This chronological and bibliometric analysis aims to explore the chronological relationships among fuzzy sets (Table 2, Step 1). Based on this aim, the following main research questions (MRQ) and sub-questions (RQ) are defined: MRQ1:

How have fuzzy sets evolved over time?

Since this research focuses on fuzzy sets without specific limitations, the main search term is “fuzzy set”. According to PICOC (Population, Intervention, Comparison, Outcomes, Context) (Kitchenham and Charters, 2007), the search is defined as follows:

Population – research papers on fuzzy sets. Intervention – currently, only a partial analysis of the evolution of fuzzy sets exists. Therefore, there is a need for a more comprehensive study of fuzzy sets evolution. Comparison – not applicable. Outcomes – the measurable outcome is not considered in the analysed papers. Context – academic context, scientific papers on fuzzy sets.

The chosen keyword was transformed into the search string and processed on the Web of Science (WoS) digital library. Table 3 presents the search limitations. The categories were limited to Computer Science to focus on this research area and to restrict the final quantity of papers, since initially 44.834 papers were found, which would be difficult to process manually. Moreover, after excluding other categories, the number of scientific papers remains big enough and suitable for the bibliometric analysis.

This study was conducted in May 2023, supplemented with the newest publications in January 2025 and additionally supplemented in September 2025. There were no restrictions by year in the search. The document type has been limited to articles, proceedings papers and reviews. The WoS digital library has been chosen for this chronological and bibliometric analysis based on the experience published in Kalibatiene and Miliauskaitė (2021). In addition, the authors of Donthu et al. (2021) recommend deciding on one suitable digital library to avoid the impact of the consolidation of different formats of references and possible human errors, which ensures the construct validity of the analysis.

The research paper selection was conducted in two branches as follows (Fig. 1):

After excluding non-relevant papers for the related works analysis, a set of 95 relevant papers was obtained for further reading of their abstracts. The same IC and EC were applied when reading abstracts. Finally, 12 research papers were selected for full reading of their text. An initial set of papers, a set of excluded papers and a set of included papers are presented in GitHub (https://github.com/Jolantux13/A-Chronological-and-Bibliometric-Analysis-Fuzzy-Sets (accessed on February 2025)). The results of the related work analysis are presented in Table 1. Moreover, such well-defined qualitative IC and EC allow us to ensure the validity of the results (Alves et al., 2010).

To answer the defined research questions and develop a keyword map on fuzzy sets, a bibliographic data mapping and visualization tool, VOSviewer (https://www.vosviewer.com/ (accessed on February 2025)) was employed. This tool implements the chosen techniques and allows us to perform co-word and co-citation analyses (Table 2).

A co-word analysis based on the developed keyword map helps enrich understanding of the thematic clusters and predict future research trends (Donthu et al., 2021). In this research, notable keywords and future research directions (RQ2, RQ3) are identified. The words in the co-word analysis were derived from “All keywords”. The co-word analysis assumes that words appearing together frequently have a thematic relationship with each other. Donthu et al. (2021) have identified downsides of co-word analysis, such as multiple contexts, a high level of abstraction, etc. To mitigate the potential downsides of co-word analysis and perform data cleaning (Van Eck and Waltman, 2019), the thesaurus (https://github.com/Jolantux13/A-Chronological-and-Bibliometric-Analysis-Fuzzy-Sets (accessed on February 2025)) was created by the authors of this paper according to the following rules:

The developed thesaurus includes 569 items for word merging or exclusion. Its inclusion into data extraction allows us to ensure the validity of the results (Ampatzoglou et al., 2019). Finally, VOSviewer identified 40.371 keywords, 374 of which are found at least 20 times in the chosen 23 712 scientific papers.

A co-authorship analysis, which is used to examine intellectual collaboration among scholars (Donthu et al., 2021; Acedo et al., 2006; Cisneros et al., 2018), was applied to analyse the relationship among countries (RQ4). The tools used for the current analysis are shown in Table 4.

