# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 4 Experiment
### Quantitative Analysis of Decision Variables  
To comprehensively evaluate the differences and advantages between the analysis method proposed in this paper and the method in LMEA, this paper makes some necessary modifications to LMEA, renaming it as LMEA-DVQA, and conducts comparative experiments:  
1. Replace the original decomposition method in LMEA with the DVQA method proposed in this paper.  
2. Crossover: During convergence optimization, only select decision variables related to convergence for crossover operations; during diversity optimization, only select decision variables related to diversity.  
3. Mutation: Adopt a mechanism oriented towards determining deviation; define the probability value of each decision variable using Equation (8), where \(R(x_{i})\) represents the characteristic value of decision variable \(x_{1}\); \(n\) represents the number of decision variables related to convergence; and CV and DV represent sets of decision variables related to convergence and diversity, respectively. Then, use the roulette method to select a variable for mutation operation during diversity optimization.  
In the final part of this subsection, LMEA and LMEA-DVQA will be subjected to experimental comparative analysis, specifically considering test cases such as DTLZ1 and DTLZ2-6 [52]. The analysis will focus on the impact of different grouping results compared to LMEA and the influence of varying degrees of \(R_{x}\) values on population search. Each test problem considers 5 and 10 objectives, as well as 100 and 500 decision variables. The following are the parameter settings:  
1. Population size: Both algorithms use the same population size for the current number of objectives.
2. Termination criteria and number of runs: Use estimated evaluations as the termination criteria. For test cases with 100 and 500 decision variables, the maximum evaluation number is set to 10,000, 6800, and 6800.
3. Other parameters: For LMEA, the resistance factor for selecting solutions within decision variable clusters is set to \(nSel=5\) and \(nPer=10\).
4. Performance metrics: IGD is used as a performance metric to compare the performance of algorithms.  
Table 2 presents the IGD metrics of the two algorithms across 10 test problems with 100 and 500 decision variables. Each algorithm was independently executed 20 times to obtain the mean and variance of the IGD as the final evaluation results. The Wilcoxon rank-sum test was employed, with a significance level of 0.05, where the symbols \"+\", \"-\", and \"\(\approx\)\" indicate results that are significantly better than LMEA-DVQA, significantly worse than LMEA-DVQA, and statistically similar, respectively. The best results for each test instance are highlighted with a gray background.  
According to the results in the table, LMEA outperforms LMEA-DVQA in seven test instances, underperforms in nine, and shows similar performance in four; thus, the overall performance of LMEA-DVQA is superior to that of the LMEA algorithm. Notably, LMEA-DVQA exhibits outstanding performance on DTLZ1 and DTLZ4-6, as the proposed decision variable quantitative analysis method fully leverages the inherent characteristics of the variables, quantifying their convergence and divergence properties. This facilitates the population's directed search for optimal solutions during the optimization phase, leading to better superior solutions. In the DTLZ5-5 test instance, the proposed LMEA-DVQA significantly outperforms LMEA in terms of IGD value. This is because the characteristic values of the convergence variables in the decision variables of the DTLZ5 problem differ, indicating that, when the convergence characteristic values of decision variables vary, the search mechanism employed in this study is more conducive to the population's search and evolution.