## 5 Computation Experiments
### Algorithm Comparison  
In order to verify the effectiveness of the two improvement strategies, we compare the results of IMHGA with those of MHGA. In MHGA, the encoding scheme and mutation operator are the same as those in IMHGA, in which the operation sequence chromosome and machine assignment chromosome adopt IPOX and MPX as the crossover operator separately. Moreover, the parameters' setting of the MHGA is the same as that of IMHGA.  
In this paper, we adopt the convergence metric proposed by [PERSON] et al. (2018) and the spacing metric proposed by [PERSON] et al. (2019) to evaluate the convergence and distribution of Pareto solution sets, respectively. The two metrics are defined as follows.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Solution** & **\(C_{\text{max}}\) (min)** & _TEC (kw/h)_ & _MHQ (kg)_ & _PE (kw/h)_ & _IE (kw/h)_ & _TE (kw/h)_ \\ \hline
13 & 22.0077 & 904.4978 & 107.4975 & 550 & 85.7541 & 268.7437 \\
14 & 21.5220 & 906.5748 & 106.9683 & 550 & 89.1541 & 267.4207 \\
15 & 22.7642 & 909.2554 & 105.9224 & 550 & 94.4495 & 264.8059 \\
16 & 22.6255 & 910.7705 & 106.9167 & 550 & 93.4788 & 267.2918 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The solutions in the trough and the peak.  
Figure 17: The Gantt chart of solution 13.  
Figure 18: The layout scheme of solution 13.  
Convergence metric: Compares the dominance relations between the pareto solutions obtained by the two algorithms to evaluate their convergence quality. It is assumed that \(X\) and \(Y\) are two pareto solutions obtained by the two algorithms. The dominance rate of \(X\) to \(Y\) can be expressed by the following formulation:  
\[C(X,Y)=\frac{|\{y\in Y\},\ \exists x\in X:x\succ y\ or\ x=y|}{|Y|}. \tag{26}\]  
In Equation (26), \(|\,Y|\,\) is the number of solutions in the \(Y\) pareto solution set. The value \(C(X,Y)=1\), which means that all solutions in \(Y\) are dominated by or equal to solutions in \(X.\) On the contrary, if \(C(X,Y)=0,\) this represents the situation when none of the solutions in \(Y\) are covered by the set \(X.\) It is noted that both \(C(X,Y)\) and \(C(Y,X)\) must be considered, since \(C(X,Y)\) is not necessarily equal to \(1-C(Y,X).\)  
Spacing metric: Assumes that \(A\) is the Pareto solution set obtained by the algorithm. The formula for calculating the spacing metric is defined as follows.  
\[S(Z)\triangleq\sqrt{\frac{1}{|Z|-1}\sum_{i=1}^{|Z|}\left(\overline{d_{Z}}-d_{zi }\right)^{2}}; \tag{27}\]  
\[d_{zi}=\underset{j}{\min}\{\sum_{m=1}^{k}\big{|}f_{m}(z_{i})-f_{m}(z_{j})\big{|} \},\ (z_{i},z_{j}\in Z;i,j=1,2,\ldots,|Z|;i\
eq j). \tag{28}\]  
In Equations (27) and (28), \(\overline{d_{z}}\) is the mean of all \(dzi,\) and \(|\,Z|\,\) is the number of solutions in the \(Z\) pareto solution set. If the metric is equal to zero, all solutions among the current Pareto front are equidistant.  
Specifically, Figure 19 shows an example including two objectives to illustrate the above convergence and spacing metric. In the example, \(A\) and \(B\) are two pareto solutions obtained by different algorithms, in which \(A=[a_{1}(1,7),\)\(a_{2}(2,4),\)\(a_{3}(3,3),\)\(a_{4}(4,2),\)\(a_{4}(7,1)]\) and \(B=[b_{1}(2,6),\)\(b_{2}(3,5  
practice. It can be seen that the Pareto solution set obtained by IMHGA is better than that by MHGA in terms of convergence and distribution, which illustrates the efficiency of MIPOX, MMPX, and the parent-offspring competition strategy.