Analysis of the Use of Similarity Coefficients in Manufacturing Cell Formation Processes

Abstract

This study investigated the application of similarity coefficients in cellular layout and group technology in industrial organizations, focusing on multicellular manufacturing. Cell formation methods and techniques were explored, ranging from similarity of operations to production volume, in addition to the main elements of group technology. Cellular layout and group technology offer tangible benefits to industrial processes, such as increased operational efficiency, reduced production costs, and improved quality of final products. The choice and implementation of techniques based on similarity take into account factors such as product variety, production volume, process complexity, and market demand. One of the techniques is the use of similarity coefficients. The purpose of this study is to analyze the use of similarity coefficients in the cell formation process. The technical contribution of this study is that now practitioners have a detailed guide to applying similarity coefficients and verifying the results of the cell formation process in manufacturing activities. A bibliometric search using convenient keywords in the Google Scholar search engine identified the incidences of twenty types of similarity coefficients. The most cited coefficient, the Jaccard coefficient, was tested in standard and non-standard application cases, and the results were compared to support a conclusion. Further research should involve quantitative techniques such as multicriteria evaluation and fuzzy logic in the cell formation process.

1. Introduction

Cellular layout is a key component of cellular manufacturing (CM) in industrial settings and involves strategically grouping machines, supply, feed and depletion systems, transport, tools, and magazines. A cellular layout ensures that each cell element fits together, entailing fluidity and productivity in the process. For example, in an assembly line, workstations are organized to efficiently assemble product components or subsystems, with each station representing a cell within the layout. Cellular layout promotes smooth and efficient movement between workstations, reducing idleness and enhancing efficiency [1].

The concept of cellular layout extends beyond industrial environments to other engineering fields, such as the design of integrated circuits and modular products. Each module can be viewed as a cell within the overall layout of a device. The underlying concept also applies to the design of scalable software, where modularization allows for the addition or removal of functionalities without affecting the system’s stability. For instance, in a business management system, modules like accounting, human resources, and logistics can be considered as cells within the overall software layout, simplifying system maintenance and evolution [2].

Cellular layout is also utilized in computer-aided design (CAD) systems, particularly in contexts involving modular components. Cells are basic building blocks organized in libraries. Software designers can select and combine building blocks to create customized subsystems, reducing design effort and simultaneously ensuring the quality of the product due to the use of previously tested modules [3].

Cellular layout is a modular and hierarchical approach to shop floor organization, streamlining industrial implementation by segmenting the workspace into independent cells, each dedicated to a specific stage of the production process. In practice, industrial facilities are organized to cluster machines, equipment, and workstations into autonomous but interconnected cells. This array ensures a fair distribution of production resources, such as raw materials, energy, and labor, which avoids bottlenecks [4].

As important as the distribution of machines into cells is the classification of parts into families based on convenient grouping criteria. The technique used to group similar parts into families is known as group technology (GT), a complementary technique to CM. While CM organizes machines into cells, GT organizes parts into families based on their similarities. GT is a systematic and structured approach to the design and production of moderately mass-produced products. Families are created based on shared characteristics, such as manufacturing process, machine requirements, production layout, materials used, or physical product attributes [5]. GT is not applicable to customized products, as it would be difficult to group unique products into families if the production strategy is product differentiation, as is the case in engineering to order (ETO). Additionally, GT is not suitable for serial products, as their product variety is low, and it does not make sense to group them. Therefore, GT is most effectively applied when a certain degree of product variety is present, but is not excessive [6].

Both CM and GT have a common element: the need for a multivariate technique to segregate elements of a set into subsets according to a multidimensional vector of attributes that describes the elements. A similarity function or a membership function for a subset is then determined. Similarity is applied to pairs of elements, while membership applies to individuals. Through heuristics or algorithms, individuals (machines in CM and parts in GT) are separated into subsets that have some similarity between them. In summary, given a universe of individuals S = [A₁ U A₂ U A₃ … U A_n], if an individual belongs to set A_i, where 1 ≥ i ≥ n, it does not belong to any other subset A ≠ Ai. In multivariate analysis, this technique is called cluster analysis and can be used to define both cells and families of parts [7].

