# Density-Based Prioritization Algorithm for Minimizing Surplus Parts in Selective Assembly
## 4 Performance Evaluation and Results  
For evaluating the effectiveness of DBP, an assembly scenario was reproduced using data collected from the actual precision ball bearing assembly facility described in Section 2.1 and the DBP algorithm was simulated. The simulation had a total of 125,447 cycles and one outer ring and one inner ring were supplied for each cycle. Thirty outer ring slots and seven ball tanks were used. The effectiveness of DBP was assessed by comparing it with the traditional algorithm. Three versions of DBP with different levels of tolerance phasing were created. The compared algorithms were as follows:  
1. Traditional: select the combination where Y is the most approximate to 0 \(\upmu\)m among combinations where Y is within \(\pm\)1.2 \(\upmu\)m.
2. DBP-I: select the combination with the highest-priority outer ring (set by DBP) among combinations where Y is within \(\pm\)1.2 \(\upmu\)m.
3. DBP-II: Select the combination with the highest-priority outer ring (set by DBP) among combinations where Y is within \(\pm\)0.6 \(\upmu\)m. If no satisfactory combination is identified, explore using \(\pm\)1.2 \(\upmu\)m.
4. DBP-III: Select the combination with the highest-priority outer ring (set by DBP) among combinations where Y is within \(\pm\)0.4 \(\upmu\)m. If no satisfactory combination is identified, explore using \(\pm\)0.8 and \(\pm\)1.2 \(\upmu\)m in sequence. \begin{table}
\begin{tabular}{c c} \hline \hline \(C_{pk}\) & **Sigma Level** \\ \hline
2 & 6\(\sigma\) \\
1.67 & 5\(\sigma\) \\
1.33 & 4\(\sigma\) \\
1 & 3\(\sigma\) \\
0.67 & 2\(\sigma\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Process capabilities and sigma levels. Table 2 shows the surplus part ratio and process capability for each algorithm in the simulation. The surplus part ratio is the ratio of the surplus outer rings to the total outer ring supply; the reduction rate compared with the traditional algorithm is the difference in the number of the surplus outer rings between the traditional and applied algorithms divided by the number of the surplus outer rings in the traditional algorithms. Findings confirm that surplus parts can be reduced using the DBP algorithm compared with the traditional algorithm. DBP-I does not generate any surplus parts but reduces \(C_{pk}\) to 1.109. As the assembly tolerance is phased under DBP-II and DBP-III, the surplus parts gradually increase, but they are better than the results of the traditional algorithm; \(C_{pk}\) is also improved. Accordingly, an appropriate assembly clearance tolerance can be set by balancing the surplus part ratio and process capability. Figure 4 shows the dimensional distributions of the outer ring diameter errors in the slots for each DBP version. The dimensional concentration phenomenon is less than that in Figure 3. Compared with DBP-I, DBP-II and DBP-III show a weaker alleviation of the dimensional concentration phenomenon as the tolerance is phased. This is because DBP operates more similarly to the traditional algorithm as the tolerance is phased. If the tolerance were phased in units of 0.1 \(\upmu\)m, which is the minimum unit, DBP would produce exactly the same results as the traditional algorithm. Therefore, the surplus part ratio and process capability are adjusted according to the level of tolerance phasing. Table 3 shows the operation times of the algorithms to evaluate their computational costs. The operation time is measured starting from the completion time of the inner ring raceway diameter measurement and ending at the decision of one combination of the parts to be assembled. The traditional algorithm always examines all combinations, so it shows a constant operation time of approximately 210 \(\upmu\)s/cycle. DBP does not have to examine the subordinated combinations if products can be assembled into high-priority combinations, so its operation time varies in some cases.