## 4 The DABC Algorithm for an EAHFSP
### Scout Bee Phase
\[c(A,B)=\frac{|\{x\in B|\exists y\in A:y\succ x\}|}{|B|} \tag{21}\]

where \(A\) and \(B\) are two non-dominated solution sets and \(y\succ x\) represents that solution \(x\) is dominated by \(y\). If \(c(A,B)>c(B,A)\), algorithm \(A\) is better than \(B\). In this study, the ABC, TICA [48], HPSO [51], and CCA [52] are selected as comparative algorithms. The ABC algorithm is chosen to demonstrate the effectiveness of new strategies. It is conducted using the traditional method. [PERSON] [48] designed a TICA for an energy-efficient HFSP, with the relative importance of objectives to minimize the makespan, total tardiness, and total energy consumption. After the newly defined Pareto dominance is removed, the TICA can be directly used to solve EAHFSPs. [PERSON] [51] investigated an HPSO to optimize the electric power costs and total tardiness in an energy-aware flexible flow shop including time-of-use and variable processing speeds. [PERSON] [52] introduced a CCA for a multi-objective hybrid flow shop problem. The CCA can be applied to solve EAHFSPs after adding the assembly operation. So, we choose these algorithms as the comparative algorithms. ### Parameter Settings

Our experiments indicate that DABC can converge fully in all instances after \(0.3\times n\times m\) seconds; moreover, all comparative algorithms also converge well when this termination time is used. All parameters except the above termination condition are chosen from the literature. There are four key parameters in the DABC: \(R\), \(N\), \(Limit\), and \(\xi\). The Taguchi method is used to decide the settings of these four parameters. The levels of each parameter are listed in Table 3. Table 4 defines the orthogonal array and \(L_{16}(4^{4})\), where \(ARV\) denotes the average value of the DABC with each combination independently run 10 times, for instance, \(80\times 8\). The rank of each parameter is given in Table 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multicolumn{4}{c}{**Factor**} & \multirow{2}{*}{**ARV**} \\ \cline{2-2} \cline{4-5}  & \(R\) & \(N\) & \(Limit\) & \(\xi\) & \\ \hline
1 & 1 & 1 & 1 & 1 & 0.314 \\
2 & 1 & 2 & 2 & 2 & 0.295 \\
3 & 1 & 3 & 3 & 3 & 0.184 \\
4 & 1 & 4 & 4 & 4 & 0.320 \\
5 & 2 & 1 & 2 & 3 & 0.084 \\
6 & 2 & 2 & 1 & 4 & 0.137 \\
7 & 2 & 3 & 4 & 1 & 0.377 \\
8 & 2 & 4 & 3 & 2 & 0.447 \\
9 & 3 & 1 & 3 & 4 & 0.159 \\
10 & 3 & 2 & 4 & 3 & 0.432 \\
11 & 3 & 3 & 1 & 2 & 0.237 \\
12 & 3 & 4 & 2 & 1 & 0.176 \\
13 & 4 & 1 & 4 & 2 & 0.190 \\
14 & 4 & 2 & 3 & 1 & 0.170 \\
15 & 4 & 3 & 2 & 4 & 0.273 \\
16 & 4 & 4 & 1 & 3 & 0.175 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The orthogonal array \(L_{16}(4^{4})\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{4}{c}{**Factor Level**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline \(R\) & 6 & 8 & 10 & 12 \\ \(N\) & 30 & 40 & 50 & 60 \\ \(Limit\) & 7 & 8 & 9 & 10 \\ \(\xi\) & 0.2 & 0.3 & 0.4 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The levels of four parameters. Figure 5: Main effect plot for metric \(IGD\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{4}{c}{**Factor Level**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline \(R\) & 6 & 8 & 10 & 12 \\ \(N\) & 30 & 40 & 50 & 60 \\ \(Limit\) & 7 & 8 & 9 & 10 \\ \(\xi\) & 0.2 & 0.3 & 0.4 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The levels of four parameters. ### Results and Comparative Analysis

The DABC, ABC, TICA, HPSO, and CCA were independently run 10 times for each instance. The reference set \(\Delta^{*}\) contains the non-dominated solutions of \(\Delta_{1}\cup\Delta_{2}\cup\Delta_{3}\cup\Delta_{4}\cup\Delta_{5}\), where \(\Delta_{1}\), \(\Delta_{2}\), \(\Delta_{3}\), \(\Delta_{4}\), and \(\Delta_{5}\) indicate the optimal solutions obtained using the DABC, ABC, TICA, HPSO, and CCA methods, respectively.