## 5 Application to Manufacturing System Design Problems  
### Production Rate Maximization Problem  
At first, the meta-models developed were applied to the BAP which maximizes production rate. The purposes of 5.1 are to estimate production rate and to see the pattern of optimal buffer allocation, when total buffer (TB), product mix, and reconfiguration strategies are predetermined. Then, this problem is generally defined as shown in Equations (9)-(12). Equation (9) defines the objective function, which is to maximize the production rate, where \(i\) is the buffer group, \(k\) is the total number of buffer positions (\(k=8\) herein), and \(n\) is the total number of factors included in the meta-model, where the value of \(z_{j}\) (\(j=k+1,\dots,n\)) is predetermined as \(v_{j}\). In this study, we developed a meta-model that is generally used for \begin{table}
\begin{tabular}{c l c c c c} \hline \hline \multicolumn{2}{c}{**Data Set**} & \multicolumn{1}{c}{**DF**} & \multicolumn{1}{c}{**Adj. SS**} & \multicolumn{1}{c}{**F**} & \multicolumn{1}{c}{\(p\)} \\ \hline \multirow{3}{*}{Model} & Linear & 12 & 3.4030 & 86,449 & \textless{}0.001 \\  & Quadratic & 8 & 0.0079 & 302.15 & \textless{}0.001 \\  & 2 way interaction & 31 & 0.0044 & 45.56 & \textless{}0.001 \\  & Total model & 51 & 3.4153 & 2671.62 & \textless{}0.001 \\ \hline \multirow{3}{*}{Residual} & Lack of fit & 230 & 0.0008 & 1.1436 \(\times 10^{26}\) & \textless{}0.001 \\  & Pure error & 6 & 0 & \\  & Total error & 236 & 0.0008 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Result of analysis of variance (meta-model for strategy 3). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Data Set**} & \multicolumn{1}{c}{**Number of Points**} & \multicolumn{1}{c}{**Minimum**} & \multicolumn{1}{c}{**Mean**} & \multicolumn{1}{c}{**Maximum**} \\ \hline \multicolumn{2}{c}{DOE points} & 288 & 0.00 & 0.29 & 2.91 \\ \hline \multirow{2}{*}{Random points} & \(\alpha=1\) & 45 & 0.00 & 0.21 & 0.57 \\  & \(\alpha=2\) & 45 & 0.03 & 1.01 & 5.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Validation of the meta-model (strategy 3). estimating the production rate; therefore, \(z_{9},\ z_{10},\ z_{11}\) and \(z_{12}\) were included in the metamodel. However, these control variables should be fixed for solving the BAP. Equation (10) shows the constraint related to the capacity of the total buffer (TB), where \(c_{i}\) is the number of buffer positions in buffer group \(i\), and \(z_{i}\) is the capacity of buffer group \(i\). Equation (11) defines the lower limit (\(LB_{i}\)) and upper limit (\(UB_{i}\)) of \(z_{i}\). In addition, Equation (2) restricts \(z_{i}\) to a non-negative integer. It is noteworthy that the meta-model developed uses the normalized variable \(x_{i}\), not the original variable \(z_{i}\). Hence, the objective function is redefined as shown in Equation (13), where \(h_{i}(z_{i})\) is the transformation function that transforms the value of \(z_{i}\) to \(x_{i}\), as defined in Equation (4). \[\max PR(\mathbf{Z})=g(z_{1},\ z_{2},\ \ldots,z_{i},\ \ldots,z_{k}\ |z_{k+1}=v_{k+1}, \ldots,z_{n}=v_{n}) \tag{9}\]  
\[\text{s.t.}\ \sum_{i=1}^{k}c_{i}z_{i}=TB \tag{10}\]  
\[LB_{i}\leq z_{i}\leq UB_{i}\leq TB,\ i=1,2\ldots,k \tag{11}\]  
\[z_{i}\ :\text{nonnegative integer},\ i=1,2\ldots,k \tag{12}\]  
\[ \begin{split}\max PR(\mathbf{X})=f(x_{1},\ \ldots,x_{i},\ \ldots,x_{k}\ |x_{k+1}=v^{\prime}_{k+1},\ldots,x_{n}=v^{\prime}_{n})\\ =f(h_{1}(z_{1}),\ldots,h_{i}(z_{i}),\ \ldots,h_{k}(z_{k})\ |x_{k+1}=h_{k+1}(v_{k+1}), \ldots,x_{n}=h_{n}(v_{n}))\end{split}  \tag{13}\]  
The problem of reconfiguration strategy 3 was solved with CPLEX(r) because the efficiency of optimization algorithm was not our concern, and the results of the optimal buffer allocations are shown in Table 8. In these experiments, \(LB_{i}\) and \(UB_{i}\) were set to 1 and 13 (except for \(UB_{8}=5\)), respectively, as defined in Table 4. The values of \(x_{9}\), \(x_{10}\), \(x_{11}\), and \(x_{12}\) were set to medium values (level = 0).