## 5 Case Study
### Parameter Settings  
The proposed NSGA-II-MOEA/D-CIS hybrid algorithm in this paper includes three key parameter groups:  
1. NSGA-II algorithm: Population size \(N_{upper}\), crossover rate \(P_{c}\), and mutation rate \(P_{m}\). 2. MOEA/D algorithm: Population size \(N_{lower}\) and neighborhood size \(T\). 3. CIS: Iteration count \(K\) and satisfaction weights \(\alpha\), \(\beta\). To determine the optimal values for these parameters, this section employs the design of experiments (DOE) method for parameter tuning. Since the tuning process is similar across all parameters, this section only presents the optimization process for the three NSGA-II parameters: \(N_{upper}\), \(P_{c}\), and \(P_{m}\). During the DOE process, each of these three parameters is assigned five different levels, as shown in Table 3. Specifically, the candidate values for population size, crossover rate, and mutation rate were selected based on commonly used settings in multi-objective evolutionary algorithm studies, as well as empirical tuning to balance solution quality and computational cost. For each parameter combination, the algorithm performance evaluation process is designed as follows:  
Step 1: Execute multiple rounds of experiments. For all parameter combinations to be evaluated, conduct 50 independent experiments. In each experiment, each parameter combination independently performs an optimization calculation, generating a corresponding non-dominated solution set. Step 2: Construct a reference set. After each round of experiments, globally merge the non-dominated solution sets produced by all parameter combinations in that round. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Order** & **Specification** & **Demand** & **Date** & **Order** & **Specification** & **Demand** & **Date** \\ \hline \multirow{6}{*}{O\({}_{7}\)} & P\({}_{7,1}\) & 30,000 & 18 December 2024 & \multirow{6}{*}{P\({}_{12,5}\)} & 24,000 & 15 January 2025 \\ \cline{2-6}  & P\({}_{7,2}\) & 3000 & 9 December 2024 & & P\({}_{12,6}\) & 3900 & 13 December 2024 \\ \cline{2-6}  & P\({}_{7,3}\) & 2000 & 13 December 2024 & & P\({}_{12,7}\) & 3000 & 13 December 2024 \\ \cline{2-6}  & P\({}_{7,4}\) & 3000 & 23 December 2024 & & P\({}_{13,1}\) & 13,500 & 23 December 2024 \\ \cline{2-6}  & P\({}_{7,5}\) & 3600 & 23 December 2024 & & P\({}_{13,2}\) & 45,000 & 20 December 2024 \\ \cline{2-6}  & P\({}_{7,6}\) & 3600 & 20 December 2024 & & P\({}_{13,3}\) & 5400 & 10 December 2024 \\ \hline \multirow{6}{*}{O\({}_{8}\)} & P\({}_{8,1}\) & 80,000 & 30 December 2024 & \multirow{6}{*}{O\({}_{14}\)} & P\({}_{14,1}\) & 90,000 & 20 December 2024 \\ \cline{2-6}  & P\({}_{8,2}\) & 6000 & 23 December 2024 & & P\({}_{14,2}\) & 8000 & 12 December 2024 \\ \cline{1-1} \cline{2-6}  & P\({}_{8,3}\) & 1000 & 2024-12-23 & & P\({}_{14,3}\) & 25,440 & 17 December 2024 \\ \cline{1-1} \cline{2-6}  & P\({}_{8,4}\) & 2400 & 27 December 2024 & & P\({}_{15,1}\) & 4000 & 20 December 2024 \\ \hline \multirow{6}{*}{O\({}_{15}\)} & P\({}_{8,6}\) & 80,000 & 30 December 2024 & \multirow{6}{*}{O\({}_{15}\)} & P\({}_{15,2}\) & 74,250 & 13 December 2024 \\ \cline{1-1} \cline{2-6}  & P\({}_{8,6}\) & 80,000 & 30 December 2024 & & P\({}_{15,2}\) & 74,250 & 13 December 2024 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Levels**} & \multicolumn{3}{c}{**Parameters**} \\ \cline{2-4}  & \(N_{upper}\) & \(P_{c}\) & \(P_{m}\) \\ \hline
1 & 50 & 0.60 & 0.05 \\
2 & 100 & 0.70 & 0.06 \\
3 & 200 & 0.80 & 0.07 \\
4 & 300 & 0.85 & 0.08 \\
5 & 500 & 0.90 & 0.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter levels of the NSGA-II algorithm. Then, filter out individuals in the merged set that are not dominated by any other solutions, forming the benchmark reference set for that round. Step 3: Compute the single-round contribution rate. For the non-dominated solution set of a given parameter combination in the current round, count the number of solutions that meet the following two conditions:  
1.