## Materials and Methods  
### Basic Input  
Different approaches for the GCF take different approaches to the definition of alternative process plans or routings. [PERSON] and [PERSON] [54] classify the approaches by sorting the processes or routings into six categories: (i) fixed routing (process route), (ii) routing with replicate machines, (iii) routing with alternate machines for some operations, (iv) several fixed routings, (v) fixed process plan, and (vi) alternative process plans. In this paper, we take the combination of manners (ii) and (iv) to define alternative process plans to model the GCF problem. To model the GCF problem, the binary part-machine incidence matrix (PMIM), which represents the association between the process plans of parts and machines, will be used as a basic input. Given \(m\) part types with a total of \(t\) process plans and \(n\) different machine types, the \(t\times n\) binary \(\text{PMIM}\,\mathbf{A}\!\left(=\left[a_{irj}\right]\right)\) is defined as follows:  
\[a_{irj}= \begin{cases}1&\text{if process plan $r$ of part $i$ is processed on machine $j$}\\ 0&\text{otherwise.}\end{cases} \]  
In addition, the following indices, notation, and decision variables will be used throughout the paper:  
Indices  
\(i=\text{part index}\);  
\(r=\text{process plan index}\);  
\(j=\text{machine or cell index}\);  
\(k=\text{copy index of replicate machine}\);  
\(c=\text{machine cell/part family index}\). Parameters  
\(m=\text{number of part types}\);  
\(R_{i}=\text{set of process plans of part type $i$}\);  
\(t=\text{total number of process plans}\);  
\(n=\text{number of different machine types}\);  
\(p=\text{number of cells}\);  
\(U=\text{upper limit on the cell size}\);  
\(DM=\text{set of replicate machine types}\);  
\(M_{j}=\text{set of copies of replicate machine type $j\in DM$}\);  
\(q=n-|DM|+\sum\limits_{j\in DM}\left|M_{j}\right|=\text{total number of machines including copies of replicate machine types (the symbol }\) denotes the cardinality of the set \(X\));  
\(gs_{j_{1},j_{2}}=\text{generalized similarity coefficient between machine types $j_{1}$ and $j_{2}$}\);  
\(c_{j_{1},j_{2}}=\text{similarity coefficient between machines $j_{1}$ and $j_{2}$}\);  
\(MC_{c}=\text{set of machines in machine cell $c$}\);  
\(PF_{c}=\text{set of parts in part family $c$}\);  
\(e=\text{total number of 1s in the block diagonal solution matrix}\);  
\(e_{0}=\text{number of exceptional elements(EEs) in the block diagonal solution matrix}\);  
\(e_{y}=\text{number of voids in the block diagonal solution matrix}\);Decision variables  
\[x_{irc}= \begin{cases}1&\text{if process plan $r$ of part $i$ is assigned to cell; $c$}\\ 0&\text{otherwise.}\\ \end{cases} \] \[y_{jkc}= \begin{cases}1&\text{if copy $k$ of replicate machine $j$ is assigned to cell; $c$}\\ 0&\text{otherwise.}\\ \end{cases} \] \[z_{j_{1},j_{2}}= \begin{cases}1&\text{if machine $j_{1}$ belongs to cell; $j_{2}(j_{1},j_{2}=1,\cdots,q)$}\\ 0&\text{otherwise.}\\ \end{cases} \]  
Then, the 0-1 GCF problem can be illustrated with a generalized 0-1 PMIM as shown in Figure 1, in which each row indicates a part and each column a machine. The manufacturing system shown in Figure 1 has eight part types with a total of 19 process plans and five different machine types with an extra copy for machine type 3. An entry of \"1\" indicates that a part is processed by its associated machine, and an entry of \"0\", which is not shown for visual convenience, indicates that it is not processed by its associated machine. The alphabetical letters after the part numbers indicate alternative process plans. Rearranging the rows and columns in such a way that only one process plan is selected for each part and only one copy is allowed for each different machine type in each cell results in a block diagonal solution matrix, as shown in Figure 1. The solution of Figure 1 shows two machine cells and two part families.