This study focuses on the industrial use of cellular layout, particularly multicellular layout, where a manufacturing facility is organized into more than one cell that must produce at least two families of components. It is not unusual for multicellular manufacturing (MCM) to produce more than one family of parts in the same cell as long as the families are compatible and a large part of the tooling can be shared. Additionally, the number of families is usually greater than the number of cells, which implies that most cells must manufacture more than one family. Among the known cluster analysis techniques, this study is particularly interested in the use of the similarity coefficient, a quantitative measure that defines the similarity between parts or processes [1]. The coefficient is calculated based on criteria including technical characteristics of products or parts, process requirements, tools, material flow, setup time, and market demand, among others. The greater the similarity between parts, the more likely they will be grouped in the same cell. At least twenty types of similarity coefficients have been proposed in the literature [8]. Based on the criterion of whether, in an optimized process, part i ⋲ [set of parts] requires or does not require machine j ⋲ [set of machines], the parameters a, b, c, and d are defined in terms of the relevance between machines and parts.

The main purpose of this study is to analyze the use of similarity coefficients in the cell formation process. The technical contribution of this study is a detailed guide to applying similarity analysis supported by a similarity coefficient in the cell formation process. The technical contribution includes a method to verify the outcomes of the process and choose the best options. The application consists of separating machines in cells, verifying if there are exceptionalities that could be amended, and evaluating the options to support the final decision. In particular, exceptional cases are continuous improvement opportunities that practitioners should not neglect when applying the suggested method. A bibliometric search was conducted using convenient keywords in the Google Scholar search engine to identify the incidences of twenty types of similarity coefficients. The most cited coefficient was tested in standard and non-standard application cases, and the results were compared to support a conclusion.

The literature presents several cases of the use of similarity coefficients for the cell formation process in manufacturing [8]. Among many others, in engine manufacturing, the authors used hierarchical clustering to allocate machines to cells [9]. In an aluminum industry plant, the authors used non-hierarchical clustering, specifically the PFA (product flow analysis) method [10]. Finally, in the auto parts industry, the authors employed non-hierarchical clustering, specifically the ALC (average linkage clustering) technique, to allocate machines to cells [11]. The use of similarity coefficients is also observed in totally disparate activities outside the scope of manufacturing, such as co-citation management in bibliometric studies [12] or risk management in regions subject to river flooding [13], which conveys the capacity of generalization in this study.

The rest of the article is structured as a review of MCM and GT, methodology and results, discussion, and conclusion.

2. Cell Formation Methods in Multicellular Manufacturing

It is possible to assess the effectiveness of cellular manufacturing clusters using various performance measures such as grouping efficiency, grouping efficacy, number of exceptional elements, grouping index, and grouping measure. Of particular importance are grouping efficiency and grouping efficacy, which are evaluated using similarity coefficient methods in the cell formation process. Various algorithms and methods have been suggested to improve the efficiency of cell formation in manufacturing systems, including similarity coefficient methods, principal component analysis (PCA), and agglomerative clustering algorithms (ACA) [14]. This approach is significant as the design of machine cells and part families, as well as the reduction of typical setup times, is crucial for the success of cellular manufacturing systems (CMS) [15].

In MCM, cell formation is defined as the organized production process that involves grouping machines into clusters. There are two criteria for cell formation: project and process attributes.

Table 1. Cell formation process possibilities. (adapted from [16]).

Method	Technique	Reference
Classification and Codification	Group Technologies	[1]
Mathematical Programming	Linear Integer Programming	[17]
	Branch and Bound	[18]
Clustering	Hierarchical Clustering (dendrogram)	[7,9]
	Non-Hierarchical Clustering (CNA, ROC, PFA)	[7,9]
Artificial Intelligence	Neural Networks	[19]
	Fuzzy Logic	[20]
Graph Theory	Maximum Flow Algorithms	[21]
	Minimal Cost Algorithms	[21]
Heuristics	Genetic Algorithms	[22]
	Ant Colony Algorithms	[23]

categorizes possibilities for the cell formation process. The table presents methods and techniques from the field of operations research that are commonly utilized in cell formation processes, along with relevant references that effectively illustrate each technique.

The greatest interest in this study’s scope lies in clustering methods and hierarchical clustering techniques based on similarity coefficients.

