The fuzzy subset A of is a fuzzy number if its r-levels are closed and nonempty intervals of and the support of A, , is limited . The family of the fuzzy subsets of with nonempty compact and convex r-levels is denoted by , while the family of fuzzy numbers is denoted by .
According ISI Web of Science, from 1965 more than one hundred thousand papers on fuzzy set theory foundations and applications have been published in journals. In spite of this huge number of published papers there has been no bibliometric study on the field.
The study reveals the main themes treated by the Spanish fuzzy set theory community from 1978 to 2008. It also allows comparison with those themes treated by other important international communities such as the USA, Canada, the United Kingdom, Germany, Japan, the Peoples Republic of China and Spain, which are collectively known as the G7 group. These seven countries were selected because they satisfy three conditions:
The first step in the process of bibliometric mapping, is the collection of raw data. In this paper, the raw data consist of a corpus containing 16,344 original papers, about fuzzy set theory produced by the G7 group from 1965 to 2008. These original papers were extracted from ISI Web of Science with the query number 1 on 7th January 2009:
Two blocks of papers were collected, one for analysing the evolution of the fuzzy set theory field in all of the above mentioned countries taken together and another block to analyse the Spanish case. From these two blocks of papers, four subsets of papers are extracted, one for each studied sub-period: 1965-1993 (although the fuzzy sets based research in Spain started in 1978), 1994-1998, 1999-2003 and 2004-2008. In co-word analysis, in a longitudinal study, it is usual for the first sub-period to be the most long-lasting to get a representative number of published papers. For this reason, in this paper, the first studied sub-period includes twenty-nine years (from 1965 to 1993) and the other three sub-periods just five years. In this way, separate bibliometric maps can be constructed for each one of the four studied sub-periods. The number of papers in each block and sub-period is shown in Figure 2.
This section analyses the evolution of the fuzzy set theory field over recent years. First, the evolution of the field in Spain is studied (including the analysis of the fuzzy sets research carried out by the G7 countries and a comparison between both Spanish and International cases). Then, a snapshot of the visibility of the Spanish fuzzy community in the ISI Web of Science is presented. Finally, the importance of the Spanish community in the fuzzy set theory field is shown.
To analyse the evolution of the fuzzy set theory field over the last years in Spain, strategic diagrams for the four studied sub-periods are shown in Figure 6. The volume of the spheres is proportional to the number of documents corresponding to each theme in each sub-period (a number is used to indicate the papers per theme). In the following the four sub-periods are described.
In the second sub-period (1994-1998), with 199 papers, CoPalRed shows nine themes, where linguistic-preference-relations (six papers) (the most central and dense theme), linguistic-modelling (six papers), fuzzy-logic (nine papers) and fuzzy-control (eight papers) were the four most studied themes.
From 1999-2003, the three most studied themes by the Spanish fuzzy sets research community were: genetic-algorithms (twenty-two papers), neural-networks (fifteen papers) and OWA-operators (fourteen papers). The themes OWA-operators and genetic-algorithms were the motor-themes of the sub-period (see Figure 6c).
From 2004 to 2008 the three most studied themes were the basic and transversal theme fuzzy-sets (with twenty-two papers), following by the most applied themes fuzzy-control (with twenty-three papers) and group-decision-making (twenty-one papers) (see Figure 6d).
To complete the description of the evolution of the fuzzy sets based research in Spain, in Appendix A, Figures 13, 14, 15 and 16, show the complete relations among keywords and themes for each sub-period. In these networks, the volume of the spheres is proportional to the number of documents corresponding to each keyword, the thickness of the link between two spheres i and j is proportional to the equivalence index eij. Keywords belonging to a theme are labelled with the same number and each number represents a theme. Different numbers are used in each sub-period. Only the strongest relations are shown.
To analyse the recent evolution of the fuzzy set theory field in the above mentioned G7 countries, strategic diagrams for the four studied sub-periods (1965-1993, 1994-1998, 1999-2003 and 2004-2008) are shown in Figure 8. The volume of the spheres is proportional to the number of documents corresponding to each theme in each sub-period (a number is used to indicate the papers per theme). In the following the four sub-periods are described.
