# Harmony Search Algorithm for Minimizing Assembly Variation in Non-linear Assembly
## 5 Numerical Illustration  
Overrunning clutch assembly (OCA), dealt with by [PERSON] et al. (2001) [23], shown in Figure 1, is considered an example product to show the efficiency of the proposed method. It consists of a hub, four rollers, and a cage. The nominal dimensions and their allocated tolerance; the minimum, maximum process tolerances; and the tolerance cost function constants of alternative processes of each component are listed in Table 1. The critical dimension Y is determined using Equation (1), and the accepted value of Y is assumed to be 7.0124 \(\pm\) 2\({}^{\circ}\). Equation (2) represents the reciprocal tolerance cost function to calculate the component's tolerance cost. The cost function constants are taken from [PERSON] et al. (2001) [23]. The cost of assembly based on allocated and maximum process tolerances is computed using Equations (3) and (4), respectively, and listed in Table 2.  
Figure 1: Overrunning clutch assembly. \[Y=\cos^{-1}\left(\frac{X1+\left(\frac{X2+X3}{2}\right)}{X4-\left(\frac{X2+X3}{2} \right)}\right) \tag{1}\]  
\[C_{i}=A_{i}+\frac{B_{i}}{t_{i}} \tag{2}\]  
\[C_{a}=\sum_{i=1}^{nc}A_{i}+\frac{B_{i}}{t_{ai}} \tag{3}\]  
\[C_{m}=\sum_{i=1}^{nc}A_{i}+\frac{B_{i}}{t_{mi}} \tag{4}\]  
where \begin{tabular}{l l l} \(X1\) & --Dimension of the hub in mm \\ \(X2\) and \(X3\) & --Dimension of the rollers in mm \\ \(X4\) & --Dimension of the cage in mm \\ \(Y\) & --Critical dimension in degrees \\ \(A_{i}\) & --Fixed cost of the \(i\)th component in \(\$\) \\ \(B_{i}\) & --Cost function constants of the \(i\)th component \\ \(t_{i}\) & --\(i\)th component's tolerance in mm \\ \(C_{i}\) & --Cost of the \(i\)th component in \(\$\) \\ \(t_{ai}\), \(t_{mi}\) & --\(i\)th component's allocated and maximum tolerance in mm \\ \(C_{a}\), \(C_{m}\) & --Cost of assembly in \(\$\) based on \(t_{ai}\) and \(t_{min}\) \\ \(nc\) & --Number of components \\ \(i\) & --Component number index \\ \end{tabular} \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{_CNO._} & \multicolumn{2}{c}{**Nominal Dimension Dimension (D.) (mm)**} & \multirow{2}{*}{\(\mathbf{t_{ai}}\) (mm)} & \multirow{2}{*}{\(\mathbf{P_{s}}\)} & \multirow{2}{*}{\(\mathbf{P_{s}}\)} & \multirow{2}{*}{\(\mathbf{P_{s}}\)} & \multirow{2}{*}{\(\mathbf{P_{s}}\)} & \multirow{2}{*}{\(\mathbf{A_{i}}\)} & \multirow{2}{*}{\(\mathbf{B_{i}}\)} & \multirow{2}{*}{\(\mathbf{t_{min}}\) (mm)} & \multirow{2}{*}{\(\mathbf{t_{max}}\) (mm)} \\ \cline{5-6}  & & & & & & & \\ \hline \multirow{2}{*}{\(X1\)} & \multirow{2}{*}{55.29} & \multirow{2}{*}{0.179806} & \multirow{2}{*}{3} & \(P1\) & 10.0 & 0.015 & 0.015 & 0.08 \\ \cline{5-6}  & & & & & \(P2\) & 5.0 & 0.500 & 0.060 & 0.15 \\ \cline{5-6}  & & & & & \(P3\) & 3.5 & 0.750 & 0.120 & 0.25 \\ \hline \multirow{2}{*}{\(X2\)} & \multirow{2}{*}{22.86} & \multirow{2}{*}{0.165358} & \multirow{2}{*}{2} & \(P1\) & 8.0 & 0.250 & 0.020 & 0.15 \\ \cline{5-6}  & & & & & \(P2\) & 3.0 & 0.650 & 0.080 & 0.30 \\ \hline \multirow{2}{*}{\(X3\)} & \multirow{2}{*}{22.86} & \multirow{2}{*}{0.120132} & \multirow{2}{*}{1} & \(P1\) & 2.5 & 0.300 & 0.040 & 0.20 \\ \cline{5-6}  & & & & & \(P2\) & 5.0 & 0.045 & 0.120 & 0.25 \\ \hline \multirow{2}{*}{\(X4\)} & \multirow{2}{*}{101.69} & \multirow{2}{*}{0.200581} & \multirow{2}{*}{3} & \(P1\) & 4.0 & 0.560 & 0.080 & 0.12 \\ \cline{5-6}  & & & & & \(P2\) & 6.0 & 0.160 & 0.150 & 0.25 \\ \cline{5-6}  & & & & & \(P3\) & 0.5 & 0.880 & 0.200 & 0.40 \\ \hline \hline \end{tabular} \(CNO._{\text{--component name}}\); \(\mathbf{P_{No.}}\)—process number.  
\end{table} 
Table 1: Manufacturing details of overrunning clutch assembly.