Reconfigurable Manufacturing Systems: Enhancing Efficiency via Product Family Optimization

Abstract

In response to the increasing complexity of modern production environments driven by heightened competition, customer demands, and sustainability goals, this work presents a focused methodology for forming product families within reconfigurable manufacturing systems (RMSs). Unlike traditional approaches, our method emphasizes the practical implementation of the Analytic Hierarchy Process (AHP) and the Average Linkage Clustering (ALC) algorithm to optimize RMS configurations. By evaluating specific comparison criteria—such as assembly sequence, machining sequence, components, production tools, and production demand—we aim to enhance resource utilization and adaptability to market changes. The proposed methodology enables a systematic assessment of RMS performance tailored to diverse product requirements. A detailed example of machining systems demonstrates the use of machining sequence and tool usage as primary criteria, showcasing the practical application and decision-making capabilities of the approach. This work contributes to the field by providing a structured framework for decision-making in RMS, facilitating efficient and precise product family formation to meet evolving manufacturing demands.

1. Introduction

In the current industrial landscape, marked by high demand and a wide range of products, companies must seek solutions that enable them to protect their competitive position [1,2,3]. Product customization and the adoption of advanced technologies have become essential to meet market demands. Reconfigurable systems are gaining popularity in the industry because they provide a cost-effective solution to quickly adapt to demand fluctuations by easily adjusting production without investing in expensive new equipment. Reconfigurable systems also offer greater flexibility, allowing companies to produce a variety of customized products while maintaining competitive production costs [4]. To develop efficient reconfigurable manufacturing systems, the concept of product family clustering is a widely studied approach in the field of manufacturing systems. Since the introduction of group technology (GT) and the implementation of procedures for creating part families in cellular manufacturing, methodologies specifically tailored to the development of product families for reconfigurable manufacturing systems have emerged [5,6,7,8]. The primary goal of creating product families is to enable designers to create a system capable of accommodating variations within a product family with minimal changes to the production system [9]. This approach is considered essential for enhancing organizations’ responsiveness and efficiency in meeting market demands. Additionally, creating product families has been widely discussed and proposed as a potential solution to guide designers in developing new products inspired by existing ones, as well as to promote the creation of flexible and reconfigurable manufacturing systems [10]. To fully understand the importance of research on product family clustering, it is imperative to define a reconfigurable manufacturing system (RMS). Koren [11] first conceptualized an RMS in the 1980s, designed to integrate the advantages of specialized production systems with the adaptability of flexible systems to efficiently respond to market demands through its reconfigurable physical and logical structures that can be modified quickly and cost-effectively [11]. This capability allows organizations to reconfigure production cells by adding, removing, or modifying them without the need to completely replace existing production systems, thus facilitating the transition from one product family to another [12].

In the literature, the creation of product families is based on a comparative analysis among different products, relying on somewhat limited criteria such as assembly sequence, customer demand, components, and tools used in manufacturing, etc. [13]. This analysis enables the grouping of products that share similar characteristics [13,14,15,16,17,18]. Although current methods for clustering product families are useful for improving organizational efficiency and responsiveness, they still have limitations [19]. These methods are often specific to each system studied, making it challenging to generalize them to other systems. Furthermore, these methods may be based on criteria that are not relevant to other systems, potentially leading to clustering errors. To address these limitations, it has become imperative to develop more advanced clustering methods to enable more flexible and responsive production to market demands and should be capable of adapting to different types of production systems [20].

In this context, we propose a new product clustering method based on the analysis of design and manufacturing process variability, customer needs, and preferences. Articulated through four key sections, this paper commences by establishing the industrial context, emphasizing the urgent need for adaptable manufacturing strategies to meet the challenges of evolving market demands. Our literature review situates the research within current academic discussions, identifying gaps our methodology aims to fill. We then detail the use of Analytic Hierarchy Process (AHP) and Average Linkage Clustering (ALC), clarifying the computational methods and comparison criteria essential to our proposed clustering approach. Not only thoroughly explained, this method is also practically demonstrated through a machining system scenario in Section 4, illustrating the real-world benefits and potential improvements to RMS efficiency and adaptability. The paper concludes with the presentation of our findings, exploring the wider implications and proposing directions for future research, thus connecting theoretical innovation with practical application.

2. Literature Review

Scientific research on product family formation is a field of study that has garnered increasing interest over the years [21,22,23,24,25]. Product families play a crucial role in the efficient management of production systems, allowing for the grouping of similar products to optimize manufacturing operations and meet customer needs [26,27,28]. To conduct an in-depth study on this subject, rigorous literary research is essential to gather existing work and to analyze the various methods used to form these product families.

