## 5 IICA-QL for EEMsMIAJSP
### Revolution Based on Hyper-Heuristic Variable Neighborhood Search
The problem scale is denoted by O/J/A_M/T/K, where O/J/A represent the number of processing operations, jobs, and assembly operations, respectively. M/T/K represent the total number of processing machines, transport vehicles, and assembly machines, respectively. Detailed information about the product structure, order information, and production parameter data used in the experiments can be found from [[https://github.com/captain-dyq/EEMsMIAJSP.git](https://github.com/captain-dyq/EEMsMIAJSP.git)]([https://github.com/captain-dyq/EEMsMIAJSP.git](https://github.com/captain-dyq/EEMsMIAJSP.git)) accessed on 22 August 2024. The experimental time \(T_{(i)}\) for each problem scale is determined based on the specific problem scale, where \(T_{(i)}=O\times M+J\times T+A\times K\). ### Performance Metrics

To comprehensively evaluate the convergence and diversity of the approximation Pareto fronts set obtained by the algorithm, the following three commonly used metrics are employed to measure the proposed IICA-QL. 1. Solution set Coverage (C-metric). This metric measures the dominance relationship between two solution sets and is defined as follows: \[C(A,B)=\frac{|x\in B|\exists y\in A:y\prec x|}{|B|}\] (33) where A and B are two non-dominated solution sets obtained by the algorithm. C(A,B) represents the proportion of solutions in set B that are dominated by at least one solution in set A. The larger the value of C(A,B), the better the performance of A. 2. Generational Distance (GD). The proximity of the Pareto front \(PF\) obtained by the algorithm to the optimal Pareto front \(PF\)* is evaluated, where \(PF\)* is the true non-dominated solution set among the non-dominated sets obtained by all algorithms. For each solution x in \(PF\), the nearest solution y in \(PF\)* is found, and their Euclidean distance is calculated. GD is the average of the shortest Euclidean distances, which is described as Equation (34). A smaller value of GD indicates better convergence and that the found \(PF\) is closer to the optimal Pareto front. \[GD=\frac{\sqrt{\sum\limits_{x\in PF}\frac{\text{mind}(x,y)^{2}}{y\in PF^{*}}}}{ |PF|}\] (34)
3. Hypervolume (HV) The HV metric is a univariate metric that represents the volume of the region in the objective space enclosed by the non-dominated solution set obtained by the algorithm and the reference point, which is set to (1.01,1.01) [34]. The hypervolume metric is a comprehensive metric for evaluating the convergence and diversity of the non-dominated solution set obtained by the algorithm. The larger the HV value, the better the overall performance of the algorithm. \[HV(E)=Leb(\cup_{x\in E}[f_{1}(x),r_{1}]\times\ldots\times[f_{m}(x),r_{m}])\] (35) where \(Leb(\cdot)\) represents the Lebesgue measure, and m represents the number of objectives in the problem. To eliminate the impact of different dimensions, the solutions in the solution set are normalized according to the following formula [30]:

\[f_{i}^{\prime}=\frac{f_{i}-f_{i}^{\text{min}}}{f_{i}^{\text{max}}-f_{i}^{\text{ min}}},\ i=1,2 \tag{36}\]

### Parameter Settings

To determine the optimal values for the parameters in IICA-QL, the Design of Experiment (DOE) method is used for parameter setting. IICA-QL has a total of seven parameters, each with three levels to choose from. The orthogonal array L27 (\(3^{\prime}\)) is selected, and the meanings and optional parameter levels for each parameter are shown in Table 2. Experiments are conducted using a medium-sized experimental scale (\(82/42/71\)\(3/3/3\)), with each parameter combination run 10 times and the runtime being \(T_{(i)}=O\times M+J\times T+A\times K\). The hypervolume (HV) metric is calculated after each run for each parameter combination, and the average of the ten HV values is used as the response value (RV). Additionally, the average RV and influence of each parameter level are shown in Table 3. According to Table 3, the optimal parameter set is \(P_{s}=10\), \(N_{im}=3\), \(P_{a}=0.8\), \(P_{r}=0.1\), \(\alpha=0.2\), \(\gamma=0.9\), and \(\varepsilon=0.6\).