# Density-Based Prioritization Algorithm for Minimizing Surplus Parts in Selective Assembly
## 2 Search-Based Selective Assembly of Flow Production Systems and Surplus Parts  
### Selective Assembly Procedure  
In this study, an actual precision single-row deep-groove ball bearing assembly process is analyzed as an example of selective assembly. The ball bearing consists of an outer ring, an inner ring, and balls, as shown in Figure 1. In this example, the dimensions of each part and the specifications of the assembly clearance are as follows:  
* Outer ring raceway diameter: \(A=40^{+0.024}_{-0.005}\) mm;
* Inner ring raceway diameter: \(B=24\pm 0.025\) mm;
* Ball diameter: \(C=8\pm 0.0005\) mm;
* Assembly clearance: \(Y=0.009\pm 0.0025\) mm.  
The assembly clearance is formed according to the dimensions of each part as follows:  
\[Y=A\ -B-2C. \tag{1}\]  
Figure 1: Precision ball bearing assembly.  
The given ball bearing assembly process consists of the structure shown in Figure 2. The outer and inner rings are supplied one by one in every process cycle by the flow production system and the balls are preproduced and placed in seven tanks. The assembly facility has thirty slots for storing the outer rings one by one. Additional types of balls are prepared to increase the possibility of assembly between the outer and inner rings. Seven types of balls have different biases, ranging from \(-6\) to \(+6\)\(\upmu\)m in increments of 2 \(\upmu\)m, with respect to the ball diameter of 8 mm, and the tolerance is \(\pm 0.5\)\(\upmu\)m, as ever. When an inner ring is supplied, the assembly facility checks the stored outer rings and balls to find a combination that can satisfy the assembly clearance tolerance. According to Equation (1), in a total of 210 cases, assembly is performed by selecting the combination whose assembly clearance is the most approximate to 9 \(\upmu\)m and the slot vacated by the selected outer ring is refilled with a newly supplied outer ring.  
The diameters of each ball in the tanks cannot be identified and recorded, thus all balls in the same tank are deemed to have the same diameter. Therefore, the allowable range of Y should be set to \(\pm 1.2\)\(\upmu\)m in consideration of the tolerance (\(\pm 1\)\(\upmu\)m) and other measurement errors. In addition, if the center value of each dimension in Equation (1) were removed, the calculation would only be possible with the error values. Based on this, the dimensional values to be used for selective assembly are redefined as follows:  
* Outer ring raceway diameter error: \(A=\pm 15\)\(\upmu\)m;
* Inner ring raceway diameter error: \(B=\pm 25\)\(\upmu\)m;
* Ball diameter bias:  
\[C_{3-}=-6\)\(\upmu\)m  
\[C_{2-}=-4\)\(\upmu\)m  
\[C_{-}=-2\)\(\upmu\)m  
\[C_{0}=0\)\(\upmu\)m  
\[C_{+}=2\)\(\upmu\)m  
\[C_{2+}=4\)\(\upmu\)m  
\[C_{3+}=6\)\(\upmu\)m;  
Figure 2: Selective assembly of precision ball bearing.  
Then, the combination whose assembly clearance is the closest to the target value is identified by finding the combination where \(Y\) is the most approximate to 0 \(\upmu\)m.  
If no combination satisfies the assembly tolerance, the outer rings in all slots will be removed and all slots will be refilled with newly supplied outer rings. The inner ring remains in the assembly facility and will be used again when assembly resumes. Therefore, the inner rings supplied to the assembly facility as good products must be assembled. The extracted outer rings are reprocessed or discarded as surplus parts. In addition, assembly is paused until all slots are refilled, so the generation of surplus parts reduces process productivity and increases production costs.