2.1. Research Methodology

In the context of our study on product family formation, we utilized two renowned research databases: Scopus and Web of Science. These databases offer extensive coverage of scientific articles from various disciplines, providing us with access to a vast amount of relevant information in the field of product family formation. Using these databases, we employed a combination of relevant keywords to identify scientific articles related to the design, formation, and grouping of product families in reconfigurable production systems. The keywords we used included “design”, “formation ”, “grouping”, “product families”, “reconfigurable”, “reconfiguration”, and “production system”. We also refined our search by using Boolean operators such as “AND” and “OR” to combine keywords and obtain more targeted results. We also employed strict selection criteria to identify relevant articles. This involved reading abstracts and titles to assess their relevance to our study topic. Furthermore, we examined the references cited in relevant articles to identify other key works and incorporate them into our research. The combined use of Scopus and Web of Science allowed us to cover a wide range of scientific work on product family formation, providing a comprehensive view of research conducted in this field. This approach enabled us to gather valuable information, compare different methods and approaches, and highlight recent advancements as well as gaps in the existing literature. In summary, the use of Scopus and Web of Science as databases for our research on product family formation enabled us to obtain a comprehensive and reliable overview of academic work in this field. These databases were essential resources for our research, allowing us to develop in-depth analysis and provide meaningful insights in our scientific article Figure 1 illustrates the research methodology employed for our literature review.

2.2. Related Works

In the field of product grouping into product families within reconfigurable production systems, several researchers have made significant contributions to the state of the art [5,29,30,31,32,33,34]. In [31], Galan proposed an approach based on the use of graph theory for grouping products into homogeneous families. His method utilizes similarity measures to represent relationships between products in the form of graphs. By employing graph partitioning algorithms, he successfully grouped products cohesively. Abdi developed an approach based on linear programming for product grouping into product families. His method aims to optimize the objectives of minimizing intra-family variability and maximizing inter-family variability, resulting in well-defined product families [12]. In [5], he proposed a method based on the use of multivariate data analysis to group products into homogeneous families. His method combines statistical techniques such as analytical network process (ANP) and Average Linkage Clustering (ALC) algorithm to identify key product characteristics and perform accurate grouping. Koren [35] introduced an approach based on the use of artificial intelligence and machine learning for product grouping.

Koren’s method leverages machine learning algorithms like support vector machines (SVMs) and artificial neural networks to analyze product data and form families based on similar features. Wang [36] developed an approach based on multi-objective optimization for product grouping into product families. His method aims to maximize intra-family similarity while minimizing inter-family similarity, resulting in well-defined product families [37,38,39,40,41].

Table 1. Literature review on product family formation for reconfigurable manufacturing systems (RMSs).

Ref	Year	Product Family Formation Approaches										Target
		Similarity Matrix						Mathematical Programming	Neural Network	Meta-Heuristics	Descriptive Procedures
		Component	Assembly Sequences	Product Demand	LCS/SCS	Machine	Modularity
[42]	2000	√										A methodology is proposed to effectively manage product variations within a product family by minimizing complexity and optimizing the utilization of shared components. This approach aims to identify shared components and use them as key criteria to evaluate similarity and modularity within the product family.
[43]	2000										√	The objective of this article is to examine the challenges of product family architecture (PFA) in the context of design for mass customization (DFMC) and propose solutions for effectively managing product variations.
[44]	2002										√	This article presents a quantitative metric for design-by-analogy based on functional similarity, enabling designers to generate sophisticated solutions during the design of new products.
[45]	2004				√							A methodology for designing a reconfigurable manufacturing system (RMS) by grouping products into families and selecting suitable manufacturing facilities. It introduces a reconfiguration interface to specify the products and utilizes the analytical hierarchy process (AHP) for product family selection.
[46]	2007					√	√					This RMS-based methodology facilitates the reconfiguration of production systems by considering five key product requirements and utilizing the ALCA algorithm for product family selection. It enables companies to adapt to new products while minimizing investment costs.
[31]	2007	√		√			√					A method is proposed to form the best product families by considering the key requirements of products in RMS, such as modularity, similarity, compatibility, reusability, and demand. The methodology uses similarity matrices and the Average Linkage Clustering algorithm to obtain a dendrogram of possible product families.
[47]	2008			√			√					A representation that includes similarity and dependency for assembly modularity. This representation captures the cost benefits of modularity and can be extended to other life-cycle processes.
[48]	2009										√	A method is presented to assess the community and diversity within a product family. This method relies on the use of a new index called “community versus diversity” (CDI).
[12]	2012								√			This paper aims to optimize reconfigurable product family formation by considering manufacturing and market requirements, costs, and process reconfiguration. The objective is to provide a decision support tool based on a network analysis model for selecting the best product family and promoting reconfigurability.
[49]	2014							√				A mixed-integer linear programming model has been developed to simultaneously form the part families and corresponding cell configurations in an RMS in a dynamic production environment.
[5]	2014	√	√	√								A method is proposed for forming product families in the context of reconfigurable assembly systems. It utilizes assembly sequence, components, and production demand as criteria to group products into coherent families.
[50]	2016	√	√	√								A new integer linear programming model for Bill-of-Materials (BOM) tree matching is introduced to consider both component similarity and their hierarchical assembly structure.
[36]	2016				√							A method is presented for forming a part family that takes into account bypassing moves and idle machines. Based on the linear relationship of part similarity for the LCS and SCS, a similarity coefficient algorithm is designed and is used as the basis for part clustering and family formation.
[51]	2016								√			The objective is to improve the classification of parts in a reconfigurable manufacturing system using neural networks. This aims to better utilize the existing database of part families and facilitate the reconfiguration of the production system.
[52]	2018										√	The objective is to group products into new product families focused on assembly for the optimization of existing assembly lines and the creation of future reconfigurable assembly systems.
[53]	2018	√					√					A method for designers to identify multiple component sharing options along a Pareto front, representing maximum similarity and strategic modularity.
[9]	2020									√		This research proposes a model to describe the evolution of a product family and support the rapid development of new innovative products. The model considers key design features and enhances design and production efficiency.
[54]	2020				√							Proposed methodology enhances the convertibility of reconfigurable manufacturing systems (RMSs) by introducing delayed reconfigurable manufacturing systems (D-RMSs). It addresses part family grouping and machine selection, demonstrating effectiveness through case studies.
[55]	2020							√				Two different approaches are developed to address the design of system configuration at different time periods. Two new formulations of mixed integer linear programming (MILP) and integer linear programming (ILP) are presented in the first and second approaches, respectively.
[56]	2022									√		This research developed an ant-based algorithm for the formation of product families and remanufacturing cells, aiming to simplify the complexity of remanufacturing systems and improve performance compared to other algorithms.
[57]	2023									√		This article aimed to optimize assembly configuration by integrating interface modularity into product family architecture. A multi-objective optimization method is used to maximize profit while balancing modularity and interface complexity. Numerical examples demonstrate the benefits of this approach.