As proposed by the guidelines of Petersen et al. (2015), Wohlin et al. (2012), this analysis covers internal, construct, external and conclusion validity. Ampatzoglou et al. (2019) presented a more extended list of validity threats for the secondary studies. Some of them are described while presenting the methods of this analysis. Here we discuss only the four main threats to validity.

Construct validity refers to the operational measures for the concept being studied, i.e. fuzzy set extensions. Since we are interested in analysing all possible extensions of fuzzy sets, a general term “fuzzy set” was chosen for the search string without any specific limitation of the studied concept. Our research questions and search string may not be able to completely cover all papers on fuzzy set extensions. However, according to Donthu et al. (2021), we have obtained a sufficient set of papers (Table 3) for the analysis, and can provide valuable findings for researchers and practitioners with its overview.

Internal validity refers to whether an experimental condition affects the results. The individual researcher’s bias influences the selection of literature reviews and surveys for the related works analysis (the second branch in Fig. 1, Section 3). To mitigate the bias of researchers, the authors of this paper clearly defined the search strategy, carefully examined titles and abstracts of primarily found papers and assessed the obtained results together. Also, the authors searched only one online digital library for the bibliometric analysis, as recommended in Donthu et al. (2021).

External validity refers to the generalizability of the results. Though we have limited the search results to Computer Science to reduce the final number of papers (the initial set contained 42 640 papers that would be manually difficult to process), it (∼58%) still remains sufficient and suitable for the bibliometric analysis and valid for other research areas. Other research areas actively involved in the development and application of fuzzy set extensions are Engineering (∼29%) and Mathematics (∼13%). However, their inclusion in the analysed set of papers only slightly influences the final results by adding only a few new fuzzy set extensions.

Conclusion validity deals with the degree to which conclusions are reasonable within the data collected, i.e. interpretive validity. In this paper, a potential threat to conclusion validity is the interpretation of the extracted data presented by keyword maps. To guarantee the conclusion’s validity to a reasonable extent, we have reviewed other related bibliometric surveys on fuzzy set extensions. The conclusions drawn do not contradict the previous studies.

To ensure the replicability of this research, the authors have explicitly defined steps performed in the chronological and bibliometric analysis, developed a thesaurus, and provided evidence about those findings in GitHub.

Figure 2 shows the chronological distribution of the scientific papers published on the topic of fuzzy sets in the period from 1990 to September 2025 (RQ1).

Although the trendline in the number of publications is increasing every year (in Fig. 2, dotted line), the growth is wavy. In the last 14 years (2012–2025), the number of publications (14 620, i.e. 60%) exceeded the total number of publications from 1990 to 2011 (i.e. 22-year interval) (9 828, i.e. 40%). This indicates the increasing interest of researchers in fuzzy sets over the last period. Also, two visible waves, with peaks in 2009 (987 publications) and 2020 (1 333 publications), are observed in Fig. 2.

A keyword map of fuzzy sets was developed and visualized with VOSviewer (Fig. 3). It should be analysed from two perspectives: 1) year perspective, and 2) occurrence perspective. From a year perspective, VOSviewer coloured and visualized keywords according to Average Publication Year (APY) of the paper, in which a keyword occurs (Van Eck and Waltman, 2019). In the Fig. 3 legend, the APY interval from the oldest to the latest is coloured from blue to yellow. Also, larger bubbles represent more frequently occurring keywords. Smaller bubbles represent keywords found less frequently in the analysed articles. Based on the colour and size of the bubbles, we can see whether the keyword is relevant in today’s context. For example, small and yellow bubbles represent newly appearing keywords in the analysed articles, while small and blue bubbles show keywords that are disappearing and have been analysed for a long time and little. Note that this explanation should be used to decode all the keyword maps presented in this paper (see Figs. 3–7).

From the occurrence perspective, VOSviewer identified, sized and visualized keywords according to their density of occurrence. Keywords with the biggest bubbles have the largest occurrence.

