Bipolar fuzzy graph representation of concept lattice
TL;DR: This work proposes an algorithm for generating the bipolar fuzzy formal concepts, a method for ( α, β ) -cut ofipolar fuzzy formal context and its implications with illustrative examples.
About: This article is published in Information Sciences.The article was published on 2014-12-20. It has received 112 citations till now. The article focuses on the topics: Fuzzy set operations & Fuzzy number.
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TL;DR: This study investigates the relationship between multigranulation rough sets and concept lattices via rule acquisition and algorithm complexity analysis is made for the acquisition of "AND" decision rules, "OR" decisionrules, granular rules and disjunctive rules.
Abstract: Transforming decision systems into formal decision contexts is studied.Relationship between "AND" decision rules and granular rules is discussed.Relationship between "OR" decision rules and disjunctive rules is investigated.Support and certainty factors of different rules are compared.Algorithm complexity of rule acquisition is analyzed. Recently, by combining rough set theory with granular computing, pessimistic and optimistic multigranulation rough sets have been proposed to derive "AND" and "OR" decision rules from decision systems. At the same time, by integrating granular computing and formal concept analysis, Wille's concept lattice and object-oriented concept lattice were used to obtain granular rules and disjunctive rules from formal decision contexts. So, the problem of rule acquisition can bring rough set theory, granular computing and formal concept analysis together. In this study, to shed some light on the comparison and combination of rough set theory, granular computing and formal concept analysis, we investigate the relationship between multigranulation rough sets and concept lattices via rule acquisition. Some interesting results are obtained in this paper: (1) "AND" decision rules in pessimistic multigranulation rough sets are proved to be granular rules in concept lattices, but the inverse may not be true; (2) the combination of the truth parts of an "OR" decision rule in optimistic multigranulation rough sets is an item of the decomposition of a disjunctive rule in concept lattices; (3) a non-redundant disjunctive rule in concept lattices is shown to be the multi-combination of the truth parts of "OR" decision rules in optimistic multigranulation rough sets; and (4) the same rule is defined with a same certainty factor but a different support factor in multigranulation rough sets and concept lattices. Moreover, algorithm complexity analysis is made for the acquisition of "AND" decision rules, "OR" decision rules, granular rules and disjunctive rules.
149 citations
TL;DR: A method is proposed to generate the component wise three-way formal fuzzy concept and their hierarchical order visualization in the fuzzy concept lattice using the properties of neutrosophic graph, neutrosophile lattice, and Gödel residuated lattice with an illustrative example.
Abstract: Recently, three-way concept lattice is studied to handle the uncertainty and incompleteness in the given attribute set based on acceptation, rejection, and uncertain regions. This paper aimed at analyzing the uncertainty and incompleteness in the given fuzzy attribute set characterized by truth-membership, indeterminacy-membership, and falsity membership functions of a defined single-valued neutrosophic set. For this purpose a method is proposed to generate the component wise three-way formal fuzzy concept and their hierarchical order visualization in the fuzzy concept lattice using the properties of neutrosophic graph, neutrosophic lattice, and Godel residuated lattice with an illustrative example.
120 citations
Cites background from "Bipolar fuzzy graph representation ..."
...Similarly, the interval-valued [23] (or bipolar [7]) fuzzy set represents the acceptation and rejection part of the attributes in the defined interval [0,1] ([-1,1])....
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...Formal concept Bělohlávek [38] Djouadi and Prade [9] Singh and Aswani Kumar[7] Qi et al....
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...of FCA is augmented with fuzzy context [4], heterogeneous context [5], interval-valued fuzzy context [6], bipolar fuzzy context [7], linked fuzzy context [8], possibility-...
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TL;DR: A model for multiple attribute decision-making problem for bipolar fuzzy Dombi aggregation operators under the bipolar fuzzy environment is developed and a practical example for selection of investment alternatives is given.
