Gradual Numbers and Their Application to Fuzzy Interval Analysis
IEEE Transactions on Fuzzy SystemsPublished 1 April 2008
Jérôme Fortin, Didier Dubois, Hélène Fargier
Citations197
SJR quartileQ1
SJR score3.61
SNIP2.80
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
TL;DR
A new way of looking at fuzzy intervals is introduced, which enables interval analysis to be directly extended to fuzzy intervals, without resorting to alpha-cuts, in agreement with Zadeh's extension principle.
Abstract
International audience
Keywords
Computer ScienceMathematicsDecision Sciences
Information SciencesThe concept of a linguistic variable and its application to approximate reasoning-III
12,148 Citations1975L. A. Zadeh
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.
Mathematics in Science and Engineering/Mathematics in science and engineeringFuzzy Sets and Systems - Theory and Applications
6,430 Citations1980Didier Dubois, Henri Prade
Society for Industrial and Applied Mathematics eBooksMethods and Applications of Interval Analysis
3,537 Citations1979Ramon E. Moore
Finite representations Finite evaluation Finite convergence Computable sufficient conditions for existence and convergence Safe starting regions for iterative methods.
International Journal of Systems ScienceOperations on fuzzy numbers
2,631 Citations1978Didier Dubois, Henri Prade
The usual algebraic operations on real numbers are extended to fuzzy numbers by the use of a fuzzification principle, and the practical use of fuzzified operations is shown to be easy, requiring no more computation than when dealing with error intervals in classic tolerance analysis.
Computers & Mathematics with ApplicationsA computational approach to fuzzy quantifiers in natural languages
1,825 Citations1983Lotfi A. Zadeh
Information SciencesA procedure for ordering fuzzy subsets of the unit interval
1,407 Citations1981Ronald R. Yager
A function to help in the ordering of fuzzy subsets of the unit interval is introduced, which is the integral of the mean of the level sets associated with the fuzzy subset.
Fuzzy Sets and SystemsThe mean value of a fuzzy number
743 Citations1987Didier Dubois, Henri Prade
Some light is shed on the debate between probability and possibility theory, and the problem of how to carry measure-theoretic notions into the field of possibility theory.
Graphs And Algorithms
621 Citations1984Michel Gondran, Michel Minoux +1 more
Presents a review of graph theory, analyzing the existing links between abstract theoretical results and their practical implications using graph theoretical models and combinatorial algorithms.
Fuzzy Sets and SystemsFuzzy weighted averages and implementation of the extension principle
558 Citations1987Weihong Dong, F. S. Wong
A computational algorithm based on the α-cut representation of fuzzy sets and interval analysis is described which provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner.
The Handbooks of fuzzy sets seriesFuzzy Interval Analysis
303 Citations2000Didier Dubois, Etienne Kerre +2 more
This chapter is an overview of past and present works dealing with fuzzy intervals and their operations, and contains a reasoned survey of methods for comparing and ranking fuzzy intervals.
European Journal of Operational ResearchFuzzy scheduling: Modelling flexible constraints vs. coping with incomplete knowledge
286 Citations2003Didier Dubois, Hélène Fargier +1 more
An overview of some fuzzy set-based approaches to scheduling is proposed,phasizing two distinct uses of fuzzy sets: representing preference profiles and modelling uncertainty distributions, and a possibility-theoretic counterpart of PERT.
Applied Fuzzy Arithmetic
256 Citations2005Michael Hanss
Journal of Mathematical Analysis and ApplicationsCompactness in fuzzy topological spaces
252 Citations1978T. E. Gantner, R. C. Steinlage +1 more
It is observed that it is possible to have degrees of compactness, which is called α-compactness (α a member of a designated lattice), and a Tychonoff Theorem is obtained for an arbitrary product of α-Compact fuzzy spaces and a 1-point compactification.
Fuzzy Sets and SystemsA new fuzzy arithmetic
246 Citations1999Ming Ma, Menahem Friedman +1 more
A new fuzzy arithmetic is defined and applied to fuzzy linear equations and fuzzy calculus based on a new parametric form presented in this paper.
Fuzzy Sets and SystemsUsing fuzzy set theory in a scheduling problem: A case study
189 Citations1979Henri Prade
Fuzzy concepts are shown to be very useful and easy to work with in this decision-aid problem where it seems interesting to use fuzzy sets.
Soft ComputingGradual elements in a fuzzy set
185 Citations2007Didier Dubois, Henri Prade
A new concept in fuzzy set theory is introduced, that of a gradual element, that embodies the idea of fuzziness only, thus contributing to the distinction between fuzziness and imprecision.
