## 3 Results
For this reason, an important result is the identification of the function of approximating the cutting width in relation to the oxygen gas pressure. In this case, the dependent variable is the cutting width. The initial function depends on the set of \(\mathrm{x_{i}}\) points, according to Table 5. The values considered were: (4900 W; 0.45 bar; 1700 mm/min), (5000 W; 0.50 bar; 1800 mm/min) and (5100 W; 0.55 bar; 1900 mm/min). The Lagrange interpolation polynomial that corresponds to a functional relation f in nodes \(\mathrm{x_{0}}\), \(\mathrm{x_{1}}\), \(\mathrm{x_{2}}\) is given by the relation:  
\[\mathrm{L(X)}=\frac{\mathrm{x-x_{1}}}{\mathrm{x_{0}}\mathrm{-x_{1}}}\frac{ \mathrm{x-x_{2}}}{\mathrm{x_{0}}\mathrm{-x_{2}}}.\mathrm{f(x_{0})}+\frac{ \mathrm{x-x_{0}}}{\mathrm{x_{1}}\mathrm{-x_{0}}}\frac{\mathrm{x-x_{2}}}{ \mathrm{x_{1}}\mathrm{-x_{2}}}.\mathrm{f(x_{1})}+\frac{\mathrm{x-x_{0}}}{ \mathrm{x_{2}}\mathrm{-x_{0}}}\frac{\mathrm{x-x_{1}}}{\mathrm{x_{2}}\mathrm{- x_{1}}}.\mathrm{f(x_{2})}, \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cost Element** & **Calculation Relationship** & **Value** \\ \hline Depreciation of Bystronic Autonom & Average annual depreciation: 12,000 Euro/year & \\
4020 equipment & (12,000 Euro/365 days/8 h) & \\ \hline Maintenance & \begin{tabular}{c} 36/2000 \(\cdot\) 33.701 Euro/h \\ (Taking into account the indication of the manufacturer: 36 h \\ maintenance at 2000 operating hours) \\ \end{tabular} & 0.606 \\ \hline Helium & 0.02 m\({}^{3}\)/h \(\cdot\)40.1 euro/m\({}^{3}\) & 0.802 \\ \hline Azote & 0.008 m\({}^{3}\)/h \(\cdot\)5.95 euro/m\({}^{3}\) & 0.047 \\ \hline CO\({}_{2}\) & 1.21/h \(\cdot\)2.02 euro/1 & 2.424 \\ \hline Oxygen & 1.2 m\({}^{3}\)/h \(\cdot\)0.45 euro/m\({}^{3}\) & 0.540 \\ \hline Laspur 208 & 0.696 euro/1 \(\cdot\)7.51 /72 h & 0.072 \\ \hline Electricity consumed by the laser & \
\begin{tabular}{c} 21 kVA\(\cdot\)0.8-0.085 euro/kWh \\ 80\%—power factor (efficiency) \\ \end{tabular} & 1.428 \\ \hline Electricity of the cooling system & 36 kW/0.8-0.085 euro/kWh & 2.448 \\ \hline Energy consumed for moving the & 5.5 kW/0.8-0.085 euro/kWh & 0.374 \\ \hline cutting equipment & 1 kW/0.085 euro/kWh & 0.085 \\ \hline Nozzle NK1515 & consumption: 7 euro for 200 h of laser operating & 0.035 \\ \hline Lens 7.5 inch & estimated 400 euro for 2000 h of laser operating & 0.200 \\ \hline Air filter &
\begin{tabular}{c} it is replaced every 300 h of operation of the installation and has \\ an average price of 150 euro \\ \end{tabular} & 0.050 \\ \hline Materials for lubrication, & consumption related to the vacuum pump, turbocharger, cooling & 0.014 \\ maintenance and cleaning & circuit were estimated at 28 euros per 2000 operating hours & 13.245 \\ \hline Total & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Determination of the cost of the laser cutting machining process. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Independent** & \(\mathrm{x_{i}}\) & \(\mathrm{x_{0}}\) = 0.45 bar & \(\mathrm{x_{1}}\) = 0.50 bar & \(\mathrm{x_{2}}\) = 0.55 bar \\ \hline Value of \(f\) function & f(\(\mathrm{x_{i}}\)) & f(\(\mathrm{x_{0}}\)) = 0.40 mm & f(\(\mathrm{x_{1}}\)) = 0.30 mm & f(\(\mathrm{x_{2}}\)) = 0.35 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Interpolation points and value of the function f(x). The polynomial passes through the points (0.45; 0.40), (0.50; 0.30), (0.55; 0.35) being described by a parabola. The Lagrange interpolation polynomial is obtained from the basic Lagrange polynomials and the value of the function f at points x\({}_{i}\). The mathematical expression has the following form:  
\[ \begin{array}{l}\mathrm{L(x)}=0.15\cdot x^{2}-0.17\times+4,\\ \mathrm{f(x)}=x^{2}-x+0.24\leftrightarrow\mathrm{initial\ function,}\\ \mathrm{R(x)}=f(x)-\mathrm{L(x)}=0.85x^{2}-0.83x-3.76\leftrightarrow\mathrm{ error,}\end{array}  \tag{5}\]  
The interpolation approximation criterion presents the best possible approximation of the function f(x) by the interpolation polynomial L(x).