## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
The best value of an objective function term is obtained when it is the only term optimized, and the worst is found when it is unaccounted for optimization. These significantly large deteriorations of unaccounted objective function terms in a large size problem greatly emphasize the need for multi-objective optimization in real industrial scheduling problems that are usually large in size. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Objective Term** & **Notation** & **Value** \\ \hline Makespan & \(Z_{1}\) & 2603.8 \\ \hline Maximum Sublot Flowtime & \(Z_{2}\) & 2487.5 \\ \hline Total Sublot Flowtime & \(Z_{3}\) & 16,560.6 \\ \hline Maximum Job flowtime & \(Z_{4}\) & 2487.5 \\ \hline Total job flowtime & \(Z_{5}\) & 9014.7 \\ \hline Maximum Sublot Finish-time Separation & \(Z_{6}\) & 1006.1 \\ \hline Total Sublot finish-time Separation & \(Z_{7}\) & 1787.1 \\ \hline Maximum Machine Load & \(Z_{8}\) & 2603.8 \\ \hline Total Machine Load & \(Z_{9}\) & 2603.8 \\ \hline Maximum Machine Load Difference & \(Z_{10}\) & 427.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Values of the objective function components. #### 5.1.3 Jointly Optimizing \(Z_{1},\dots,Z_{10}\)

In the previous section, the best and the worst values of the various objective function terms were determined when only one term was optimized at a time. In this section, we attempted to illustrate the ability of the proposed algorithm to jointly optimize all the terms and achieve values close to their best-known ones. In doing so, we first provide a plot of the values of the various objective function terms of Problem-4 in Figure 10a when makespan \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Term Optimized as a Single**} & \multicolumn{6}{c}{**Objective Function Term Evaluated**} \\ \cline{2-11} \multicolumn{1}{c}{**Objective Function**} & \multicolumn{1}{c}{\(Z_{1}\)} & \(Z_{2}\) & \(Z_{3}\) & \(Z_{4}\) & \(Z_{5}\) & \(Z_{6}\) & \(Z_{7}\) & \(Z_{8}\) & \(Z_{9}\) & \(Z_{9}\) \\ \hline Makespan & \(Z_{1}\) & 26,604 & 265.511 & 2,500,084 & 26,594 & 1,009,491 & 850 & 96,070 & 26,593 & 660,550 & 715 \\ Maximum Sublot Flowtime & \(Z_{2}\) & 28,756 & 24,014 & 2,495,091 & 28,367 & 1,016,441 & 9426 & 110,413 & 28,175 & 665,522 & 3063 \\ Total Sublot Flowtime & \(Z_{3}\) & 28,822 & 27,996 & 2,115,095 & 27,977 & 951,313 & 13,437 & 173,636 & 27,964 & 667,816 & 2747 \\ Maximum Job Flowtime & \(Z_{4}\) & 28,619 & 25,165 & 2,520,020 & 25,161 & 900,089 & 8029 & 71,921 & 28,155 & 663,042 & 2989 \\ Total Job Flowtime & \(Z_{5}\) & 27,316 & 26,770 & 2,312,532 & 26,846 & 868,391 & 6431 & 41,727 & 27,071 & 652,393 & 2169 \\ Maximum Sublot Finish-Time Separation & \(Z_{6}\) & 37,849 & 37,323 & 3,417,121 & 37,608 & 1,337,687 & 709 & 192,320 & 39,096 & 695,833 & 6275 \\ Total Sublot Finish-Time Separation & \(Z_{7}\) & 30,966 & 30,475 & 2,887,497 & 356,111,064 & 1788 & 10,813 & 29,629 & 609,055 & 4700 \\ Maximum Machine Load & \(Z_{8}\) & 44,197 & 43,469 & 4,040,945 & 43,658 & 1,580,503 & 14,786 & 158,042 & 25,793 & 643,747 & 227 \\ Total Machine Load & \(Z_{9}\) & 43,242 & 41,933 & 3,926,277 & 42,356 & 1,494,746 & 15611 & 76,124 & 28,123 & 573,164 & 10,040 \\ Maximum Machine Load Difference & \(Z_{10}\) & 36,943 & 36,288 & 3,314,121 & 36,671 & 1,348,060 & 15,639 & 161,671 & 27,729 & 693,034 & 15 \\ \hline Minimum (Best Value) & & 26,604 & 24,014 & 2115,095 & 25,161 & 868,391 & 709 & 10,813 & 25,793 & 573,164 & 15 \\ Maximum (Worst Value) & & 44,197 & 43,469 & 4,004,945 & 43,658 & 1,586,503 & 15,639 & 173,636 & 30,936 & 695,833 & 10,040 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Values of the objective function terms in Problem-4 when only one objective function term is optimized. Figure 9: Values of the objective function terms in Problem-1 when only one objective function term is optimized. is the only objective function optimized. In this figure, the values of the objective function terms are plotted on a scale between 0 and 1, corresponding to the best and worst values, respectively.