## 4 Results
### Multi-Objective Optimization  
Before starting the optimization process, choosing the best combination of the RPM and WOB is necessary. As demonstrated in the previous section, the graphical response for RPMs of 110 and 120 [rev/min] presents a considerable similarity to Figure 2. For this reason, these are the RPMs chosen for the further development of the multi-objective optimization in this study. The next necessary step is to correlate the response variables with the WOB through regressions. The objective is to obtain a multi-objective optimization that is able the combine the maximum ROP and the minimum MSE, concomitantly. The ROP minimum response will be based on Equation (3) (LTB), while the MSE will be minimized according to Equation (4) (STB), both presented in Section 3. A multi-objective optimization is carried out using Microsoft Excel software version 14,072,685,000 as a base, as shown in Figure 7.  
The generalized reduced gradient (GRG) approach is used to solve nonlinear optimization problems and depends on changing the values of the considered variables gradually while monitoring the governing conditions, until the partial derivatives of the target function are equal to zero [60]. The GRG nonlinear solver method was used in this analysis to solve the nonlinear model that combines the maximum ROP and the minimum MSE. To obtain the optimal points, the option \"use multiple starts\" was used. The following tables (Tables 1-7) show the results obtained from the multi-objective optimization for RPMs of 110 and 120 [rev/min] and WOB groups of 3, 5, and 7 [klb]. The tables are characterized by different values of weights for the response variables, allowing the operator to choose the best scenario for the current drilling process. Table 1 shows the results of the multi-objective optimization for the grouping of 3 [klb]. Upon comparing the values shown in scenario 1 of Table 1, it is noted that the maximum values reached for the ROP and MSE are 19.14 [ft/h] and 81.99 [kpsi] for an RPM of 110 [rev/min], and 21.14 [ft/h] and 158.30 [kpsi] for an RPM of 120 [rev/min]. Note that an increase in the ROP of 2.00 [ft/h] leads to an increase in the MSE of 76.31 [kpsi] and a decrease in the WOB of 5.78 [kpsi]. This improvement of 2.00 [ft/hr] in the ROP is considerably small for the large increase in the MSE used. Scenario 11 for an RPM of 120 [rev/min] shows good results. For this, an ROP of 20.10 [ft/h] is obtained when applying a WOB of 11.61 [klb]. By comparing these values with scenario 1 for an RPM of 110 [rev/min], it is possible to obtain a better ROP using half the WOB value; however, this leads to an MSE increase of 26.16 [kpsi]. Figure 6: WOB groups of 7 [klb]: (**a**) ROP vs. WOB plot; (**b**) MSE vs. WOB plot. Therefore, it can be concluded that the combination of an RPM of 120 [rev/min] and lower WOB values (when compared to an RPM of 110 [rev/min]) resulted in better ROP values; however, it caused high MSE values. And high MSE values are not desired, and one should seek the lowest achievable MSE. Table 2 shows the results of the multi-objective optimization for the grouping of 5 [klb]. Through Table 2, it can be seen that the WOB, ROP, and MSE value ranges for an RPM of 110 [rev/min] vary from 21.24 to 22.79 [klb], 21.37 to 21.53 [ft/h], and 63.95 to 69.22 [kpsi], respectively. These results show a good fit of the parameters. Again, a difference in WOB and MSE values is seen when the values presented for an RPM of 110 [rev/min] and for an RPM of 120 [rev/min] are compared.