## 2 Materials and Methods
### Stage 2: Analysis of the Key Process, Encapsulation  
During the research stay, the process was characterized, and 10 activities were identified for the development of the final product. The focus was on analyzing the encapsulation process, as it was determined to be the key process affecting the production system, acting as a bottleneck. The process flow diagram was developed following the elements established [95]. Analysis of technology used in the process: The production process was identified, and each of its components was analyzed through interviews with staff and non-participant observation to understand the processes and equipment in detail [96]. Process simulation: For the development of the simulation, the methodology of [97] was used (Figure 2), using Arena Software (Simulation software version 16, Rockwell Automation, Milwaukee, WI, USA). Data collection: For data sampling, a record of encapsulation process times was made at different times during a workday, recorded in minutes. To collect these time data, visits were made over a period of 90 days, during which 50 samples were taken according to the different activities required for the encapsulation production process (Table 1). Figure 1: Location of the study site. Data Sampling: The sampling frame consisted of sampling units equivalent to the minutes and seconds of a day of production. The unit of observation for quality characteristics corresponded to the time at each workstation, according to the production process flow diagram. It was decided, in conjunction with the factory personnel, to take the times using systematic sampling [99] at 10 sampling points on the production line, at 30-min intervals, forming a pilot sample of 50 records, which correspond to 20% of the shift time. We began with the first sampling point randomly during one shift using the following procedure:  
1. The sampling frame was divided into segments, where \(n\) is the desired sample size. The size of these segments will be \(K=N/n\), where \(K\) is called the interval or elevation coefficient;
2. Starting number: A random integer, \(r\), less than or equal to the interval was obtained. This number corresponds to the first subject we will select for the sample within the first segment into which we have divided the population;
3. Selection of the remaining \(n-1\) individuals: The individuals were selected starting from the randomly selected individuals, using an arithmetic sequence, selecting individuals from the remaining segments into which the sample was divided, occupying the same position as the initial subject. This is equivalent to selecting individuals with the following equation: \[r,r+K,r+2K,r+3K,\ldots,r+(n-1)K\] (1) \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Sampling Technique: Random Sampling**} & \\  & **Instrument: Time Recording Table** & & \\ \hline Type of instrument & Time Sheet & Target Population & Units produced in the production process \\ Study Representativeness & Local & Sample Frame & Number of daily lots, 24 h \\ Number of Sampling Points & 10 & Sampling Method Unit & Systematic \\ Number of Data per Point & 50 & Data Collection Method & Digital Stopwatch \\ Total Number of Samples & 50 & Dates of Data Collection & Phase1 March (2022) \\ Sampling Design, & E.D.P., E.A.B.-T., R.A.M.-A, & & Phase 2 April (2022) \\ Measurement Instrument, & J.M.C.D., M.J.H.-R., M.E.G.-R & Confidence Level\% & 95\% \\ Capture, and Analysis. & and O.B.-S. & & \\ Field operation, supervision & E.D.P. & Margin of error \% & 5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Methodological record for time recording. Figure 2: Development of simulation models [97,98]. This method was chosen because it is commonly used in inspection and quality control due to its high speed and low variance [100]. The optimal sample size was determined to be 50 data points taken during a full day of work. Given the nature of the process, it was not feasible to select large samples during the process [101].