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07-11-2019 | Original Paper

Gradual interval arithmetic and fuzzy interval arithmetic

Authors: Reda Boukezzoula, Laurent Foulloy, Didier Coquin, Sylvie Galichet

Published in: Granular Computing | Issue 2/2021

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Abstract

This paper proposes an analysis of and a reflection on interval arithmetic (IA) and its extension to gradual interval arithmetic (GIA). Through this reflection, an overview of a part of IA that is directly related to fuzzy interval arithmetic (FIA) is analyzed, compared, and categorized according to two main families of IA: standard interval arithmetic (SIA) and instantiated interval arithmetic (IIA). Furthermore, SIA and IIA visions represent two viewpoints of computation that are different and they will cause modifications in interval interpretation and manipulation. This vision is essential in understanding the philosophy of IA and GIA computational mechanisms. The contribution of this paper is twofold. First, according to SIA and IIA visions, an analysis and a classification of a part of IAs are given. Equivalences and links between these IAs are analyzed and established. Second, an extension of IA to the gradual context is proposed. The GIA extension provides a new interpretation of FIA according to the gradual representation.

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Let us note that ⊗ is used to represent the cartesian product in place of the conventional symbol ×, which is used in this paper for the multiplication operator.

Allahviranloo T, Ghanbari M, Hosseinzadeh AA, Haghi E, Nuraei R (2011) A note on fuzzy linear systems. Fuzzy Sets Syst 177:87–92MATH

Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349MathSciNetMATH

Barros LC, Pedro FS (2017) Fuzzy differential equations with interactive derivative. Fuzzy Sets Syst 309:64–80MathSciNetMATH

Bodjanova S (2003) Alpha-bounds of fuzzy numbers. Inf Sci 152:237–266MathSciNetMATH

Boukezzoula R, Galichet S (2010) Optimistic arithmetic operators for fuzzy and gradual intervals-part I: interval approach, part II: fuzzy and gradual interval approach. IPMU conference, June 2010, Dortmund, Germany

Boukezzoula R, Foulloy L, Galichet S (2012) model inversion using extended gradual intervals arithmetic. IEEE Trans Fuzzy Syst 1:82–95MATH

Boukezzoula R, Galichet S, Foulloy L, Elmasry M (2014) Extended gradual interval (EGI) arithmetic and its application to gradual weighted averages. Fuzzy Sets Syst 257:67–84MathSciNetMATH

Boukezzoula R, Jaulin L, Foulloy L (2019) Thick gradual intervals: an alternative interpretation of type-2 fuzzy intervals and its potential use in type-2 fuzzy computations. Eng Appl Artif Intell 85:691–712

Burkill JC (1924) Functions of intervals. Proc London Math Soc 22:275–336MathSciNetMATH

Cabral VM, Barros LC (2015) Fuzzy differential equations with completely correlated parameters. Fuzzy Sets Syst 265:86–98MathSciNetMATH

Carlsson C, Fuller R (2005) On additions of interactive fuzzy numbers. Acta Polytech Hung 2:59–73

Chalco-Cano Y, Lodwick WA, Bede B (2014) Single level constraint interval arithmetic. Fuzzy Sets Syst 257:146–168MathSciNetMATH

Chen S-M, Yang M-W, Yang S-W, Sheu T-W, Liau C-J (2012a) Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets. Expert Syst Appl 39(15):12085–12091

Chen S-M, Lee L-W, Liu H-C, Yang S-W (2012b) Multiattribute decision making based on interval-valued intuitionistic fuzzy values. Expert Syst Appl 39(12):10343–10351

Costa TM, Chalco-Cano Y, Lodwick WA, Silva GN (2015) Generalized interval vector spaces and interval optimization. Inf Sci 311:74–85MathSciNetMATH

Cuso I, Dubois D (2014) Statistical reasoning with set-valued information: ontic vs. epistemic views. Int J Approx Reason 55(7):1502–1518MathSciNetMATH

Dimitrova N, Hayes N, Markov S (2010) Motion 12.03: inner addition/subtraction over intervals (IEEE interval arithmetic standard WG). http://grouper.ieee.org/groups/1788. Accessed 2010

Dong WM, Wong FS (1987) Fuzzy weighted averages and implementation of the extension principle. Fuzzy Sets Syst 1:183–199MathSciNetMATH

Dubois D (2011) Ontic vs. epistemic fuzzy sets in modeling and data processing tasks. Keynote Lecturer, Int. Joint Conference on Computational Intelligence-IJCCI, France

Dubois D (2014) On various ways of tackling incomplete information in statistics. Int J Approx Reason 55(7):1570–1574MATH

Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New YorkMATH

Dubois D, Prade H (1988) Possibility theory. An approach to computerized processing of uncertainty. Plenum Press, New YorkMATH

Dubois D, Prade H (2008) Gradual elements in a fuzzy set. Soft Comput 12:165–175MATH

Dubois D, Kerre E, Mesiar R, Prade H (2000) Fuzzy interval analysis. In: Dubois, Prade (eds) Fundamentals of fuzzy sets, the handbooks of fuzzy sets series. Kluwer, Boston, pp 483–581MATH

Dutta P, Doley D (2019) Fuzzy decision making for medical diagnosis using arithmetic of generalised parabolic fuzzy numbers. Granul Comput. https://doi.org/10.1007/s41066-019-00192-4CrossRef

Dutta P, Saikia B (2019) Arithmetic operations on normal semi elliptic intuitionistic fuzzy numbers and their application in decision-making. Granul Comput. https://doi.org/10.1007/s41066-019-00175-5CrossRef

Esmi E, Barros LC, Wasques VF (2019) Some notes on the addition of interactive fuzzy numbers. In: Fuzzy techniques: theory and appl, IFSA/NAFIPS. Springer, pp 246–257

