## 3 Results and Discussion
### Multi-Objective Optimization  
Obtaining high reliability of joints made by the FSW method requires the selection of optimal parameters of the slide diamond burnishing. Proper selection of parameters should ensure the possibility of introducing high compressive stresses on the joint surface, hardening of the joint, and reduction of surface roughness to remove fatigue micro-notches that may reduce the fatigue strength of the joint. This selection must guarantee, apart from the required parameters of the surface layer, also high stability and efficiency of the process. This requires multi-criteria optimization and finding a compromise solution that meets the presented conditions. The following notations were adopted for the presented multi-criteria problem:  
* \(E\subset R^{m}\)--a set of feasible solutions--the range of the SDB process parameters. * \(x=(x_{1},x_{2},\ldots,x_{m})\in E\)--an acceptable solution. * \(f_{i}\):\(E\)\(\rightarrow\)\(R\)--\(i\)-th objective function (\(i=1,2,\ldots,k\)). * \((x)=(f_{1}(x),f_{2}(x),\ldots,f_{7}(x)\)--objective function of a multi-criteria problem. The problem of the multi-criteria optimization of the selection of the slide diamond burnishing process parameters can be written in the form (11): \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **R\({}^{2}\)** & **Intercept** & \(f\) & \(P\) & \(fP\) & \(P^{2}\) \\ \hline \(H_{V}\) & 0.9293 & 0.0245 & 0.8187 & 0.0068 * & 0.6635 & **0.0062 * \\ \hline \end{tabular} Legend: * significance important.  
\end{table} 
Table 10: R\({}^{2}\) value and \(p\)-value for individual structural elements of the regression model for \(H_{V}\). Figure 12: Plot of the regression functions of microhardness. \[\left\{ \begin{array}{l}f_{1}(x)=\sigma_{x}\to min,\\ f_{2}(x)=\sigma_{y}\to min\\ f_{3}(x)=t\to min\\ f_{4}(x)=S_{a}\to min\\ f_{5}(x)=S_{ku}\to min\\ f_{6}(x)=S_{pc}\to min\\ f_{7}(x)=H_{v}\to max\\ x\in E\end{array} \right. \tag{11}\]  
In addition, the analysis took into account the process execution time \(t\), which was determined as the quotient of the path traveled by the tool and the feed rate. For the feed rate of \(f=500\) mm/min, the specimen processing time was 2 min, while for the feed rate of \(f=1500\) mm/min 0.66 min. A one-criteria problem written as (12) (in Table 11 was presented results):  
\[f_{i}(x)\to ekstremum,\ z\in E \tag{12}\]  
is the \(i\)-th partial issue, while the vector \(x^{io}\in E\) in which the \(i\)-th objective functionachieves the extremum is the \(i\)-th partial solution. Vector (13)  
\[\varphi^{o}=(f_{1}(x^{1o}),f_{2}(x^{2o}),\ldots,f_{2}(x^{7o})) \tag{13}\]  
is a vector called the ideal (utopian) solution in the evaluation space, whereas:  
\[x^{o}=x^{1o},x^{2o},\ldots,x^{7o} \tag{14}\]  
is the solution to the ideal problem. The set of efficient solutions usually contains many solutions. Therefore, the purpose of the presented problem was to select one solution from the set of effective solutions, called a compromise solution (compromise-optimal). For this purpose, issues (13) were reduced to a single-criterion form, with a scalarizing function \(s:R^{k}\to R\) in the form of (15):  
\[\min(s(f_{1}(x),f_{2}(x),\ldots,f_{7}(x)):z\in E) \tag{15}\]  
The scalarization of the issue was carried out using the evaluation weighting method. The weight values \(u_{i}>0\) individual \(f_{i}\) criteria were adopted (meeting the condition \(u1+u2+u3+u4=1\)), and then the optimal solution of the problem was determined (16):  
\[\min(\sum_{i=1}^{k}u_{i}f_{i}(x):x\in E) \tag{16}\]  
Creating the function \(\varphi(x)=\sum_{i=1}^{k}u_{i}f_{i}(x)\) is possible only when all objective functions are expressed in the same units and scales of values. Since, in the considered case, the objective functions were expressed in different scales of values, they were transformed into