Abstract
Neutrosophic set, initiated by Smarandache, is a novel tool to deal with vagueness considering the truth-membership T, indeterminacy-membership I and falsity-membership F satisfying the condition \(0\le T+I+F\le 3\). It can be used to characterize the uncertain information more sufficiently and accurately than intuitionistic fuzzy set. Neutrosophic set has attracted great attention of many scholars that have been extended to new types and these extensions have been used in many areas such as aggregation operators, decision making, image processing, information measures, graph and algebraic structures. Because of such a growth, we present an overview on neutrosophic set with the aim of offering a clear perspective on the different concepts, tools and trends related to their extensions. A total of 137 neutrosophic set publication records from Web of Science are analyzed. Many interesting results with regard to the annual trends, the top players in terms of country level as well as institutional level, the publishing journals, the highly cited papers, and the research landscape are yielded and explained in-depth. The results indicate that some developing economics (such as China, India, Turkey) are quite active in neutrosophic set research. Moreover, the co-authorship analysis of the country and institution, the co-citation analysis of the journal, reference and author, and the co-occurrence analysis of the keywords are presented by VOSviewer software.
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1 Introduction
To dispose uncertain or vague information in decision making, Zadeh (1965) presented the fuzzy set (FS) that characterized by a membership function which assigns to each target a membership value ranging between 0 and 1. Alcantud and Díaz (2017) defined the notion of sequential application of fuzzy choice functions, and investigated its normative implications under related concepts of rationalizability. Alcantud and Calle (2017) discussed the problem of collective identity in a fuzzy environment. Intuitionistic fuzzy set (IFS), initially proposed by Atanssov (1986), is associated with each element of a universe not only take membership function but also non-membership (whose sum is less than or equal one). Hence it can describe more precisely and definitively than fuzzy set. However, it can only deal with incomplete and uncertainty information but not the indeterminate and inconsistent information which exists commonly in real-life. Therefore, Smarandache (1998) originally proposed the concept of a neutrosophic set (NS) from philosophical point of view. According to the definition of a NS presented by Smarandache, a NS A in a universal set X is characterized independently by a truth-membership function \(T_A(x)\), an indeterminacy-membership \(I_A(x)\) and a falsity-membership \(F_A(x)\). The functions \(T_A(x), I_A(x)\), and \(F_A(x)\) in X are real standard or nonstandard subsets of \(]^-0,1^+[\), i.e., \(T_A(x):X\rightarrow ]^-0,1^+[, I_A(x):X\rightarrow ]^-0,1^+[,\) and \(F_A(x):X\rightarrow ]^-0,1^+[\). Smarandache (1999) and Wang et al. (2010) further proposed a single valued neutrosophic set (SVNS), by modifying the conditions \(T_A(x), I_A(x)\) and \(F_A(x)\in [0, 1]\) and \(0\le T_A(x) + I_A(x) + F_A(x)\le 3\), which are more suitable for solving scientific and engineering problems.
Neutrosophic set has attracted the attention of numerous scholars in a short period of time because of its wide scope of description cases are very common in diverse real-life issue, and this new set boosts the management of vagueness caused by neutrosophic scope. A deep revision of the specialized literature shows the rapid growth and serviceability of NS, which has been expanded to diverse point of visual angle, quantitatively and qualitatively.
Given the neutrosophic-related research has been lasted for 20 years and is increasingly attracting researcher’s interests, it is necessary for us to make a comprehensive overview toward this domain to seek for some potential patterns or scientific development path over the NS research. Bibliometric analysis is a widely used method to depict the development of a certain field (Merigó et al. 2016). Although there is a survey related to NS (Nguyen et al. 2017; El-Hefenawy et al. 2016; Rivieccio 2008), it only focused on reviewing the neutrosophic set in biomedical diagnoses. Meanwhile, it did not provide any bibliometric analysis for NS-related research. Therefore, in this paper, we conduct a bibliometric analysis on NS-related research to fill in this gap.
The paper is organized as follows. Section 2 reviews seven main research points for NS. Section 3 depicts the patterns and dynamics of neutrosophic research along with six aspects: (1) annual trends; (2) country level; (3) institutional level; (4) publishing journals; (5) highly cited papers; and (6) research landscape. Moreover, (1) the co-authorship analysis of the country and institution; (2) the co-citation analysis of the journal, reference and author; (3) the co-occurrence analysis of the keywords are presented by VOSviewer software. Conclusions with some findings are drawn in the last section.
