## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(m\) & \(r\) & \(j\) & \(s\) & \(o\) & SB & SE/PB & PE \\ \hline
1 & 1 & 3 & 1 & 1 & 840.0 & 920.0 & 1342.0 \\  & 2 & 3 & 1 & 2 & 1342.0 & 1422.0 & 1783.8 \\  & 3 & 3 & 1 & 3 & 1783.8 & 1883.8 & 2165.2 \\  & 4 & 3 & 3 & 3 & 2165.2 & 2175.2 & 2456.5 \\  & 5 & 3 & 2 & 3 & 2456.5 & 2466.5 & 2603.8 \\ \hline
2 & 1 & 2 & 3 & 1 & 0.0 & 120.0 & 303.0 \\  & 2 & 2 & 1 & 1 & 303.0 & 313.0 & 494.6 \\  & 3 & 2 & 2 & 1 & 494.6 & 504.6 & 640.0 \\  & 4 & 2 & 2 & 2 & 640.0 & 720.0 & 1058.5 \\  & 5 & 2 & 2 & 3 & 1098.5 & 1178.5 & 1652.5 \\  & 6 & 2 & 3 & 3 & 1652.5 & 1667.5 & 2308.1 \\  & 7 & 2 & 2 & 4 & 2308.1 & 2428.1 & 2597.4 \\ \hline
3 & 1 & 2 & 3 & 2 & 303.0 & 403.0 & 792.0 \\  & 2 & 2 & 1 & 2 & 792.0 & 812.0 & 1197.8 \\  & 3 & 2 & 1 & 3 & 1237.8 & 1317.8 & 1907.8 \\  & 4 & 2 & 1 & 4 & 1907.8 & 2007.8 & 2257.4 \\  & 5 & 2 & 3 & 4 & 2338.1 & 2348.1 & 2599.8 \\ \hline
4 & 1 & 4 & 2 & 1 & 120.0 & 220.0 & 520.0 \\  & 2 & 4 & 1 & 1 & 520.0 & 530.0 & 830.0 \\  & 3 & 4 & 1 & 2 & 830.0 & 930.0 & 1167.5 \\  & 4 & 4 & 2 & 2 & 1167.5 & 1177.5 & 1415.0 \\  & 5 & 4 & 2 & 3 & 1415.0 & 1515.0 & 1577.5 \\  & 6 & 1 & 1 & 2 & 1577.5 & 1817.5 & 2092.5 \\  & 7 & 1 & 1 & 3 & 2112.5 & 2212.5 & 2587.5 \\ \hline
5 & 1 & 1 & 1 & 1 & 0.0 & 100.0 & 750.0 \\  & 2 & 3 & 2 & 1 & 750.0 & 960.0 & 1185.5 \\  & 3 & 3 & 3 & 1 & 1185.5 & 1200.5 & 1662.8 \\  & 4 & 3 & 3 & 2 & 1662.8 & 1762.8 & 2104.4 \\  & 5 & 3 & 2 & 2 & 2104.4 & 2114.4 & 2281.1 \\  & 6 & 4 & 1 & 3 & 2281.1 & 2521.1 & 2583.6 \\ \hline \hline \end{tabular} SB = Setup Begins; SE = Setup Ends; PB = Processing Begins; and PE = Processing Ends.

\end{table} 
Table 7: Operation scheduling for Problem-1 with reference to the machines. \begin{table}
\begin{tabular}{c c c} \hline \hline \(m\) & **Workload** & **Utilization** \\ \hline
1 & 2603.8 & 100.0 \\
2 & 2557.4 & 98.2 \\
3 & 2176.1 & 83.6 \\
4 & 2567.5 & 98.6 \\
5 & 2583.6 & 99.2 \\ \hline \hline \end{tabular} Maximum workload = 2603.8; Total workload = 12488.4; and Maximum workload difference = 427.7.

\end{table} 
Table 8: Machine workload related performance (\(\hat{l}_{max}\) and \(\hat{l}_{max}-\hat{l}_{min}\)). However, when another term alone is optimized, makespan greatly deteriorates. For instance, when only \(Z_{2}\) alone is optimized, the value of the makespan increases to 5067 (94% increase). Figures (b)b is a plot of the maximum sublot flowtime (\(Z_{2}\)) when \(Z_{1}\) through \(Z_{10}\) are optimized one at a time. Its minimum value is 1468 when \(Z_{2}\) alone is optimized. This value increases to 2431 when \(Z_{1}\) alone is optimized. A single objective optimization of \(Z_{6}\) through \(Z_{10}\) has significantly negative impacts on \(Z_{2}\). A similar phenomenon is observed on all the other objective function terms, as can be seen from Figures (c)c-j. Here, it is important to note that the magnitude of the severity of a single objective optimization on the objective function terms that are not incorporated increases as the problem size increases. To exemplify this fact, we conducted a similar analysis on a relatively large problem (Problem-4), and the result is compiled in Table 10. For instance, when the total machine load (\(Z_{9}\)) was the only objective function, its value was 573,164 min (see at row-\(Z_{9}\) column-\(Z_{9}\)), which increases by 84,386 min when makespan (\(Z_{1}\)) is the only objective function optimized (see at row-\(Z_{1}\) column-\(Z_{9}\)). This increment was only 721 min in a similar analysis in Problem-1. The total sublot flow time in Problem-4 was equal to 2,115,095 when it was the only objective function (see row-\(Z_{3}\) column-\(Z_{3}\)). This value increases to 4,004,945 when minimizing the maximum machine load is the only objective (see row-\(Z_{8}\) column \(Z_{3}\)). The last two rows of Table 10 show the best and the worst observed values of each objective function term. From these rows, we can see a considerably large gap between the best and the worst values of an objective function term.