## 4 Case Study
### Case Information
It fluctuates around the optimal solution in a certain range and does not converge to a fixed value. This is because the problem is a combinatorial optimization problem, and the objective function is not convex. Because of the particularity of the problem, a heuristic algorithm is chosen to solve it. From the running time of the algorithm in Figure 6, it can be seen that the genetic algorithm can obtain the near-optimal solution in a relatively short time, which shows that the genetic algorithm has a better solving ability when the objective function is not convex. The optimal solution of operators assignment is shown in Table 14: \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline
**Staff Job** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline
1 & 0.0175 & 0.0156 & 0.0163 & 0.0173 & 0.0155 & 0.0161 & 0.0169 & 0.0179 & 0.0167 & 0.0176 & 0.0173 & 0.0179 \\
2 & 0.0181 & 0.0167 & 0.0172 & 0.0180 & 0.0167 & 0.0171 & 0.0177 & 0.0184 & 0.0175 & 0.0182 & 0.0184 & 0.0170 \\
3 & 0.0173 & 0.0153 & 0.0160 & 0.0171 & 0.0151 & 0.0158 & 0.0166 & 0.0177 & 0.0164 & 0.0174 & 0.0171 & 0.0177 \\
4 & 0.0173 & 0.0153 & 0.0160 & 0.0171 & 0.0151 & 0.0158 & 0.0166 & 0.0177 & 0.0164 & 0.0174 & 0.0171 & 0.0177 \\
5 & 0.0171 & 0.0149 & 0.0157 & 0.0169 & 0.0148 & 0.0155 & 0.0164 & 0.0176 & 0.0162 & 0.0172 & 0.0169 & 0.0176 \\
6 & 0.0180 & 0.0165 & 0.0170 & 0.0178 & 0.0164 & 0.0168 & 0.0175 & 0.0183 & 0.0173 & 0.0181 & 0.0178 & 0.0183 \\
7 & 0.0172 & 0.0150 & 0.0158 & 0.0169 & 0.0149 & 0.0155 & 0.0164 & 0.0176 & 0.0163 & 0.0173 & 0.0169 & 0.0176 \\
8 & 0.0176 & 0.0157 & 0.0164 & 0.0173 & 0.0156 & 0.0162 & 0.0169 & 0.0179 & 0.0168 & 0.0176 & 0.0173 & 0.0179 \\
9 & 0.0178 & 0.0161 & 0.0167 & 0.0176 & 0.0160 & 0.0165 & 0.0172 & 0.0181 & 0.0171 & 0.0179 & 0.0176 & 0.0181 \\
10 & 0.0179 & 0.0163 & 0.0169 & 0.0177 & 0.0162 & 0.0167 & 0.0174 & 0.0182 & 0.0172 & 0.0180 & 0.0177 & 0.0182 \\
11 & 0.0184 & 0.0172 & 0.0176 & 0.0182 & 0.0171 & 0.0175 & 0.0180 & 0.0186 & 0.0179 & 0.0184 & 0.0182 & 0.0186 \\
12 & 0.0176 & 0.0158 & 0.0164 & 0.0174 & 0.0157 & 0.0162 & 0.0170 & 0.0180 & 0.0168 & 0.0177 & 0.0174 & 0.0180 \\ \hline \end{tabular}
\end{table} 
Table 12: Downtime data of some the j-th operator in the i-th job. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline
**P J** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline
1 & 1.6294 & 1.8116 & 0.2540 & 1.8268 & 1.2647 & 0.1951 & 0.5570 & 1.0938 & 1.9150 & 1.9298 & 0.3152 & 1.9412 \\
2 & 1.5094 & 0.5521 & 1.3594 & 1.3102 & 0.3252 & 0.2380 & 0.9967 & 1.9195 & 0.6808 & 1.1705 & 0.4476 & 1.5025 \\
3 & 0.0238 & 0.6742 & 0.3244 & 1.5886 & 0.6224 & 1.0571 & 0.3313 & 1.2040 & 0.5259 & 1.3082 & 1.3784 & 1.4963 \\
4 & 0.2466 & 0.3678 & 0.4799 & 0.8345 & 0.0993 & 1.8054 & 1.8896 & 0.9817 & 0.9785 & 0.6754 & 1.8001 & 0.7385 \\
5 & 1.0170 & 1.0215 & 1.6353 & 1.5897 & 1.2886 & 0.7572 & 1.6232 & 1.0657 & 0.7015 & 1.8780 & 1.7519 & 1.1003 \\
6 & 1.4607 & 0.9772 & 1.1571 & 0.4745 & 0.9177 & 1.9262 & 1.0936 & 1.0423 & 0.4632 & 0.9778 & 1.2481 & 1.3583 \\
7 & 0.9798 & 0.3359 & 1.9574 & 1.4254 & 1.0009 & 0.9422 & 0.1192 & 1.3639 & 0.0849 & 0.1429 & 1.0433 & 0.1935 \\
8 & 1.4757 & 0.5382 & 0.8457 & 1.0957 & 1.8855 & 0.8355 & 1.9661 & 0.6029 & 1.4022 & 1.3327 & 1.0783 & 1.3962 \\
9 & 1.6355 & 0.5215 & 1.1887 & 0.0450 & 0.8505 & 0.6254 & 0.3230 & 0.3575 & 0.8458 & 0.1885 & 1.1970 & 0.9418 \\
10 & 0.4481 & 1.3357 & 1.6888 & 0.6889 & 1.5610 & 1.3507 & 0.0134 & 1.2043 & 0.7735 & 1.8320 & 0.0023 & 0.9249 \\
11 & 0.9175 & 1.3239 & 1.5406 & 0.7004 & 1.3240 & 0.8323 & 1.6839 & 1.6658 & 0.5129 & 1.2269 & 1.1645 &Figure 5: Evolution process of the solution. Figure 6: Running time graph.