## 2 Materials and Methods
The DOE generates an orthogonal matrix that can be discussed and evaluated mathematically and theoretically based on data obtained for Kerf. The dimensions of the semi-finished product used in the experiment are 300 mm, 220 mm and 8 mm for length, width and thickness, respectively. The 5.181 kg plate was cut, as shown in Figure 1, to perform a complete factorial experiment, based on an experimental plan, to determine the optimal value of the input parameters that influence the cutting width. The factorial design consists of 27 tests, followed by four replicas. There were two semi-finished products containing 135 pieces. Test processing was done to determine the combinations of parameters that provide the least satisfactory results, especially that still ensure cutting. The extreme combinations were tested, and the average values of laser power, gas pressure and cutting speed were chosen. Following the experimental design, design 33 was built which offers all possible combinations of the values of the cutting parameters. The complete factorial plan allows the study of the common influence of the input parameters on the variation of the cutting width. The complete design of the independent test block was established with respect to the single central test point, maintaining a balance of increase and decrease in the values of the independent parameters, resulting in an orthogonal matrix. The piece with a contour consisting of three straight segments and a circular profile, was designed to be cut to the dimensions: length 20 mm, width 20 mm, with a semi-circular side, from a 20 mm side square, effective area 0.00036 m\({}^{2}\) and mass 0.02803 kg. The sketch of the resulting piece is shown in Figure 1.  
Figure 1: Cutting plan and dimensions of the semi-finished product. A complete factorial experiment was performed to determine the most suitable combinations that decrease the cutting width and improve the production cost. The ByAutonom 4020/6000 W laser cutting equipment, manufacturer: Bystronic Switzerland, was used to cut H400 sheets (Figure 2). The wavelength of the CO\({}_{2}\) laser beam is 10.6 \(\upmu\)m. The diameter of the laser beam D at the entrance to the lens is 20 mm, and the diameter of the spot is 0.2 mm. The conical nozzle through which the gas is directed has a diameter of 1.5 mm. Starting from the source, the laser beam reaches the lens with focal length of 190.5 mm, and is concentrated in the focus of the lens, where the incident intensity on the piece increases 103-104 times, compared to the value recorded before entering the lens, approximately 162.5 \(\times\) 105 W/cm\({}^{2}\). The focusing position of the laser spot was kept constant at +1 mm above the upper level of the blank, taking into account that the assistant gas is O\({}_{2}\). The research of the influencing factors on the dependent variable, the cutting width, was based on performing preliminary tests to identify the central point, then the cutting experiments were run, the selection of parameters being a condition to obtain correct results. After these tests, the reference values were established: laser power \(p\) = 5000 W, auxiliary gas pressure \(p\) = 0.50 bar and cutting speed v = 1800 mm/min. The factorial experiment has three influencing factors, and each factor has three levels. Compared to the central values, the experiment predicted the variation of the input parameters as follows: laser power \(\pm\) 100 W, auxiliary gas pressure \(\pm\) 0.05 bar, cutting speed \(\pm\) 100 mm/min, so that the levels for laser power reached the values of 4900 W, 5000 W and 5100 W; the auxiliary gas pressure was 0.45, 0.50 and 0.55 bar; the cutting speed was 1700, 1800 and 1900 mm/min. The process parameters of the laser installation are shown in Table 2.