Currently, the newest fuzzy set extensions (RQ2) found in the keyword map of fuzzy sets (Fig. 3) are as follows: Fermatean fuzzy set (32, APY = 2021.32), T-spherical fuzzy set (25, APY = 2021), probabilistic hesitant fuzzy set (21, APY = 2020.7), spherical fuzzy set (71, APY = 2020.51) and q-rung orthopair fuzzy set (91, APY = 2020.30). Their occurrence comparing to the whole number of the analysed papers (23 712) is rather small, however, observable, which shows the future direction of the analysed topic (RQ3). Also, among the newest keywords there are multi-criteria decision making (MCDM) methods, like WASPAS, EDAS, COPRAS and Best-Worst Method (BWM), which shows a future direction in the application of fuzzy sets (RQ3).

From this initial keyword map of fuzzy sets, we have extracted 65 extensions of fuzzy sets (RQ2), which are visualized by VOSviewer in Fig. 4 and summarized in Table 5. From this analysis, we can see that five most occurring fuzzy sets are as follows: intuitionistic fuzzy set (1 513, APY = 2015.22), interval type-2 fuzzy set (501, APY = 2016.20), type-2 fuzzy set (463, APY = 2014.22), hesitant fuzzy set (431, APY = 2017.97), and interval-valued fuzzy set (290, APY = 2012.18). These sets are presented by the biggest bubbles. Their APYs are older (i.e. green coloured), since they are the main and oldest extensions of fuzzy sets used in the research papers until now.

It is also worth paying attention to the medium and smaller bubbles in blue. These bubbles represent fuzzy set extensions, which APY tend to be oldest, and they are less mentioned in current papers, like convex fuzzy set (30, APY = 2001.97), eigen fuzzy set (8, APY = 2005.5), balanced fuzzy set (6, APY = 2005.5), l-fuzzy set (53, APY = 2006.74), three-dimensional fuzzy set (6, APY = 2008.83), and normal fuzzy set (5, APY = 2010.2), etc. They have been less used in current papers or have evolved into other fuzzy set extensions.

Table 5 presents all found fuzzy set extensions and the most occurring keywords with them. Keywords, like multi-criteria decision making (MCDM), multi-attribute decision making (MADM), and group decision making, are found with the majority of fuzzy set extensions. Therefore, they are generalized into a “decision-making” keyword for the sake of simplicity. The whole version of Table 5 with relevant references is presented in GitHub (https://github.com/Jolantux13/A-Chronological-and-Bibliometric-Analysis-Fuzzy-Sets (accessed on February 2025)).

In Table 5, a “decision making” (dm) keyword was found within 41 fuzzy set extensions. Additionally, 19 particular MCDM methods are observed 81 times (i.e. several MCDM methods can be repeated within the same fuzzy set extension). A keyword “select” variations, like “supplier selection”, “facility location selection”, etc., are found within 14 fuzzy set extensions. Similarly, a keyword “diagnos” variations, such as “medical diagnosis” (found 8 times), “fault diagnosis” (found 1 time), “diagnosis” (found 1 time), are observed within 10 fuzzy set extensions. The most popular keywords found within fuzzy set extensions are as follows: decision making (41), topsis (31), correlation coefficient (18), vikor (15), todim (11), medical diagnosis (8) and supplier selection (8).

Also, according to the fuzzy set extensions occurrence (see Fig. 3), the most commonly occurring fuzzy set extensions, like type-2 fuzzy set, Pythagorean fuzzy set, intuitionistic fuzzy set, etc., have large communities of keywords (Table 5). These rich keyword communities show the popularity of these fuzzy set extensions and their wide applicability. Contrary, less occurred fuzzy set extensions, such as Fermatean fuzzy set, Pythagorean fuzzy soft sets, etc., have fewer related keywords. These poor or middle keywords communities indicate the lower applicability of these fuzzy set extensions.

We can summarize that MCDM tasks remain the most important and significant real-life problems with uncertain features of input data due to the complex information collection process and influence of different interests of stakeholders (Beg et al., 2022). So, to present the information and handle with the imprecision of data, different fuzzy set extensions and their theories have recently been developed.