In manufacturing studies, assessing the similarity between different components involves looking at a variety of factors, including parts, processes, machinery, material flow, and production volume [24]. Recent research [25] has introduced a coefficient that takes into account the type of batch movement within and between parts when evaluating similarity. When considering part-based similarity, the focus is on grouping parts that require similar or identical operations based on characteristics such as shape, size, material, and required operations. This approach aids in defining production cells based on the common process operations for each family of parts. The process-based similarity is similar to a job shop layout, where each cell is assigned a specific process, such as washing, machining, assembly, welding, or painting. Machinery similarity involves grouping parts based on their machine, tool, and equipment requirements. Parts with similar machine, tool, or equipment needs are grouped together [26]. Flow- and volume-based similarities help to define cells based on material flow and demand, which in turn reduce transportation distances and setup times, respectively [27].

It is important to take into account that each manufacturing method has its distinct advantages and disadvantages. The most suitable method to select depends on specific operational characteristics, such as the variety of products, production volume, and process complexity, among other factors. In many cases, a combination of methods may offer the most benefits. For example, cells can be defined based on production volume. Accordingly, the machines or processing work centers (PWCs) within each cell can be sequenced according to production flow. It is also common to define cells based on processes and then sequence multicells according to flow [28]. Once cells are defined, they can adopt different typologies, including U-shaped, ladder, and centered-robot typologies. Variants such as the U-shaped combination plus centered robot, intermediate storage structures, two inputs, and one output, as well as multicellular layouts, are also widespread in the industry [29]. The focus of this study is particularly located on multicellular layouts. Figure 1, Figure 2 and Figure 3 display some relevant configurations aligned with this study. The configurations are, respectively, robot-centered multicells, robot-centered multicells with buffers, and U-shaped multicells with robot-centered cells, buffers, two inputs, and one output.

3. Support Methods to Group Technology

GT is a management theory founded on the principle that tasks or processes exhibiting similar characteristics should be executed uniformly. This theory posits that products requiring analogous operations and sharing a common set of resources should be organized into product families, with these resources subsequently reallocated into specialized production subsystems [30]. GT is recognized as a strategic approach aimed at optimizing industrial production by clustering similar products into families. The primary objective is to increase production efficiency and operational flexibility and simultaneously improve overall product quality by streamlining setup times, maximizing machine utilization, and reducing intermediate inventory levels [31].

GT has seen extensive application across various manufacturing environments, including industries such as the automotive industry, electronics, and consumer goods, where production processes benefit from a reduction in complexity and resource redundancy [32]. The key components of a GT-based system include several critical elements, as outlined by [33], which serve as the foundation for its implementation and operational success:

• Family Identification: Products that share similar characteristics or require similar manufacturing processes are identified, which involves analyzing technical specifications, manufacturing methods, and market requirements.
• Classification and Coding: Families are classified and coded according to criteria such as materials, manufacturing processes, and physical dimensions.
• Process Grouping: Like products, processes are grouped so that similar operations are performed together, reducing setup times.
• Component Standardization and Reuse: TG allows for the standardization of components and increased reuse, reducing inventory costs.

Figure 4 provides an example of part separation in multicellular manufacturing.

The figure demonstrates how a product is broken down into subsystems and parts, which are then grouped into families. Parts are allocated to families based on shared characteristics related to the project’s features, including shape, size, surface specifications, and the types of holes and grooves present. Parts are also allocated according to the specific processing requirements for their fabrication, encompassing operations such as milling, drilling, deburring, and surface treatments. Some heuristics mix both sets of attributes to achieve an optimized solution, regardless of the subsystems to which the components belong. The organization of the resulting parts optimizes resource utilization and reduces production complexity. The figure underscores the importance of strategic part and subsystem organization in achieving the operational benefits of multicellular manufacturing [34].

At least three methods are most commonly used in GT: the OPITZ, KKK, and Vuoso Praha methods.

The Opitz Method was first proposed by Herwart Opitz from Aachen University in 1970. It is a systematic approach that identifies and classifies components into coherent, homogeneous families. The method considers technical attributes such as geometrical configuration, dimensionality, constituent materials, and fabrication processes. Once classified, components are separated into families with similar characteristics. The method requires an evaluation of machine capacity and setup requirements prior to suggesting an optimal allocation of components to production cells with congruent configurations. This method is particularly beneficial on the shop floor, especially when resource efficiency is a competitive criterion [35].