2 Literature review
Just as denoted by the distinguished British philosopher and Nobel Laureate, Russell (1923), “ All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence,” the relationship between precision and uncertain has puzzled scholars and philosopher for centuries. Lukasiewicz, born in Polish, introduced the multi-valued logic that extended the range of truth values to all real numbers in [0, 1] and thus led to an inexact reasoning technique called possibility theory (Lukasiewicz 1930). Later, Black (1937) defined the first simple fuzzy set and outlined the basic ideas of fuzzy set operations. Zadeh (1965) rediscovered fuzziness and extended the work on possibility theory into a formal system of mathematical logic. Nearly 30 years later, Smarandache stated that “Neutrosophy is a new branch of philosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra” (Smarandache 1998). Neutrosophy is a multiple value logic that specifies classical logic, fuzzy logic, and imprecise probability. Neutrosophy is closer to human rational as it describes the imprecision of knowledge or linguistic inaccuracy established by several observers. Every event in the neutrosophy theory has certain degree of truth, falsity degree, and an indeterminacy degree, which should be considered independently from each other. The realism of the neutrosophic paradigm is justified by well-established social phenomena that include different sports (win–tie–defeat) and voting situations (yes-abstention-no). Alcantud and Laruelle (2014) gave real examples and an axiomatic basis for the voting interpretation of the truth-indeterminacy-falsity setting. As he gave a systematic paradigm to use and operate over NS, Professor Smarandache is deemed as the “master of neutrosophic logic.”
In the following, we main focus on seven parts to review the whole development of NS.
2.1 The extensions of neutrosophic set
The core idea of modeling such a neutrosophic situation has been expanded together with the previous methods and tools to the following new cases:
to handle the neutrosophic in qualitative environments in which information is linguistic form
to manage the truth-membership, indeterminacy-membership and falsity-membership that are not exactly defined but expressed by interval-values, intuitionistic fuzzy sets, triangular fuzzy sets, cubic sets, bipolar fuzzy set, trapezoidal fuzzy sets, or hesitant fuzzy set
to deal with the inadequacy of the parameterized by combining soft set
to cope with the lower and upper approximations by fusing with rough set
These extensions are further detailed in Table 1.
In the future, although some extensions may be proposed, they will not be published in a famous journal. Because of combining NS with other mathematics tools will not obtain special or novel results. Hence, the research directions will focus on the existing fundamental extensions of NS such as SVNS, INS, SNS.
2.2 Aggregation operators
Multitudinous aggregation operators used for decision making are based on the geometric mean, arithmetic mean, and integrals. A number of popularized operators have been developed to aggregate diverse kinds of evaluation information. We will follow with interest in the aggregation operators under neutrosophic set and its extension environment.
2.2.1 Algebraic aggregation operators
In real world decision situation, the aggregation problems in the MCDM are solved using the scoring techniques such as the weighted aggregation operator based on multi attribute theory. The classical weighted aggregation is usually known by the weighted average (WA) or weighted geometric (WG) or simple additive weighting method. A very common aggregation operator is the ordered weighted averaging (OWA) operator or ordered weighted geometric (OWG) which provides a parameterized family aggregation operator between the minimum, the maximum, the arithmetic average, and the median criteria whose originally introduced by Yager (1988).
The related neutrosophic Algebraic aggregation operators are shown in Table 2.
2.2.2 Bonferroni mean aggregation operators
The Bonferroni mean (BM) was originally introduced by Bonferroni (1950). The classical Bonferroni mean is an extension of the arithmetic mean and its generalized by some researchers based on the idea of the geometric mean (Sun and Sun 2012). The BM differs from the other classic means such as the arithmetic, the geometric and the harmonic because this mean reflect the interdependent of the individual criterion meanwhile on the classic means the individual criterion is independent, which makes BM very useful in various application fields.
The related neutrosophic Bonferroni mean aggregation operators are shown in Table 3.
2.2.3 Einstein aggregation operators
The related neutrosophic Einstein aggregation operators are shown in Table 4.
2.2.4 Power aggregation operators
Power average (PA), originally proposed by Yager (2001), uses a non-linear weighted average aggregation tool and a power ordered weighted average (POWA) operator to provide aggregation tools which allow exact arguments to support each other in the aggregation process. The weighting vectors of the PA operator and the POWA operator depend on the input arguments and allow arguments being aggregated to support and reinforce each other. In contrast with most aggregation operators, the PA and POWA operators incorporate information regarding the relationship between the values being combined. Recently, these operators have received much attention in the literature.
The related neutrosophic Power aggregation operators are shown in Table 5.
2.2.5 Hamacher aggregation operators
The related neutrosophic Hamacher aggregation operators are shown in Table 6.
2.2.6 Cloud aggregation operators
The normal cloud (NC) model, which is based on probability theory and fuzzy set theory (Yang et al. 2014), was originally proposed by Li et al. (1995, 2004) as a novel cognition model of uncertainties in response to the randomness of membership functions. Wang et al. (2014) defined several aggregation operators, including the cloud weighted arithmetic averaging (CWAA) operator, cloud weighted geometric averaging (CWGA) operator, cloud-ordered weighted arithmetic averaging (COWA) operator, and cloud hybrid aggregation operator in order to develop a linguistic decision-making approach.
The related neutrosophic Cloud aggregation operators are shown in Table 7.
