# Modified Coral Reef Optimization Methods for Job Shop Scheduling Problems
## 4 Experiment Results and Discussion
### Experiment Results
#### 4.3.1 Search Performance on CRO-Based Algorithm with Different Reef Sizes
All \(\chi^{2}\) values are larger than the critical value \(\chi^{2}_{0}=5.991\), and all \(\rho\) values are less than 0.05, demonstrating statistically significant differences in the performance of three CRO-based algorithms. Following [PERSON]'s test made a significant result, we performed [PERSON]'s signed-rank test as a posthoc analysis to determine the difference between three CRO-based algorithms in pairwise group. The search findings of three reef sizes will be averaged to feed the [PERSON]'s test with a statistical significance of 0.05. Table 6 provides the statistical results of the [PERSON]'s test. Using the \(Z\) distribution table, we can find the critical value of \(Z\) is 1.96 at a statistical significance of 0.05. According to Table 6, the statistical findings of two pairs of algorithms, CRO-CROLS1 and CRO-CROLS2, are nearly equivalent as \(Z=-3.516\) (\(\mid Z\mid>1.96\)) and \(p=0.000438<0.05\). So, the null hypothesis is rejected at statistical significance of 0.05; the advantage of local search approaches on CROLS1 and CROLS2 compared to the original CRO algorithm is statistically significant. In contrast, the values \(Z=-1.533\) (\(\mid Z\mid<1.96\)), and \(p=0.125153>0.05\) indicate no significant performance difference between the CROLS1 and CROLS2 hybrid algorithms. 3.2 Comparison of the Computational Result of CRO-Based Algorithms with Other Implemented Local Search Technique Algorithms  
This experiment aims to test the algorithm's efficiency in finding the optimal value of the problem. We choose MA and HGA as two algorithms employing local search strategies to compare the performance of algorithms that use such techniques. Five algorithms are mentioned in Table 7, including the original algorithm CRO and two proposed hybrid algorithms CROLS1 and CROLS2. Each CRO algorithm is executed 30 times with three different reef sizes and takes the best value obtained in Table 7. The results of HGA and MA were consulted from the research of [PERSON] et al. [44] and [PERSON] et al. [45], respectively. \begin{table}
\begin{tabular}{c c c c} \hline  & **CRO-CROLS1** & **CRO-CROLS2** & **CROLS1-CROLS2** \\ \hline \(Z\) & \(-3.516\) & \(-3.516\) & \(-1.533\) \\ \hline \(p\) & 0.000438 & 0.000438 & 0.125153 \\ \hline \end{tabular}
\end{table} 
Table 6: [PERSON]’s test statistic of three CRO-based algorithms. \begin{table}
\begin{tabular}{c c c} \hline  & \(\mathbf{X^{2}}\) & \(\rho\) \\ \hline Reef Size 10 \(\times\) 10 & 25.733 & 0.000003 \\ \hline Reef Size 20 \(\times\) 20 & 16.625 & 0.000245 \\ \hline Reef Size 30 \(\times\) 30 & 21.556 & 0.000021 \\ \hline \end{tabular}
\end{table} 
Table 5: [PERSON]’s test statistic of three CRO-based algorithm. It can be seen that the two-hybrid algorithms CROLS1 and CROLS2 can discover the optimal solution in the majority of instances with the reef size set to 20 \(\times\) 20 and 30 \(\times\) 30. However, the original algorithm CRO can only find the optimal solution in a few cases with a small number of jobs and machines. Compared with the other two metaheuristics, the performance of the two-proposed hybrid algorithm is superior when the reef size is 20 \(\times\) 20 or 30 \(\times\) 30 in some instances of the problem. 3.3 Comparison of the Computational Result of CRO-Based Algorithms with Other Contemporary Algorithms  
To further verify the effectiveness of CRO and two proposed hybrid algorithms, we performed extensive experiments on the 12 LA instances mentioned before, [PERSON] and [PERSON] instances [46]: FT06, FT10, FT20; [PERSON] and [PERSON] instances [47]: ORB01-ORB09; and five of [PERSON] et al. instances [48] denoted as ABZ05 to ABZ09.