## 2 Materials and Methods  
### Machine Performance Studies  
Machine performance studies are short-term analyses, typically, used to detect and evaluate the influence of the machine on the manufacturing process without including other possible influences related to the operator, the processing conditions, the material, or the environment. The term \"machine performance\" is used instead of \"machine capability\" in accordance with ISO 22514-3 [34] because this type of analysis does not require the product characteristic to be in control. The main objectives of a performance study are to identify the probability distribution of a product characteristic, to estimate the corresponding distribution parameters, and to determine if the results are acceptable according to a quality specification. The specification for a dimensional characteristic is defined by means of an upper specification limit (\(U\)), corresponding to the highest acceptable value and a lower specification limit (\(L\)), corresponding to the lowest acceptable value. Accordingly, the \(Tl\) is the range between \(U\) and \(L\), whereas the target value (\(T\)) is the preferred or reference value for the characteristic. In a machine performance study, a series of parts are manufactured and measured. Each characteristic (\(X\)) is considered independently at a time, and the measured values for every \(i\) part (\(X_{i}\)) are used to determine the distribution model that better fits the results of the product characteristic. Once the statistical distribution has been determined, a reference interval is chosen for the expected result's characteristic, following the common industrial convention of having a 99.73% of the measures within limits [41]. This interval corresponds to a to a \(\pm 3\sigma\) range for a normal distribution of  
Figure 1: Basic steps of the proposed quality improvement methodology. values, with mean \(\mu\) (\(X_{50\%}\)) and standard deviation \(\sigma\). Such range would be bounded by the 99.865% distribution quantile (\(X_{99.865\%}\)) and the 0.135% distribution quantile (\(X_{0.135\%}\)), that is: between \(\mu-3\sigma\) and \(\mu+3\sigma\). The machine performance index (\(P_{m}\)) evaluates the ratio between the \(Tl\) and the reference interval (Equation (1)). \[P_{m}\ =\ \frac{U-L}{X_{99.865\%}-X_{0.135\%}} \tag{1}\]  
Nevertheless, \(P_{m}\) does not take into account if the process is centred or not, since it only compares the width of the intervals. Accordingly, \(P_{m}\) provides an estimation of the maximum theoretical performance that would be obtained if the mean value for the product characteristic matches perfectly the target value. Consequently, this index is sometimes referred to as the \"potential\" performance index. Situations where the mean value of the obtained distribution model and the target value are significantly different can be better described through the upper (\(P_{mkl}\)) (Equation (2)) and the lower (\(P_{mkl}\)) (Equation (3)) machine performance indexes. \[P_{mkl}\ =\ \frac{U-X_{50\%}}{X_{99.865\%}-X_{50\%}} \tag{2}\]  
\[P_{mkl}\ =\ \frac{X_{50\%}-L}{X_{50\%}-X_{0.135\%}} \tag{3}\]  
Once both indexes have been obtained, the critical (minimum) machine performance index (\(P_{mkl}\)) can be calculated as the minimum of \(P_{mkl}\) and \(P_{mkl}\). Although this is the usual approach regarding the application of capability studies in AM processes, there are some circumstances to be considered. Firstly, in mass production, several identical parts would be manufactured in the same tray depending on the size of the part, on the batch size, and on the working volume of the machine. If the position of each part within the manufacturing tray has an influence upon the measured values of the characteristic, this could suppose a violation of the repeatability conditions. This circumstance is related in [34] as a possibility in multiple fixture set-up, multiple-cavity, or multi-stream processes. In those cases, the reference standard to be applied would be the ISO 22514-8 [40].