## 3 The Algorithm
### SaNSDE Algorithm
We analyzed the outcomes by comparing performance. The results indicate that the proposed SaNSDE algorithm is not sensitive to the learning period parameter \(LP\) for Case 1 through Case 4, Case 9 and Case 10. The sensitivity of the average fitness values obtained by the proposed SaNSDE algorithm is less than 5% for most cases, with the exception of Case 7. Another observation of the results is that \(LP\) should be large enough to obtain better average fitness values. If the parameter \(LP\) is too small, it is difficult to learn the best strategy within the learning period. As a result, the probability of creating a potential candidate solution using the best strategy will be low. In this case, the performance will be degraded. If \(LP\) is big enough, the probability of learning the best strategy within the learning period will be higher. Hence the probability of creating a potential candidate solution by learning the best strategy will be higher. In this case, the performance tends to be improved. The results presented in Section 4.3 indicate that for the algorithm to perform better, \(LP\) must be large enough. This poses an interesting and challenging future research direction to prove that the performance can be improved by setting \(LP\) to a large value through theoretical analysis. To study the effects of population size on the performance of the proposed algorithm, we performed three series of experiments for all test cases for \(NP=10\), 30 and 50. The \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case** & **I** & **K** & **DE1** & **DE2** & **DE3** & **DE4** & **DES5** & **DE6** \\ \hline
1 & 4 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\
2 & 5 & 5 & 0 & 1.5811 & 0 & 0 & 0 & 1.5811 \\
3 & 5 & 5 & 3.1622 & 12.1586 & 7.3098 & 17.1298 & 7.9728 & 16.1386 \\
4 & 10 & 10 & 0 & 5.0596 & 25.5757 & 20.6812 & 8.2535 & 5.0596 \\
5 & 10 & 10 & 35.323 & 98.8249 & 16.0407 & 92.1441 & 106.6155 & 153.1257 \\
6 & 10 & 20 & 54.118 & 56.1074 & 69.6333 & 80.8964 & 117.6617 & 144.5901 \\
7 & 20 & 20 & 202.1124 & 486.4097 & 361.1803 & 510.4664 & 539.5477 & 486.2034 \\
8 & 30 & 10 & 24.7512 & 292.6231 & 18.0027 & 24.708 & 228.503 & 38.1465 \\
9 & 30 & 20 & 41.3173 & 73.7039 & 0 & 47.8294 & 61.3681 & 56.0769 \\
10 & 40 & 10 & 16.1551 & 107.1078 & 21.7411 & 26.8692 & 242.5954 & 193.7009 \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Standard deviation of fitness function values for DE1, DE2, DE3, DE4, DE5 and DE6 with \(NP=50\) and \(LP=1000\). results show that the average fitness values of Case 1, Case 2 and Case 3 are the same. For Case 4 through Case 10, the differences between the average fitness values for \(NP=\) 10, 30 and 50 are small in terms of percentage. This indicates that the proposed SaNSDE algorithm is not sensitive to population size. In summary, 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. There are two problems formulated in this paper. The first problem formulation is for a general planning problem, whereas the second one is a special case of the first problem formulation for a sequential processes planning problem. As sequential processes are found in many production systems, we first started with the development of an effective problem solver for the second problem formulation and verification for sequential processes in this paper. As the solution algorithms take the same form, with the exception of the fitness function, the solution algorithm for the second problem (for sequential processes) can be tailored for other types of production processes. This includes the development of efficient methods to compute a penalty due to the violation of constraints \(\Re(x,B,R)\), a penalty due to the violation of constraints \(\T(x,B)\) and a penalty due to the violation of constraints \(\Psi(x,B)\). Based on the discussion above, the proposed algorithm has the potential to be extended to deal with other types of production processes.