# List-Based Threshold Accepting Algorithm with Improved Neighbor Operator for 0-1 Knapsack Problem
## 5 Competitiveness of the LBTA Algorithm
### Competitiveness of the LBTA Algorithm on the Second Set of Instances  
First, we made a comparison on 25 small instances where the size varied between 8 and 24. Details of these 25 KPs were obtained from [PERSON]'s research [38]. To evaluate the effectiveness and accuracy of LBTA, it is assessed in relation to recent studies, including BEO [35], BinEHO [36], BMPA [39], BinRSA [31], BSMA [32], BHHO [34], BTSA [33], and BAOA [40]. In Table 3, the average results and gap values of each algorithms are provided. The proposed method demonstrates superior or equal performance compared with other algorithms across all 25 problems, as indicated by the gap results. Notably, LBSA outperforms all competing algorithms, achieving the best solution in 100% of the instances.  
Figure 1: Performance comparison of LBTA with the different numbers of bit-flip operators.  
While algorithms such as BMPA, BSMA, and BEO also obtain the best solution in 100% of the 24 instances, their performance falls slightly short of LBSA on specific problems--BMPA and BEO on the P24 instance and BSMA on the P9 instance. Additionally, although BinRSA achieves the best average in 20 instances, and BinEHO, BHHO, BTSA, and BAOA secure the best results in 17 out of 25 instances, none match the consistent and optimal performance of LBSA across all test cases.  
Furthermore, the LBTA algorithm was evaluated on large-scale instances from [PERSON]'s optimization codes [41]. The datasets used in this study fall into three different categories: datasets 1-6 are high dimensional uncorrelated, datasets 7-12 are high dimensional weakly correlated, and datasets 13-18 are high dimensional strongly correlated. In Table 4, a comparison of the performance of the LBTA algorithm with that of IGA-SA [42], BinRSA [31], BSMA [32], BTSA [33], BAOA [40], and BHHO [34] is presented. The results, as shown in Table 4, demonstrate that LBSA performs similarly as or better than other algorithms in 14 out of 18 problems according to the mean results, matching or exceeding the number of optimal solutions obtained by the other algorithms. While the BinRSA algorithm achieves the same number of optimal solutions, our proposed LBTA ranks first overall when all algorithms are considered, with BinRSA ranking third behind BSMA. The results clearly show that LBTA performs only slightly worse than BinRSA in the KP1_5000, KP2_5000, and KP3_5000 instances, and its performance in the KP1_2000 instance is marginally worse than those of BSMA, BAOA, and BHHO. However, LBTA outperforms all algorithms in the median items for the KP2_500 instance and matches the best value achieved by other algorithms in 13 other instances.  
In conclusion, the LBTA algorithm demonstrated strong performance across multiple large-scale instances, particularly on high-dimensional weakly correlated and strongly correlated datasets, showcasing its robust competitiveness. Although it performed slightly less effectively than certain algorithms on a few specific instances, it ranked first overall, proving the effectiveness and robustness of the LBTA algorithm in solving 0-1 KPs.