Abstract: The operations of t-norm and t-conorm introduced by Dombi was known as Dombi operations can have a lead of good flexibility with the general parameter. The Dombi operations have so far not yet been applied for bipolar fuzzy sets. Motivated from Dombi operations, we have proposed bipolar fuzzy Dombi weighted averaging operator, bipolar fuzzy Dombi order weighted averaging operator, bipolar fuzzy Dombi hybrid weighted averaging operator, bipolar fuzzy Dombi weighted geometric operator , bipolar fuzzy Dombi order weighted geometric operator and bipolar fuzzy Dombi hybrid weighted geometric operator as these operators have good advantage of flexibility in operational behavior. Many properties of these operators are investigated. Then, we developed a model for multiple attribute decision-making problem for bipolar fuzzy Dombi aggregation operators under the bipolar fuzzy environment. Finally, a practical example for selection of investment alternatives is given to demonstrate for the utility and application of the proposed work.
112 citations
TL;DR: It is concluded that Li's reduction maintaining the non-redundant decision rules of a formal decision context is coarser than others, and the results obtained are beneficial for users to select an appropriate reduction method for meeting their requirements.
82 citations
TL;DR: This paper aims to provide an understanding of the necessary mathematical background for each extension of FCA like FCA with granular computing, a fuzzy setting, interval-valued, possibility theory, triadic, factor concepts and handling incomplete data.
Abstract: In recent years, FCA has received significant attention from research communities of various fields. Further, the theory of FCA is being extended into different frontiers and augmented with other knowledge representation frameworks. In this backdrop, this paper aims to provide an understanding of the necessary mathematical background for each extension of FCA like FCA with granular computing, a fuzzy setting, interval-valued, possibility theory, triadic, factor concepts and handling incomplete data. Subsequently, the paper illustrates emerging trends for each extension with applications. To this end, we summarize more than 350 recent published after 2011 research papers indexed in Google Scholar, IEEE Xplore, ScienceDirect, Scopus, SpringerLink, and a few authoritative fundamental papers.
80 citations
References
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Book•
04 Dec 1998
TL;DR: This is the first textbook on formal concept analysis that gives a systematic presentation of the mathematical foundations and their relation to applications in computer science, especially in data analysis and knowledge processing.
Abstract: From the Publisher:
This is the first textbook on formal concept analysis. It gives a systematic presentation of the mathematical foundations and their relation to applications in computer science, especially in data analysis and knowledge processing. Above all, it presents graphical methods for representing conceptual systems that have proved themselves in communicating knowledge. Theory and graphical representation are thus closely coupled together. The mathematical foundations are treated thoroughly and illuminated by means of numerous examples.
4,757 citations
"Bipolar fuzzy graph representation ..." refers background in this paper
...The operators ("; #) are known as Galois connection [26] and extensively studied with fuzzy [14,15,42] as well as with interval-valued fuzzy setting [20–22,30] by the researchers....
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...Formal Concept Analysis (FCA) was introduced by Wille [52] for knowledge discovery and representation tasks....
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...Generalized concepts contain more objects while Specialized concepts contain more attributes [26,52]....
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...FCA became more popular in scientific community when its mathematical foundation was established by Ganter and Wille [26]....
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...Formal context Wille [52] Burusco and Gonzales [19] Djouadi Prade [21,22] Formal concept Ganter Wille [26] Belohlavek [12–14] Alcalde et al....
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12 May 2009
TL;DR: Restructuring lattice theory is an attempt to reinvigorate connections with the authors' general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattices theory.
Abstract: Lattice theory today reflects the general Status of current mathematics: there is a rich production of theoretical concepts, results, and developments, many of which are reached by elaborate mental gymnastics; on the other hand, the connections of the theory to its surroundings are getting weaker and weaker, with the result that the theory and even many of its parts become more isolated. Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theory.
2,407 citations
"Bipolar fuzzy graph representation ..." refers background in this paper
...[4,5] Lattice Wille [52] Belohalvek and Vychodil [15] Djouadi [20] Bloch [17,18] Concept lattice with graph Berry and Sigayret [16] Ghosh et al....
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...Formal Concept Analysis (FCA) was introduced by Wille [52] for knowledge discovery and representation tasks....
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...Generalized concepts contain more objects while Specialized concepts contain more attributes [26,52]....
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...FCA became more popular in scientific community when its mathematical foundation was established by Ganter and Wille [26]....
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...Formal context Wille [52] Burusco and Gonzales [19] Djouadi Prade [21,22] Formal concept Ganter Wille [26] Belohlavek [12–14] Alcalde et al....
