## 4 Computational Experiments and Statistical Analysis  
For experimentation purposes, a set of 180 benchmark instances proposed by [PERSON] et al. (2018) were selected to evaluate the effects of different probability distributions (PDs) and coefficients of variation (CVs) of the processing and setup times in the objective functions. Specifically, the 180 instances were 10 instances for each combination of 20, 50, and 100 jobs with 5, 10, and 20 machines and a size of sequence-dependent setup times of 50% and 125%. Two PDs, lognormal, uniform, and three CVs, 0.3, 0.4, and 0.5, were selected to model both the stochastic processing and setup times. This corresponds to 6480 NDS archives. The proposed method was implemented in Java and run on an Intel Core i7-4770 with a 3.4 GHz processor and 8 GB of RAM. The stopping criterion for MC-SIM-GRASP was established as \(numberOf jobs\cdot numberOf Machines\cdot 1.0\) s. The best-qualified solution of each NDS archive was selected by using the AHP method. This results in a total of 51,840 solutions, each of which has an AHP score and a value for the four objective functions. Four ANOVAs were conducted to jointly analyze the effect of the eight factors on the four selected objective functions (E[E/T], SD[E/T], E[CI], and E[PI]). The factors selected for the experimental design are: probability distribution of processing times (PDPT), coefficient of variation of processing times (CVPT), probability distribution of setup times (PDST), coefficient of variation of setup times (CVST), the vectors of criteria weights of the AHP (WV), number of jobs, number of machines, and generation size of the standard deviation of sequence-dependent setup times (SDST). The factors and their levels are presented in Table 7. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **E[E/T]** & **SD(E/T)** & **CI** & **PI** & **Resulting Weight Vector** \\ \hline
**E[E/T]** & 1 & 1/3 & 3 & 5 & **25.56\%** \\
**SD(E/T)** & 3 & 1 & 5 & 9 & **57.64\%** \\
**CI** & 1/3 & 1/5 & 1 & 3 & **11.72\%** \\
**PI** & 1/5 & 1/9 & 1/3 & 1 & **5.07\%** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Example of AHP scores and the resultant priority vector. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Objective Function**} & \multirow{2}{*}{**1**} & \multirow{2}{*}{**2**} & \multicolumn{6}{c}{**Weights Vector**} \\  & & & & **4** & **5** & **6** & **7** & **8** \\ \hline
**E[E/T]** & 57.64\% & 25.56\% & 5.07\% & 5.07\% & 11.72\% & 11.72\% & 25.56\% & 57.64\% \\
**SD(E/T)** & 25.56\% & 57.64\% & 57.64\% & 25.56\% & 5.07\% & 5.07\% & 11.72\% & 11.72\% \\
**CI** & 11.72\% & 11.72\% & 25.56\% & 57.64\% & 57.64\% & 25.56\% & 5.07\% & 5.07\% \\
**PI** & 5.07\% & 5.07\% & 11.72\% & 11.72\% & 25.56\% & 57.64\% & 57.64\% & 25.56\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Criteria weights for Pareto solutions. The results showed that all main effects are statistically significant on the four objective functions except PDPT for E[CI] and E[PI] and the PDST and CVST for E[PI]. Nevertheless, some double and triple interactions that include the PDPT, PDST, and CVST have a significant effect on E[CI] and E[PI]. In fact, for at least one of the four objective functions, the double interaction effects are also significant (see Table 8). This shows that the WV discriminates among the Pareto solutions, helping the decision-maker select a solution from the Pareto frontier. We identified in addition that as the CVPT and CVST increase, the expected value of earliness (E[E/T]) and the standard deviation of earliness tardiness indicator (SD[E/T]) augment as well. The same occurs with E[CI], but not to the same degree. Additionally, the measures tend to be greater for the lognormal distribution than for the uniform distribution. This shows the importance of accurately fitting the PD to obtain adjusted robust measures. Figures 3 and 4 show the main effect plots of the factors WV, PDST, PDPT, CVST, and CVPT on E[E/T] and SD[E/T], respectively.