Based on Fig. 4, the newest fuzzy set extensions (RQ2) with APY > 2020 are presented in Table 6. Their occurrences comparing to the whole number of the analysed papers (21 379) are rather small, which shows a future direction and new extensions of fuzzy sets (RQ3). Moreover, the newest APYs of those fuzzy set extensions show a greater interest in them. Among fuzzy set extensions that have engaged greater interest from scientists in the analysed papers, we observe the following sets: q-rung orthopair fuzzy set, spherical fuzzy set and Fermatean fuzzy set.

Links in the keyword map show relationships between the keywords. These links can be characterized by the distance and the strength, showing the number of papers in which two keywords occur together. Thicker lines and shorter distances represent stronger relationships between the keywords.

Based on the developed keyword map of fuzzy sets (Fig. 4), it was determined that among all found extensions of fuzzy sets, there are only 8.60% relationships and their link strength falls in the interval [ 1 ; 51 ]. In addition, 44.02% of existing links have a strength equal to 1,47.85% have strengths ranging in the interval ( 1 ; 10 ], and only 8.13% have strengths ranging in the interval ( 10 ; 51 ]. Therefore, for better visualization, three co-occurrence matrices of fuzzy set extensions are developed as follows: 1) with link strengths 1 (Fig. 5); 2) having link strengths in the interval ( 1 ; 10 ] (Fig. 6); and 3) with link strengths in the range ( 10 ; 51 ] (Fig. 7). Note that these three matrices are symmetric. However, for better visualization, only one part of the symmetry is presented in Fig. 5, Fig. 6 and Fig. 7. In Fig. 5, the axes present fuzzy set extensions, and the bubbles – their links with strength 1. It is observed that such links exist between fuzzy set extensions with different occurrences and APY, which are shown in the brackets after the title of a fuzzy set extension. The fuzzy set extensions with the most links having a link strength of 1 are as follows: neutrosophic set (7), hesitant fuzzy set (6), type-2 fuzzy set (6), q-rung orthopair fuzzy set (6), Pythagorean fuzzy set (6). This finding shows the greater popularity and development of the found sets in the analysed papers.

In Fig. 6, the axes present fuzzy set extensions, the bubbles – links with strengths in the range ( 1 ; 10 ]. The bubbles are sized and coloured according to the link strength (Fig. 6).

It is observed that such links exist between fuzzy set extensions with different occurrences and APY, which are shown in the brackets after the title of a fuzzy set extension. As can be seen, the majority (43%) of the bubbles represent a link strength equal to 2. The size distribution of the bubbles is as follows: link strength equal to 3–24%; link strength equal to 4–9%; link strength equal to 5–8%; link strength equal to 6 and 7–4%; link strength equal to 8–5%; link strength equal to 9–2%; and link strength equal to 10–1%. From Fig. 6, we found that the following fuzzy sets have the biggest link strength in the range ( 1 ; 10 ]: intuitionistic fuzzy set – q-rung orthopair fuzzy set (10), pythagorean fuzzy set – q-rung orthopair fuzzy set (9), intuitionistic fuzzy set – intuitionistic fuzzy soft set (9), intuitionistic fuzzy set – fuzzy rough set (8), intuitionistic fuzzy set – fuzzy soft set (8), intuitionistic fuzzy set – interval neutrosophic set (8), spherical fuzzy set – t-spherical fuzzy set (8), type-1 fuzzy set – type-2 fuzzy set (8). The found link strengths between fuzzy sets in most cases can be explained by the fact that some fuzzy sets are extensions of other fuzzy sets, like intuitionistic fuzzy set and intuitionistic fuzzy soft set, spherical fuzzy set and t-spherical fuzzy set, etc.

