# Analysis of the Use of Similarity Coefficients in Manufacturing Cell Formation Processes
## 4 The Research
### Application: Standard Real Cases
The visual final representation illustrates the similarity among the machines within a cell, solving the cell formation process that validates the effectiveness of the use of the Jaccard similarity coefficient. After defining the cells, it is necessary to assess the solution's quality. Several alternatives have been suggested in previous studies. A recent study [25] describes global efficiency (evaluating all cells), group efficiency (assessing each group separately), and group technology efficiency (considering machines and batches together). This study focuses solely on global efficiency. More comprehensive studies primarily center on the use of manufacturing technology that can utilize the other two types. Group efficiency can aid in enhancing product design by adjusting requirements or eliminating functions. In contrast, technology efficiency can improve the manufacturing process by incorporating multiple machine functions to minimize gaps and inter-cell transfers [39]. A further study [40] introduced Equation (1) for global efficiency _Eff_. \[Eff=1-\frac{e_{0}}{e_{1}} \tag{1}\]  
where:  
\(e_{1}=\) total number of 1s in the matrix;  
\(e_{0}=\) number of exceptional elements (outside the cells) in the matrix. Although the balance is not flawless, as there are voids that could cause bottlenecks or disruptions in the production process, the overall efficiency is still calculated at 100% due \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Machines/** & **A** & **D** & **H** & **C** & **E** & **I** & **B** & **F** & **G** \\
1 & & & & & & & & & \\
5 & & & & & & & & & \\
2 & & & & & & 1 & 1 & & \\
3 & & & & & 1 & 1 & 1 & & \\
6 & & & & & & 1 & & & \\
4 & & & & & & & & & \\
7 & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Reordering columns to highlight the solution. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Machines/** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** & **I** \\ \hline
2 & & & & & & 1 & & & 1 \\
3 & & & 1 & 1 & & & 1 \\
6 & & & 1 & & & & 1 \\
7 & & & & & & & & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Reordering rows to highlight the solution. to the absence of exceptional elements. According to two further studies [41; 42], neglecting the voids in the diagonal renders the initial expression inadequate. They propose another expression, Equation (2), which they refer to as grouping efficiency _Efc_. \[Efc=\frac{e_{1}-e_{0}}{e_{1}+e_{v}}=\frac{1-\frac{e_{0}}{e_{1}}}{1+\frac{e_{0}}{ e_{1}}} \tag{2}\]  
where:  
\(e_{1}=\) total number of 1s in the matrix;  
\(e_{0}=\) total number of exceptional elements in the matrix;  
\(e_{v}=\) total number of voids in the matrix. Equation (2)'s second form emphasizes the two main sources of efficiency loss: the percentage of exceptional elements and voids in the cells. The calculated grouping efficiency is _Efc_ = \((1-0/17)/(1+4/17)\) = 81%. This considers the number of operations outside the cell as potential voids, indicating possible bottlenecks. Voids in the cell signal the need to redesign the product or process to balance equipment usage better.