## 4 Comprehensive Experiments
### Workshop Example Simulation
It is crucial to recognize that in the realm of multi-objective optimization, a single \"optimal\" solution encompassing all objectives is generally unattainable. The improvement of one objective often comes at the expense of another, highlighting the intrinsic trade-offs involved. Utilizing a satisfaction function, constructed based on fuzzy membership degree theory, provides decision-makers with a range of viable solutions. This flexibility allows for a balanced evaluation of the economic, environmental, and production-related benefits, enabling the selection of an optimal strategy tailored to specific priorities. Simultaneously, the model accounts for completion time, energy efficiency, and processing quality. In doing so, it ensures that production deadlines and quality standards are met while also striving to fulfill green objectives such as energy conservation and emissions reduction. (2) Comparison of the algorithm results  
The algorithm proposed in this paper and the above-mentioned four comparison algorithms were independently run 20 times in two instances to avoid the chance of solving the algorithm. Figure 8 shows the convergence of the optimization process, and Figure 9 shows the box plot results of each algorithm run 20 times. Figure 8: Convergence curves of different algorithms. Figure 9: Boxplot of 20-times solution results. The above results show that the improved algorithm proposed in this paper had the best effect on the two instances, and its advantages are embodied in the following aspects:  
1. Model convergence: From the convergence curve, the proposed IWOA algorithm had good global search performance and local information mining ability. The blue curve shows the iteration process of the IWOA. In the early stage of iteration, the curve jump distance is large, which indicates that the algorithm has better global detection ability. At the later stage of iteration, it still shows good search ability and frequent small distance down hops until finally stable, without falling into the local optimum. Formally, the convergence speed of the IWOA was better than IWOA(2) and WOA-LEDE on \(5\times 5\) instances. Although IWOA(1) and NSG-II converged before IWOA, their optimization effect was not good. In addition, in \(5\times 15\) instances, the convergence speed of IWOA was significantly better than that of the other four comparison algorithms. Therefore, it can be concluded that compared with small-scale scheduling problems, the proposed algorithm has stronger performance in large-scale numerical examples. 2. Model solution quality: According to the convergence curve, the loss function of the IWOA was 0.250 and 0.245, respectively, in two instances of different sizes, which is better than the loss of the comparison algorithm, and the multi-objective solution with the maximum satisfaction degree was obtained. In addition, it can be seen from Figure 9 that the mean value of loss function of IWOA in the \(5\times 5\) and \(15\times 5\) instances was 0.265 and 0.258, respectively, and its upper and lower limits were also much better than its comparison algorithm. 3. Stability of the model: Figure 9 shows that the loss function fluctuations of IWOA in the two instances were [0.251, 0.275] and [0.245, 0.273] respectively, which were significantly smaller than the loss fluctuation interval of the comparison algorithm. The algorithm shows good stability, and the stability performance on large-scale instances is superior. 4. Statistical analysis We performed statistical analysis on the experimental results. The Wilcoxon's test was used to verify whether the differences between models with different strategies were significant. We took IWOA as the control algorithm and tested whether the difference between the proposed algorithm and the comparison algorithm was significant with 95% confidence.