## Materials and Methods
### Mathematical Models  
#### 2.3.1 Exact Model  
Several authors have provided exact formulations maximizing the grouping efficacy in the 0-1 SCF problem with a single copy of each machine type [11; 12; 13; 14; 15; 16; 17; 18]. In this subsection, we present an exact formulation extended for the 0-1 GCF problem with multiple copies of replicate machine types. To construct the exact formulation which directly maximizes the GE, the values of \(e\), \(e_{0}\), and \(e_{v}\) should be calculated. They are given by the following equations:  
\[e=\sum\limits_{c=1}^{p}\sum\limits_{i=1}^{m}\sum\limits_{r\in R_{i}}\sum \limits_{j=1}^{n}\sum\limits_{k\in M_{j}}a_{irj}x_{irc}y_{jkc}. \tag{2}\]  
\[ \begin{array}{c}e_{0}=\sum\limits_{c=1}^{p}\sum\limits_{i=1}^{m}\sum\limits _{r\in R_{i}}\sum\limits_{j=1}^{n}a_{irj}x_{irc}\left[\sum\limits_{k\in M_{j}} \left(1-y_{jkc}\right)-\left(\left|M_{j}\right|-1\right)\right]\\ =\sum\limits_{c=1}^{p}\sum\limits_{i=1}^{m}\sum\limits_{r\in R_{i}}\sum \limits_{j=1}^{n}a_{irj}x_{irc}\left[1-\sum\limits_{k\in M_{j}}y_{jkc}\right] \\ =\sum\limits_{c=1}^{p}\sum\limits_{i=1}^{m}\sum\limits_{r\in R_{i}}\sum \limits_{j=1}^{n}a_{irj}x_{irc}-\sum\limits_{c=1}^{p}\sum\limits_{i=1}^{m} \sum\limits_{r\in R_{i}}\sum\limits_{j=1}^{n}\sum\limits_{k\in M_{j}}a_{irj}x _{irc}y_{jkc}\\ \end{array}  \tag{3}\]  
\[e_{v}=\sum\limits_{c=1}^{p}\sum\limits_{i=1}^{m}\sum\limits_{r\in R_{i}}\sum \limits_{j=1}^{n}\sum\limits_{k\in M_{j}}\left(1-a_{irj}\right)x_{irc}y_{jkc}. \tag{4}\]  
Note that once a process plan \(r\) of part \(i\) is processed on a copy \(k\) of replicate machine type \(j\) in cell \(c\), the remaining \(\left(\left|M_{j}\right|-1\right)\) elements with \(a_{irj}=1\) processed by different copies of that replicate machine type in other cells are not the EEs. Then, the mathematical model directly maximizing the GE is formulated as follows:  
(Model 1)  
\[ \begin{array}{c}\text{Maximize }\Gamma=\frac{\sum\limits_{c=1}^{p}\sum \limits_{i=1}^{m}\sum\limits_{r\in R_{i}}\sum\limits_{j=1}^{n}\sum\limits_{k \in M_{j}}a_{irj}x_{irc}y_{jkc}}{\sum\limits_{c=1}^{p}\sum\limits_{i=1}^{m} \sum\limits_{r\in R_{i}}\sum\limits_{j=1}^{n}\sum\limits_{k\in M_{j}}a_{irj}x_ {irc}+\sum\limits_{c=1}^{p}\sum\limits_{i=1}^{m}\sum\limits_{r\in R_{i}}\sum \limits_{j=1}^{n}\sum\limits_{k\in M_{j}}\left(1-a_{irj}\right)x_{irc}y_{jkc}} \end{array}  \tag{5}\]  
Subject to  
\[\sum\limits_{c=1}^{p}\sum\limits_{r\in R_{i}}x_{irc}=1,\ i=1,\cdots,m \tag{6}\]  
\[\sum\limits_{c=1}^{p}\sum\limits_{k\in M_{j}}y_{jkc}=1,\ j=1,\cdots,m \tag{7}\]\[\sum_{k\in M_{j}}^{p}y_{jkc}\leq 1,\ j\in DM,\ j=1,\cdots,n;c=1,\cdots,p \tag{8}\]  
\[\sum_{j=1}^{n}\sum_{k\in M_{j}}y_{jkc}\leq U,c=1,\cdots,p \tag{9}\]  
\[x_{irc},y_{jkc}=0\ \text{or}\ 1,r\in R_{i},i=1,\cdots,m;k\in M_{j},j=1,\cdots,n;c=1, \cdots,p. \tag{10}\]  
The objective function in Equation (5) maximizes the GE. Constraint (6) ensures that only one process plan of each part is assigned to only one cell. Constraint (7) ensures that each machine belongs to exactly one machine cell. Constraint (8) ensures that at most a single copy of a replicate machine type is assigned to each cell. Constraint (9) ensures that the number of machines in each cell does not exceed \(U\) machines. Constraint (10) ensures the binary restriction of variables.