## 5 The Computational Results and Analysis
### Performance Analysis of IQABC Algorithm without Learning and Deteriorating Effects  
To prove the performance of the integrated QL-driven ABC algorithm without learning and deterioration effects (IQABC), we compared it with three different competing algorithms for the same problem and constraints without considering the learning and deteriorating effects. The comparative algorithms are : (1) a population-based metaheuristic, which is the ABC algorithm of [54] with random neighborhood selection; (2) a unique solution-based metaheuristic, which is the Variable Neighbor Search (VNS) algorithm of [83]; (3) and finally, INEH, our newly designed constructive heuristic. The comparison was carried out by means of 1100 runs of the test benchmark. The resulting ARPD values reported in Table 1 were calculated based on the \(C_{max}\) value given in Formula (3), with no consideration of the learning and deteriorating effects, and with respect to the first and second maintenance modes (\(M1\) and \(M2\)). Table 1 shows that the INEH heuristic produces competitive ARPD results compared to the VNS metaheuristic, even though the INEH is a constructive method. In fact, for small instances (20x5, 20x10, 20x20), INEH achieves the best ARPD values: for Mode 1, an average of 16.76 against 20.90 for VNS; for Mode 2, an average of 19.14 for INEH against 22.59 for VNS. Nonetheless, it fails to reach the results obtained by the VNS algorithm for large instances (50x5-200x20): for Mode 1, an average of 9.60 against 8.03 for the VNS; and \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{4}{c}{**M1**} & \multicolumn{4}{c}{**M2**} \\ \cline{2-9}  & **ABC** & **VNS** & **INEH** & **IQABC** & **ABC** & **VNS** & **INEH** & **IQABC** \\ \hline
20x5 & 4.69 & 11.52 & 6.77 & 3.11 & 3.43 & 10.61 & 8.87 & 3.34 \\
20x10 & 13.34 & 23.01 & 16.22 & 11.45 & 13.97 & 24.20 & 19.09 & 12.33 \\
20x20 & 25.35 & 28.19 & 27.31 & 23.16 & 27.03 & 32.98 & 29.46 & 24.84 \\
50x5 & 2.02 & 1.72 & 2.66 & 0.54 & 2.83 & 1.93 & 2.83 & \(-\)0.001 \\
50x10 & 9.93 & 8.24 & 12.11 & 5.91 & 12.62 & 9.95 & 15.79 & 6.91 \\
50x20 & 19.79 & 18.5 & 21.94 & 15.12 & 24.65 & 19.28 & 25.01 & 16.93 \\
100x5 & 1.16 & 1.35 & 1.47 & 0.93 & 1.40 & 1.46 & 1.93 & 0.96 \\
100x10 & 4.34 & 5.09 & 5.56 & 2.90 & 6.37 & 5.48 & 10.27 & 3.82 \\
100x20 & 13.41 & 15 & 17.36 & 11.40 & 16.31 & 15.61 & 21.53 & 13.22 \\
200x10 & 1.77 & 3.13 & 3.10 & 1.94 & 2.58 & 3.43 & 6.45 & 2.68 \\
200x20 & 7.31 & 11.21 & 12.6 & 6.58 & 9.53 & 12.55 & 16.92 & 8.94 \\
500x20 & 2.99 & - & - & 3.08 & 2.98 & - & - & 3.64 \\ \hline average & 8.84 & 11.54 & 11.55 & 7.18 & 10.30 & 12.50 & 14.38 & 8.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance offset from [PERSON]’s upper bounds of IQABC, ABC, VNS, and INEH algorithms for maintenance mode 1 and 2.  
for Mode 2, an average of 12.59 for INEH against 8.71 for VNS. The great advantage of the INEH heuristic is its ability to obtain near-optimal solutions in a reasonable computation time (on average 6.99 time units (TU) compared to 3120.89 TU for VNS). Although the ABC metaheuristic of [54] gave better solutions than the VNS algorithm, which is considered an efficient metaheuristic for the problem [83; 84], the IQABC algorithm outperforms it, as can clearly be seen from Table 1 for the two maintenance modes. This is proved by the employed bees' local search procedures and the QL-driven neighborhood selection. To further verify the statistical significance of the difference in ARPD values between ABC and IQABC for the two maintenance modes, we performed a [PERSON]'s two-way analysis of variance test [85]. The result summary in Table 2 as well as the mean plots displayed in Figure 2 show the significant differences between IQABC and ABC. Therefore, the IQABC algorithm is significantly superior to the ABC, VNS, and INEH algorithms. However, the ABC and IQABC algorithms are population-based methods, which means an important amount of time is used to effectively explore the search space.