## Materials and Methods
### Mathematical Models
#### 2.3.5 Illustrative Example  
The proposed procedures for part assignment and improperly assigned EM/RM reassignment are illustrated with a hypothetical GCF incidence which includes seven machine types, 15 part types, and 35 process plans as shown in Table 1, which lists the routing information of parts. Machine types 3 and 4 have two and one extra copies, respectively. After model 3 is solved under the condition of \(p=3\) and \(U=4\), the following machine cells are obtained: MC\({}_{1}\) = {1, 3, 6}, MC\({}_{2}\) = {2,4,3}, and MC\({}_{3}\) = {5,7,3,4}.  
Once these machine cells are determined, partial assignment steps of parts 1, 2, and 3 are presented in Table 2, where the column (1) indicates the number of EEs generated by the process plan assigned to a specific cell. The symbol * indicates the entries corresponding to the candidate process plan and cells selected by the criterion of that step. The symbol ** indicates the entries corresponding to the best process plan and the cell selected by the criterion of the final step. If the best candidate plan is determined, the associated best process plan and cell can be identified by scanning the columns from  
Figure 4: Flow chart identifying the type of EM and RM.  
left to right. After all parts are assigned, no RMs and improperly assigned EMs are found. Figure 5 is the resulting block diagonal solution matrix with a GE of 77.19%.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Part No.** & **Process Plan** & **Machines** & **Part No.** & **Process Plan** & **Machines** \\ \hline
1 & a & \(1,5\) & \(9\) & a & \(5,7\) \\  & b & \(1,3,4,6\) & & b & \(2,4\) \\ \hline
2 & a & \(3,5,7\) & \(10\) & a & \(1,3,4\) \\  & b & \(2,3\) & & b & \(1,2,5\) \\ \hline
3 & a & \(2,3,4\) & \(11\) & a & \(2,3,4,6\) \\ \hline
4 & a & \(2,3,4,5\) & & b & \(1,2,5\) \\  & b & \(1,2,4,5\) & & c & \(5,6,7\) \\  & c & \(2,5,6\) & & d & \(2,3,4,5,7\) \\ \cline{3-6}  & d & \(2,3,4,7\) & \(12\) & a & \(3,5\) \\ \hline
5 & a & \(3,4,5,6\) & & b & \(2,6\) \\  & b & \(2,7\) & & c & \(2,3,4\) \\ \cline{3-6}  & c & \(1,3,4,5,7\) & \(13\) & a & \(1,2,3,6\) \\ \hline
6 & a & \(2,4\) & \(14\) & a & \(2,3,4,6\) \\  & b & \(1,3,6\) & & b & \(1,2,3,4\) \\ \hline
7 & a & \(3,5\) & \(15\) & a & \(2,7\) \\  & b & \(4,5,7\) & & b & \(3,4\) \\ \hline
8 & a & \(2,3,5\) & & & \\  & b & \(4,6,7\) & & & \\  & c & \(1,2,3,6\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Routing information of parts.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Part No.** & **Process Plan** & **Cells Assigned** & \multicolumn{2}{c}{**Criteria**} & \multirow{2}{*}{**Part Type**} & \multicolumn{2}{c}{**Criteria**} \\ \cline{3-6}  & & & **1** & **2** & & **4** & **5** & **6** \\ \hline
1 & a * & \(1\)* & \(1\)* & \(2\) & & & \\  & & \(2\) & \(2\) & \(3\) & & & \\  & & \(3\)* & \(1\)* & \(3\) & & & \\ \cline{3-6}  & b ** & \(1\)** & \(1\)* & \(0\)* & & & \\  & & \(2\) & \(2\) & \(1\) & & & \\  & & \(3\) & \(2\) & \(2\) & & & \\ \hline
2 & a ** & \(1\) & \(2\) & \(2\) & & & \\  & & \(2\) & \(2\) & \(2\) & & & \\  & & \(3\)** & \(0\)* & \(1\)* & & & \\ \cline{3-6}  & b * & \(1\) & \(1\) & \(2\) & & & \\  & & \(2\)* & \(0\)* & \(1\)* & & & \\  & & \(3\) & \(1\) & \(3\) & & & \\ \hline
3 & a ** & \(1\) & \(2\) & \(2\) & & & \\  & & \(2\)** & \(0\)* & \(0\)* & \(1\)* & & \\  & & \(3\) & \(1\) & \(2\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Assignment of parts 1, 2, and 3 to cells.