## 3 Results
\[X_{5}=X_{1}X_{2} \tag{29}\]  
The coded form of the plan is presented in Table 5.  
For the first fractional plan, the function describing the response surface is a linear function and is calculated as:  
\[Z=6.814+0.644X_{1}-0.544X_{2}-1.040X_{3}+0.170X_{4}+0.137X_{5} \tag{30}\]  
The values of the coefficients indicate that the influence of the factors \(X_{1}\), \(X_{2}\) and \(X_{3}\) are significant; however, the influence of the factors \(X_{4}\) and \(X_{5}\) cannot be neglected. Table 6 presents the results of simulations compared with the results approximated by the obtained function. Comparing the results obtained from the numerical simulation and those approximated by the linear function, the differences between these results were found to be slight, with a maximum difference of 0.09 (approximately 2.05%). Therefore, it should be considered that the function of the object was calculated correctly. However, in order to be sure that the object can be considered to be linear (within the assumed range of parameters), one must perform the analysis for the two-level fractional plan in which one of the contrasts is equal to \(-1\). Therefore, for the second two-level fractional plan, the contrast for the fifth factor was changed to \(I=(-1)\) and the other values remained unchanged. Furthermore, for this plan, the calculation of values using a linear function with coefficients calculated for the first plan was carried out--the results are shown in the 'Approximation' column in Table 7. The function describing the response surface was calculated as:  
\[Z=6.811+0.637X_{1}-0.537X_{2}-1.037X_{3}+0.158X_{4}-0.224X_{5} \tag{31}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Maximum Recoiling Assembly Velocity (m/s)**} \\ \cline{2-3} \(i\) & **Simulation** & **Function** \\ \hline
1 & 7.67 & 7.72 \\
2 & 9.16 & 9.08 \\
3 & 6.61 & 6.70 \\
4 & 7.97 & 7.92 \\
5 & 6.03 & 5.98 \\
6 & 6.56 & 6.65 \\
7 & 4.37 & 4.28 \\
8 & 6.13 & 6.18 \\
**Mean** & 6.81 & 6.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The results of simulations compared to the results approximated by the obtained function— for the first plan. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**N = 5; n = 25\({}^{-2}\) = 8**} \\ \hline \(i\) & \(X_{1}\) & \(X_{2}\) & \(X_{3}\) & \(X_{4}\) = \(X_{1}X_{2}X_{3}\) & \(X_{5}\) = \(X_{1}X_{2}\) \\ \hline
2 & 1 & \(-1\) & \(-1\) & \(1\) & \(-1\) \\
3 & \(-1\) & \(1\) & \(-1\) & \(1\) & \(-1\) \\
4 & \(1\) & \(1\) & \(-1\) & \(-1\) & \(1\) \\
5 & \(-1\) & \(-1\) & \(1\) & \(1\) & \(1\) \\
6 & \(1\) & \(-1\) & \(1\) & \(-1\) & \(-1\) \\
7 & \(-1\) & \(1\) & \(1\) & \(-1\) & \(-1\) \\
8 & \(1\) & \(1\) & \(1\) & \(1\) & \(1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The first two-level fractional plan. Although the function gives a good estimation of the results from simulations, the maximum difference value is 0.09 (about 2.01%), the approximated values are significantly different. The recoiling assembly velocity values for the second plan, according to the coefficients determined for the first plan, compared to the values from the simulation show a discrepancy of 5.7%. It is also puzzling that the coefficient at \(X_{5}\) has changed its sign. Values of coefficients at other factors have not changed considerably. These findings may indicate that the influence of the parameters is not linear. The third plan is formed by the connection of the first and second two-level fractional plan and so the number of treatments is 16 (Table 8). For the third plan, the function describing the response surface was calculated as:  
\[Z=6.812+0.641X_{1}-0.540X_{2}-1.039X_{3}+0.164X_{4}-0.044X_{5} \tag{32}\]  
For the third plan, the maximum difference value is 0.24 (about 4.13%). The functions of the object obtained for all three plans are summarized in Table 9.