# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 4 Experiment
### Comparison of Algorithm Experimental Results and Analysis on Different Test Problems  
We selected six comparison algorithms--NSGA-II, MOEA/D, NSGA-III, LMEA, FDV [53], and S3 CMAES [54]--to experimentally validate the performance of DVA-TPCEA in solving LaMaOPs. The following is a brief introduction:  
* NSGA-II: A widely used evolutionary algorithm based on non-dominated sorting and crowding distance for multi-objective optimization, which emphasizes both convergence and diversity. * MOEA/D: A decomposition-based algorithm that transforms a multi-objective problem into multiple scalar optimization subproblems, solving them concurrently to approximate the Pareto front. * NSGA-III: An extension of NSGA-II designed for many-objective optimization, incorporating a set of reference points to improve performance in high-dimensional objective spaces. * LMEA: Another decomposition-based algorithm, specifically crafted for large-scale multi-objective problems, optimizing high-dimensional decision spaces. * S3 CMAES: This algorithm employs a scalable evolution strategy, integrating subpopulation mechanisms to enhance adaptability in complex, large-scale environments. * FDV: A framework introducing fuzzy decision variables, facilitating decision making under uncertainty by leveraging fuzzy logic to improve robustness. Among these, NSGA-III, MOEA/D, and LMEA are decomposition-based algorithms; S3 CMAES introduces a subpopulation-scalable evolutionary strategy; and FDV proposes a fuzzy decision variable framework. The experiments will be conducted on three multi-objective benchmark test sets: DTLZ, WFG, and LSMOP. The test functions we adopted are DTLZ4-7, WFG-6-9, and LSMOP1-5. The DTLZ (Deb-Thompson-Leyton-Zhang) functions, particularly DTLZ4 and DTLZ5, are characterized by their complex Pareto fronts, testing the algorithms' convergence and diversity. The WFG (Walking Fish Group) functions, such as WFG6 and WFG9, are designed to assess an algorithm's effectiveness in handling multiple objectives and feature various local optima, challenging global search capabilities. LSMOP (large-scale multi-objective problem) functions, including LSMOP1 to LSMOP5, focus on large-scale optimization, providing problems with high-dimensional decision variables to evaluate the robustness and efficiency \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Problem** & **M** & **D** & **LMEA** & **LMEA-DVQA** \\ \hline \multirow{4}{*}{DTLZ1} & 5 & 100 & 5.99e-2 (4.22e-4) & 5.48e-2 (3.114e-4) \\  & 5 & 500 & 6.11e-2 (5.16e-4) & 5.52e-2 (2.855e-5) \\  & 10 & 100 & 1.63e-1 (4.87e-3) & 1.32e-1 (2.726e-3) \\  & 10 & 500 & 1.59e-1 (4.05e-3) & 1.32e-1 (4.381e-3) \\ \hline \multirow{4}{*}{DTLZ3} & 5 & 100 & 1.89e-1 (2.14e-3) & 2.22e-1 (2.954e-3) \\  & 5 & 500 & 1.90e-1 (4.44e-3) & 2.08e-1 (1.188e-4) \\  & 10 & 100 & 5.53e-1 (3.56e-2) & 4.97e-1 (6.959e-3) \\  & 10 & 500 & 5.51e-1 (1.66e-2) & 4.80e-1 (5.159e-4) \\ \hline \multirow{4}{*}{DTLZ4} & 5 & 100 & 2.64e-1 (1.55e-2) & 3.40e-1 (1.860e-1) \\  & 5 & 500 & 2.72e-1 (2.46e-2) & 2.07e-1 (4.354e-4) \\ \cline{1-1}  & 10 & 100 & 5.08e-1 (2.47e-2) & 4.77e-1 (2.238e-3) \\ \cline{1-1}  & 10 & 500 & 5.27e-1 (3.98e-3) & 5.49e-1 (1.005e-1) \\ \hline \multirow{4}{*}{DTLZ5} & 5 & 100 & 5.06e-2 (7.520e-5) & 5.61e-3 (2.436e-4) \\  & 5 & 500 & 5.17e-2 (8.511e-6) & 5.66e-3 (6.554e-4) \\ \cline{1-1}  & 10 & 100 & 2.39e-3 (6.95e-5) & 7.86e-3 (3.857e-4) \\ \cline{1-1}  & 10 & 500 & 2.27e-3 (4.47e-5) & 5.14e-1 (1.081e-1) \\ \hline \multirow{4}{*}{DTLZ6} & 5 & 100 & 5.07e-2 (1.601e-4) & 6.41e-3 (4.407e-5) \\ \cline{1-1}  & 5 & 500 & 5.49e-2 (4.450e-3) & 5.26e-3 (4.841e-4) \\ \cline{1-1}  & 10 & 100 & 6.93e-2 (2.425e-3) & 9.32e-3 (9.276e-4) \\ \cline{1-1}  & 10 & 500 & 8.38e-3 (7.313e-4) & 8.94e-3 (3.807e-4) \\ \hline \multicolumn{4}{c}{\(+/-/=\)} & 7/9/4 \\ \end{tabular}
\end{table} 
Table 2: IGD metrics of the two algorithms on the DTLZ1 and DTLZ3-6 problems. of algorithms in complex scenarios. Together, these functions comprehensively test the strengths and weaknesses of multi-objective optimization approaches.