## 4 Computational Experiments and Statistical Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Factor**} & **Levels of Factor** & **N** & **Expected Median** & **Sum of Ranks** & **Overall Median** & **DF** & \(p\)**-Value** \\ \hline PDPT & Lgn & 25,920 & 132,275 & 39,378 & 132,234 & 1 & 0 \\  & Unf & 25,920 & 132,194 & 38,382 & & & \\ \hline CVPT & 0.3 & 17,280 & 129,225 & 27,574 & 131,165 & 2 & 0 \\  & 0.4 & 17,280 & 130,886 & 33,622 & & & \\  & 0.5 & 17,280 & 133,385 & 42,484 & & & \\ \hline PDST & Lgn & 25,920 & 132,331 & 39,465 & 132,301 & 1 & 0 \\  & Unf & 25,920 & 132,271 & 38,295 & & & \\ \hline CVST & 0.3 & 17,280 & 130,486 & 29,026 & 131,757 & 2 & 0 \\  & 0.4 & 17,280 & 131,744 & 34,544 & & & \\  & 0.5 & 17,280 & 133,040 & 40,110 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Friedman test for E[E/T] versus the factors PDPT, CVPT, PDST, and CVST. Table 12 shows the minimum difference in the MID and SNS between the lognormal and uniform distributions for the processing times, under the same coefficient of variation. Nevertheless, as the coefficient of variation of the processing times increases, independently of the probability distribution, the MID and SNS augment. That means that the quality of the Pareto frontier is worse as the coefficient of variation of the processing times increases. This suggests that production managers should encourage a continuous improvement of the production processes to reduce the variability of the process insofar as that is possible. Moreover, it is interesting to see that the NPS decreases as the coefficient of variation of the processing times increments, showing that high variability scenarios present fewer possible solutions to choose for the decision-maker. The measures presented in Table 13 show a large difference in the SNS between the lognormal and uniform distributions of the sequence-dependent setup times, under the same coefficient of variation of the setup times, which is a different behavior than that presented for the coefficient of variation of the processing times. Moreover, the NPS also presents high differences between the uniform and lognormal distributions of the setup times, whereas for the coefficient of variation of the processing times, the NPS values were very close. This allowed us to conclude that each input parameter can cause different performances of the solutions, and thus, the AHP selects the corresponding solution of the Pareto frontier to fulfill the decision-maker's expectations. \begin{table}
\begin{tabular}{l l r r r} \hline \hline
**Number of Jobs** & **Machines** & **NPS** & **MID** & **SNS** \\ \hline
20 & 5 & 182.59 & 18,412.20 & 915.68 \\  & 10 & 169.80 & 19,660.78 & 1386.57 \\  & 20 & 183.03 & 23,370.81 & 2142.31 \\ \hline
**Total 20** & & **178.47** & **20,481.26** & **1481.52** \\ \hline
50 & 5 & 262.71 & 123,904.5 & 2784.69 \\  & 10 & 254.54 & 138,695.12 & 2920.58 \\  & 20 & 247.28 & 153,472.48 & 3480.98 \\ \hline
**Total 50** & & **254.84** & **138,690.7** & **3062.08** \\ \hline
100 & 5 & 210.46 & 500,238.28 & 7475.30 \\  & 10 & 227.30 & 550,636.3 & 7165.95 \\  & 20 & 250.03 & 601,917.09 & 7166.19 \\ \hline
**Total 100** & & **229.26** & **550,930.55** & **7269.15** \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Performance results of the proposed approach. \begin{table}
\begin{tabular}{l l r r r} \hline \hline
**PDPT** & **CVPT** & **NPS** & **MID** & **SNS** \\ \hline lgn & 0.3 & 273.64 & 232,650.34 & 3884.93 \\  & 0.4 & 194.92 & 236,692.91 & 3928.33 \\  & 0.5 & 144.34 & 241,086.47 & 4045.65 \\ \hline
**Total lgn** & & **204.3** & **236,809.91** & **3952.97** \\ \hline unf & 0.3 & 297.71 & 232,730.09 & 3925.04 \\  & 0.4 & 225.32 & 236,299.54 & 3862.35 \\  & 0.5 & 189.23 & 240,745.7 & 3979.21 \\ \hline
**Total unf** & & **237.42** & **236,591.77** & **3922.20** \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Performance multi-objective metrics for the PDPT and CVPT.