2.2.7 Exponential aggregation operators
Some optimization models cannot deal with the NSs directly, and also cannot make full use of the original neutrosophic information. To overcome this issue and avoid the loss of decision information in the aggregation and modelling process, some exponential operational law for SVNS, INS and SNS are developed. The related neutrosophic Exponential aggregation operators are shown in Table 8.
2.2.8 Prioritized aggregation operators
In practical situations, decision-makers usually consider different criteria priorities. To deal with this issue, Yager (2008) proposed prioritized average (PA) operators by modeling the criteria priority on the weights associated with the criteria, which depend on the satisfaction of higher priority criteria. The PA operator has many advantages over other operators. For example, the PA operator does not need to provide weight vectors and, when using this operator, it is only necessary to know the priority among the criteria.
The related neutrosophic Prioritized aggregation operators are shown in Table 9.
2.2.9 Choquet integral aggregation operators
One of the popular aggregation operator fuzzy integrals is the Choquet integral which is introduced by Choquet (1953). Choquet integral is defined as a subadditive or superadditive to integrate functions with respect to the fuzzy measures (Murofushi and Sugeno 1989).
The related neutrosophic Choquet integral aggregation operators are shown in Table 10.
2.2.10 Heronian aggregation operators
Heronian mean (HM) operator is an important aggregation operator which has the characteristic of capturing the correlations of the aggregated arguments. Beliakov et al. (2007) had firstly proved that Heronian mean was an aggregation operator, but he did not do further researches. Further works are extended to the generalized Heronian means, and discussed two special cases of them. Meanwhile, combining Heronian means and neutrosophic set with its extensions, some related neutrosophic Heronian aggregation operators are shown in Table 11.
2.2.11 Correlated aggregation operators
The related neutrosophic Correlated aggregation operators are shown in Table 12.
2.2.12 Frank aggregation operators
The related neutrosophic Frank aggregation operators are shown in Table 13.
2.2.13 Dombi aggregation operators
Dombi (1982) developed the operations of the Dombi T-norm and T-conorm, which have the advantage of good flexibility with the operational parameter. Hence, Liu et al. (2018) extended the Dombi operations to IFSs and proposed some intuitionistic fuzzy Dombi Bonferroni mean operators and applied them to multiple attribute group decision-making (MAGDM) problems with intuitionistic fuzzy information.
The related neutrosophic Dombi aggregation operators are shown in Table 14.
2.2.14 Maclaurin symmetric mean aggregation operators
The related neutrosophic Maclaurin symmetric mean aggregation operators are shown in Table 15.
From above 14 kinds of aggregation operators, we can know that the final destination is to make decision. Meanwhile, for some real applications, different aggregation operators have different application scenes. In the future, some research points are shown as follows.
- (1)
Extended the 14 kinds of aggregation operators into diverse extensions of NS;
- (2)
Combined novel aggregation operators out of above 14 kinds with NS or its extensions;
- (3)
Applied one kind of aggregation operators to solve a decision making problem in certain filed;
- (4)
Combined some existing aggregation operators for obtaining new aggregation operators (still in above 14 kinds) such as neutrosophic prioritized power aggregation operators (prioritized \(+\) power).
2.3 Information measures
In this subsection, the axiomatic skeleton of information measures (distance measure, similarity measure, entropy measure, inclusion measure/subsethood measure, correlation coefficients) are reviewed.
2.3.1 Similarity measure
The similarity measure indicates the similar degree of two objects. Wang (1983) initially proposed the concept of fuzzy sets’ similarity measure and gave a computation formula. It has been applied to different settings such as intuitionistic fuzzy set (Peng et al. 2015b), hesitant fuzzy set (Xu and Xia 2011), Pythagorean fuzzy set (Peng et al. 2017d; Peng and Dai 2017b; Peng and Ganeshsree 2018), interval-valued fuzzy soft set (Peng and Yang 2017; Peng and Garg 2018).
In the following, some related similarity measures for NS and its extensions are reviewed, which is shown in Table 16.
2.3.2 Distance measure
Distance measure is an important tool for measuring the vague information which describes the difference between two objects and has become a hot topic in decision making, machine learning, and pattern recognition. In the following, some related distance measures for NS and its extensions are reviewed, which is shown in Table 17.
2.3.3 Entropy measure
Entropy is used to measure the uncertain degree of two objects and has been widely used in diverse domains. Several scholars have studied it from different points of view. For example, Luca and Termini (1972) developed some axioms which captured human’s intuitive comprehension to describe the fuzziness degree of a fuzzy set.
In the following, some related entropy measures for NS and its extensions are reviewed, which are shown in Table 18.
2.3.4 Correlation coefficients
Correlation coefficient is employed to explore the nature of the relations between the variables, and also may be used to make inferences about any one of the variables on the basis of the others. Based on these concepts and their axiomatic definitions, some existing correlation coefficients are shown in Table 19.
