## 3 Results
### Single and Multi-Objective Optimization  
Most of the research on this topic found in the literature and also in the industry proposes a single-objective optimization approach, where the goal is to achieve the highest ROP. However, searching for drilling mechanical parameters that simply provide the highest ROP can result in reduced process efficiency and ultimately excessive vibration, among other issues. In fact, parameters that allow drilling a phase in the shortest possible time must be sought, while maintaining an environmentally friendly and continuously safe operation. Attaching the applied analysis of MSE in it makes it possible to obtain drilling  
Figure 8: Surface plots: (**a**) MSE vs. WOB and RPM; (**b**) MSE vs. WOB and TOR; (**c**) MSE vs. WOB and FLOW; (**d**) MSE vs. RPM and TOR; (**e**) MSE vs. RPM and FLOW; (**f**) MSE vs. FLOW and TOR. parameters that help to optimize the process targeting the overall implicit efficiency. It is important to emphasize that, for an efficient drilling process, it is not only a question of maximizing ROP but the combination of the highest ROP together with the lowest MSE. Table 4 shows the single-objective optimization, using the larger-the-best (LTB) desirability function to obtain the highest ROP. As seen in Table 4, the input variable values (WOB, RPM, TOR, and FLOW) are obtained by the single-objective optimization. These values allow the highest ROP, i.e., response variable, to be obtained. Thus, to obtain the maximum ROP, 23.99 ft/hr, the input variables must be adjusted independently. However, it is important to note that TOR is a variable that depends on other mechanical parameters, for example, on WOB and RPM. Next, multi-objective optimization is performed to compare the results with a single-objective optimization (Table 4). The multi-objective optimization was used to combine the maximum ROP and the minimum MSE simultaneously. Table 5 shows the multi-objective optimization, using the LTB and smaller-the-best (STB) desirability functions, leading to obtaining concomitantly the highest ROP and the lowest MSE, respectively. The multi-objective optimization process is performed applying different values of weights for the response variables (ROP and MSE), allowing the operator to select the best scenario for the current drilling process. From Table 5, it can be seen that RPM and FLOW need to be kept at low and high values, respectively. The increase in WOB value from 11.75 to 13.88 klb caused an increase in ROP from 21.57 to 23.99 ft/h (desirable); however, this increase in WOB also caused a large increase in the MSE, going from 11.36 to 156.43 kpsi (undesirable). In addition, it is important note that this increase of 2.13 klb in WOB also caused the increase of 15.66 klb-ft in TOR. From this data, it can be verified that the increase in ROP as a standalone does not always reflect the most efficient scenario. The increase of 2.42 ft/h may not be practical due to the consequence implicit in it, i.e., increasing the MSE to 145.07 kpsi. The best scenario shown in Table 5 is scenario 8, where, using multi-objective optimization, it is \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Variables** & **Values** \\ \hline \multirow{3}{*}{Input Variables} & WOB klb & 14.49 \\  & RPM rev/min & 109.36 \\  & TOR klb-ft & 19.00 \\  & FLOW gpm & 803.32 \\ \hline Response Variable & ROP ft/h & 23.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Single-objective optimization.