\) very well, rGD(\(t\)) can measure both the diversity and convergence of \(Q(t)\). To have a low value of rGD(\(t\)), \(Q(t)\) must be very close to the \(POF^{*}(t)\), and cannot miss any part of the \(POF^{*}(t)\). Therefore, the value of rGD(\(t\)) is smaller, which means that the convergence and diversity of the obtained Pareto front is better. \(d_{i}^{\prime}\) is the minimum Euclidean distance between point \(i\) of \(Q(t)\) and the points in \(|POF^{*}(t)|\). GD(\(t\)) measures the distance of \(Q(t)\) to \(POF^{*}(t)\). The value of GD(\(t\)) is smaller, which means that the convergence and diversity of the obtained Pareto front is better. The perfect situation is that GD(\(t\)) = 0, which means that all solutions in \(Q(t)\) are Pareto optimal solutions. ### Setting of Experimental Parameters

Individuals used real vector encoding in all test problems. Differential evolution (DE) crossover and polynomial mutation operators [30] are adopted in the five compared algorithms (i.e., dMOEAD-\(L_{p}\), dMOEA/D-TCH, dMOEA/D-WS, dMOEA/D-PBI and DNSGA-II) [16], where crossover rate \(CR=0.9\), scaling factor \(F=0.5\). The whole clone (clone ratio of 5) and non-uniform mutation is adopted in QICCA [7]; diversified introduction ratio (\(\zeta\)) was set to 0.2 in DNSGA-II [7]; memory pool size was set to 100. All of number of neighbors in four compared algorithms (dMOEAD-\(L_{p}\), dMOEA/D-TCH, dMOEA/D-WS, and dMOEA/D-PBI) was set to 20; _Bsize_ was set to 5 for 2-objective problems, and to 15 for 3-objective problems; the penalty parameter (\(\theta\)) was set to 5 in dMOEA/D-PBI. For parameters used in the environmental monitoring operator, we took \(k=2\), \(\tilde{\delta}=0.002\) for problems with two objectives, and \(k=3\), \(\tilde{\delta}=0.006\) for 3-objective problems. Table 2 displays the common parameter settings for the compared algorithms, where \(n\) is the dimension of the decision variable. To study the effects of various dynamic changes in uncertain environments, different combinations of \((\tau_{T},n_{T})\) were set according to different DMOPs. Thirty independent runs were done for each combination of the various problems. In each experiment, various algorithms uniformly

\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Problems** & **Objective Functions** & **Variable Bounds** & \(n\) \\ \hline \multirow{4}{*}{FDA5} & \multirow{4}{*}{\(\begin{cases}f_{1}(x)=(1+g(x_{1l}))\prod_{j=1}^{m-1}\cos(0.5\pi y_{j})\\ f_{l}(x)=(1+g(x_{1l}))\prod_{j=1}^{m-k}\cos(0.5\pi y_{j})\sin(0.5\pi y_{m-k+1}),k=2:m-1\\ f_{m}(x)=(1+g(x_{1l}))\sin(0.5\pi y_{1})\\ g(x_{1l})=G(t)+\sum_{k\in\Omega(t)}\left(x_{l}-G(t)\right)^{2}},G(t)



tracked 100 times of environmental change. The number of generations was the combination of the environmental change time of each algorithm and the frequency of environmental change. To choose a proper \(p\) for each test instance, we executed dMOEA/D-\(L_{p}\) on FDA1-FDA5 with different \(p\) values: \(p=0\), \(p=1/3\), \(p=1/2\), \(p=1\), \(p=2\). Experimental results are shown in Table 3. The best performance is highlighted with bold font. It can be seen from the table that \(p=2\) generally performEd the best for dMOEA/D-\(L_{p}\) in dealing with the FDA series problems. This may be because the POFs curvature of FDA1-FDA5 are quadratic, and \(p=2\) in the \(L_{p}\) function was the most suitable according to the geometric shape of the \(L_{p}\) function in Figure 1.

### Experimental Results and Analysis

In this subsection, we carry out the two parts of the experimental results and analyses. In the experiments, the statistics of the performance metrics were based on 30 independent runs. (1) Compare dMOEA/D-\(L_{p}\) with dMOEA/D-WS, dMOEA/D-TCH, and dMOEA/D-PBI

To test the effectiveness of the \(L_{p}\) decomposition method, dMOEA/D-\(L_{p}\) was compared with dMOEA/D-WS, dMOEA/D-TCH, and dMOEA/D-PBI on the FDA series problems.