In the first sub-period (1965-1993), the longer one, in which 2,397 papers were published, the most important themes in the international context (Considering the countries: USA, Canada, UK, Germany, Spain, Japan and Peoples Republic of China), in relative weight in terms of number of documents, were: uncertainty and expert-systems (both with thirty-eight papers), applications (twenty-three papers) and decision-making (with twenty papers) (see Figure 8a). Because of their strategic situation (lower-right quadrant), uncertainty and expert-systems and decision-making are considered as general basic themes, with high centrality, although with low internal development. In this first sub-period, the most central theme was uncertainty, the fuzzy set theory revolve around this concept.
To complete the description of the evolution of fuzzy sets research in the G7 group, in Appendix A, Figures 17, 18, 19 and 20, the whole relations among keywords and themes for each sub-period are shown. In these whole networks, the volume of the spheres is proportional to the number of documents corresponding to each keyword, the thickness of the link between two spheres i and j is proportional to the equivalence index eij. Keywords belonging to a theme are labelled with the same number and each number represents a theme. Different numbers are used in each sub-period. Only the strongest relations are shown.
From 1994-1998, whereas in the G7 group, the theme connectives (with forty-eight papers) was the most central theme (upper-right quadrant), in the Spanish research, connectives was situated in the lower-left quadrant (with just three papers). In the G7 group, the most studied theme was fuzzy-logic (with ninety-three papers), in the Spanish community just nine papers were published on this topic (see Figures 8b and 6b).
In Figure 11 we observe an increasing number of publications each year with more than 100 papers a year in recent years. In Figure 12 we observe that the number of citations shows a similar increasing trend in recent years. All these data can allow us to say the field of fuzzy set theory has now reached a stage of maturity after the earliest papers published at 1978; there are also many basic issues yet to be resolved and there is an active and vibrant worldwide community of researchers working on these issues.
Spain is one of the most active countries researching on fuzzy set theory; it is in seventh position in the ranking of the top ten most productive countries (as it could be observed in the ISI Web of Science using query number 1 on 7th January 2009). Spain is also one countries working on the topic for the longest period of time. Its first paper is from 1978, whereas other important countries on the topic published their first papers some years after, e.g., Peoples Republic of China (1980) or Taiwan (1983).
With respect to the institutions researching fuzzy set theory, in Table 3 below we can see that two Spanish institutions are in the top-twenty of the most productive. One of the Spanish institutions, the University of Granada, is the most active institution researching on the topic in the G7 group. Two Spanish institutions (University of Granada and University of Oviedo) were always active on the topic, as can be observed in Appendix A in Tables 9, 10, 11 and 2. In spite of Spain having published its first paper thirteen years after the first paper on the topic was published (1965), one Spanish institution reached seventh position of the most productive institutions in the first studied sub-period (1965-1993) (see Table 9 in Appendix A). In addition, the University of Granada and the University of Oviedo were two of the three European institutions in those rankings (including the University of Sheffield).
In this paper, we have presented the first bibliometric study of fuzzy set theory research, focusing on the Spanish fuzzy sets research community. More than 16,344 original research papers have been processed. Based on this analysis, we have drawn the visual structure of fuzzy set theory research carried out by seven of the top-ten most productive countries on the topic.
Finally, we have to remark that the analysis is not without problems because of the bias implied. The most important caveat is that co-word analysis concentrates on priority themes and inevitably excludes those that have an anecdotal appearance. On the contrary, the analysis will legitimatize discussion about general tendencies accepted by the majority of the scientific community. So, experts and novices can use these results and maps to know the current situation of the fuzzy set theory field.
Alsina, C. and Trillas, E. and Valverde, L. (1983) [136 citations ]. On some logical connectives for fuzzy-sets theory, Journal of Mathematical Analysis and Applications, 93(1), 15-26.
Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, interpreting, and using data and information that are vague and lack certainty. 2b1af7f3a8