The KKK Method (Koizumi–Kishida–Kameda), which originated in Japan, emphasizes flexibility and adaptability in the orchestration of manufacturing cells. The method assigns components to families based on technical and process attributes while specifying cells that maximize the capacity to handle a given diversity of components. Such requirements involve multifunctional machinery and operators, facilitating swift reconfiguration to align with fluctuating production requirements. The KKK method is ideally suited to dynamic production environments in which adaptability to shifting demands constitutes a competitive priority [36].

Finally, the Vuoso Praha Method (Production Harmony), originated in the Czech Republic, focuses on equilibrating the workload between machinery and operators by categorizing components into families grounded in process characteristics and production flow. The method reduces waiting and idle times and searches for an uninterrupted component flow through the manufacturing process. It is particularly efficacious in contexts where preserving a consistent production flow is indispensable for operational efficacy [37].

4. The Research

The theoretical basis of this study is formed by [8,26,38], who offer a comparative list of similarity coefficients (SC). A previous study analyzes the structure of a similarity metric and lists properties that a similarity relation should have. According to [38], there are two types of SC, Jaccardian and non-Jaccardian. In the first type, the SC is a metric that represents the proportion of hits that a solution presents in relation to the expected or maximum number of hits. The SC varies between 0 and 1. The second type considers an additional term proportional to the misses, which is subtracted from the number of hits before dividing by the expected number of hits. The SC varies between −1 and 1. A similarity relation S_ij should have the following properties:

• If two objects are completely similar, that is, every piece that visits i also visits j, and every piece that visits j also visits i, then S_ij = 1.
• If two objects are completely dissimilar, that is, no piece visits i and j at the same time, S_ij = 0 or −1, depending on the type of the coefficient.
• If i is more similar to j than to k, S_ij > S_ik.
• If i is less similar to j than to k, S_ij < S_ik.

This study examines twenty similarity coefficients sourced from the existing literature [8,38]. Each coefficient was analyzed to determine its influence on the cellular manufacturing process literature.

The research method begins with a bibliometric search in the Google Scholar search engine. Google Scholar fits with the scope of this study even if it does not curate data as other databases do. It was chosen due to the following advantages adequate to this study: (i) full-text searching, not only abstracts, titles, and keywords; (ii) fully open access, which allows triangulation and replication over the years; and (iii) broad coverage, which includes articles published in secondary journals, proceedings, books, theses and dissertations, technical reports, and patents. The keywords were “similarity coefficient” AND “cellular manufacturing”. Only pages in English were considered. Any part of the text counts. Next, the study identified the six most influential types of coefficient and, by a standard application, the most influential one, which was applied to a non-standard, exceptional case. The results of the applications were discussed, and a conclusion was applied. Figure 5 synthesizes the method.

4.1. The Most Influential Similarity Coefficients

Given machines i and j, S_ij is the similarity between them, a is the number of parts that require both, b is the number of parts that require only i, b is the number of parts that require only j, and d is the number of parts that require neither.

Table 2. Incidences of similarity coefficients (adapted from [8,38]).

