## 3 Preliminary Process Analysis
### Calculation of the Preliminary Process Capability  
Understanding the structure of the process and quantifying its performance are essential for improving it and successfully implementing and executing the Six Sigma model. Therefore, process capability analysis is an extremely important and well-defined tool of statistical process control, serving the function of continuous quality improvement [59]. Process capability is its ability to produce a product that meets specifications. A process will be capable when all products are produced within the given tolerances based on objective evidence regarding process performance [1]. In other words, process capability refers to a process that can produce a product within specific tolerance intervals for certain quality characteristics [59]. Process capability is defined as the range that encompasses all potential values of specified quality characteristics produced by the process under defined conditions [60; 61]. The Six Sigma methodology evaluates process performance by considering the shift (\(\sigma\)) within the process; however, assessing process capability through Process Capability Indices (PCIs) offers a more comprehensive understanding of the process. PCIs focus on key aspects such as process consistency, uniformity, and losses within the process [62; 63]. Preliminary process capability assessment is performed at the beginning of the process or after a relatively short period of process monitoring (with a sample size of at least 100 units). In this context, the term 'performance' is used instead of 'capability' in the index nomenclature, but the indices are calculated in the same way as process capability indices. Process capability, measured using the \(C_{p}\) index, relates to the variation in the process around the mean value. Therefore, the index measures potential capability, assuming that the process mean is equal to the midpoint of the specification limits and that the process operates under statistical control [1; 60]. If the process is centered within the tolerance limits, i.e., when the \(\bar{\mathfrak{x}}=(LSL+USL)/2\), the index is equal to:  
\[C_{p}=\frac{USL-LSL}{6\sigma} \tag{3}\]As the minimum acceptable value for this index, a value of 1.33 is taken, meaning that the width of the range of acceptable values or process tolerance should not exceed 75% of the USL-LSL span. Since the process is not centered within the tolerance limits but is closer to one of the limits, i.e., the mean is often not at the midpoint, the resulting measure is the \(C_{pk}\) index. \[C_{pk}=Min\left[\frac{USL-\bar{\pi}}{3\sigma},\frac{\bar{\pi}-LSL}{3\sigma}\right] \tag{4}\]  
It is also considered acceptable if its value is at least 1.33. For the purposes of this study, the following preliminary process capability indices were calculated (Table 4):  
\(\bullet\)_Pr_--potential process capability  
\(\bullet\)_Pr_--capability ratio  
\(\bullet\)_Ppk_--demonstrated performance  
\(\bullet\)_Ppl_--lower potential capability  
\(\bullet\)_Ppu_--upper potential capability  
\(\bullet\)_K_--non-centring correction factor  
The data required to calculate the _Pp_ values, including the sample means, are listed in Table 2 and were used for the calculations shown in Table 4.  
The value of the _Pp_ index should be greater than 1.33, but it is evident that the variables _K80_, _C100_, _M100_, and _Y100_ have a lower value than this threshold. Since the _Pr_ index is the reciprocal of the _Pp_ index, deviations in the same variables are expected, meaning the index value is greater than 1. The _Ppl_ and _Ppk_ indices, representing the lower and upper process capabilities, are also below the defined values for these variables. The _Pp_ and _Pr_ indices alone do not indicate how the process is positioned relative to the specification limits, which can be determined by comparing these two indices.