## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & **Objective Function Term Weight** & & & & \\
**Case** & \(W_{1}\) & \(W_{2}\) & \(W_{3}\) & \(W_{4}\) & \(W_{5}\) & \(W_{6}\) & \(W_{7}\) & \(W_{8}\) & \(W_{9}\) & \(W_{10}\) \\ \hline
2 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 \\
3 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\
4 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
5 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
6 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
7 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\
8 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 \\
9 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
10 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Values of the weights of the objective function terms in Problem-4 in eleven different cases. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & **Objective Function Terms** & & & & \\
**Case** & \(Z_{1}\) & \(Z



1,017,100), (23,470 and 1,195,014), and (24,578 and 1,013,725), respectively. In shifting from Case-5 to Case-6, \(Z_{2}\) deteriorates by 4.7%, and \(Z_{3}\) improves by 15%. Thus, minimizing \(Z_{2}\) alone results in an unfavorable value of \(Z_{3}\) and vice versa. By adopting Case-1, \(Z_{2}\) deteriorates only by 0.9% from its value in Case-5, and \(Z_{3}\) deteriorates only by 0.33% from its value in Case-6. Thus, instead of minimizing the maximum or the total sublot flowtime alone, it is preferable to minimize both of them simultaneously. By examining Case-2, Case-7, and Case-8, we can also arrive at a similar conclusion regarding \(Z_{4}\) and \(Z_{5}\). In the literature, workload balancing in FJSP has been handled either by minimizing the workload of the most loaded machine (maximum workload \(Z_{8}\)) or by minimizing the total workload (\(Z_{9}\)). Accordingly, Case-9 minimizes \(Z_{8}\), and Case-10 minimizes \(Z_{9}\). However, in both cases, we can see that the difference between the workloads of the most loaded and the least loaded machines (maximum workload difference, \(Z_{10}\)) is significant compared to all the cases from Case-1 to Case-8 where \(Z_{10}\) is also minimized along with other objective function terms. Thus, for better workload balancing, it is desirable to minimize \(Z_{10}\) along with \(Z_{8}\) and \(Z_{9}\). The minimization of \(Z_{10}\) to improve workload balancing is reported for the first time in this paper. ### Performance Evaluation of RGA and 2 SGA

#### 5.2.1 Initial Solution Quality

[PERSON] et al. [44] and [PERSON] et al. [45] argued that the quality of the initial population is an important factor in determining the abilities of evolutionary algorithms to find acceptable solutions with minimal execution times. With this in mind, [PERSON] and [PERSON] [1] illustrated that one of the key factors for the success of their two-stage GA is its ability to find initial solutions with greatly improved makespan. In this paper, we further illustrate the ability of 2 SGA in finding an improved initial population not only with respect to the makespan but also with many other performance metrics of the multi-objective FJSP lot streaming presented in Section 3.2. Table 13 provides the means and the standard deviations of the objective functions \(Z_{1}\) to \(Z_{10}\) in the initial population of 2000 individuals in Problem-1 and Problem-4. From this table, it can be clearly seen that the mean and the standard deviation of values of the various objective functions in the initial population are greatly improved as we move from RGA to 2 SGA. For instance, the mean and standard deviation of the maximum sublot flowtime (\(Z_{2}\)) improve by 41% and 57%, respectively, in Problem-1 and by 47% and 79%, respectively, in Problem-4. The histogram for the weighted sum of all the objective function terms of the initial population is displayed in Figure 11. The histogram shows that 2 SGA results in highly improved initial solution quality in solving the proposed multi-objective FJSP lot streaming problem.