2.3.5 Inclusion measure/subsethood measure
The inclusion measure (subsethood measure) of fuzzy sets indicates the degree to which a fuzzy set is contained in another fuzzy set. Zadeh (1965) initially developed the definition of a fuzzy set inclusion and pointed out that inclusion was a crisp relation. That is to say, a fuzzy set is either included or not included in another fuzzy set. After that, many scholars study the inclusion measure in diverse environment by the axiomatic approach. In the following, some related inclusion measures/subsethood measures for NS and its extensions are reviewed, which are shown in Table 20.
From above 5 kinds of information measures, we can know that the most of final destinations are to decision making. Also, some are used for image processing, medical diagnosis, pattern recognition. In the future, some research points are shown as follows.
- (1)
Proposed some novel information measures (similarity measure, distance measure, entropy measure, inclusion measure/subsethood measure, correlation coefficients) formulae under the corresponding 4 axiomatic definitions;
- (2)
Utilized some existing formulae for decision making, image processing, medical diagnosis, pattern recognition;
- (3)
Suggested the systematic transformation of information measures for achieving their fundamental properties.
2.4 MCDM methods
Decision making is one of the most important and complex tasks for individuals or organizations and is an interdisciplinary research area attracting researchers from almost all fields from psychologists, economists, to computer scientists (Zhan and Alcantud 2018). As an important research branch of decision-making theory, multiple criteria decision making (MCDM) has gained great success. MCDM methods cover a wide range of quite distinct approaches. MCDM methods can be broadly classified into two categories: discrete MADM (multi-attribute decision making) and continuous MODM (multi-objective decision making) methods.
In MODM problems, the number of alternatives is effectively infinite, and the trade-offs among design criteria are typically described by continuous functions. MADM problems are distinguished from MODM problems, which involve the design of a best/optimal alternative by considering the trade-offs within a set of interacting design constraints and a set of quantifiable objectives. MADM refers to making selections among some courses of action in the presence of multiple, usually conflicting, attributes. Although Philosophers often make a distinction between properties and attributes, it is common that many scholars take MCDM and MADM as interchangeable and use MCDM to represent the discrete MCDM.
Table 21 shows frequency of both neutrosophic MCDM tools and approaches. Based on results presented in this table, a total of 163 studies have employed MCDM tools and approaches. This table shows that aggregation operators method (51 papers) has been used more than other tools and approaches. The second one is the method of information measures (32 papers) and traditional hybrid MCDM (16 papers) is the third in this ranking. Tables 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34 and 35 show implementation of each neutrosophic MCDM tools and approaches.
2.5 Image processing
2.5.1 Medical imaging processing
Image processing is processing of images using mathematical operations by using any form of signal processing for which the input is an image, a series of images, or a video, such as a photograph or video frame; the output of image processing may be either an image or a set of characteristics or parameters related to the image. Generally, the neutrosophic set (NS) approaches were applied successfully into image processing including image denoising, image thresholding, image classification, image clustering, and image segmentation . In the following, we will show the concrete details in Tables 36, 37, 38, 39 and 40.
2.5.2 Medical diagnosis
Medical diagnosis is the process of determining which disease or condition explains a person’s symptoms and signs. It is most often referred to as diagnosis with the medical context being implicit. The information required for diagnosis is typically collected from experts’ examination of the person seeking medical care. Often, one or more diagnostic procedures, such as diagnostic tests, are also done during the process. Sometimes Posthumous diagnosis is considered a kind of medical diagnosis. Diagnosis is often challenging, because many signs and symptoms are fuzzy. For accurate medical diagnosis, researchers are interested in developing new algorithms to handle the modalities variety output. Recently, a new trend is to use the NS approaches in the processing stages to achieve precise diagnoses from the captured images.
In the following, we will show the concrete details in Table 41.
2.6 Graph
A graph is a convenient way of representing information involving relationship between objects. The objects are represented by vertices and the relations by edges. When there is vagueness in the description of the objects or in its relationships or in both, it is natural that we need to designe a fuzzy graph Model. The extension of fuzzy graph theory (Gani and Ahamed 2003) have been developed by several researchers including intuitionistic fuzzy graph (Akram and Davvaz 2012) considered the vertex sets and edge sets as intuitionistic fuzzy sets. Interval valued fuzzy graphs (Akram 2012) considered the vertex sets and edge sets as interval valued fuzzy sets. In the following, we will review some neutrosophic graphs in Table 42.
From above references of graph, we can know that the most of final destination are widely used in diverse domains. In the future, some research points are shown as follows.
- (1)
Discussed some basic properties of neutrosophic graph or its extensions;
- (2)
Applied graph theory into more areas for solving more issues.
2.7 Algebraic structures
Algebraic structure is a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. In the following, we only review some neutrosophic algebraic structures or its extensions in Table 43.