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TL;DR: Through a human centered design project focused on an information science problem, students will gain experience and a better understanding of the process to develop an innovative solution addressing a societal need.
Abstract: Courses IS 100 Exploring the iSchool with a Human-Centered Lens credit: 1 Hour. (https://courses.illinois.edu/schedule/terms/IS/100/) This course introduces students to the School of Information Sciences (iSchool). Students will explore career and professional development within information sciences, building their leadership and collaborative skills, and building a network within and beyond the iSchool. Through a human centered design project focused on an information science problem, students will gain experience and a better understanding of the process to develop an innovative solution addressing a societal need. Prerequisite: Restricted to Majors Only; First Semester Freshman, Intercollegiate and Off-Campus Transfer Students Only.
1,029 citations
TL;DR: A theorem is presented characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context, Dedekind–MacNeille completion of a partial fuzzy order and results provide foundations for formal concept analysis of vague data.
436 citations
"Bipolar fuzzy graph representation ..." refers background in this paper
...For any L-set A2 L of objects, and B2 L of attributes we can define L-set A 2 L of attributes and L-set B 2 L of objects as follows [13,14]:...
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...Each relation e Rðx; yÞ 2 L represents the membership value at which the object x 2 X has the attribute y 2 Y in [0,1] (L is a support set of some complete residuated lattice L) [14]....
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...The operators and! are defined distinctly by Lukasiewicz, Godel, and Goguen t-norms and their residua as given below [14,15,27]; Lukasiewicz:...
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...The operators ("; #) are known as Galois connection [26] and extensively studied with fuzzy [14,15,42] as well as with interval-valued fuzzy setting [20–22,30] by the researchers....
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18 Dec 1994
TL;DR: It is proved that /spl alpha/-level fuzzy number-based bipolar operations can be converted to interval-based and then real-valued bipolar operations and the conversions lead to significant computational simplification on bipolar fuzzy relations.
Abstract: A bipolar fuzzy set theory is presented for cognitive modeling and multiagent decision analysis. Firstly, notions of bipolar fuzziness are introduced. Secondly an interval-based bipolar fuzzy logic is defined which generalizes a real-valued bipolar fuzzy logic by allowing interval-based linguistic variables x and y to be substituted into (S, =, V, /spl otimes/) or (S, =, /spl cup/, /spl odot/) where S={/spl forall/(x,y)|(x,y)/spl isin/([-1,0]/spl times/[0,+1])}. Thirdly, a fuzzy number-based bipolar logic is presented which. Further generalizes the interval-based model by allowing /spl alpha/-level fuzzy number-based linguistic variables x and y to be substituted into (S, =, V, /spl otimes/) or (S, =, /spl cup/, /spl odot/), S={/spl forall/(x,y)|(x,y) maps ([-1,0]/spl times/[0,+1]) to [0,1]}. Bipolar fuzzy set operations of disjunction composition (V-/spl otimes/), union-composition (/spl cup/, /spl odot/), are proved commutative and associative; V respect to /spl otimes/ and /spl cup/ respect to /spl odot/ are proved distributive. It is shown that a interval-based bipolar variable is a nesting of a real-valued bipolar variable; a trapezoidal-fuzzy number-based bipolar variable is an 2-level nesting of an interval-based bipolar variable; and an /spl alpha/-level (/spl alpha/ is an integer) fuzzy number-based bipolar variable is an /spl alpha/+1 level nesting of an interval-based bipolar variable. Based on the nesting features, it is proved that /spl alpha/-level fuzzy number-based bipolar operations can be converted to interval-based and then real-valued bipolar operations. The conversions lead to significant computational simplification on bipolar fuzzy relations. Major advantages of the bipolar fuzzy set theory include: (1) it formalizes a unified approach to polarity and fuzziness; (2) it captures die bipolar or double-sided (negative and positive, or effect and side effect) nature of human perception and cognition; and (3) it provides a basis for bipolar cognitive modeling and multiagent decision analysis. >
342 citations
"Bipolar fuzzy graph representation ..." refers background in this paper
...An another extension of fuzzy set (called as bipolar fuzzy set) represents the bipolar information more precisely [58]....
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