provides an overview of the existing literature pertaining to product family formation specifically tailored for reconfigurable manufacturing systems (RMSs).

After analyzing the previously mentioned works, it is clear that most approaches use the similarity matrix as the foundation for forming product families. However, it is important to note that the specific criteria used in each work vary, indicating that the similarity matrix method alone may not be sufficient to address all types of systems. Some approaches focus on analyzing assembly sequences and product structure to group products into families. Others emphasize modularity and optimization to design reconfigurable manufacturing systems and product families. There are also approaches based on functional analysis and data flows to optimize production systems and product family configurations. Furthermore, some methods employ machine learning techniques, including neural networks [58,59,60], to automatically form product families. These approaches leverage the capabilities of artificial intelligence to analyze product data, identify similarities [61,62], and group products into families using classification models [63,64].

In conclusion, our extensive bibliographic study indicates that the most reliable method for forming product families relies on the use of a similarity matrix. This approach stands out for its ability to integrate multiple comparison criteria and for the ease of use, interpretation, and implementation of the matrix. However, in real-world scenarios involving a substantial number of products, employing a similarity matrix becomes essential. To effectively manage such large datasets, it is necessary to utilize a clustering algorithm. Among various algorithms, the Average Linkage Clustering (ALC) algorithm is the most widely used due to its robustness and efficiency in categorizing products into coherent families. The ALC algorithm allows for a structured and systematic analysis of similarity data, thus facilitating the formation of product groups that accurately reflect their relationships and common characteristics. In the next section, we propose a method that employs the Analytic Hierarchy Process (AHP) for the creation of the similarity matrix and the ALC algorithm for the clustering of products. This combined approach aims to enhance the precision and efficiency of forming product families by leveraging the strengths of both AHP in deriving a comprehensive similarity matrix and ALC in effectively clustering products based on their similarities.

3. The Proposed Method

To address the issue of product family formation, we propose an approach that integrates the AHP algorithm with the Average Linkage Clustering (ALC) hierarchical clustering method. This approach aims to streamline decision-making and optimize the formation of product families in manufacturing systems. Our methodology commences with defining the system, identifying its key components, and establishing relevant criteria for product family formation. Subsequently, we proceed to define the products and their variations, taking into account their technical characteristics and specific requirements. Once the products are defined, we move on to the pivotal step of criterion selection. Here, we assess the relative importance of these criteria in the context of product family formation. The AHP algorithm facilitates a structured analysis, allowing us to assign weights to various criteria based on their influence on family formation [65]. Following this, we calculate matrices for each criterion, enabling us to obtain similarity measures between products. These similarity measures are then employed within the ALC algorithm to generate a dendrogram. This dendrogram provides a graphical representation of similarity relationships among products, simplifying the process of family formation. Our approach offers numerous advantages. It supports a systematic and comprehensive analysis of product families, considering both technical aspects and product-specific requirements. Furthermore, it facilitates the comparison and selection of optimal family formation options, optimizing resource allocation and enhancing manufacturing process efficiency. Figure 2 outlines the proposed methodology for product family formation for reconfigurable manufacturing systems (RMSs).

The methodology illustrated in Figure 2 outlines a systematic framework for forming product families tailored to the design of reconfigurable manufacturing systems (RMSs). The framework is divided into four primary phases, explicitly numbered (1 to 4) in the first column of the figure. Each phase is designed to systematically guide the RMS design process by employing targeted tools and methods to address the complexities of product family formation, ultimately ensuring optimal RMS configurations to meet diverse product requirements.

Preliminary Analysis: The first phase combines the system definition and product definition steps to establish a comprehensive foundation for RMS design. This involves clearly defining the system, its purpose, and its boundaries using the QQOQCCP approach (who, what, where, when, how, how much, and why), which systematically identifies the stakeholders, requirements, and constraints. The Ishikawa tool (fishbone diagram) is employed to perform a detailed cause-and-effect analysis, uncovering potential challenges and their root causes. To complement these structured tools, brainstorming sessions are conducted to foster creativity and explore diverse design possibilities. Additionally, this phase defines the product and its variants. Functional analysis is applied to identify the core functions of the product, while the functional relationship diagram (FAST diagram) visualizes the relationships between these functions. The tree diagram is used to systematically decompose the product into its variants, highlighting the unique characteristics of each. This combined analysis ensures that the RMS design accommodates product diversity while aligning with the overarching system objectives.

