# A Guideline for Implementing a Robust Optimization of a Complex Multi-Stage Manufacturing Process
## 4 Statistical Modeling and Non-Destructive Testing Results
It must be noted that in the logit model, we included several discrete variables (Table 2): amount of blade-use at five levels (estimated with regard to #1); dicing-saw at three levels (estimated with regard to #2); workers employed at Stage 9 (four workers; estimates are performed with regard to #1); workers employed at Stage 11 (three workers; estimates are performed with regard to #1). The main relevant results show that workers and the time spent among stages are significant, or almost highly significant, while the rest of the variables play a relevant role (blade, dicing-saw, and resin). The statistical analysis provided enough suggestions, that implied the need to implement a design of experiments by considering the first five stages of the process, in which the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable Description** & **Symbol** & **Estimate** & **St. Error** & \(p\)**-Value (Wald’s Test)** \\ \hline Intercept & \(\theta_{0}\) & \(-11.50\) & 4.63 & 0.0131 \\ \#blade-usage-1 & \(\theta_{11}\) & 0.0 & & \\ \#blade-usage-2 & \(\theta_{12}\) & \(-0.39\) & 0.86 & 0.6525 \\ \#blade-usage-3 & \(\theta_{13}\) & 0.92 & 0.89 & 0.3061 \\ \#blade-usage-4 & \(\theta_{14}\) & 1.90 & 1.12 & 0.0907 \\ \#blade-usage-5 & \(\theta_{15}\) & \(-2.81\) & 2.08 & 0.1782 \\ Dicing-Saw-1 & \(\theta_{21}\) & 0.0 & & \\ Dicing-Saw-3 & \(\theta_{22}\) & 3.01 & 2.56 & 0.2401 \\ Dicing-Saw-4 & \(\theta_{23}\) & 2.79 & 2.59 & 0.2820 \\ Worker-St9-\#1 & \(\theta_{31}\) & 0.0 & & \\ Worker-St9-\#2 & \(\theta_{32}\) & 3.46 & 1.86 & 0.0631 \\ Worker-St9-\#3 & \(\theta_{33}\) & \(-2.66\) & 1.76 & 0.1302 \\ Worker-St9-\#4 & \(\theta_{34}\) & 4.10 & 1.86 & 0.0282 \\ Worker-St11-\#1 & \(\theta_{41}\) & 0.0 & & \\ Worker-St11-\#2 & \(\theta_{42}\) & 4.89 & 2.29 & 0.0325 \\ Worker-St11-\#3 & \(\theta_{43}\) & 4.81 & 2.28 & 0.0349 \\ \#days-St5-6 & \(\theta_{5}\) & \(-0.36\) & 0.21 & 0.0792 \\ \#days-St10-11 & \(\theta_{6}\) & 3.56 & 1.42 & 0.0125 \\ Epoxy resin-St3 & \(\theta_{7}\) & 1.23 & 0.97 & 0.1976 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Logit model results. Response Y = binary response variable; Formulas (4) and (6). mechanical dicing is considered the highest critical level of risk. The statisticians supported the engineers in the manuscript [70] and the statistical study led to implementing SAM analysis campaign able to detect potential latent failure by inspecting the US probe during the construction (Figure 5) at stage #3 (bonding piezoelectric element on backing). In this context, a method for the detection and the visualization of latent defects on US probes for medical imaging was carried out by means of SAM (Figure 6). Therefore, we decided to scan 12 workpieces by means of X-scan analysis consisting in multiple C-scan (Figure 6a) generating a stack of equidistant 2D-images (Figure 6b) by simulating the potential introduction of this NDT method in-line of the manufacturing process. Measurements were performed by means of a commercial acoustic microscope IS-350 manufactured by Insight kk, Tokyo, Japan [71]. Samples were aligned parallel to the (x-y) working plane. The exam direction, obtained through the application of SAM, was conducted by a focused ultrasound transducer that was aligned vertically to this plane along the z-axis, and it could be moved along this axis for focusing the sound energy either onto the sample surface or inside the sample volume. The workpiece and the SAM transducer were plunged into the coupling liquid, i.e., deionized water, whose temperature was regulated and stabilized to 24.0 \(\pm\) 0.5 \({}^{\circ}\)C. Ultrasound pulses with frequency of 30 MHz were generated and their reflection received by a 500 MHz bandwidth dual pulser-receiver system (DPR500, JSR Ultrasonics, Imaginant Inc., Pittsford, NY, USA). The SAM campaign on the 12 workpieces revealed that four of them were affected by internal cracks in the bulk of the piezoelectric element (blue arrows in Figure 7).