## 5 Testing Study
### Evaluating the MD/ABC Algorithm  
This section aims to evaluate the performance of the MD/ABC algorithm by comparing it to other algorithms [40; 41; 42; 43; 44]. These algorithms include NSGAII, NSGAIII, MOEA/D, iMOEA/D, EAMOA, and MOABCD. iMOEA/D and EAMOA are used to solve energy-efficient HFSP problems, while NSGAII, NSGAIII, and MOEA/D are the three most popular algorithms for solving a wide range of optimization problems. nSGAII and NSGAIII use a fast nondominated sorting strategy, while MOEA/D uses a decomposition strategy. iMOEA/D is an improved version of MOEA/D for solving combinatorial optimization problems by a strategy that compensates for the efficiency problem of the decomposition strategy. MOABCD is a multi-objective variant of the ABC algorithm for solving combinatorial optimization problems, also based on a decomposition strategy. Both NSGAII and NSGAIII frameworks are similar, yet they employ different selection mechanisms. NSGAII ranks solutions by their crowding distances, while NSGAIII incorporates widely distributed reference points for selection. All compared algorithms use [PERSON]'s method to set the parameters appropriately. The encoding and decoding of the solutions are kept consistent with MD/ABC, and specific operators are adapted  
\begin{table}
\begin{tabular}{c c c c} \hline
**Problem** & **MD/ABC** & **MD/ABCa** & **MD/ABCs** \\ \hline
20 \(\times\) 3 & 0.074 & **0.072** & 0.124 \\
20 \(\times\) 5 & 0.08  
==WARNING: Truncated because of repetitions==
22 & 47 & 27 & 31 & 11 \\ \(20\times 8\) & **43** & 21 & 36 & 16 & 33 & 27 & 8 \\ \(20\times 10\) & **52** & 26 & 38 & 8 & 50 & 36 & 17 \\ \(40\times 3\) & **92** & 16 & 21 & 23 & 81 & 82 & 32 \\ \(40\times 5\) & **51** & 12 & 8 & 16 & 37 & 31 & 16 \\ \(40\times 8\) & **42** & 9 & 11 & 9 & 34 & 40 & 14 \\ \(40\times 10\) & **49** & 6 & 19 & 7 & 34 & 36 & 23 \\ \(60\times 3\) & **87** & 39 & 32 & 25 & 72 & 74 & 26 \\ \(60\times 5\) & **71** & 17 & 24 & 8 & 69 & 59 & 10 \\ \(60\times 8\) & **54** & 13 & 31 & 18 &
+++  
According to the Spread metrics in Table 7, MD/ABC performs the best among the 20 problems, showing its robustness and ability to generate uniformly distributed solutions. As seen from the ANOVA results in Figure 10, the MD/ABC algorithm significantly outperforms the other algorithms, which is attributed to its angle-based selection strategy and scouting mechanism, which helps to perform an adequate search along the neighboring weights to obtain a uniformly distributed Pareto solution. Furthermore, it is observed that algorithms using decomposition strategies, such as MD/ABC, MOEA/D, iMOEA/D, and  
Figure 11: ANOVA-based effect plot on GD. Figure 10: ANOVA-based effect plot on Spread. MOABCD, usually outperform NSGAII, NSGAIII, and EAMOA, prompting the adoption of decomposition strategies to solve the problem. According to the GD metrics in Table 8, MD/ABC performs best in 19 out of 20 problems, reaching the minimum overall value, indicating that it is capable of obtaining solutions closest to the true Pareto frontier. The advantage of the MD/ABC algorithm is more obvious as the problem size increases. Therefore, it can be concluded that MD/ABC has better global exploration ability and can jump out the local optimal solution efficiently. As can be seen from Figure 11, the MD/ABC algorithm is significantly better than the other algorithms. In terms of IGD metrics, it can be observed from Table 9 and Figure 12 that the MD/ABC algorithm again outperforms the other algorithms based on the superiority of the Spread and GD metrics, suggesting that MD/ABC has better diversity and convergence in terms of the quality of the solution set. In terms of NOS metrics, the NOS values are given in Table 10, with MD/ABC obtaining the maximum average overall problems, producing a larger overall average, supporting the previous conclusions.