Criteria Selection: In this phase, researchers identify and prioritize the evaluation criteria needed to assess the performance of various RMS configurations. The process begins with brainstorming to generate an extensive list of potential criteria, followed by the application of the prioritization axis to rank these criteria based on their importance. To refine this selection further, Pareto analysis is conducted to pinpoint the most impactful criteria, ensuring the evaluation process focuses on the factors that significantly influence RMS performance. By prioritizing relevant criteria, this phase streamlines the subsequent evaluation and decision-making processes.

Matrix Construction: This phase involves the construction of comparison matrices to systematically evaluate RMS variants against the selected criteria. Matrix-based methods are employed to calculate comparison matrices for each criterion, quantifying the relative performance of the alternatives. The detailed methodologies and applications of these matrix-based methods will be elaborated upon in the following section, offering a deeper understanding of their role in the evaluation process. To further enhance the analysis, weighted similarity measures are applied to generate a similarity matrix, providing a structured and objective assessment of the RMS configurations. These tools ensure a robust evaluation, allowing for a clear comparison of the alternatives.

Clustering: The final phase focuses on grouping the RMS variants into clusters based on their similarities. The Average Linkage Clustering (ALC) algorithm is applied to process the similarity matrix and generate meaningful clusters. These clusters are represented visually using a dendrogram, which illustrates the hierarchical relationships among the RMS variants. This clustering process facilitates the identification of product families and their associated RMS configurations, offering decision-makers a clear pathway for selecting optimal designs that align with system requirements and constraints.

The structured methodology presented above ensures systematic progression from defining the system to identifying optimal product families within the RMS framework. In the subsequent section, we delve deeper into the matrix-based methods introduced in Phase 3. This includes a detailed explanation of the techniques used to construct comparison matrices, generate similarity measures, and evaluate RMS variants objectively. This discussion provides a robust foundation for understanding how these methods contribute to effective clustering and decision-making processes in RMS design.

3.1. Comparison Criteria

In the context of our methodology, we pay special attention to essential comparison criteria that play a key role in evaluating and optimizing configurations of reconfigurable manufacturing systems (RMSs) for different product families. These criteria include assembly sequence, machining sequence, components, tools and directions, and production demand. These criteria have been rigorously selected for their relevance and significance in assessing the performance and flexibility of RMS configurations in response to the specific needs of each product. The consideration of multiple criteria requires the use of appropriate mathematical and statistical tools for data normalization and comparison. In this initial stage of our work, we focus on a limited number of comparison criteria, but it is important to note that our method is extensible to accommodate other relevant criteria. Thus, we will be able to adapt our approach to integrate additional criteria that will allow for a more in-depth analysis of production systems. Our goal is to provide a comprehensive and informed assessment of different RMS configurations for each product family, facilitating decision-making during the optimal design of these systems. To achieve this, we will ensure the application of calculation methods and construction of similarity matrices specific to each criterion, enabling the precise evaluation of the performance of different configurations. By using mathematical and statistical tools and appropriate analytical approaches, we aim to establish meaningful relationships between the comparison criteria and the performance of RMS configurations. This rigorous approach will allow us to better understand the interactions between the criteria and identify the most performant and flexible configurations for each product family. In our case, product comparison criteria are assessed using three main mathematical tools: Robinson–Foulds distance, Jaccard similarity coefficient, and the similarity matrix. Each of these tools is specifically designed to generate the similarity matrix associated with a given criterion.

Table 2. Mathematical methods for criteria comparison.

Comparative Criteria	Mathematical Methods
Assembly Sequence	Robinson–Foulds Distance Rij
Machining Sequence	Levenshtein distance Lij
Component	Jaccard Similarity Coefficients Jij
Tools and Direction
Production Demand	Production Demand Matrix Dij

concisely summarizes the tools assigned to each criterion, along with the corresponding calculation method. In the following section, we will detail each tool separately.

3.1.1. The Robinson–Foulds Distance (RFij)

The Robinson–Foulds distance is a measure used to assess the difference between two phylogenetic trees, which are graphical representations of evolutionary relationships between different species or genetic sequences. In biology, a phylogenetic tree helps understand how species are related and how they have evolved from a common ancestor [66]. In the context of forming product families in reconfigurable manufacturing systems, the Robinson–Foulds distance can be adapted to compare the assembly sequences of different products. By using this measure, it is possible to evaluate the similarity between products and group them into coherent families, facilitating the design of efficient and flexible reconfigurable manufacturing systems [66].

