Measuring incompatibility between Atanassov’s intuitionistic fuzzy sets☆
Introduction
It is well-known that Atanassov’s intuitionistic fuzzy sets [3], [4] were introduced as an extension of Zadeh’s fuzzy sets [32]. Due to the additional degree of freedom, they are a tool for more human-consistent reasoning. Despite the major controversies that their similarities and differences to other set theory extensions modelling imprecision have sparked off within the scientific community, these sets have spawned a great deal of research. In this respect, some authors have examined the relationships between the different mathematical models in an attempt to clarify what role intuitionistic sets play within imprecision modelling framework [11].
Many researchers have investigated intuitionistic set theory and its application to several fields of science like decision making [20], [25], [29], [30], pattern recognition [14], [28], [31] or image processing [5] and so on. In these papers, they use measures or indices to estimate the level of similarity or difference between intuitionistic sets. These measures take into account what is known as the hesitation margin [23], [24], [26]. In this article, we present a way of measuring the extent to which intuitionistic sets differ, in this case, how inconsistent they are.
The need to avoid inconsistencies in a process output is an evident requirement in any reasoning, that is, not only to elicit preferences according to certain criteria or for decision optimization but also in inference processes. Nevertheless, there is more than one type of inconsistency. For example, it is possible to gather contradictory or incompatible information. These two concepts are equivalent in classical logic, but they are not so in either fuzzy logic or in Atanassov logic. So they have to be dealt with separately. If an output contains two incompatible Atanassov fuzzy sets, one possibility is to change the reasoning method and choose the right rules, connectives, etc. But there is another possible solution. In fuzzy logic, as in the real world, most information is not precisely known. It would be unreasonable then to reject two incompatible fuzzy sets out of hand. Indeed, we should accept some degree of incompatibility. So there is a need to measure the extent to which the incompatibility is satisfied and determine an acceptable level of tolerance, that is, a threshold.
This paper deals with this problem by establishing an axiomatic definition of incompatibility measure in the framework of Atanassov’s intuitionistic fuzzy sets and providing some families of such measures. The paper is organized as follows. The preliminaries (Section 2) present the background regarding both fuzzy connectives and Atanassov’s sets. Section 3 introduces the axiomatic definition of incompatibility measure for a given intuitionistic t-norm. Section 4 sets out the main results. We construct some families of incompatibility measures for intuitionistic t-norms that can be represented by t-norms and t-conorms, and we obtain some relations between some particular cases of the different families. Section 5 translates this study to the case of non-representable intuitionistic t-norms. Finally, Section 6 sets out some conclusions.
Section snippets
Preliminaries
This section establishes the essential requirements for the frameworks the paper deals with and on which our work is based. The first requirement applies to fuzzy connectives, which are common knowledge. The second refers to intuitionistic fuzzy sets that are characterized by both their membership and non-membership function.
An axiomatic definition of incompatibility measures on AIFS
From a logical viewpoint, two (precise) predicates A and B in a referential universe X are incompatible if their conjunction is false, that is, . Thus, considering the extension of these predicates to sets and , sets A and B are incompatible if . In the same way, we can define incompatible fuzzy sets regarding a fixed t-norm or incompatible AIFSs regarding an intuitionistic t-norm. To be precise, Definition 3.1 Given , if is an intuitionistic t-norm, [6]
Constructing incompatibility measures for TS-representable intuitionistic t-norms
This section looks at constructing incompatibility measures for TS-representable intuitionistic t-norms using mainly geometrical methods. To do this, we first examine certain regions of related to the incompatibility between two AIFSs. Next, we construct a function based on these regions and distances showing that it is an incompatibility measure. Moreover, the idea of this construction leads to more incompatibility measures. Finally, we establish some relations among theses measures.
Constructing incompatibility measures for non-representable intuitionistic t-norms
This section shows how the ideas underlying the construction of the above incompatibility measures can be translated into the context of some non-representable t-norms. To be precise, we consider, for each t-norm T, the intuitionistic t-norm (see [6], [9]) defined for each bySo, are -incompatible if and only if or or , and . Hence, given and
Conclusions
In this paper an axiomatic model has been proposed to measure the incompatibility between Atanassov’s intuitionistic fuzzy sets. Firstly we established the minimal axioms that any function should satisfy to measure this property on Atanassov’s sets. Then, we constructed incompatibility measures for representable intuitionistic t-norms and non-representable intuitionistic t-norms . The measures were constructed in two ways: considering a geometrical method and using a property of the
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions on future lines of research.
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This paper is partially supported by CICYT (Spain) under Project TIN2008-06890-C02-01 and by UPM-CAM under Project CCG07-UPM/003167.