## 2 Selective Assembly Employing Multiple-Slot Rows in Flow Production Systems
### Selective Assembly Process  
In the investigated bearing assembly process, the component specifications are defined as follows (Figure 1):  
* Outer-ring raceway diameter: A = 40.017 \(\pm\) 0.020 mm;
* Inner-ring raceway diameter: B = 24 \(\pm\) 0.010 mm;
* Ball diameter: C = 8 \(\pm\) 0.0005 mm;
* Assembly clearance: Y = 0.017 \(\pm\) 0.005 mm. To achieve precision in the assembly process, it is important to calculate the assembly clearance Y. Tolerance (clearance) Y is the target parameter, which is commonly adopted in general selective assembly algorithms to evaluate component compatibility [25]. In this case, Y is determined using the following equation:  
\[\mathrm{Y=A-B-2C} \tag{1}\]  
The above equation indicates the interdependence of the component dimensions to achieve the target assembly clearance. Given the specified tolerances for A, B, and C, the above equation assesses component compatibility and ensures that the assembled bearing meets the required precision standards. Y represents the cumulative effect of variations in the component dimensions within its corresponding tolerances. Given these tolerances, for simplification, the basic dimensions are offset and omitted during the selective assembly process. Therefore, the tolerances are redefined without specifying the nominal dimensions; this is a common practice when focusing on component variability and compatibility rather than its absolute dimensions:  
* Outer-ring raceway diameter tolerance: A = \(\pm\)20 \(\upmu\)m;
* Inner-ring raceway diameter tolerance: B = \(\pm\)10 \(\upmu\)m;
* Ball diameter tolerance: C = \(\pm\)0.5 \(\upmu\)m;
* Assembly clearance tolerance: Y = \(\pm\)5 \(\upmu\)m.  
The assembly facility employs 2-slot rows including 10 slots for the outer rings and 6 slots for the inner rings along with 3 ball tanks, which are categorized according to the diameter biases of the balls. The diameter biases of the balls allocated to each tank are as follows:  
\[\mathrm{C_{-}=-3\pm 0.5\ \upmu m}\]  
\[\mathrm{C_{0}=0\pm 0.5\ \upmu m}\]  
\[\mathrm{C_{+}=3\pm 0.5\ \upmu m}\]  
The three sizes of the balls in the tanks--each with a different diameter bias (\(\mathrm{C_{-},C_{0}}\), and \(\mathrm{C_{+}}\))--are strategically selected during the selective assembly process to correct the  
Figure 1: Precision ball-bearing assembly. assembly clearance, thus increasing the probability of a successful assembly. This setup enables the system to allow for component dimensional variations using balls with slight size variations, thus improving the compatibility among outer rings, inner rings, and balls. The bearing assembly process, which is depicted in Figure 2, follows a structured flow production system. In each production cycle, one outer ring and one inner ring are supplied to the assembly facility. The balls, which are preproduced with specific diameter biases, are stored in three separate tanks. During assembly, potential combinations based on the dimensions of the outer rings, inner rings, and balls stored in the tanks are calculated. Using Equation (1), the assembly system evaluates each combination of rings and balls to identify those that satisfy the Y = \(\pm\)5 \(\upmu\)m clearance. If multiple combinations satisfy this tolerance, the system selects the combination with the smallest deviation from the ideal 0.017 mm clearance. This selection criterion ensures that the resulting tolerance is as close as possible to the ideal tolerance, thus optimizing the precision of the assembled product. Upon selecting the best combination, the assembly facility completes the assembly process, and the now-vacant slots are refilled with the newly supplied outer and inner rings. This continuous refilling process ensures that the outer and inner ring slots remain fully occupied, allowing the assembly process to proceed without interruption according to the production system flow. If no combination meets the assembly tolerance, all outer rings in the slots are emptied and replenished with the newly supplied outer rings.