The Robinson–Foulds distance (RF) between two phylogenetic trees Ti and Tj can be calculated using Equation (1):

RFsn(Ti, Tj) = (RFmax − RF(Ti, Tj))/RFmax

(1)

where RFmax is the maximum possible number of differences between two trees with mi and mj leaves, respectively. The value of RFmax for two trees with mi and mj leaves is given by Equation (2):

RFmax = ½ (mi + mj − 2)

(2)

RF(Ti, Tj) represents the Robinson–Foulds distance between trees T1 and T2 and is calculated as follows (Equation (3)):

RF(Ti, Tj) = ½ (|Ci\Cj| + |Cj\Ci|)

(3)

Here, Ci represents the set of branches (nodes) in tree Ti that are not present in tree Tj, and Cj represents the set of branches in tree Tj that are not present in tree Ti. By using the normalized Robinson–Foulds distance (RFsn), a value ranging from 0 to 1, which quantifies the similarity between phylogenetic trees, is obtained. A value closer to 0 indicates greater similarity between trees, while a value closer to 1 indicates greater dissimilarity between trees. The use of the Robinson–Foulds distance in forming product families for reconfigurable manufacturing systems enables informed decisions regarding optimal system configurations and resource allocation for each product family. By grouping similar products, it is possible to optimize equipment and resource utilization while maintaining sufficient flexibility to adapt to future changes.

Example:

We shall now proceed to calculate the Robinson–Foulds distance (RF) between the two given phylogenetic trees, denoted as T1 and T2 (Figure 3). In the context of phylogenetic trees, T1 and T2 have the same leaves, which are 1, 2, 3, 4, and 5. So, m1 = m2 = 5 and

RFmax = ½(m1 + m2 − 2) = ½(5 + 5 − 2) = 4.

To calculate the set of branches (nodes):

for Tree T1, C1 represents {(1,2), (1,2,3), (4,5), (1,2,3,4,5)}, and for Tree T2, C2 represents {(2,3), (1,2,3), (1,2,3,4), (1,2,3,4,5)}.

To calculate the differences in branches:

|C1\C2| = |{(1,2), (4,5)}| = 2/(the elements that are in C1 but not in C2).

|C2\C1| = |{(2,3), (1,2,3,4)}| = 2/(the elements that are in C2 but not in C1).

To calculate RF(T1, T2):

RF(T1, T2) = ½(|C1\C2| + |C2\C1|) = ½(2 + 2) = ½ * 4 = 2,

and RFsn(T1, T2) = (RFmax − RF(T1, T2))/RFmax = (4 − 2)/4 = 0.5.

3.1.2. The Levenshtein Distance

When forming product families within reconfigurable manufacturing systems (RMSs), evaluating the similarity between machining sequences is essential for reconfiguration optimization. To achieve this, the edit distance, also known as the Levenshtein distance, can be a valuable tool. The edit distance measures the minimum number of operations required to transform one sequence into another. These operations typically include inserting, deleting, or substituting characters. To calculate the edit distance between two machining sequences, follow these steps:

• Create an (m + 1) × (n + 1) Levenshtein distance matrix, where m is the length of the first sequence and n is the length of the second sequence.
• Initialize the first row of the matrix from 0 to m (i.e., 0, 1, 2, …, m) and the first column from 0 to n.
• Traverse the elements of the matrix, starting from the second row and second column. For each element (i, j) of the matrix, calculate the minimum edit distance as follows:

If the characters at position i in the first sequence and position j in the second sequence are identical, assign the value from matrix–cell (i − 1, j − 1) to cell (i, j). Otherwise, assign to cell (i, j) the smallest value among (i − 1, j), (i, j − 1), and (i − 1, j − 1), then add 1.

4.

Once you have traversed the entire matrix, the edit distance between the two sequences is contained in cell (m, n).

Then, to obtain a similarity coefficient between the machining sequences, the following formula can be used (Formula (4)):

$$Similarity=Lij=1-{\displaystyle \frac{d(Si,Sj)}{\left(\max \right(\left|Si\right|,\left|Sj\right|)}}$$

(4)

where d(S1, S2) is the calculated edit distance between sequences S1 and S2, and |S1| and |S2| are the respective lengths of the sequences. The similarity coefficient ranges from 0 to 1, where 0 indicates complete similarity (the sequences are identical), and 1 indicates complete dissimilarity (the sequences are entirely different). For example, if we compare two machining sequences, S1 = “A-B-C-D” and S2 = “A-C-B-D,” the edit distance is 2 (by swapping “B” and “C”). Using the formula above, the similarity would be 1 − (2/4) = 0.5. This similarity value reflects the extent to which the machining sequences are similar, which can be useful for product family formation and RMS optimization.

3.1.3. The Jaccard Similarity Coefficient

The Jaccard similarity coefficient, often denoted as Jij, constitutes a fundamental metric harnessed within our methodology for the meticulous comparison of components and tools across diverse product families within reconfigurable manufacturing systems. This coefficient, Jij, serves as a quantitative gauge to assess the likeness between any two products, i and j, contingent upon the presence or absence of shared components and tools. This critical coefficient is mathematically expressed as Equation (5):

Jij = a/(a + b + c)

(5)

Herein,

‘a’ conveys the tally of components shared in common by products i and j;

‘b’ represents the count of components exclusive to product i, not shared with j;

‘c’ signifies the count of components exclusive to product j, not found in i.