In Fig. 7, the axes present fuzzy set extensions, and the bubbles – links with strengths in the range ( 10 ; 51 ]. The bubbles are sized and coloured according to the link strength. It is observed that such links exist mainly between fuzzy set extensions with large occurrence and older APY. Exceptions are as follows: intuitionistic fuzzy rough set with small occurrence, and picture fuzzy set and Pythagorean fuzzy set with newer APY. Therefore, their link strengths are weaker compared to other link strengths. Moreover, in Fig. 7 we can observe the same trend – the link between fuzzy sets in most cases is explained by the fact that some fuzzy sets are extensions of other fuzzy sets, such as type-2 fuzzy sets, interval type-2 fuzzy sets (Snyder, 2019).

VOSviewer has created a country/region occurrence map from the analysed papers as presented in Fig. 8. VOSviewer has identified 124 countries, of which 93 countries have published at least 5 papers on the analysed topic. Note that WoS indicates not only countries but also regions. For the sake of simplicity, below we write “countries”.

The five most popular countries found in the map are as follows: China (6 464, APY = 2014.3), India (2 281, APY = 2015.31), USA (2 125, APY = 2008.48), Taiwan (1 383, APY = 2010.23) and Spain (1 369, APY = 2012.27), where the number near the country indicates the number of papers published by the country and APY. The APY of those countries is older, since they have started publishing papers on fuzzy sets earlier. The newer countries-contributors to the fuzzy set topic are as follows: Yemen (18, APY = 2021.44), Iraq (70, APY = 2019.61), Palestine (7, APY = 2019.5), Pakistan (730, APY = 2019.02), Ecuador (9, APY = 2018.87), Nigeria (39, APY = 2018.5), Estonia (6, APY = 2018.5), Saudi Arabia (471, APY = 2018.37), and Peru (6, APY = 2018.2). As can be seen in Fig. 8, links represent relationships between countries. Thicker links and shorter distances present stronger relationships. Based on the developed country map of fuzzy sets (Fig. 10), all link strengths range in the interval [ 1 ; 263 ] as follows: 38.04% – equal to 1, 46.82% – ( 1 ; 10 ], 13.12% – ( 10 ; 50 ], and 2.02% – ( 50 ; 263 ].

The co-occurrence of countries with link strengths in the interval (50; 263] is presented in Fig. 9. They can be understood as strong relationships between countries in fuzzy set research. It can be seen that the three strongest relationships are between the USA and China (263), Poland and Canada (229), and Canada and China (172). Links with strength ( 10 ; 50 ] can be treated as developing relationships, for example, between South Korea and Poland (40), Spain and Italy (31), Turkey and England (29), etc. Links with strength ( 1 ; 10 ] can be understood as weak, and it is not clear how they will develop. Also, from these relationships, we can see that the countries with the most links are as follows: USA (66), China (68), England (62), India (61), and Spain (60).

The detailed chronological analysis of 15 mostly publishing countries participating in the fuzzy set research is presented in Fig. 10 (RQ1, RQ4). According to this visualization, all 15 countries can be classified into three classes according to their involvement into the research on fuzzy sets as follows: 1) from the beginning of the period and having small increase of papers during the entire period (like England, USA, Japan); 2) from the beginning of the period and having significant increase of papers at a certain point of time (like China, India); 3) from a certain point of the period and having some increase of papers (like Pakistan, Spain). Fig. 10 shows that the intensive participation of China has a significant impact on the overall publication of papers on fuzzy sets (Fig. 2).

