## 4 Parameter Tuning
### Parameter Tuning of VBIH Algorithm  
In this section, we again present a DOE approach for parameter setting of the VBIH algorithms. In order to carry out experiments, we generate random instances with the method proposed in [20]. In other words, random instances are generated for each combination of \(n\in\{20,50,100,200,500\}\) and \(m\in\{5,10,20\}\). Five instances are generated for each job and machine combination, respectively. Ultimately, we obtained 75 instances for each pair of job and machine size. We consider three parameters in the DOE approach. These are maximum block size (\(bS_{max}\)), maximum block insertion move size (\(mS_{max}\)); and temperature adjustment parameter (\(\tau P\)). We have taken the maximum block size with four levels as \(bS_{max}\in(4,8,12,16)\); the maximum block insertion move size with three levels as \(mS_{max}\in(0.1*(n-bS_{max})\,,\,0.2*(n-bS_{max})\,,\,0.3*(n-bS_{max}))\,;\) and the temperature adjustment parameter with five levels as \(\tau P\in(0.1,\,0.2,\,0.3,\,0.4,\,0.5)\). We conducted a full factorial design of experiments resulting in \(4\times 3\times 5=60\) treatments. For each job and machine combination, each instance is run for 60 treatments with a maximum CPU time equal to \(T_{max}=10\times n\times m\) milliseconds. The relative percent deviation is calculated as follows:  
\[RPD=\sum_{i=1}^{60}\left(\frac{C_{i}-C_{min}}{C_{min}}\right)*100. \tag{12}\]  
where \(C_{i}\) is the total flowtime generated by each treatment and \(C_{min}\) is the minimum total flowtime found amongst 60 treatments. This is repeated for five instances and RPD values are averaged for each treatment. Then the response variable is obtained by averaging the RPD values of 15 different job and machine combinations for each treatment. The DOE is coded in Visual C++13 and carried out on an Intel (R) Core (TM) i7-2600 CPU with 3.40 GHz PC with 8.00 GB memory.  
Figure 9: Main effects plot.  
Once the response variable was determined for each treatment, we analyze the main effects plots of the parameters, which are given in Figure 10. Figure 10 suggests that the \(bS_{max}\) should be taken as \(bS_{max}=16\); \(mS_{max}\) should be taken as \(mS_{max}=0.1*(n-bS_{max})\); and \(\tau P\) should be taken as \(\tau P=0.2\).  
The main effects plot might not be meaningful when there are significant interactions between the parameters. For this reason, the ANOVA table should be analyzed to see whether interactions are significant or not. The ANOVA results are given in Table 2. From Table 2, it can be seen that \(bS_{max}+mS_{max}\) interaction is found to be significant because of the very high magnitude of \(F\) ratio and the \(p\)-value being less than \(\alpha=0.05\) level. For these reasons, we look at the \(bS_{max}*mS_{max}\) interaction plot, which is given in Figure 11.  
As seen in Figure 11, \(bS_{max}=16\) with \(mS_{max}=0.3*(n-bS_{max})\) generated the lowest ARPD. For this reason, we decided to take the parameters as follows: \(bS_{max}=16\); \(mS_{max}=0.3*(n-bS_{max})\); and \(\tau P=0.2\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline _Source_ & _DF_ & _Seq SS_ & _Adj SS_ & _Adj MS_ & \(F\) & \(p\) \\ \hline \(bS_{max}\) & 3 & 0.095370 & 0.095370 & 0.031790 & 98.55 & 0.00 \\ \(mS_{max}\) & 2 & 0.019873 & 0.019873 & 0.009937 & 30.80 & 0.00 \\ \(tP\) & 4 & 0.000676 & 0.000676 & 0.000169 & 0.52 & 0.72 \\ \(bS_{max}*mS_{max}\) & 6 & 0.039930 & 0.039930 & 0.006655 & 20.63 & 0.00 \\ \(bS_{max}*tP\) & 12 & 0.002395 & 0.002395 & 0.000200 & 0.62 & 0.81 \\ \(mS_{max}*tP\) & 8 & 0.001585 & 0.001585 & 0.000198 & 0.61 & 0.76 \\ \(Error\) & 24 & 0.007742 & 0.007742 & 0.000323 & - & - \\ \(Total\) & 59 & 0.167569 & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: ANOVA table.  
Figure 10: Main Effects Plots of Parameters.