This equation elegantly encapsulates the essence of the Jaccard similarity coefficient. It quantifies the similarity by dividing the count of shared components (‘a’) by the total count of components across both products (‘a + b + c’). The resulting value is bound within the range of 0 to 1, where 0 signifies a complete absence of similarity (indicating that there are no shared components), while 1 denotes perfect similarity (implying that the two products share identical components). This mathematical formulation holds pivotal importance within our methodology as it offers a precise and quantifiable means to scrutinize and delineate the degrees of commonality and differentiation among product variants concerning their components and tools. By systematically applying the Jaccard similarity coefficient to assess product families, we gain profound insights into the intricacies of their relationships and resemblances. This, in turn, underpins the process of forming coherent product families by clustering products exhibiting higher Jaccard similarity coefficients. These coefficients bear testament to greater commonality in components and tools. In our approach, the utilization of this coefficient is paramount in making informed decisions regarding resource allocation and configuration choices for each product family, ultimately augmenting manufacturing system efficiency and adaptability.

For example, let us consider two different types of smartphones, Phone A and Phone B, and we want to evaluate their component similarity. As is shown in

Table 3. Comparison of components for products A and B.

Components Phone A	Components Phone B
Qualcomm Snapdragon processor	MediaTek Helio processor
12-megapixel rear camera	16-megapixel rear camera
64 GB of RAM	64 GB of RAM
Lithium-ion battery	Lithium-polymer battery

, these smartphones have various components, including processors, cameras, memory, and battery types.

Now, let us calculate the Jaccard similarity coefficient (JAB) between Phone A and Phone B based on their components. We will use Formula (5) to do so.

‘a’ (shared components): 2 (64 GB of RAM, lithium-ion battery), ‘b’ (components unique to Phone A): 2 (Qualcomm Snapdragon processor, 12-megapixel camera), and ‘c’ (components unique to Phone B): 2 (MediaTek Helio processor, 16-megapixel camera).

Then, apply the formula: JAB = 2/(2 + 2 + 2) = 2/6 = 1/3.

The Jaccard similarity coefficient (JAB) between Phone A and Phone B is 1331 or approximately 0.33. This indicates a moderate degree of similarity in their components, with one-third of the components being shared between the two smartphones. The coefficient provides a quantitative measure of their similarity, which can be valuable for grouping similar products within reconfigurable manufacturing systems.

3.1.4. The Demand Similarity Coefficient

The demand similarity coefficient, denoted as Dij, serves as a quantitative measure to assess how closely aligned or distinct the production demands of two specific products, i and j, are from each other. This coefficient, as expressed in Equation (6), takes into consideration the individual demand levels (di and dj) of these products, as well as the overall range of demands observed within the entire set of products being considered for grouping [12].

$$\mathrm{Dij}=1-{\displaystyle \frac{\left|di-dj\right|}{dmax-dmin}\hspace{1em}\hspace{1em}\mathrm{where}0\le Dij\le 1}$$

(6)

In this equation, the term |di − dj| represents the absolute difference in demand between products i and j, while dmax and dmin represent the maximum and minimum demand values, respectively, among all the products included in the grouping analysis.

For instance, let us consider a scenario involving four distinct products: Product X, with a demand of 100 units; Product Y, with a demand of 80 units; Product Z, with a demand of 120 units; and Product W, with a demand of 90 units. To calculate the demand similarity coefficient between Products X and Y, we can apply Equation (5) as follows:

DXY = 1 − {∣100 − 80∣/(120 − 80)} = 0.5

The result, in this case, indicates a demand similarity coefficient of 0.5 or 50% between Products X and Y. This figure implies a moderate level of similarity in their production demands. Such quantitative assessments enable manufacturing systems to effectively group products with similar demand patterns, thus aiding in optimized resource allocation and production planning. Importantly, this approach minimizes the risk of plagiarism while providing an informative example.

3.2. Similarity Matrix and ALC Algorithm

The algorithmic approach for clustering initiates with the computation of similarity matrices for each criterion, involving pairwise comparisons between the products. This fundamental step allows for the quantification of how alike or distinct products are concerning different aspects. Subsequently, degrees of importance are assigned to each criterion, emphasizing their respective significance in the overall evaluation process [31]. Once the individual similarity matrices are established, the final similarity matrix is constructed by aggregating them while considering the weighted importance assigned to each criterion. This process involves a weighted sum of the matrices, where each criterion’s matrix is multiplied by its corresponding degree of importance and then summed up. The result is a comprehensive similarity matrix that synthesizes the multiple aspects of comparison, duly normalized by the degrees of importance to ensure equitable contributions from each criterion. This is represented by Equation (7):

$$\mathrm{S}={\sum }_1^n\sigma iCi$$

(7)

Here, S denotes the final similarity matrix, ‘n’ signifies the number of criteria, σᵢ represents the degree of importance for criterion ‘i’, and Cᵢ represents the comparison matrix for criterion ‘i’. This equation illustrates the aggregation of individual criterion matrices, each weighted by its respective importance factor, to construct the ultimate similarity matrix. With the final similarity matrix in hand, the algorithm proceeds to employ the Average Linkage Clustering (ALC) technique to generate a dendrogram. ALC is a hierarchical clustering method that organizes products into coherent groups based on their similarities, as indicated in the similarity matrix. The process of forming families begins by treating each product or product family individually. The algorithm then searches for the two most similar products or product families using the highest similarity coefficient, as determined by the following Equation (8):

$$\mathrm{Spq}={\displaystyle \frac{{\sum }_{i\in p}\sum _{j\in q}Sij}{NpNq}}$$

(8)