From above neutrosophic algebraic structures, we can know that the most used sets are SVNS and INS due to their special properties. In the future, the main research point is to explore more algebraic structures such as group-like (semigroup, group, Abelian group, Quasigroup), ring-like structures (semiring, near-ring, lie ring, boolean ring, field), lattice structures (complete lattice, bounded lattice, complemented lattice, modular lattice, distributive lattice ), Hybrid structures (topological group, Lie group, ordered groups, ordered rings, ordered fields, Archimedean group).
3 Analyses
3.1 Data and method
3.1.1 Web of Science\(^\mathrm{TM}\) core collection: all
Science Citation Index Expanded (SCIE), Social Sciences Citation Index (SSCI), Conference Proceedings Citation Index-Science (CPCI-S), Conference Proceedings Citation Index-Social Science & Humanities (CPCI-SSH) , four widely used citation indexes of Thomson Reuters Web of Science, were chosen for raw bibliographic data collection in this research compared with only SCIE/SSCI. In order to retrieve the neutrosophic-related publications, we used the following search strategy. We retrieved our data on June 22, 2017, and the search returned 212 hits.
3.1.2 Web of Science\(^\mathrm{TM}\) Core Collection: SCIE/SSCI
In order to retrieve the neutrosophic-related publications by SCIE/SSCI, we used the following search strategy. We retrieved our data on June 22, 2017, and the search returned 137 hits.
In the following paper, we mainly take SCIE/SSCI into consideration.
3.2 Annual trends and possible explanations
Figure 1 plots the annual trends of neutrosophic-related publications. Since the classical reference (book) of Smarandache (1998), the neutrosophic-related research obtained no essential journals’ papers over the first 7 years (most papers are conference papers shown in Fig. 2 when 2001 and 2005). However, when entering into the 2008, more and more scholars paid attentions to this area. This leds to the steady and stable increase in neutrosophic-related publications. As the neutrosophic set theory became more and more influential in scientific community, the publication records even received exponential increase at the beginning of the year 2013. There are many possible reasons resulting in this strong increase. Firstly, the great development of economic and Internet makes it easy to obtain different kinds of references and materials on neutrosophic theory. Meanwhile, more and more scholars worldwide, especially in China, have joined into this research field. The widely spread of neutrosophic publications shows the success of neutrosophic theory in practical applications. In addition, the increase in neutrosophic publications should also owe to the creation of journals and other related ones that were recently accepted neutrosophic publications which is indexed by SCIE/SSCI in Web of Sciences. This is apparently reflected in Fig. 1. Observe that many journals have expanded their issues to accommodate more papers. This also leads to the increase in neutrosophic records.
It should be noted that the number of publications will be far more than 48 at the end of 2017 (The current publications are 27).
Annual trends of neutrosophic-related publications by SCIE/SSCI
Annual trends of neutrosophic-related publications by ALL
Figure 2 is more than the number of papers in Fig. 1 for the conference papers. That is to say, in the early period, conference papers are main way due to its fast publish.
3.3 Country level
Our data shows that researchers from over 15 countries/territories were involved in the neutrosophic-related knowledge production during our total study period, but over \(80\%\) of the publications were contributed by authors from the top 4 active countries/territories. China, as a rising power in scientific research, is the largest producer of neutrosophic publications. Researchers from China have published 69 publications in this domain with a share of \(50.36\%\). The India is the second prolific producer in this field with \(11.68\%\) of the world total publications; however, its share is far behind its Chinese counterpart. Following China and the India, Turkey, and USA are also prolific actors ranked between 3rd and 4th with the publication shares of \(10.95\%\) and \(10.22\%\), respectively.
Table 44 depicts the top 15 most prolific regions of neutrosophic-related knowledge production. It is quite surprising that some small or developing economies such as India, and Turkey are among the top players in this research domain. The main actors in this research domain are significantly different from those in other fields such as nanotechnology (Chen et al. 1991), biomass-based bioenergy (Liu et al. 2014b). To further explore this issue, we calculate the shares of all SCIE/SSCI publications in all research fields in 2017 named as Share B in Table 44 for these 15 regions.
The shares of neutrosophic publications of all the study period (Share A) and the shares of all SCIE/SSCI publications in 2017 (Share B) are quite different for almost all the top regions. Lithuania contributed 0.11\(\%\) to the world total SCIE/SSCI publications in 2017; however, 26.55\(\%\) of the neutrosophic publications were authored from the affiliations in Lithuania. This demonstrates that researchers from Lithuania are relatively active in publishing in this domain compared to their contribution to all the research fields. Turkey, Pakistan, and Serbia are also similar to Lithuania. On the contrary, researchers from some developed economies such as the USA, Spain, and Australia are relatively unlikely to contribute to this domain than to other research fields.
