# Analysis of the Use of Similarity Coefficients in Manufacturing Cell Formation Processes
## 4 The Research
### Application: Standard Real Cases  
Two practical examples from the metal-mechanical industry can effectively demonstrate the application of the most commonly used coefficients in the literature. Table 3 showcases the initial real-life scenario in MCM, in which batches [A, \(\ldots\), I] are ideally processed through machines [1, \(\ldots\), 7]. In the matrix, a value of 1 denotes that batch X needs machine \(n\) in an optimized process, while a blank or 0 indicates the opposite. Table 4 displays the non-zero similarity coefficients based on five Jaccardian and one non-Jaccardian metric commonly referenced in the literature. The Jaccard coefficient is the most distinctive, with a 50% difference between the largest and smallest values, followed by the Phi and Sorensen coefficients. Previous studies [9, 10, 11] have confirmed the relevance of the Jaccard similarity coefficient in manufacturing applications. In the example, the C&R coefficient failed to distinguish between the machines. Therefore, this study, from now on, focuses only on the Jaccard coefficient. Figure 6 displays a dendrogram representing a solution built with the aid of the PAST software, version 4.03. The dendrogram uses the Jaccard coefficient and produces consistent qualitative results, particularly showcasing the distribution of machines in cells. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Coefficient** & **Sites (in 2024)** & **Equation \(S_{ij}\)=** & **Range** \\ \hline Sokal and Sneath 2 & 38 & \(\frac{a}{a+2b+2c}\) & 0–1 \\ Sokal and Sneath 4 & 24 & \(\frac{\big{(}\frac{a}{a+b}\big{)}+\big{(}\frac{a+b}{a+b}\big{)}+\big{(}\frac{ a}{a+b}\big{)}}{\big{(}\frac{a}{a+b+b+d}+\sqrt{a}}\) & 0–1 \\ Chandrasekharan/Rajagopalan & 433 & \(\frac{a}{min((a+b),(a+c)]}\) & 0–1 \\ MaxSC & 28 & \(max\Big{[}\frac{a}{(a+b)},\frac{a}{(a+c)}\Big{]}\) & 0–1 \\ Baker and Maropoulos & 7 & \(\frac{a}{max[(a+b),(a+c)]}\) & 0–1 \\ Ravichandran and Rao & 36 & \(\frac{a(a+d)}{\big{(}a\sqrt[2]{a+d}+b+c\big{)}\big{(}\sqrt[2]{a+d}\big{)}}\) & 0–1 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Machines/Batches** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** & **I** \\ \hline
2 & & & & & 1 & & & & 1 \\
3 & & & 1 & & 1 & & & & 1 \\
4 & & 1 & & & & 1 & & & 1 \\
5 & 1 & & & & & & & 1 \\
6 & & & 1 & & & & & & 1 \\
7 & & 1 & & & & 1 & & & \\ \hline \end{tabular}
\end{table} 
Table 3: Example of application of similarity coefficients. The horizontal line positioned below 0.5 divides the dendrogram into three sections, assigning each to a distinct cell. A line placed between 0.6 and 0.5 would allocate the machines into four cells. Remaining at three cells, the three sections of the dendrogram indicate the following overall multicellular distribution:  
* cell 1: machines 1 and 5 processing batches A, D, and H;
* cell 2: machines 2, 3, and 6 processing batches C, E, and I;
* cell 3: machines 4 and 7 processing batches B, F, and G. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Parameter**} & \multicolumn{4}{c}{**Coefficient**} \\ \cline{2-10} \(S_{ij}\) & **a** & **b** & **c** & **d** & **Jaccard** & **C\&R** & **Phi \({}^{*}\)** & **SM** & **S\&S** & **Sorensen** \\ \hline  
\(\Delta\%\)= & & & & & & 50\% & 0\% & 35\% & 13\% & 7\% & 33\% \\ \hline \hline \end{tabular} \({}^{*}\) Non-Jaccardian method.  
\end{table} 
Table 4: The most common similarity coefficients for the application. Figure 6: Dendrogram of the first application from the Jaccard coefficient. To verify the indication, Tables 5 and 6 reordered the rows and columns according to the cell designation. In Table 5, the lines representing the machines were grouped together according to the cell allocation. In Table 6, similarly, the rows representing parts were grouped together, also according to the cell allocation.