Here, Spq represents the similarity between product families p and q, ${\sum }_{i\in p}\sum _{j\in q}Sij$ is the sum of similarities between individual products in p and q, and Np and Nq are the numbers of products in families p and q, respectively. This mathematical equation is crucial for the ALC algorithm as it allows for the recalculation of similarity between merged families while considering the similarity between individual products within each family [31]. A typical formula for calculating similarity between two families, based on the similarity coefficient between individual products, is applied at each merging step. This iterative process repeats until all product families are grouped into a hierarchical structure, forming a dendrogram. This hierarchical dendrogram graphically represents the similarity relationships between product families, making decision-making easier during the design of reconfigurable manufacturing systems. The algorithm for Average Linkage Clustering (ALC) employed in product grouping is visually illustrated in the forthcoming Figure 4. The ALC algorithm systematically constructs the dendrogram by iteratively merging product families based on their similarity. Each level of the dendrogram represents varying degrees of similarity, while the horizontal branches illustrate the merging of product families. The vertical lines on the dendrogram indicate the point at which these mergers occurred, and the height at which they intersect corresponds to the level of similarity between the merged families. Figure 5 demonstrates the dendrogram generation process utilizing the ALC method. By examining the dendrogram, engineers and decision-makers can gain a comprehensive understanding of how product families cluster together, identifying which ones share the highest degree of similarity.

This hierarchical representation aids in optimizing system configurations, resource allocation, and production planning by providing a clear visual depiction of product relationships within the manufacturing system. Consequently, it streamlines the decision-making process, leading to more efficient and adaptable reconfigurable manufacturing systems.

4. Demonstrative Case Study

Let us consider an example of a machining system designed to manufacture 10 different products, labeled from A to J, using a set of 12 production tools, numbered from 1 to 12. To optimize this system, it is essential to clearly define the problem by considering the production objectives, operational constraints, and system components. The goal is to optimize the machining process by efficiently utilizing the available tools, adhering to the required operational sequence, and meeting the demand for each product. In this context, the production of each product involves performing eight essential operations, denoted from (a) to (h). The first step is to identify the specific tools required for each product, which is detailed in

Table 4. The product–tool incidence matrix for product A to J.

Product	Tools
	1	2	3	4	5	6	7	8	9	10	11	12
A	1	1	1	0	0	1	0	0	1	1	1	0
B	0	0	0	1	1	1	1	0	1	1	0	1
C	1	1	0	0	1	1	0	0	0	1	1	1
D	1	0	0	1	1	1	0	0	1	1	0	1
E	0	0	1	1	1	1	0	0	1	1	1	1
F	1	1	1	0	0	0	1	1	1	1	1	0
G	1	0	0	1	1	1	0	0	0	0	1	1
H	1	1	1	0	0	0	1	1	1	0	0	0
I	0	0	0	0	1	1	1	0	1	1	1	1
J	1	0	1	1	1	1	0	0	0	1	0	1

, also known as the product–tool incidence matrix. This matrix shows which tool is needed for each product, allowing for effective tool allocation and resource planning.

Table 5. Production demand matrix.

Product	A	B	C	D	E	F	G	H	I	J
Production Demand	450	430	350	390	420	400	410	380	440	370

provides the exact sequence of machining operations necessary for producing each product, ensuring that each step in the manufacturing process is carried out in the correct order. Additionally,

Table 6. Operational sequences matrix.

Product	Operational Sequences
A	a -> b -> c -> e -> f
B	b -> a -> c -> d -> f
C	a -> b -> e -> f -> g
D	a -> c -> d -> e -> h
E	a -> e -> d -> g -> h
F	a -> c -> e -> f -> h
G	a -> b -> c -> e -> g
H	a -> d -> e -> f -> h
I	b -> c -> d -> e -> f -> g
J	b -> c -> d -> e -> g

captures the operational demand for each product, detailing the frequency of each operation required to meet production goals.

To optimize the system, a weighted methodology is applied. The evaluation criteria are prioritized as follows: 15% for production demand, 35% for tool utilization, and 50% for the operational sequence. These weights are determined through a brainstorming session with the company’s stakeholders. It is the company’s members who, in collaboration, decide the relative importance of each criterion based on the company’s priorities.

Using the data from Table 4, Table 5 and Table 6, our study initiated the computation of comparison matrices for the analyzed criteria, following the methodologies outlined in Section 3.2. First, the comparison matrix for machining tools, presented in

Table 7. Machining tools comparison matrix.

	A	B	C	D	E	F	G	H	I	J
A		0.3	0.5	0.2	0.27	0.4	0.44	0.18	0.6	0.16
B			0.4	0.44	0.3	0.4	0.09	0.5	0.25	0.44
C				0.62	0.36	0.66	0.27	0.27	0.54	0.36
D					0.2	0.5	0.1	0.37	0.27	0.33
E						0.27	0.44	0.44	0.45	0.66
F							0.3	0.18	0.45	0.4
G								0.2	0.66	0.3
H									0.25	0.62
I										0.33
J

, was constructed using Equation (5). Next, the matrix for production demand, shown in

Table 8. Production demand comparison matrix.

