# A Hybrid Crow Search Algorithm for Solving Permutation Flow Shop Scheduling Problems
## 5 Experimental Results
### Comparison of CSA Algorithms Incorporated with the SA-Based Local Search  
NEHLIJP1 combined with the CSA algorithm provided the best BRPD and ARPD values. In this section, we validate the effectiveness of the algorithm on the number of iterations in the SA local search. We compared two different iterations of the local search, 1000 and \(jobs\times(jobs-1)\)[49], using the BRPD and ARPD values and the average computation time \(t_{a}vg\), where \(jobs\) indicates the number of jobs of the benchmark and the values in bold font represent the algorithms that achieve superior performance.  
The computed BRPD and ARPD results are listed in Tables 6 and 7. The \(jobs\times(jobs-1)\) strategy achieved better BRPD and ARPD values for small jobs (0.54 and 0.69, respectively). However, the BRPD  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & & **NEH** & **NEHD** & **NEHKK1** & **NEHKK2** & **CL** & **NEHLIJP1** \\ \hline \hline \multicolumn{2}{l}{Problem} & & & & & \\ ta01 & BRPD & 0.70 & 1.49 & **0.47** & 1.48 & 0.70 & 1.48 \\  & ARPD & 0.73 & 1.55 & 1.36 & 1.48 & **0.71** & 1.48 \\ ta11 & BRPD & 2.52 & 2.84 & **2.14** & 2.46 & 2.72 & 2.66 \\  & ARPD & 3.28 & 3.66 & 4.79 & 3.73 & 4.08 & **3.12** \\ ta21 & BRPD & 3.22 & 2.82 & 2.39 & **1.92** & 2.05 & 2.66 \\  & ARPD & 4.22 & 2.82 & 3.93 & **2.42** & 2.73 & 3.37 \\ ta31 & BRPD & 0.18 & **0.00** & 1.02 & 0.18 & 0.18 & 0.18 \\  & ARPD & **0.18** & 0.39 & 1.98 & 0.21 & 0.25 & **0.18** \\ ta41 & BRPD & 4.74 & 3.81 & **3.51** & 4.54 & 4.54 & **3.51** \\  & ARPD & 5.05 & 5.62 & **4.33** & 4.54 & 5.48 & 4.57 \\ ta51 & BRPD & **3.45** & 5.61 & 4.52 & 4.55 & 4.50 & 4.16 \\  & ARPD & **3.65** & 7.20 & 5.21 & 4.96 & 4.71 & 4.38 \\ ta61 & BRPD & 0.38 & 0.38 & 1.36 & 0.43 & **0.04** & 0.09 \\  & ARPD & 0.38 & 0.43 & 1.37 & 0.43 & **0.04** & 0.45 \\ ta71 & BRPD & 1.06 & 1.87 & **0.88** & 0.92 & 0.74 & **0.88** \\  & ARPD & 1.19 & 2.10 & 1.06 & 0.94 & 1.14 & **0.98** \\ ta81 & BRPD & 2.85 & 3.14 & 3.64 & **2.77** & 3.19 & 3.17 \\  & ARPD & **3.21** & 4.14 & 3.93 & 3.31 & 3.56 & 3.31 \\ ta91 & BRPD & 0.74 & **0.20** & 0.72 & 0.52 & 0.68 & 0.75 \\  & ARPD & 0.75 & **0.49** & 0.72 & 0.61 & 0.68 & 0.75 \\ ta101 & BRPD & 2.46 & **1.57** & 2.13 & 2.76 & 2.28 & 1.89 \\  & ARPD & 2.66 & **1.69** & 2.24 & 2.88 & 2.59 & 2.24 \\ ta111 & BRPD & 1.47 & 1.59 & 1.54 & 1.22 & 1.52 & **1.15** \\  & ARPD & 1.79 & 1.69 & 1.70 & 1.43 & 1.77 & **1.38** \\ \hline \hline \multicolumn{2}{l}{Average} & BRPD & 1.98 & 2.11 & 2.03 & 1.98 & 1.93 & **1.88** \\  & ARPD & 2.26 & 2.65 & 2.72 & 2.25 & 2.31 & **2.18** \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of different variants of initialization algorithms. ARPD: average percentage relative deviation; BRPD: best percentage relative deviation.  
and ARPD values were almost the same for large jobs. Additionally, the computation times were almost identical for two different numbers of iterations. In summary, the \(jobs\times(jobs-1)\) strategy achieved the most desirable BRPD and ARPD values for all benchmarks, specifically 0.61 and 0.81, respectively.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline  & & **1000** & \(jobs\times(jobs-1)\) \\ \hline Problem & & & \\ ta01 & BRPD & 0.00 & 0.00 \\  & ARPD & 0.00 & 0.00 \\  & \(t_{avg}\) & 1.43 & 1.43 \\ ta11 & BRPD & 0.00 & 0.00 \\  & ARPD & 0.06 & **0.00** \\  & \(t_{avg}\) & **1.67** & 1.65 \\ ta21 & BRPD & 0.00 & 0.00 \\  & ARPD & 0.27 & **0.18** \\  & \(t_{avg}\) & **2.26** & 2.23 \\ ta31 & BRPD & 0.00 & 0.00 \\  & ARPD & 0.00 & 0.00 \\  & \(t_{avg}\) & **3.08** & 3.21 \\ ta41 & BRPD & 1.47 & **1.24** \\  & ARPD & 2.29 & **1.70** \\  & \(t_{avg}\) & **3.75** & 3.82 \\ ta51 & BRPD & 2.18 & **1.95** \\  & ARPD & 2.42 & **2.25** \\  & \(t_{avg}\) & **5.22** & 5.31 \\ \hline Average & BRPD & 0.61 & **0.54** \\  & ARPD & 0.84 & **0.69** \\  & \(t_{avg}\) & **2.90** & 2.94 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of different iterations of the local search for small job sizes.  
Figure 3: Comparison of the results for all different NEH-based algorithms.