## 3 Materials and Methods
### Experimental Design
#### 3.3.1 Data Collection  
Two distinct sets of data were used to evaluate the performance of the proposed model: synthetically generated data and data from a real-world case study. The synthetic data were generated to assess the model's ability to perform across different configurations. The real-world data, drawn from a case study in the automotive sector, provided instead a practical application context for validating the model's effectiveness in a real-world setting.  
Different cardinalities of the sets described in Section 3.2 were defined for the synthetic data. Table 2 reports the range of cardinality examined for each set.  
For the real-world case study, data were collected from an automotive company involved in quality control operations. The cardinalities of the sets in this case study are shown in Table 3. Specifically, the set H related to the possible inspection technologies was composed in the case study of three alternatives: human inspections, human inspection assisted with digital tools, and machine vision systems.  
The values of the parameters \(q_{s,p,h,k}^{time}\) and \(det_{s,p,h,k}^{prob}\) used in the quality control case study are instead reported in Figure 2a,b. Each panel is arranged as a grid of heatmaps, where each subplot corresponds to a specific product and technology combination. In every heatmap, the horizontal axis represents the inspection stations, while the vertical axis corresponds to the charac  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Sets** & **Cardinality** \\ \hline P & [1, 5] \\ S & [10, 20, 30, 40,5 0] \\ K & [10, 50, 100, 200] \\ T & [1, 2, 3, 4, 5] \\ \hline \hline \end{tabular}
\end{table}
Table 2: The cardinality of the different sets adopted to generate synthetic data.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Sets** & **Min Cardinality** \\ \hline P & 3 \\ S & 24 \\ H & 3 \\ K & 96 \\ T & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Cardinality of the different sets in the real-world case study.  
Figure 2a displays the inspection time (\(q_{s,p,h,k}^{time}\)) required for each station-characteristic pair. The color intensity suggests that inspections at later stations are typically more time-consuming due to physical layout constraints, workflow accumulation, and technological limitations.  
Figure 2b presents the probability of detecting a defect for each combination of station and characteristic. In contrast to the inspection time patterns, detection probabilities generally decrease with higher station indices. This implies that earlier stations tend to be more effective at identifying defects due to better visibility and less interference from downstream processing. Moreover, several station-characteristic combinations show zero probability and a very high inspection time, indicating that inspections are not feasible at those positions. These infeasibilities reflect predefined structural limitations of the production system.  
A similar pattern to that reported in Figure 2b can also be noticed in Figure 3a,b, representing, respectively, the evolution of the parameter \(C_{s,p,k}^{rep}\) and \(rep_{s,p,k}^{time}\). According to the chart, also in this case, the cost and the time to repair a defect for specific characteristics increase as products progress over the stations.  
Lastly, Figures 4 and 5 report, respectively, the evolution of the parameter \(def_{p,k}^{prob}\), and \(D_{p,t}\) in the examined case study.  
Figure 3: (**a**) Variation in repair cost based on product, station, and characteristic. (**b**) Variation in repair time based on product, station, and characteristic.  
Figure 2: (**a**) Variation in detection probability as a function of technology, product, station, and characteristic; (**b**) variation in inspection time as a function of technology, product, station, and characteristic.