Coefficient	Sites (in 2024)	Equation Sij=	Range
Jaccard	510	${\displaystyle \frac{a}{a+b+c}}$	0–1
Hamann	61	${\displaystyle \frac{\left(a+d\right)-(b+c)}{\left(a+d\right)+(b+c)}}$	−1 to 1
Yule	43	${\displaystyle \frac{\left(ad-bc\right)}{\left(ad+bc\right)}}$	−1 to 1
Simple matching	74	${\displaystyle \frac{a+d}{a+b+c+d}}$	0–1
Sorenson	66	${\displaystyle \frac{2a}{2a+b+c}}$	0–1
Rogers and Tanimoto	53	${\displaystyle \frac{a+d}{a+2(b+c)+d}}$	0–1
Sokal and Sneath	117	${\displaystyle \frac{2(a+d)}{2(a+d)+b+c}}$	0–1
Russell and Rao	34	${\displaystyle \frac{a}{a+b+c+d}}$	0–1
Baroni-Urbani and Buser	46	${\displaystyle \frac{a+\sqrt[2]{ad}}{a+b+c+\sqrt[2]{ad}}}$	0–1
Phi	76	${\displaystyle \frac{ad-bc}{\sqrt[2]{(a+b)(a+c)(b+d)(c+d)}}}$	−1 to 1
Ochiai	46	${\displaystyle \frac{a}{\sqrt[2]{(b+a)(c+a)}}}$	0–1
PSC	40	${\displaystyle \frac{a^2}{(b+a)(c+a)}}$	0–1
Dot-product	57	${\displaystyle \frac{a}{2a+b+c}}$	0–1
Kulczynsky	36	${\displaystyle \frac{\left(\frac{a}{a+b}\right)+\left(\frac{a}{a+c}\right)}{2}}$	0–1
Sokal and Sneath 2	38	${\displaystyle \frac{a}{a+2b+2c}}$	0–1
Sokal and Sneath 4	24	${\displaystyle \frac{\left(\frac{a}{a+b}\right)+\left(\frac{a}{a+c}\right)+\left(\frac{d}{b+d}\right)+\left(\frac{d}{c+d}\right)}{4}}$	0–1
Relative matching	41	${\displaystyle \frac{a+\sqrt[2]{ad}}{a+b+c+d+\sqrt[2]{ad}}}$	0–1
Chandrsekharan/Rajagopalan	433	${\displaystyle \frac{a}{\mathit{min}[\left(a+b\right),(a+c)]}}$	0–1
MaxSC	28	$max\left[{\displaystyle \frac{a}{\left(a+b\right)}},{\displaystyle \frac{a}{\left(a+c\right)}}\right]$	0–1
Baker and Maropoulos	7	${\displaystyle \frac{a}{\mathit{max}[\left(a+b\right),(a+c)]}}$	0–1
Ravichandran and Rao	36	${\displaystyle \frac{a(a+d)}{(a\sqrt[2]{a+d}+b+c)(\sqrt[2]{a+d)}}}$	0–1

presents the twenty coefficients considered and the results of the search.

The six most prevalent coefficients, including Jaccard, Chandrasekharan and Rajagopalan (C&R), Sokal and Sneath, Phi, Simple Matching, and Sorensen, collectively represent over 65% of the occurrences. With the exception of Phi, all are of the Jaccardian type. Therefore, researchers and practitioners are advised to prioritize these six coefficients for future applications, as they are the most widely used in the literature.

4.2. 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. Example of application of similarity coefficients.

Machines/Batches	A	B	C	D	E	F	G	H	I
1	1			1				1
2					1				1
3			1		1				1
4		1				1
5	1							1
6			1						1
7		1				1	1

showcases the initial real-life scenario in MCM, in which batches [A, …, I] are ideally processed through machines [1, …, 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. The most common similarity coefficients for the application.

	Parameter				Coefficient
Sij	a	b	c	d	Jaccard	C&R	Phi *	SM	S&S	Sorensen
12	0
13	0
14	0
15	2	1	0	6	0.667	1.000	0.756	0.889	0.941	0.800
16	0
17	0
23	2	0	0	6	1.000	1.000	1.000	1.000	1.000	1.000
24	0
25	0
26	1	1	0	6	0.500	1.000	0.655	0.875	0.933	0.667
27	0
34	0
35	0
36	2	1	0	6	0.667	1.000	0.756	0.889	0.941	0.800
37	0
45	0
46	0
47	2	0	1	6	0.667	1.000	0.756	0.889	0.941	0.800
56	0
57	0
67	0
Δ%=					50%	0%	35%	13%	7%	33%

* Non-Jaccardian method.

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.

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.

To verify the indication,

Table 5. Reordering rows to highlight the solution.

Machines/ Batches	A	B	C	D	E	F	G	H	I
1	1			1				1
5	1							1
2					1				1
3			1		1				1
6			1						1
4		1				1
7		1				1	1

Green: cell 1; yellow: cell 2; blue: cell 3.

and

Table 6. Reordering columns to highlight the solution.

Machines/ Batches	A	D	H	C	E	I	B	F	G
1	1	1	1
5	1		1
2					1	1
3				1	1	1
6				1		1
4							1	1
7							1	1	1

Green: cell 1; yellow: cell 2; blue: cell 3.

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. 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.

$$Eff=1-{\displaystyle \frac{e_0}{e_1}}$$

(1)

where:

e₁ = total number of 1s in the matrix;

e₀ = 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 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.

$$Efc={\displaystyle \frac{e_1-e_0}{e_1+e_v}}={\displaystyle \frac{1-\frac{e_0}{e_1}}{1+\frac{e_v}{e_1}}}$$

(2)

where:

e₁ = total number of 1s in the matrix;

e₀ = 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.