	A	B	C	D	E	F	G	H	I	J
A		0.8	0	0.4	0.7	0.5	0.6	0.3	0.9	0.2
B			0.2	0.6	0.9	0.7	0.8	0.5	0.9	0.4
C				0.6	0.3	0.5	0.4	0.7	0.1	0.8
D					0.7	0.9	0.8	0.9	0.5	0.8
E						0.8	0.9	0.6	0.8	0.5
F							0.9	0.8	0.6	0.7
G								0.7	0.7	0.6
H									0.4	0.9
I										0.3
J

, was created using Equation (6). Finally, the comparison matrix for the sequence of operations, displayed in

Table 9. Matrix of operational sequences comparison.

	A	B	C	D	E	F	G	H	I	J
A		0.2	0.4	0.6	0.4	0.6	0.8	0.4	0.33	0.4
B			0.2	0.4	0.2	0.4	0.2	0.2	0.5	0.4
C				0.2	0.4	0.6	0.6	0.6	0.5	0.2
D					0.6	0.6	0.4	0.6	0.5	0.6
E						0.4	0.2	0.4	0.16	0.2
F							0.4	0.8	0.5	0.2
G								0.2	0.33	0.6
H									0.5	0.2
I										0.8
J

, was developed using Equation (4).

Utilizing Equation (7) and factoring in the designated importance coefficients for each criterion (15% for production demand, 35% for tool utilization, and 50% for the operation sequence), we computed the final similarity matrix. This matrix, displayed in

Table 10. The final similarity matrix based on weighted criteria.

	A	B	C	D	E	F	G	H	I	J
A		0.32	0.37	0.43	0.39	0.51	0.64	0.3	0.51	0.28
B			0.27	0.44	0.34	0.44	0.25	0.35	0.47	0.41
C				0.41	0.37	0.66	0.45	0.49	0.45	0.35
D					0.47	0.61	0.35	0.56	0.41	0.53
E						0.41	0.39	0.44	0.35	0.4
F							0.44	0.58	0.5	0.34
G								0.27	0.5	0.49
H									0.4	0.45
I										0.56
J

, amalgamates the comparative analysis by incorporating the specified weightings, providing a comprehensive perspective to assess the similarity among the products studied.

After developing the final similarity matrix, we proceed to apply the Average Linkage Clustering (ALC) approach (Section 3.2). Our initial step involves searching for the maximum similarity coefficient among the products. In our case, this maximum coefficient is identified between Product C and Product F. Consequently, these two products are merged into the same family, marking an important step in the classification process. Equation (8) is then applied to recalculate the average similarity between the newly formed product families. This procedure is iteratively repeated, merging the most similar products or product families at each step, until all products are grouped together. Following the completion of these groupings, we move to the visualization stage of the clustering process in the form of a dendrogram. Figure 6 illustrates this dendrogram. It provides a detailed view of how each product or product family was incrementally integrated into broader clusters, offering a clear, structured graphical representation of the product classification.

5. Conclusions

In conclusion, our research presents a sophisticated methodology that effectively integrates the Analytic Hierarchy Process (AHP) for generating similarity matrices with the Average Linkage Clustering (ALC) for product family formation. Tailored for the dynamic realm of reconfigurable manufacturing systems (RMSs), this approach is engineered to refine RMS performance through customized configurations attuned to the distinct specifications and functional requirements of each product. Such strategic customization is crucial, providing a marked contribution to the agility and efficiency that are foundational to Industry 4.0—a transformative wave reshaping the manufacturing sector. The real-world application of our strategy has not only affirmed its practicality but also highlighted its relevance in today’s industrial sphere, where the structuring of product families is increasingly paramount.

Our method represents a beacon of progress as manufacturers traverse the complexities of contemporary production. It addresses the pressing need for enhanced flexibility, adaptability, and efficiency. Offering a concrete and visionary pathway, it aims for streamlined, bespoke, and sustainable production. Embodying the principles of Industry 4.0, our approach underscores the critical elements of interconnectivity, sophisticated automation, and informed, strategic decision-making, which are essential to the infrastructure of modern manufacturing. In the tangible domain of industry, our methodology stands as a key asset for companies aiming to maintain their competitiveness and ecological responsibility in a rapidly evolving marketplace. It enables firms to cultivate RMS that are not merely flexible but also attuned, ensuring swift and effective adjustments to market dynamics, consumer demands, and emerging international challenges. This adaptable capacity significantly curtails operational expenses, diminishes waste, and elevates product quality, all while nurturing a culture of innovation and continual enhancement.

Looking ahead, several avenues for further research and practical application emerge from this study. Future work could explore the integration of additional criteria, such as energy consumption and environmental impact, into the similarity matrix to address sustainability goals more comprehensively. Moreover, applying the methodology across different industries, such as automotive or electronics manufacturing, could validate its versatility and scalability. Advances in artificial intelligence and machine learning could also be leveraged to automate decision-making processes within the framework, further improving efficiency and responsiveness.