## 3 The Algorithm
### SaNSDE Algorithm
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case** & **I** & **K** & **DE1** & **DE2** & **DE3** & **DE4** & **DE5** & **DE6** \\ \hline
1 & 4 & 3 & 0 & 0 & 0 & 1.8973 & 1.8973 & 1.8973 \\
2 & 5 & 5 & 0 & 0 & 1.5811 & 1.5811 & 0 & 0 \\
3 & 5 & 5 & 0 & 5.1639 & 5.986 & 5.1639 & 7.8145 & 4.8304 \\
4 & 10 & 10 & 0 & 0 & 5.0596 & 0 & 21.3987 & 0 \\
5 & 10 & 10 & 22.1143 & 112.8819 & 16.5798 & 30.6306 & 113.6875 & 72.9427 \\
6 & 10 & 20 & 64.5453 & 96.4862 & 81.4278 & 169.2747 & 156.7446 & 149.8784 \\
7 & 20 & 20 & 219.7454 & 607.2852 & 244.3599 & 404.5649 & 565.1306 & 582.2639 \\
8 & 30 & 10 & 0 & 339.5753 & 13.2199 & 308.6382 & 298.7199 & 58.4618 \\
9 & 30 & 20 & 37.8306 & 50.2354 & 0 & 182.6812 & 99.4105 & 25.1714 \\
10 & 40 & 10 & 14.9627 & 85.3661 & 21.5254 & 53.2015 & 183.8628 & 41.6898 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Standard deviation of fitness function values for DE1, DE2, DE3, DE4, DE5 and DE6 with \(NP=30\) and \(LP=1000\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **I** & **K** & **SaNSDE** & **NSDE** & **PSO** \\ \hline
1 & 4 & 3 & 0 & 1.8973 & 0 \\
2 & 5 & 5 & 0 & 2.1081 & 0 \\
3 & 5 & 5 & 0 & 4.8304 & 0 \\
4 & 10 & 10 & 0 & 0 & 0 \\
5 & 10 & 10 & 15.3291 & 17.8063 & 13.9583 \\
6 & 10 & 20 & 29.247 & 18.7675 & 27.3229 \\
7 & 20 & 20 & 76.4957 & 39.7672 & 74.9163 \\
8 & 30 & 10 & 0 & 0 & 32.1102 \\
9 & 30 & 20 & 0 & 0 & 115.8959 \\
10 & 40 & 10 & 0 & 0 & 90.0067 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Standard deviation of fitness function values for discrete SaNSDE, NSDE and PSO with population size \(NP=50\) and \(LP=1000\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **I** & **K** & **SaNSDE** & **NSDE** & **PSO** \\ \hline
1 & 4 & 3 & 0 & 0 & 0 \\
2 & 5 & 5 & 0 & 0 & 0 \\
3 & 5 & 5 & 0 & 3.1622 & 0 \\
4 & 10 & 10 & 0 & 21.3987 & 0 \\
5 & 10 & 10 & 10.3323 & 24.6737 & 20.0383 \\
6 & 10 & 20 & 27.0563 & 16.9036 & 35.6426 \\
7 & 20 & 20 & 40.7241 & 44.3883 & 65.0658 \\
8 & 30 & 10 & 0 & 0 & 51.3917 \\
9 & 30 & 20 & 0 & 0 & 89.8749 \\
10 & 40 & 10 & 0 & 0 & 111.3943 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Standard deviation of fitness function values for discrete SaNSDE, NSDE and PSO with population size \(NP=30\) and \(LP=1000\). The above results show that the proposed algorithm can ensure that the variation in solutions is smaller than in the other algorithms compared in this study with only a few exceptions regardless of the population size. ## 5 Discussion

In this study, we proposed a SaNSDE algorithm for planning processes in sustainable CPSs and conducted experiments to illustrate its effectiveness and properties. These included a comparative study with other competitive algorithms to illustrate the effectiveness of the proposed self-adaptive algorithm. In addition, we studied the sensitivity of the proposed self-adaptive algorithm with respect to algorithmic parameters, including the learning period parameter, \(LP\), and the population size parameter, \(NP\). Several other algorithms, including the PSO algorithm, six standard DE algorithms and a variant of the neighborhood search-based DE algorithm, were applied to solve ten instances. The results obtained by these algorithms were compared with those obtained by applying the SaNSDE algorithm to the same set of instances. Three series of experiments were conducted based on a small population size of 10, a moderate population size of 30 and a larger population size of 50 in this study. In terms of the number of best solutions found, the SaNSDE algorithm outperformed the other algorithms regardless of whether the population size was set to 10, 30 or 50. To study the impacts of the learning period parameter, \(LP\), on the performance, we conducted several experiments by changing the values of \(LP\). To understand the effects of the learning period parameter, we performed three series of experiments based on the same set of test cases by setting the learning period parameter, \(LP\), to 10, 1000 and 2000, respectively.