# Analysis of the Use of Similarity Coefficients in Manufacturing Cell Formation Processes
## 4 The Research
### Application: Non-Standard Real Cases  
The first example demonstrates an application that achieved distribution without extracellular incidences, meaning there is no need to transport parts between cells, only within the cells. However, in most cases, it is necessary to distribute machines in cells to minimize extracellular transport when it is impossible to eliminate it. This type of inadequacy can lead to so-called exceptional elements or extracellular incidents [40]. The examination of an application that showcases outstanding features and necessitates a reduction in extracellular transportation is important for the scope of this study. Table 7 illustrates a real-life example of an MCM (metal-cutting machine) in the metal-mechanical industry, which offers valuable insights into such scenarios. Figure 7 shows the dendrogram provided by Past software version 4.17. The dendrogram used the Jaccard coefficient to classify the machines. Based on the dendrogram, it is advisable to adopt the following cell distribution:  
* cell 1: machine 1 isolated;
* cell 2: machines 4, 5, and 6;
* cell 3: machines 2 and 3 to cell 3. Following this definition for the cell formation process, there are two primary options for assigning batches:  
* Assign batches A, B, and C to machines 2 and 3 (cell 1), batches G, F, H, and D to machines 4, 5, and 6 (cell 2), and batch E to machine 1 (cell 3);
* Assign batches A and B to machines 2 and 3 (cell 1), batches C, G, F, H, and D to machines 4, 5, and 6 (cell 2), and batch E to machine 1 (cell 3). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Machines/** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** \\ \hline
2 & 1 & 1 & 1 & & & & 1 & \\
3 & 1 & & 1 & & & & 1 & \\
4 & & 1 & 1 & & 1 & 1 & 1 & 1 \\
5 & & 1 & & 1 & 1 & 1 & 1 & 1 \\
6 & 1 & & & 1 & & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Example of application with exceptional elements. Equations (3) to (6) replicate the efficiency and effectiveness calculation of groupings for both options:  
\[\textit{Eff}\ 1=[1-(7/24)]=71\% \tag{3}\]  
\[\textit{Eff}\ 2=[1-(6/24)]=75\%; \tag{4}\]  
\[\textit{Efc}\ 1=[(24-7)/(24+2)]=65\% \tag{5}\]  
\[\textit{Efc}\ 2=[(24-6)/(24+2)]=69\% \tag{6}\] \begin{table}
\begin{tabular}{c|c c c c c c c c} \hline \hline
**Machines/ Batches** & **A** & **C** & **B** & **G** & **F** & **H** & **D** & **E** \\ \hline
2 & **1** & **1** & **1** & & & & & & \\
3 & **1** & **1** & & & 1 & & & & \\
6 & **1** & & & & 1 & 1 & 1 & 1 & \\
4 & & **1** & & 1 & 1 & 1 & 1 & & \\
5 & & & & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & & & & & & 1 & & & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: First alternative for the formation of cells with exceptional elements. \begin{table}
\begin{tabular}{c|c c c c c c c c} \hline \hline
**Machines/ Batches** & **A** & **C** & **B** & **G** & **F** & **H** & **D** & **E** \\ \hline
2 & **1** & **1** & **1** & & & & & & \\
3 & **1** & **1** & & & 1 & & & & \\
6 & **1** & & & 1 & 1 & 1 & 1 & & \\
4 & & **1** & & 1 & 1 & 1 & 1 & & \\
5 & & & & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & & & & & 1 & & & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Second alternative for the formation of cells with exceptional elements. Figure 7: Dendrogram of the second example, also calculated by the Jaccard coefficient. The second option is more efficient and effective at grouping than the first option. However, other evaluation methods are reported in the literature. In 2001, a further study collected input from other authors and suggested a grouping effectiveness measure (GM), as shown in Equation (7) [40]. \[GM=\frac{e}{e+e_{v}}-\frac{e_{0}}{e_{1}} \tag{7}\]  
where:  
\(e=\) total number of 1s in the diagonal;  
\(e_{\rm v}=\) total number of voids in the diagonal  
\(e_{0}=\) total number of exceptional elements in the matrix;  
\(e_{1}=\) total number of 1s in the matrix;  
Equations (8) and (9) calculate the GM for both alternatives. As before, the second alternative outperforms the first.