3.4 Institutional level
After identifying the top producers from the country level, we further recognize the top actors in the institutional level. Shaoxing University from China leads with 28 publications and a share of \(30.10\%\) of all the global publications. Following Shandong University of Finance and Economics University and Central South University also from China, are shared as the second producers with 10 publications. Table 45 lists the details of the top 20 most prolific institutions who are active in neutrosophic-related knowledge production. Among the top 20 active institutions, China (8 institutions), Turkey (5 institutions), India (2 institutions), USA (1 institution), Lithuania (1 institution), Pakistan (1 institution), Spain (1 institution), and Australia (1 institution) are the main countries/territories that these institutions affiliated to. This echoes the previous finding that these countries/territories are active in neutrosophic research.
3.5 Publishing journals
Over 50 SCIE/SSCI journals have published neutrosophic research work. Journal of Intelligent and Fuzzy Systems is the largest outlet which has published 36 neutrosophic publications, followed by Neural Computing and Applications (12), Applied Soft Computing (6), International Journal of Machine Learning and Cybernetics (4), and Kybernetes (4). The leading journal, Fuzzy Sets and Systems, also publishes 1 neutrosophic-related papers. Table 46 lists the top 52 publication outlets for neutrosophic research.
We further detect the publication outlet preference of the top 6 productive countries/territories among these important outlets as shown in Table 46. Chinese researchers have published in more than 25 journals for their 69 publications. Journal of Intelligent and Fuzzy Systems, with relatively low impact factors in recent years, was Chinese scholars’ first choice to publish its outputs. 36 out of 69 publications of China were published in this journal. Among all the 36 publications in this journal, 28 papers were published in recent 3 years (2015–2017). This may due to the quick rising of publication volume of this journal in recent 3 years. Besides, Neural Computing and Applications (7), International Journal of Fuzzy Systems (5), and International Journal of Systems Science (3) were also Chinese researchers’ main outlets to publish. India also has published 12 different journals which is narrower than their Chinese counterparts, but is wider than other countries.
The publication outlet preference of researchers from the Spain was quite different from other regions. Although with only 4 publications, Spain scholars have published in one journal (Kybernetes).
From 2013, a specialized neutrosophic journal named “Neutrosophic Sets and Systems” has been created by Smarandache. Although it has not been indexed by SCIE, it has a certain influence in neutrosophic domain. We also believe that it will be indexed by SCIE in our neutrosophic researchers’ endeavour.
3.6 Highly cited papers
To roughly identify the most influential scientific minds in neutrosophic-related research, we select the top 11 highly cited papers and 1 hot paper of neutrosophic publications from Web of Science which are ranked by total number of citations. Table 47 illustrates these highly cited papers in neutrosophic decision research in terms of author(s), region, journal name, publication year, title and citations.
The highest cited neutrosophic reference (Google Scholar with 470 citations), written by Smarandache (1998) (who is the “father of neutrosophic logic”), can be seen as the pioneering work for neutrosophic set as it firstly introduced the neutrosophic logic theory and thus opened a new research direction.
From Table 47, we can find an interesting phenomenon that highly cited papers are all from China. This phenomenon is credited with the numerous Chinese scholars and pioneering works. The top 1 paper, originally by Zhang et al. (2014), can be seen as the pioneering work for netrosophic aggregation operators. This paper published in the Scientific World Journal (open access journal) and gained much more citations than others highly cited papers. All the following highly cited papers developed different types of aggregation operators or information measures for decision making, such as aggregation operators (in the 4th, 6th, 7th, 8th, 9th, 10th and 12th papers), information measures (in the 2nd, 3rd, 5th and 11th papers). These methods and techniques were regarded as the indispensable and integral parts of neutrosophic decision research and thus gained more and more citation frequency. It is noted that all these highly cited papers are published in famous journals. It is also interesting to note that the authors of these highly cited papers mainly come from two teams (Ye, Jun and Wang, Jianqiang). It is too hard to find the highly cited papers from other authors or teams.
3.7 Research landscape
The neutrosophic research is not limited to the field of “Computer Science” or “Mathematics,” but covers over 20 Web of Science categories. This indicates the wide applications of neutrosophic theories and methods in various fields. “Computer Science, Artificial Intelligence” is the largest category with nearly one-second of all the neutrosophic-related publications. The followings are “Computer Science, Interdisciplinary Applications” and “Computer Science, Theory Methods,” each with over 10 publications and sharing of 12.409% and 8.029%, respectively.
Table 48 lists the main Web of Science categories that neutrosophic-related publications belong to. Besides computer science and mathematics-related categories, the neutrosophic-related publications were also found to be widely appeared in engineering-, management-, neurosciences, optics, and imaging science related categories. It shows the extensive applications of neutrosophic set in these fields.
3.8 The keywords analysis of research hot spots on NS
In this subsection, we explore the research hot spots by analyzing the distribution of keywords. The keywords co-occurrence network map, the top 10 keywords in NS publications and the keywords density visualization map will be presented. Keywords co-occurrence can effectively reflect the research hot spots in the discipline fields, offering efficacious support for scientific research (Liao et al. 2018). In all the 137 NS-related publications, we achieved 587 keywords altogether.