Fuzzy Sets and SystemsFuzzy cardinality and the modeling of imprecise quantification
182 Citations1985Didier Dubois, Henri Prade
A systematic investigation of the cardinality of a fuzzy set is performed which unifies and improves previous attempts and the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized.
International Journal of Approximate ReasoningBelief functions on real numbers
166 Citations2005Philippe Smets
The TBM (transferable belief model) is generalized to the case where the frame of discernment is the extended set of real numbers R=[-~,~], under the assumptions that 'masses' can only be given to intervals.
Journal of Mechanical DesignImprecision in Engineering Design
161 Citations1995Erik K. Antonsson, Kevin Otto
The Method of Imprecision (MoI), a formal method, based on the mathematics of fuzzy sets, for representing and manipulating imprecision in engineering design, can provide formal information upon which to base decisions during preliminary engineering design and can facilitate set-based concurrent design.
Fuzzy Sets and SystemsFuzzy weighted average: An improved algorithm
157 Citations1992Tian‐Shy Liou, Mao‐Jiun J. Wang
A steepest descend/ascend method is proposed to facilitate steps (3) and (4) of the FWA to find the minimum/maximum of f(x1,x2,…,xn,w1,w2,….,wn) = Σi = 1n wixi/Σi=1n wi.
Medical Entomology and ZoologyComputation Over Fuzzy Quantities
154 Citations1994Milan Mareš
The Structure of Fuzziness - Heuristic Views List of Main Symbols Literature Bibliography Index
Fuzzy Sets and SystemsAn efficient algorithm for fuzzy weighted average
137 Citations1997Dong Hoon Lee, Daihee Park
This paper proposes an efficient algorithm to compute fuzzy weighted average, which turned out to be superior to the previous works by reducing the number of comparisons and arithmetic operations to O(n log n).
Fuzzy Sets and SystemsFuzzy addition in the L-fuzzy real line
103 Citations1982S.E. Rodabaugh
A binary operation ⊕ is constructed on the L-fuzzy real line R (L) which reduces to the usual addition on R if ⊆ is restricted to the embedded image of R in R ( L), which yields a partially ordered, abelian cancellation semigroup with identity, and which is jointly fuzzy continuous on R (l).
North-Holland mathematics studiesLinear Algebra in Dioids: A Survey of Recent Results
102 Citations1984Michel Gondran, Michel Minoux
European Journal of Operational ResearchOn latest starting times and floats in activity networks with ill-known durations
101 Citations2003Didier Dubois, Hélène Fargier +1 more
A rigorous treatment of the problem of determining latest starting dates and slack times of activities in fuzzy activity networks in the framework of possibility theory is proposed.
Fuzzy Set Theory
90 Citations1996R. Löwen
Fuzzy Sets and SystemsImplementing fuzzy arithmetic
84 Citations1995A. M. Anile, S. Deodato +1 more
This paper presents the authors' efforts towards the development of an effective and friendly working tool for fuzzy arithmetic and describes the two programming libraries currently produced.
Computers & Mathematics with ApplicationsFuzzy weighted average: A max-min paired elimination method
80 Citations1996Yuh-Yuan Guh, Cheng-Chung Hon +2 more
By the use of max-min paired elimination, an efficient method for obtaining the fuzzy weighted average (FWA) was developed and is much more efficient than either the approach developed by Dong and Wong or the improved method developed by Liou and Wang.
Fuzzy Sets and SystemsOn computing the latest starting times and floats of activities in a network with imprecise durations
76 Citations2004Paweł Zieliński
New polynomial algorithms for determining the intervals of the latest starting times in general networks are proposed and some complexity results for floats are presented and some polynomially solvable cases are described.
Reliable ComputingIdempotent Interval Analysis and Optimization Problems
69 Citations2001Grigori Litvinov, Andreĭ Sobolevskiĭ
A generalized vertex method for computing with fuzzy intervals
66 Citations2005Didier Dubois, Hélène Fargier +1 more
A new method for computing functions of fuzzy intervals under various monotonicity assumptions on the concerned functions makes exact computation for all possibility degrees, without resorting to /spl alpha/-cuts.
Fuzzy Sets and SystemsCalculating functions of fuzzy numbers
54 Citations1993Hong Quan Yang, Yao Hua +1 more
It is shown that there are many cases for which the FWA, and all other published methods, will give wrong results, and an alternative approach is suggested which will work in all cases.
Fuzzy Sets and SystemsFuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic
52 Citations1987H. Ulrich
The Dedekind-MacNeille completion of a partially ordered L-valued set is established, and specializing this situation to the set of all crisp rational numbers the authors obtain in a unique way the fuzzy Dedeksind real numbers.