To answer this research question, we have explored the found keywords from Fig. 3 and Table 4 from the artificial intelligence perspective. The obtained results are presented in Fig. 11. The developed Fig. 11 shows significant relationship between fuzzy sets and artificial intelligence (found in 60 papers of 712 analysed in this topic), machine learning (found in 109 papers of 712 analysed in this topic), neural networks (found in 28 papers of 712 analysed in this topic) and deep learning (found in 37 papers of 712 analysed in this topic). The relationship between fuzzy sets and artificial intelligence is long-standing and old since its (APY = ∼2013). However, currently this concept is converting into explainable artificial intelligence (API = ∼2023). The same is a long-standing and old relationship exists between fuzzy sets and machine learning (API = ∼2016) and neural networks (API = ∼2018), where the newest development is fuzzy neural networks (API = ∼2024). The relationship between fuzzy sets and deep learning is a new one and current (API = ∼2023). In this context, they all form the new concept of digital transformation (API = ∼2023). Alkan and Kahraman (2023b) present the concept of a “digital supply chain” by proposing an interval-valued Fermatean fuzzy analytic hierarchy process method for the digital transformation in the supply chain. Otay et al. (2023) introduced the Interval Valued Circular Intuitionistic Fuzzy Analytic Hierarchy Process method and applied it to evaluate digital transformation projects with a complex nature. In Yang et al. (2024), authors proposed to perform the evaluation of digital transformation effectiveness in retail enterprises through multi-attribute group decision making, employing a spherical fuzzy number EDAS and utilising the Hamming distance and logarithmic distance measures. Mishra et al. (2024) proposed to rank the alternatives by applying a picture fuzzy extension of the alternative ranking order model, accounting for a two-step normalization approach. The applicability of their approach was illustrated by assessing digital transformation in higher education institutions.

Also, from Fig. 11 we can observe that the conjunction of fuzzy sets, artificial intelligence, and decision making is highly related. These two clusters of artificial intelligence and decision making in the form of different MCDM methods can be viewed in Fig. 3, also. The most popular MCDM methods used with fuzzy sets are as follows in Table 7. We have found the main 17 MCDM methods, which are extended with a particular type of fuzzy set, to handle the uncertainty and imprecision inherent in many decision-making problems. As can be seen, the most used and extended with fuzzy sets are TOPSIS, AHP, VIKOR, TODIM and ELECTRE. Those methods have the biggest number of different extensions with fuzzy sets, while others have been less extended because of their narrower applicability. Moreover, some methods are used in conjunction to solve some decision-making problems, like COPRAS and EDAS (Ganie et al., 2024a), MOLTIMOORA and BWM in Yang et al. (2020), Nemati (2024); SAW and ARAS in Gül (2021b); AHP and TOPSIS in Alkan and Kahraman (2023a), Swethaa and Felix (2023), etc.

Emerging information technologies, such as cloud computing, big data, and artificial intelligence, have changed MCDM methods by evolving them into data-driven MCDM (DDMCDM) methods, which apply different machine learning techniques to process a large amount of data when making decisions (Fu et al., 2021). Combination of MCDM methods with machine learning techniques faces two main challenges (Fu et al., 2021) as follows: 1) how to select the appropriate machine learning algorithm for a specific MCDM problem, and 2) how to build a bridge between the appropriate machine learning algorithm and the evidential reasoning approach.

Also, authors Ma and Li (2024) stressed the challenges of the supplier quality evaluation because of a large amount of qualitative and quantitative data, multi-dimensional attributes, a large number of items and suppliers, etc. Even though a number of combinations of MCDM methods (Ma and Li, 2024; Zavadskas et al., 2016) have been proposed, not all existing aggregation methods are efficient/effective for the supplier evaluation problem with high dimensions and big data (Ma and Li, 2024). Machine learning techniques may be applicable here to overcome the drawbacks of existing MCDM methods, especially when a large quantity of real business data is available (Ma and Li, 2024). Therefore, Ma and Li (2024) proposed to evaluate the alternatives according to the selected criteria by applying six MCDM methods, like TOPSIS, VIKOR, PROMETHEE with a usual type preference function, PROMETHEE with a Gaussian type preference function, MOORA with the ratio system, and MOORA with the reference approach, and then, construct an aggregated model, in which disagreements are integrated with the help of loss functions from machine learning, like L2 loss, L1 loss, Log-cosh loss, and M-estimator loss. Finally, authors make a comparison analysis to find the appropriate loss function and the final aggregated evaluation.

The presented bibliometric study on fuzzy sets can offer several practical implications for practitioners, especially in AI applications and decision-making systems. They are as follows: 1.

Toward explainable AI, practitioners can extend their AI-based system design by adding a supplementary fuzzy logic component to better explain the obtained inference results.