4.3. 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. Example of application with exceptional elements.

Machines/ Batches	A	B	C	D	E	F	G	H
1					1	1
2	1	1	1
3	1		1				1
4		1	1			1	1	1
5		1		1	1	1	1	1
6	1			1		1	1	1

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).

Table 8. First alternative for the formation of cells with exceptional elements.

Machines/ Batches	A	C	B	G	F	H	D	E
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

and

Table 9. Second alternative for the formation of cells with exceptional elements.

Machines/ Batches	A	C	B	G	F	H	D	E
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

depict both options.

Equations (3) to (6) replicate the efficiency and effectiveness calculation of groupings for both options:

Eff 1 = [1 − (7/24)] = 71%

(3)

Eff 2 = [1 − (6/24)] = 75%;

(4)

Efc 1 = [(24 − 7)/(24 + 2)] = 65%

(5)

Efc 2 = [(24 − 6)/(24 + 2)] = 69%

(6)

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].

$$GM={\displaystyle \frac{e}{e+e_v}}-{\displaystyle \frac{e_0}{e_1}}$$

(7)

where:

e = total number of 1s in the diagonal;

e_v = total number of voids in the diagonal

e₀ = total number of exceptional elements in the matrix;

e₁ = 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.

GM1 = 17/(17 + 2) − 7/24 = 60.03%

(8)

GM2 = 18/(18 + 2) − 6/24 = 65.0%

(9)

5. Conclusions

In contemporary manufacturing systems, cellular manufacturing has emerged as a relevant concept, especially in the context of layout design and its relationship with the production mix and flexibility of part fabrication [43]. In the current context, the adoption of a cellular layout may be necessary for manufacturers striving to maintain a competitive edge. Cell formation methods can effectively organize production cells, providing the capacity for adaptation to the specific needs of customers [44].

This study explored the application of the similarity coefficient in the cell formation process in multicellular manufacturing systems. This research delved into the utilization of the similarity coefficient in the context of cell layout formation and the implementation of group technology in multicellular manufacturing, highlighting their role in augmenting efficiency, adaptability, and quality within production processes. It examined various techniques for cell formation and the key components of GT, underscoring their role in managing efficiency, flexibility, and quality in production processes. In manufacturing processes, both CM and GT techniques offer an organized method for classifying machines and parts in order to balance part flow and reduce resource idleness and time to order completion.

According to the bibliometric search, six of the twenty similarity coefficients evaluated are more frequently cited. The Jaccard coefficient was the most cited in the search. However, regardless of the selected coefficient, the cell formation method should consider the distinct features of every industrial environment, including factors such as product diversity, production volume, and process intricacy. In essence, this study explored the utilization of cellular layout and cluster technology in manufacturing, emphasizing the use of the Jaccard coefficient for similarity calculation.

This article makes a technical contribution, as it introduces a systematic approach to supporting a cell formation process in manufacturing, utilizing a similarity coefficient. The initial step calculates the similarities between all machines by using a matrix [machines X parts] that allocates parts to machines. Next, based on the similarities, a dendrogram is created manually or supported by software. The dendrogram is a visual tool that provides multiple options for cell formation according to the desired number of cells. Once a specific cell configuration is selected, the matrix’s rows and columns are rearranged to cluster relevant incidences around the diagonal, promoting more organized groupings of machines and parts.

It should be recognized that the resultant classification might contain exceptions—cases where specific occurrences do not fit within the defined categories. These exceptions are opportunities for continuous improvement in both the production process and the project. Finally, the paper discusses how to evaluate the efficiencies of the feasible solutions, supporting the final choice for the cell formation process.

As a suggestion for future research, alternative similarity coefficients and methods for assessing cluster efficiency in the same case should be compared. It is also suggested that multiple similarity coefficients be combined into a single optimized index, utilizing multicriteria methods to assess the weights [45] when project and process attributes hold differing levels of significance in the multicell formation process. Additionally, exploring methods grounded in fuzzy logic [38] presents an avenue for future research. Another possibility is to evaluate how the Jaccard similarity coefficient performs in comparison with other similarity coefficients regarding computational complexity and ease of implementation in diverse manufacturing systems. Finally, a last possibility is to examine the influence of variety and volume in the choice of the similarity coefficient. Even if they are not independent variables, their combination may influence the applicability of the various similarity coefficients.