The Empirical Variance of a Set of Fuzzy Intervals
47 Citations2005Didier Dubois, Hélène Fargier +1 more
The profile method is adapted to the computation of the empirical variance of a tuple of fuzzy intervals, viewed as pairs of fuzzy bounds, and yields exact results without applying interval analysis to alpha-cuts.
Fuzzy Sets and SystemsThe set of fuzzy rational numbers and flexible querying
40 Citations2005Daniel Rocacher, Patrick Bosc
Applied to flexible querying of databases, Q"f and the notion of division on fuzzy numbers allow to generalize the relational division and define a sound basis for queries involving ratios between quantities.
Vaguely Defined Objects
38 Citations1996Maciej Wygralak
Lecture notes in computer scienceCalculus with Fuzzy Numbers
37 Citations2005Witold Kosiński, Piotr Prokopowicz +1 more
Algebra of ordered fuzzy numbers (OFN) is defined to handle with fuzzy inputs in a quantitative way, exactly in the same way as with real numbers.
Lecture notes in computer scienceInterval Analysis in Scheduling
26 Citations2005Jérôme Fortin, Paweł Zieliński +2 more
This paper reconsiders the most basic scheduling problem, that of minimizing the makespan of a partially ordered set of activities, in the context of incomplete knowledge, and provides a complete solution to the problem of finding the latest starting times and floats of activities.
Journal of Intelligent ManufacturingComputational methods for determining the latest starting times and floats of tasks in interval-valued activity networks
24 Citations2005Didier Dubois, Hélène Fargier +1 more
An efficient algorithm based on path enumeration to compute optimal intervals for latest starting times and floats in the general case, and a simpler polynomial algorithm in the case of series-parallel activity networks are proposed.
CERN Document Server (European Organization for Nuclear Research)Vaguely Defined Objects: Representations, Fuzzy Sets and Nonclassical Cardinality theory
23 Citations2013Maciej Wygralak
This chapter discusses the development of nonclassical Cardinality Theory for Vaguely Defined Objects and its application to Arithmetical Operations.
Studies in fuzziness and soft computingAverage level of a fuzzy set
23 Citations2002Dan A. Ralescu
The average level set of a fuzzy set is defined on the basis of the Kudo-Aumann integral of a set-valued mapping and its properties as well as its particularization for some special cases are analyzed.
Lecture notes in computer scienceFuzzy Calculator – Useful Tool for Programming with Fuzzy Algebra
22 Citations2004Roman Koleśnik, Piotr Prokopowicz +1 more
First version of the program in the Delphi environment is created that uses algorithms dedicated to trapezoidal-type membership relations (functions) and more useful implementation is a Fuzzy Calculator which allows counting with OFN’s of general type membership relations and is equipped with a graphical shell.
International Journal of Uncertainty Fuzziness and Knowledge-Based SystemsA TOTAL ORDER FOR THE GRADED NUMBERS USED IN DECISION PROBLEMS
16 Citations1999José A. Herencia, M. Teresa Lamata
A definition of a total order for the fuzzy numbers is considered, which is based upon two subjective aspects: the degree of optimism/pessimism and the liking for risk/safety.
International Journal of Intelligent SystemsImplementation of piecewise linear fuzzy quantities
15 Citations1995Hans Steyaert, Filip Van Parys +2 more
This article describes an implementation of piecewise linear fuzzy quantities and how to perform fuzzy operations on them.
Lecture notes in computer scienceAbout Zf, the Set of Fuzzy Relative Integers, and the Definition of Fuzzy Bags on Zf
13 Citations2003Patrick Bosc, Daniel Rocacher
This paper shows that fuzzy relative integers (Zf) offer a well-founded framework in which the difference of two fuzzy bags is always defined.
HAL (Le Centre pour la Communication Scientifique Directe)On distances between fuzzy points and their use for plausible reasoning
13 Citations1983Didier Dubois, Henri Prade
HAL (Le Centre pour la Communication Scientifique Directe)Solving Fuzzy PERT using gradual real numbers
10 Citations2006Jérôme Fortin, Didier Dubois
It is shown that algorithms in the interval--valued case can be adapted to fuzzy intervals considering them as crisp intervals of gradual numbers, that represents soft boundaries of a fuzzy interval.
Fuzzy Real Numbers
8 Citations1996R. Löwen
A account of the most common and most important types of fuzzy real numbers, together with the operations defined on them, are given.
Lecture notes in computer sciencePiecewise linear fuzzy quantities : A way to implement fuzzy information into expert systems and fuzzy databases
8 Citations1988R Baekeland, Etienne E. Kerre
It is shown that most operations on piecewise linear fuzzy quantities yield a fuzzy quantity of the same type and the given proofs indicate sufficient information for writing appropriate computerprograms.