The keyword co-occurrence network of NS(see Fig. 3) was established by the VOSviewer software. The size of the nodes and words in Fig. 3 denotes the weights of the nodes. The bigger the node and word are, the larger the weight is. The distance between two nodes reports the strength of the relation between two nodes. A shorter distance usually indicates a stronger relation. The line between two keywords denotes that they have appeared together. The thicker the line is, the more co-occurrence they have. The nodes with the same color classified as a cluster. VOSviewer divided the keywords of NS-related publications into 5 clusters. The keyword “intuitionistic fuzzy set” has a highest frequency of 53. Other keywords with a high frequency include “neutrosophic set” (41), “aggregation operators” (30), “entropy” (27), “similarity” (25), and “multicriteria decision-making” (21).
Keywords co-occurrence network of NS-related publications
The link strength between two nodes denotes the frequency of co-occurrence. It can be used as a quantitative index to describe the relationship between two nodes (Pinto et al. 2014). The total link strength of a node is the sum of link strengths of this node over all the other nodes. The node, “intuitionistic fuzzy set”, has thicker lines with “neutrosophic set” (14), “aggregation operators” (20), “entropy” (15), “similarity” (16), “multicriteria decision-making” (13), and “correlation coffeicient”(15). These are all the nodes whose link strengths are more than 13. The relationships between “intuitionistic fuzzy set” and “neutrosophic set” imply the close integration of extension. The relationships between “intuitionistic fuzzy set” and “entropy”, “similarity” and “correlation coffeicient” reflect that the neutrosophic set study needs the support from some information measure techniques. The relationships between “intuitionistic fuzzy set” and “aggregation operators” as well as “multicriteria decision-making” show the development trends of application environments. The top 10 keywords with their frequencies and total link strengths are shown in Table 49.
VOSviewer can have density visualization (see Fig. 4). Each node in the keywords density visualization plat has a color that relies on the density of items at that node. That is to say, the color of a node depends on the number of items in the neighborhood of the node. The keywords in red color area appear more frequently. On the contrary, the keywords in yellow color area appear less frequently. Density visualization are quite useful for understanding the overall structure of a map and drawing attention to the most important areas in the map. From Fig. 4, we can see the research focuses of neutrosophic set study intuitively. “intuitionistic fuzzy set”, “neutrosophic set”, “aggregation operators”, “entropy” turn out to be important. These keywords are the core keywords in the NS study.
Keywords density visualization map of NS-related publications
3.9 The co-authorship analysis on NS
It is hard for people to accomplish a research on a certain subject individually. Most of research projects or work need collaborative strength to fulfill. Co-authorship research is an important content of bibliometrics and the level of research collaboration is an index to evaluate the current status of research in a specific domain (Reyes et al. 2016). In this subsection, we mainly give the country co-authorship analysis and the institute co-authorship analysis on NS-related publications. We make the co-authorship network by means of the VOSviewer software.
3.9.1 The country co-authorship analysis
Country co-authorship analysis is an important form of co-authorship analysis which can report the degree of communication between countries as well as the influential countries in this field. The country co-authorship network of NS-related publications is presented in Fig. 5. There are different colors in the map, which shows the diversification of research directions. The big nodes denote the influential countries. The links between nodes indicate the cooperative relationships among countries. The distance between the nodes and the thickness of the links denotes the level of cooperation among countries. In Fig. 5, we can easily know that the research center in the field of NS is in the China. The link strength between the China and USA is 7, between the China and Turkey being 5. While the link strength between Turkey and Pakistan is 1. It demonstrates that geographical advantage is not the key factor that influences the cooperative relationship in country level.
The country co-authorship network of NS-related publications
3.9.2 The institute co-authorship analysis
The institute co-authorship network is shown in Fig. 6. The Shaoxing University from the China is the top influential institutes of the NS-related publications. Although so many institutes have published their papers, the relationship among all institutes has not been well or effectively linked. It indicates that the cooperative relationships among institutes have not been well formed.
The institute co-authorship network of NS-related publications
3.10 The co-citation analysis on NS-related publications
When two items (such as documents, journals and authors) are cited in a citing item’s reference list, they have a co-citation relationship. Small (1973) developed a co-citation analysis to investigate the relationship and structure of academic domains. Since then the co-citation analysis has been extensively used to reveal the relationship and structure of authors, articles and journals in academic fields. In this subsection, the reference co-citation analysis, the journal co-citation analysis, and the author co-citation analysis are shown.
