# A Heuristics-Guided Simplified Discrete Harmony Search Algorithm for Solving 0-1 Knapsack Problem
## 5 Comparative Experiments
### Experiment on the Second Set of Large-Scale 0-1 KP Instances  
Using the fifteen large-scale 0-1 KP instances introduced in Section 4, the SDHS algorithm is compared with six state-of-the-art metaheuristics, i.e., the hybrid cuckoo search algorithm with global harmony search (CSGHS) [9], binary monarch butterfly optimization (BMBO) algorithm [27], chaotic monarch butterfly optimization (CMBO) algorithm [29], opposition-based learning monarch butterfly optimization (OMBO) algorithm [28], list-based SA (LBSA) algorithm [47], and nosing method (NM) [48]. As in the LBSA algorithm and the NM, the maximum number of iterations is set to 40,000. The comparisons are listed in Tables 10-12, where the best results are highlighted in bold. Because the CSGHS algorithm and the BMBO algorithm are not tested on instances with 2000 items, the corresponding cells of those two algorithms are labeled as '-'. Table 10 shows that the CSGHS algorithm and the BMBO algorithm cannot find the best-known solution on any instance. Both the CMBO algorithm and the OMBO algorithm cannot find the best-known solution on the KP03 instance. The NM cannot find the best-known solution on the KP01 instance. Both the LBSA algorithm and the SDHS algorithm can find the best-known solutions on all uncorrelated instances, but only the SDHS algorithm can always find best-known solutions. Table 11 shows that the CSGHS algorithm and the BMBO algorithm cannot find the best-known solution on any instance. The CMBO algorithm cannot find the best-known solution on the KP08 instance. Both the LBSA algorithm and the NM cannot find the best-known solution on the KP09 instance. Both the OMBO algorithm and the SDHS algorithm can find the best-known solutions on all weakly correlated instances, but only the SDHS algorithm can always find best-known solutions. Table 12 shows that the CSGHS algorithm and the BMBO algorithm cannot find the best-known solution on any instance. The CMBO algorithm cannot find the best-known solution on the KP12 instance. All the other four algorithms can find the best-known solutions on all strongly correlated instances, but only the NM, the LBSA algorithm and the SDHS algorithm can always find best-known solutions. In summary, only the SDHS algorithm can always find the best-known solutions on all 15 instances. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Algorithm** & **Rank** & **Gap (\%)** & \(p\)**-Value** & \(p\)**-Holm** & **Sum of +/**=**/** \\ \hline SDHS & 1 & - & - & - & - \\ NGHS & 3.81 & 2.128 & \(\mathbf{4.31\times 10^{-9}}\) & \(\mathbf{1.29\times 10^{-8}}\) & 0/0/16 \\ ABHS & 3.19 & 1.565 & \(\mathbf{9.80\times 10^{-6}}\) & \(\mathbf{1.96\times 10^{-5}}\) & 0/0/16 \\ SBHS & 2 & 0.015 & 0.126 & 0.126 & 0/0/16 \\ \hline \hline \end{tabular} Note: Bold values in the \(p\) columns indicate statistical significance (\(p<0.05\)).  
\end{table} 
Table 9: The performance summary between SDHS, NGHS, ABHS, and SBHS; the average ranks are computed by the Friedman test and the significance is assessed by [PERSON]’s post hoc method. The comparison results between SDHS and the average values of the competitive algorithms, including CSGHS, BMBO, CMBO, OMBO, NM, and LBSA, are detailed in Table 13. These findings are derived from the Friedman test and [PERSON]'s post hoc test, which provide a robust statistical framework for evaluating algorithmic performance. According to the Friedman test, SDHS achieves the highest ranking with a value of 1.71, outperforming all other algorithms. Notably, SDHS exhibits a statistically significant advantage over CSGHS and BMBO. In contrast, when compared to CMBO, OMBO, NM, and LBSA, the performance differences are not statistically significant; nevertheless, SDHS consistently secures a superior ranking. The gap values further corroborate the dominance of SDHS: the smallest differences against NM and LBSA are 0.0005, while larger gaps of 0.932 and 1.071 are observed against CSGHS and BMBO, respectively.