## 1 Introduction
However, even though using strategies such as these in the case of this paper could be useful to an extent, their complexity and the nature of what is intended to be investigated in this research would not yield the expected results; thus, these methods would not be the correct option given that, as mentioned before, the authors are not looking for the best value of an infinite continuous population of levels allowing for achieving the best response, but instead they are strictly investigating specific levels of interest, and thus finding any other level value other than the studied ones may be irrelevant to what is intended to be demonstrated and investigated with this research. Even though the authors of this paper could continue to mention and review several other methods they found in the state of the art and the practice, and that could offer some value to this research and contribute to the goal of the paper, they generally believe that the well-known design of experiments (DOE) technique, and more specifically the factorial designs within it, is the method that best fits this research's needs. These factorial design techniques are very often used in experiments consisting of several factors where it is necessary to analyze the joint effect of the factors on a given response, and this is precisely the case of this research. As for the use of statistical methods concerning DOE and factorial designs in the specific area of the cold tube drawing process, also known as free tube sinking, or simply tube drawing, there has not been a significant number of publications in recent years. For example, in the case of [22], the authors focused on the tube sinking process using a conical die for the case of tubes made of polymeric materials. In their paper, they implemented a finite element study and designed an experiment where they studied the influence of factors such as the drawing speed, tube wall thickness, die/tube friction, die reduction, and die semi-cone angle on only a single response variable, namely, the drawing loads. Similarly, the authors of [23] also presented a somewhat similar and common take on the analysis of the cold tube drawing process by first conducting some mathematical simulations and subsequently using the resulting data for the experimental analysis. These authors mainly focused on studying and searching for the best geometries of the die and plug in order to reduce the drawing force, and although they claimed to have presented a random factorial analysis, they did not offer much proof of it or properly explain the randomness of the factors' levels, or the scientific, mathematical/statistical, and experimental base behind their conclusions and results of the experimental analysis. On the other hand, the authors of [24] presented a three-dimensional finite element model which was developed to help them calculate the change in the wall thickness, eccentricity, ovality, and residual macro-stress state of the tubes, produced by the same cold drawing process, in the presence or absence of a plug. These authors also carried out an experiment in order to validate their simulation results; however, they did not carry it out in the way it is conceived within this paper given that they missed or did not need a proper design for the experiment itself. As mentioned in [20], out of the many special cases of factorial designs, the most important factorial design is that having k factors, each at only two levels, i.e., the 2\({}^{\text{k}}\) type of experiment. These levels may be quantitative, such as two values of temperature, pressure, or time, or they may be qualitative, such as two machines, two operators, the \"open\" and \"close\" levels of a factor, or perhaps the presence and absence of a factor. In the case of this paper, these factors refer to: (1) the wall thickness of the pre-tube, (2) the diameter of the pre-tubes (also known as draft tubes), and, finally, (3) the diameter of the die.