## 4 Parameter Tuning
### Parameter Tuning of PFT_NEX(x) Heuristic  
As mentioned before, since the number \(x\) of sequences generated are not so costly until \(n\leq 200\) in terms of CPU times, we determined it as \(if\ \left(n\leq 200\right)\ then\ x=n\) else \(x=20\). Then, PFT_NEX(x) heuristic has two important parameters. Namely, \(\mu\) and \(\delta\). To determine \(\mu\) and \(\delta\) parameters, we carry out a design of experiments (DOE) ([PERSON] and [PERSON], 2006). To do it, we generate random instances with the method proposed in ([PERSON] and [PERSON], 2006). 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 two parameters, namely, the weight of the cost function \(\mu\) and the size of the partial NEH heuristic \(\delta\). We have taken \(\mu\) with 21 levels as \(\mu=0.00,\ 0.05,\..,1.00\) and \(\delta\) with 10 levels as \(\delta=5\), 10, 15, 20, 25, 30, 35, 40, 45, 50. We conducted a full factorial design of experiments resulting in \(21\times 10=210\) treatments. For each job and machine combination, each instance is run for 210 treatments (210 PFT_NEX(x) heuristics with \(\mu\) and \(\delta\) values). The relative percent deviation is calculated as follows:  
\[RPD=\sum_{i=1}^{210}\left(\frac{C_{i}-C_{min}}{C_{min}}\right)*100. \tag{11}\]  
where \(C_{i}\) is the total flowtime generated by each treatment and \(C_{min}\) is the minimum total flowtime found amongst 210 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.  
Once the response variable was determined for each treatment, we analyze the main effects plot of parameters, which is given in Figure 9. Figure 9 suggests that \(\mu\) (parW) and \(\delta\) (parNEH) should be taken as 0.35 and 15, respectively.  
However, Figure 9 suggests that \(\mu\) values with 0.25, 0.30, 0.35, 0.40, 0.55 and 0.60 with \(\delta=15\) generate very good and diversified results on random benchmark instances. Table 1 summarizes the computational results of those \(\mu\) values as well as the CPU times in seconds. It can be seen in Table 1 that \(\mu=0.30\) and \(\mu=0.35\) provided the best results as Figure 9 suggested. However, \(\mu=0.25,\ 0.40,\ 0.55\ and\ 0.60\) provide some diversified solutions, too. Note that the PFT_NEH(x) heuristics are substantially better than the HPF2 heuristic presented in ([PERSON] and [PERSON], 2006) since it generated the ARPDs of 1.473 and 1.468, respectively when compared to the ARPD of 3.287. In addition, the same authors proposed GRASP-based algorithms as well as NHFP1 and NHFP2 heuristics for the initial solution procedures in [40]. PFT_NEH(x) heuristics are also substantially better than NHPF1 and NHPF2 heuristics since the ARPDs of 1.473 percent and 1.468 percent are much better than those ARPDs of 3.018 percent and 2.797 percent. In addition to above, PFT_NEH(x) heuristics were able to further improve some larger instances in about 0.760 s on overall average.