3.10.1 The reference co-citation analysis
When two papers emerged simultaneously in the third paper’s citations, it is considered that the two papers built a co-citation relationship (Tang et al. 2018). Reference co-citation analysis is a significant way to investigate the structure and evolution path of a specific filed. Co-citation analysis is a kind of citation network analysis method. It is different from another citation analysis method, that is to say, the citation quantity analysis method. The citation quantity analysis method is to evaluate the quality of the subjects (journal, author, country, document, type of document, etc.) by the number of citations. Co-citation analysis chooses some representative literatures as the analysis object, and then employs the network analysis method to divide these literatures into several clusters. In this way, we can get the structure and characteristics of a specific filed. In the reference co-citation network, the importance of nodes does not reveal the high number of citations, but illustrates the research themes that are closely related to NS-related research. Figure 7 presents the reference co-citation network in the field of NS study. From Fig. 7, we can easily see that the biggest node is Atanssov (1986). His paper entitled “Intuitionistic fuzzy sets” published in FSS (Fuzzy Sets and Systems) proposed that the novel extension of fuzzy sets may be an important way to deal new extension of intuitionistic fuzzy set (neutrosophic set).
The reference co-authorship network of NS-related publications
Table 50 lists the top 10 most co-cited documents related to NS study.
3.10.2 The journal co-citation analysis
The journal co-citation analysis is not only an effective way to explore the structure and characteristics of a subject, but also reveals the overall structure of the subject and the characteristics of a journal (Hu et al. 2011). The VOSviewer software is used to plot the journal co-citation network. Figure 8 presents the journal co-citation network with 45 nodes. The size of node denotes the activity of the journal and the number of published papers. The distance between two nodes is also quite important. Generally speaking, the smaller the distance between two nodes is, the higher the citation frequency is. As the visualization illustrated in Fig. 8, each cluster has a color that denotes the group to which the cluster is allocated. It can be easily seen that all these journals are divided into four clusters. The blue cluster contains Fuzzy Sets and Systems, Information Sciences and IEEE Transactions on Fuzzy Systems, etc. This cluster represents top journals. The red cluster contains Journal of Intelligent and Fuzzy Systems, Neural Computing and Applications and Applied Mathematical Modelling. This cluster denotes science and technology journals. The green cluster represents information journals.
The journal co-authorship network of NS-related publications
3.10.3 The author co-citation analysis
Author co-citation analysis is an important and efficacious citation analysis method, since it was initially developed in 1981 (White and Griffith 1981), it has received wide attention and researches from scholars (Wang et al. 2016b). By drawing out the co-citation relations between the authors of the academic literature, author co-citation network can be obtained and used to guide the scientific research (Koseoglu et al. 2015). In the following, VOSviewer was adopted to draw out the author co-citation map on NS researches and it was shown in Fig. 9. It consists of 30 nodes and 400 edges. Unsurprisingly, the node demonstrates that Ye J is the biggest one among all the nodes. Furthermore, the nodes indicate that Ye J and some others with purple rings express high centrality in NS researches.
The author co-authorship network of NS-related publications
4 Conclusions
We focus on seven parts (extensions style, aggregation operators, information measures, MCDM methods, image processing, graph, algebraic structures) to review the whole development of NS and discuss their future directions. Meanwhile, a total of 137 neutrosophic set publication records from Web of Science (WoS) are analyzed. Many interesting results with regard to the annual trends, the top players in terms of country level as well as institutional level, the publishing journals, the highly cited papers, and the research landscape are yielded and explained in-depth. Moreover, the co-authorship analysis of the country and institution, the co-citation analysis of the journal, reference and author, and the co-occurrence analysis of the keywords are presented by VOSviewer software. It has yielded the following results:
- (1)
Our analyses have demonstrated that the academic publications in neutrosophic research domain have fluctuated at low level during the initial periods of 1998–2008, but have grown rapidly over the last ten year.
- (2)
Quite different from other research domains, some small or developing economies such as India, and Turkey were also among the largest contributors.
- (3)
Our data have also showed that the scholars from China, and India were relatively active in publishing in this domain compared to their contribution to all the research fields.
- (4)
The highly cited papers were mainly published in famous journals and contributed all by authors from China.
- (5)
The most frequently cited work in neutrosophic set area is Atanssov (1986). FSS-Fuzzy Sets and Systems is most influential in neutrosophic set domain.
- (6)
Through the analysis of keywords, we have found that intuitionistic fuzzy set is the most core keyword. At the same time, the technical support of neutrosophic set study is the key direction that people need to combine the two kinds of extention of fuzzy set. They can share with common decision making methods, aggregation operators, information measure and so on.
- (7)
In neutrosophic set domain, the phenomenon of cooperation among multiple authors is widespread. More than 66\(\%\) publications with the highest number of citations were completed with more than one author. However, the international cooperation is not universal. The future research can focus more on the impact of the research in this field and probe the reasons why some small economies are keen on academic research in this field.
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Acknowledgements
This work is sponsored by the National Natural Science Foundation of China (No. 61462019) and the General Project of Shaoguan University (No. SY2016KJ11).
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Peng analyzed the existing data and wrote the manuscript; Dai drew the beautiful figures.
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Peng, X., Dai, J. A bibliometric analysis of neutrosophic set: two decades review from 1998 to 2017. Artif Intell Rev 53, 199–255 (2020). https://doi.org/10.1007/s10462-018-9652-0
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DOI: https://doi.org/10.1007/s10462-018-9652-0