## 4 Performance Evaluation and Results  
### Testing Environment Evaluation Criteria  
The PPS algorithm was tested using two-week real process data provided by a precision ball-bearing manufacturer. The raw data contained metadata including the timestamp of the received signal, the dimensions of the inner and outer rings, and the tolerance measurements. Additionally, the data contained information about the dimensions of the components within the slots at the time of observation. The algorithm for testing was written using Python 3.10. The dimensions of the parts put into the assembly machine recorded in the actual process time were assumed to be the input values of the function, and the method of calculating the combination of inner ring, outer ring, and ball was implemented and tested according to the methodology of PPS after fixing other conditions. To validate the PPS algorithm, we compared the reconstituted ordinary process algorithm with two cases of the PPS algorithm using stride. Testing the algorithm in actual process situations was not acceptable due to the ball-bearing manufacturer's circumstances, so we replaced the experiment by simulating it in a situation where the process environment was reproduced. The similarity of process environment reproduction was confirmed by the manufacturer.  
The operating principles of the ordinary process and PPS algorithms were omitted because they have been reported in previous studies. The results showed that the outer-inner slot was 106 and the ball was classified into three types of barrels. The evaluation criteria included computation time, matching rate, \(C_{\text{pk}}\), and adjusted matching rate; these are described below:  
1. Computation time: it should be minimized to avoid time delay from combination selection to assembly.
2. Matching rate: whenever the process cycle is in progress, either the matching or the unmatching count increases, depending on whether the right group has been found. The matching rate is the percentage of the matching and unmatching ratios.
3. \(C_{\text{pk}}\): \(C_{\text{pk}}\) (adjusted process capability index) is used to evaluate the consistency and quality of the assembled product according to the following equations: \[C_{p}=\frac{\textbf{U}\textbf{S}\textbf{L}\textbf{S}\textbf{L}}{6\sigma}\] (2) \[C_{p\textbf{k}}=\min\left(\frac{\textbf{U}\textbf{S}\textbf{L}-\mu}{3 \sigma},\frac{\mu-\textbf{L}\textbf{S}\textbf{L}}{3\sigma}\right)\] (3)  
* USL: upper specification limit;
* LSL: lower specification limit;
* \(\sigma\): standard deviation;
* \(\mu\): clearance mean.
4. Adjusted matching rate: This metric indicates the probability of defective components identified at the end of the process clearance stage, accounting for the inherent mechanical limitations. Due to limitations in the sensor accuracy, some dimensional errors in individual components may not be detected, which cannot be directly observed. To estimate this, potential errors (e.g., sensor errors, ball diameter errors, etc.) are randomly generated following a normal distribution and added to the process clearance variance. This simulates the latent errors during the process. Then, the adjusted matching rate, which represents the probability of a potential error, is calculated for a low \(C_{\text{pk}}\) considering the number of components that fall outside the clearance specification limits after incorporating these potential errors.