## 2 Materials and Methods
### Experimental Procedure: Case Study  
A case study was conducted to exemplify the proposed compensation strategy. A feature of linear size (FoLS) [43] consisting of the external surface of a hollow right circular cylinder, with a target diameter of 40 mm. The part was designed in a parametric CAD software, considering a 30 mm internal diameter and a 40 mm part height. Test specimens were manufactured in a BCN3D SIGMA material extrusion equipment. This machine uses removable borosilicate glass trays with a 297 mm \(\times\) 210 mm area. A polylactic acid (PLA) was selected as the model material. This material was preferred in this case study to other thermoplastics like ABS or PA because of its lower health risk, given that the equipment has an open workspace and no particle filters. The configuration parameters used in this study are provided in Table 1.  
The layer thickness was selected after Gunay [35], whereas the 25% infill is a frequently used value [44; 45; 46]. Deposition speed and print temperatures have been adapted after the recommendations of the Ultimaker and the actual performance of the BCN3D. This process configuration delivered a stable operation through all the tests. Following the requirements of the machine performance analysis, this configuration was not modified in any way during the manufacturing operations reflected in the present work. The parts were orientated with the cylinder axis parallel to the \(Z\) coordinate of the cartesian reference system to avoid the use of support structures and prevent the increase of the volumetric error associated to slight deviations from the vertical [47]. Part location within the tray \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Configuration** \\ \hline Material & Ultimaker Though PLA Black (NFC) \\ Filament diameter & 2.85 mm \\ Print head temperature & 225 \({}^{\circ}\)C \\ Print bead temperature & 65 \({}^{\circ}\)C \\ Deposition speed & 50 mm/s \\ Nozzle diameter & 0.4 mm \\ Layer thickness & 0.200 mm \\ Wall thickness & 1.2 mm \\ Top/Bottom thickness & 1 mm \\ Infill pattern & Grid \\ Infill density & 25\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Processing configuration. was considered as a possible intrinsic factor that should to be analyzed under a multi-state perspective. Since an undetermined number of possible locations could be defined for the test part, it was decided that the machine performance would consider thirteen possible states (\(j=13\)), arranged according to Figure 3. The cartesian reference system used for part placement -according to the definition in ISO/ASTM 52921:2013 [48] with the origin in the geometrical center of the tray. The experimental procedure was divided in four steps. First, \(\hat{T}I_{0}\) was calculated under a multi-state perspective, considering four samples (\(n=4\)) per state and a total 52 samples (\(N=52\)). Each sample was obtained from a different tray. Second, \(\hat{T}I_{0}\) was directly calculated under the single-state perspective to illustrate the differences between both approaches using the same parts (52). Third, two additional trays, with 13 parts each, were manufactured to calculate the \(C_{j}\) and the design was optimized following the procedure in Figure 2. Fourth, an optimized set of 52 parts was manufactured and the resultant \(\hat{T}I_{2}\) was finally calculated. All the parts manufactured in this work were verified in a DEA Global Image 09-15-08 Coordinate Measurement Machine (CMM), calibrated according to ISO 10360-2:2009 [49].