## Materials and Methods
### Mathematical Models
#### 2.3.2 PMP-Type Model  
Since the exact model 2 has too many binary variables and constraints to optimally solve large-sized GCF incidences within a reasonably short computation time, we use the PMP-type model as a better alternative formulation to maximize the GE indirectly. However, to develop the PMP-type model of GCF, we need the definition of similarity coefficients incorporating both alternative process plans and replicate machines. In this paper, we use a modified version of [PERSON] and [PERSON]'s similarity coefficient [59] defined between pairs of machine types to formulate the mathematical model of the GCF. [PERSON] and [PERSON]'s similarity coefficient based on the binary PMIM is a generalization of the Jaccard similarity coefficient used for the SCF problem. Their similarity coefficient \(gs_{j_{1},j_{2}}\) between two machine types \(j_{1}\) and \(j_{2}\) is defined by  
\[g^{s}_{j_{1},j_{2}}=\frac{\sum_{i=1}^{m}\beta(i,j_{1},j_{2})}{\sum_{i=1}^{m} \alpha(i,j_{1})+\sum_{i=1}^{m}\alpha(i,j_{2})-\sum_{i=1}^{m}\beta(i,j_{1},j_{2 })} \tag{16}\]  
where  
\[\alpha(i,j_{1})=\begin{cases}1&\text{if $a_{i\hat{r}_{j_{1}}}=1$ for some $r\in R_{i}$}\\ 0&\text{otherwise},\end{cases}\]  
\[(1)\beta(i,j_{1},j_{2})=\begin{cases}1&\text{if $a_{i\hat{r}_{j_{1}}}=a_{i\hat{r}_{ j_{2}}}=1$ for some $r\in R_{i}$}\\ 0&\text{otherwise}.\end{cases}\]  
However, since the above similarity coefficient does not consider the replicate machines, we use [PERSON] and [PERSON]'s similarity coefficient [48] to formulate the PMP-type model of GCF. The similarity coefficient \(gs_{j_{1},j_{2}}\) between two machines \(j_{1}\) and \(j_{2}\) is defined as follows:  
\[c_{j_{1},j_{2}}=\begin{cases}-\infty&\text{if both $j_{1}$ and $j_{2}$ belong to the same replicate machine type}\\ gs_{j_{1},j_{2}}&\text{otherwise}.\end{cases} \tag{17}\]  
Based on the similarity coefficient defined in Equation (13), the PMP-type model of GCF can be formulated as follows:  
(Model 3)  
\[\text{Maximize }\sum_{j_{1}=1}^{q}\sum_{j_{2}=1}^{q}c_{j_{1},j_{2}}z_{j_{1},j _{2}} \tag{18}\]  
Subject to  
\[\sum_{j_{2}=1}^{q}z_{j_{1},j_{2}}=1,\ j_{1}=1,\cdots,q \tag{19}\]  
\[\sum_{j_{1}=1}^{q}z_{j_{1},j_{1}}=p \tag{20}\]  
\[\sum_{j_{2}=1}^{q}z_{j_{1},j_{2}}\leq Uz_{j_{1},j_{2}}=1,\ j_{2}=1,\cdots,q \tag{21}\]  
\[\sum_{j_{1}\in M_{j_{3}}}^{q}z_{j_{1},j_{2}}\leq 1,\ j_{2}=1,\cdots,q;\ j_{3}\in DM \tag{22}\]  
\[z_{j_{1},j_{2}}=0\text{ or }1,\ j_{1},j_{2}=1,\cdots,q. \tag{23}\]The objective function in Equation (18) maximizes the sum of similarities among all pairs of machines including replicate machines. Constraint (19) ensures that each machine belongs to exactly one machine cell. Constraint (20) specifies the required number of machine cells. Constraint (21) ensures that the number of machines in each cell does not exceed \(U\) machines. Constraint (22) ensures that at most a single copy of a replicate machine type is assigned to each cell. Constraint (23) ensures the binary restriction of variables.  
Since model 3 is a linear model and the number of machine types is usually much lower than the number pf process plans, moderately large-sized instances can be solved optimally within a reasonable computation time regardless of the total number of process plans. On the contrary, the number of process plans critically affects the model size in model 2. To solve the example incidence addressed in Section 2.3.1, 12,100 binary variables and 497 constraints are needed. Clearly, model 3 is very economical since it contains much fewer binary variables and/or constraints than model 2. We will show that even a large GCF incidence can be solved optimally within only one second using the LINGO solver.