## 6 Flow Shop Scheduling--A Case Study
### Comparative Study: Pure GA vs. Improved-ART-1-Guided GA
#### 6.3.1 Small Prototype Problem  
Initially, a small prototype example of FFSP is considered for the MILP model. This prototype problem consists of eight jobs (_n_) that need to be processed in a total of four stages (_i_) using a maximum of two machines (_m_) in each stage. For instance, Tables 3-5 show the details of the small problem considered for FFSP. Table 3 shows that each stage is assigned two parallel machines, i.e., the work assignment can be carried out using one of the machines, depending on the calculated optimum processing time. We have also considered the machine release date \(F_{m,i}\) in the proposed prototype problem. Machine release date is important in the production environment [51], since the available machine's selection can be influenced by the release date. Table 4 illustrates the processing time required by the machines, where a total of eight jobs were considered with various batch sizes that need to be completed at different stages. For example, job 1 (_n_) is completed in three stages (i.e., stage 1, 2, 4, and skipped stage 3), which involves three operations (_o_) considering a batch size (_Qn_) of 20. Here, \(A_{n,i}\) indicates the machine's state as a setup property. If \(A_{n,i}\) is 1, then this setup is called attached or non-anticipatory. It also indicates that the machine setup will be determined once the job arrives at the machine. On the other hand, when \(A_{n,i}\) is 0, it is called detached or anticipatory, which indicates that the machine's setup must be completed before it arrives at the job on the machine. In the proposed case study, we cons  
Figure 8: Illustration of single point crossover: (**a**) Step-1, and (**b**) Step 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Stage (_i_)** & **Number of Parallel Machine (_M\({}_{i}\)_)** & **Machine (_m_)** & **Release Date of Machine _m_** \\  & & & **at Stage _i (_F\({}_{m,i}\)_)** \\ \hline
1 & 2 & 1 & 0 \\  & & 2 & 40 \\
2 & 2 & 1 & 40 \\  & & 2 & 80 \\
3 & 2 & 1 & 0 \\  & & 2 & 120 \\
4 & 2 & 1 & 80 \\  & & 2 & 240 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A small example of case study consists of two machines and four stages. detached setup conditions. The data for each eligible machine's sequence-dependent setup time for different jobs at various stages are presented in Table 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Job (\(n\))** & **Batch Size (\(Q_{n}\))** & **Operation (\(o\))** & **Stage (\(i\))** & **Setup Property (\(A_{n,i}\))** & **Processing Time on Eligible Machine, (\(m\)), (\(m\), \(T_{n,m,i}\))** \\ \hline
1 & 20 & 1 & 1 & 0 & (1, 6) \\  & & 2 & 2 & 1 & (1, 4)(2, 3.5) \\  & & 3 & 4 & 0 & (1, 6)(2, 6.5) \\
2 & 50 & 1 & 1 & 0 & (1, 4.5)(2, 4) \\  & & 2 & 2 & 1 & (1, 6.5)(2, 5.5) \\  & & 3 & 3 & 1 & (1, 6.5)(2, 6.5) \\
3 & 20 & 1 & 1 & 1 & (1, 3.5)(2, 4) \\  & & 2 & 2 & 0 & (1, 3)(2, 3) \\  & & 3 & 3 & 1 & (1, 3)(2, 4.5) \\  & & 4 & 4 & 1 & (2, 5.5) \\
4 & 20 & 1 & 1 & 1 & (1, 4)(2, 3) \\  & & 2 & 2 & 0 & (1, 4.5)(2, 4) \\
5 & 40 & 1 & 1 & 0 & (2, 3.5) \\  & & 2 & 2 & 1 & (1, 3)(2, 4.5) \\  & & 3 & 3 & 0 & (1, 3.5)(2, 4) \\
6 & 50 & 1 & 1 & 1 & (1, 4) \\  & & 2 & 2 & 1 & (1, 5)(2, 5.5) \\  & & 3 & 3 & 0 & (1, 5.5)(2, 6.5) \\
7 & 30 & 1 & 1 & 0 & (1, 5)(2, 6) \\  & & 2 & 2 & 1 & (1, 6.5)(2, 5.5) \\  & & 3 & 4 & 1 & (1, 3) \\
8 & 30 & 1 & 1 & 1 & (1, 6.5)(2, 6.5) \\  & & 2 & 2 & 0 & (1, 3.5)(2, 3.5) \\  & & 3 & 3 & 1 & (1, 5)(2, 5) \\  & & 4 & 4 & 0 & (1, 5)(2, 6.5) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The case study includes eight jobs with different batch sizes and processing times for each eligible machine.