Generation of linear orders for intervals by means of aggregation functions
TL;DR: This work proposes a method to build admissible orders in terms of two aggregation functions and proves that some of the most used examples of total orders that appear in the literature are specific cases of this construction.
About: This article is published in Fuzzy Sets and Systems.The article was published on 2013-06-01. It has received 280 citations till now. The article focuses on the topics: Interval order & Fuzzy set.
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Book•
01 Aug 1996
TL;DR: Fuzzy sets as mentioned in this paper are a class of classes in which there may be grades of membership intermediate between full membership and non-membership, i.e., a fuzzy set is characterized by a membership function which assigns to each object its grade of membership.
Abstract: The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...
885 citations
TL;DR: The TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) technique is extended to solve MCGDM problems within the context of interval type-2 fuzzy sets (IT2FSs) and presented its application to green supplier selection problem.
479 citations
TL;DR: An overview on hesitant fuzzy sets is presented with the aim of providing a clear perspective on the different concepts, tools and trends related to this extension of fuzzy sets.
Abstract: The necessity of dealing with uncertainty in real world problems has been a long-term research challenge that has originated different methodologies and theories. Fuzzy sets along with their extensions, such as type-2 fuzzy sets, interval-valued fuzzy sets, and Atanassov's intuitionistic fuzzy sets, have provided a wide range of tools that are able to deal with uncertainty in different types of problems. Recently, a new extension of fuzzy sets so-called hesitant fuzzy sets has been introduced to deal with hesitant situations, which were not well managed by the previous tools. Hesitant fuzzy sets have attracted very quickly the attention of many researchers that have proposed diverse extensions, several types of operators to compute with such types of information, and eventually some applications have been developed. Because of such a growth, this paper presents an overview on hesitant fuzzy sets with the aim of providing a clear perspective on the different concepts, tools and trends related to this extension of fuzzy sets.
405 citations
Cites background from "Generation of linear orders for int..."
...Hesitant Fuzzy Sets: State of the Art and Future Directions...
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TL;DR: The definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature are reviewed and the relationships between them are analyzed.
Abstract: In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used.
386 citations
Cites background or methods from "Generation of linear orders for int..."
...fore, it is necessary to study the conditions that determine the choice of one order or another [33]....
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...In [33], some methods to build linear orders among intervals, for those applications where such orders are needed, are explained....
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TL;DR: This position paper studies the necessity of hesitant fuzzy sets and provides a discussion about current proposals including a guideline that the proposals should follow and some challenges of HFSs.
212 citations
References
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TL;DR: Much of what constitutes the core of scientific knowledge may be regarded as a reservoir of concepts and techniques which can be drawn upon to construct mathematical models of various types of systems and thereby yield quantitative information concerning their behavior.
12,530 citations
TL;DR: This paper develops some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionists fuzzy ordered weighted geometric(IFOWG)operator, and the intuitionism fuzzy hybrid geometric (ifHG) operators, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzz sets.
Abstract: The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets.
1,928 citations
Additional excerpts
...Focusing on interval-valued fuzzy sets, in most part of the literature, only one linear order, namely, that of Xu and Yager [13] is considered....
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...(iii) Consider the order Y X on L([0, 1]) introduced by Xu and Yager in [13]: [a, b] Y X [c, d] ⇔ a + b c + d ∨ a + b = c + d ∧ b − a≤d − c....
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...Observe that the Xu and Yager order Y X corresponds to the order 0.5+....
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...The main goal of the present work is to present a general method that allows to build different linear orders, and that in particular, covers the most widely known and used linear orders in the literature, such as Xu and Yager’s or the lexicographical ones....
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...(iii) Consider the order Y X on L([0, 1]) introduced by Xu and Yager in [13]: [a, b] Y X [c, d] ⇔ a + b < c + d ∨ a + b = c + d ∧ b − a≤d − c....
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Book•
31 Oct 1994
TL;DR: This dissertation aims to provide a history of web exceptionalism from 1989 to 2002, a period chosen in order to explore its roots as well as specific cases up to and including the year in which descriptions of “Web 2.0” began to circulate.
Abstract: Introduction. 1. Fuzzy logical connectives. 2. Valued binary relations. 3. Valued preference modelling. 4. Similarity relations and valued orders. 5. Aggregation operations. 6. Ranking procedures. 7. Multiple criteria decision making. 8. Summary, perspectives and open problems. Index.
1,886 citations
Additional excerpts
...In decision making, for instance, [6] a set of alternatives {A1, ....
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Book•
01 Aug 1996
TL;DR: Fuzzy sets as mentioned in this paper are a class of classes in which there may be grades of membership intermediate between full membership and non-membership, i.e., a fuzzy set is characterized by a membership function which assigns to each object its grade of membership.
Abstract: The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...
885 citations
563 citations
Additional excerpts
...However, this fact is in contradiction with the classical result of Brouwer [2] concerning the isomorphism of the unit square and the unit interval, excluding the continuous bijections....
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