Fahmi A, Abdullah S, Amin F (2019) Aggregation operators on cubic linguistic hesitant fuzzy numbers and their application in group decision-making. Granul Comput. https://doi.org/10.1007/s41066-019-00188-0CrossRefMATH

Fortin J, Dubois D, Fargier H (2008) Gradual numbers and their application to fuzzy interval analysis. IEEE Trans Fuzzy Syst 16(2):388–402

Fuller R, Majlender P (2004) On interactive fuzzy numbers. Fuzzy Sets Syst 143(3):355–369MathSciNetMATH

Gardenes E, Trepat A (1980) Fundamentals of SIGLA, an interval computing system over the completed set of intervals. Computing 24:161–179MathSciNetMATH

Gardenes E, Mielgo H, Trepat A (1986) Modal intervals: reason and ground semantics. In: Nickel K (ed) Interval mathematics, vol 212. Lecture notes in computer science. Berlin, Heidelberg, pp 27–35MATH

Giachetti RE, Young RE (1997) A parametric representation of fuzzy numbers and their arithmetic operators. Fuzzy Sets Syst 91(2):185–202MathSciNetMATH

Gomes LT, Barros LC (2015) A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst 280:142–145MathSciNetMATH

Guerra ML, Stefanini L (2005) Approximate fuzzy arithmetic operations using monotonic interpolations. Fuzzy Sets Syst 150:5–33MathSciNetMATH

Hanss M (2005) Applied fuzzy arithmetic. Springer, BerlinMATH

Hukuhara M (1967) Intégration des applications mesurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj 10:205–223MathSciNetMATH

Kaucher E (1973) Über metrische und algebraische Eigenschaften einiger beim numerischen Rechnen auftretender Räume. Dissertation, Universität Karlsruhe

Kaucher E (1980) Interval analysis in the extended interval space IR. Comput Suppl 2:33–49MathSciNetMATH

Kaufmann A, Gupta MM (1991) Introduction to fuzzy arithmetic: theory and applications. Van Nostrand Reinhold Company Inc., New YorkMATH

Klir GJ (1997) Fuzzy arithmetic with requisite constraints. Fuzzy Sets Syst 91(2):165–175MathSciNetMATH

Klir GJ, Pan Y (1998) Constrained fuzzy arithmetic: basic questions and some answers. Soft Comput 2:100–108

Kulpa Z (2001) Diagrammatic representation for interval arithmetic. Linear Algebr Appl 324(1–3):55–80MathSciNetMATH

Liu X, Mendel JM, Wu D (2012) Analytical solution methods for the fuzzy weighted average. Inf Sci 187:151–170MathSciNetMATH

Lodwick WA (1999) Constrained interval arithmetic. CCM Report 138

Lodwick WA (2007) Interval and fuzzy analysis: an unified approach. Adv Imaging Electron Phys 148:75–192

Lodwick WA, Dubois D (2015) Interval linear systems as a necessary step in fuzzy linear systems. Fuzzy Sets Syst 281:227–251MathSciNetMATH

Lodwick WA, Jenkins OA (2013) Constrained intervals and interval spaces. Soft Comput 17:1393–1402MATH

Markov SM (1977) Extended interval arithmetic. Compt Rend Acad Bulg Sci 30(9):1239–1242MathSciNetMATH

Markov SM (1979) Calculus for interval functions of a real variable. Computing 22:325–337MathSciNetMATH

Markov SM (1995) On directed interval arithmetic and its applications. J Univers Comput Sci 1(7):510–521MathSciNetMATH

Markov SM (1997) Isomorphic embeddings of abstract interval systems. Reliab Comput 3:199–207MathSciNetMATH

Markov SM (2001) The mystery of intervals. Reliab Comput 7(1):63–65MathSciNetMATH

Mendel JM, John RI, Liu F (2006) Interval type-2 fuzzy logic systems made simple. IEEE Trans Fuzzy Syst 14(6):808–821

Moore RE (1962) Interval arithmetic and automatic error analysis in digital computing. PhD Thesis, Department of Computer Science, Stanford University

Moore RE (1966) Interval analysis. Prentice-Hall, NJMATH

Moore R, Lodwick WA (2003) Interval analysis and fuzzy set theory. Fuzzy Sets Syst 135(1):5–9MathSciNetMATH

Moore RE, Yang CT (1959) Interval analysis I. Technical report space div. Report LMSD285875

Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, CambridgeMATH

Ortolf HJ (1969) Eine Verallgemeinerung der Intervallarithmetik. Berichte der Geselschaft für Mathematik und Datenverarbeitung, Bonn 11:1–71MathSciNetMATH

Oussalah M, DeSchutter J (2003) Approximated fuzzy LR computation. Inf Sci 153:155–175MathSciNetMATH

Pedrycz W, Skowron A, Kreinovich V (2008) Handbook of granular computing. John Wiley & Sons, Chichester

Piegat A, Landowski M (2017) Is an interval the right result of arithmetic operations on intervals? Int J Appl Math Comput Sci 27(3):575–590MathSciNetMATH

Popova ED (2001) Multiplication distributivity of proper and improper intervals. Reliab Comput 7:129–140MathSciNetMATH

Popova ED, Markov SM (1997) Towards credible implementation of inner interval operations. In: 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics

Qin J, Liu X (2015) Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment. Inf Sci 297:293–315MathSciNetMATH

Qin J, Liu X, Pedrycz W (2017) A multiple attribute interval type-2 fuzzy group decision making and its application to supplier selection with extended LINMAP method. Soft Comput 21(12):3207–3226MATH

Ratschek H, Rokne J (1995) Interval methods. In: Horst, Pardalos (eds) Handbook of global optimization. Kluwer Academic Publishers, Boston, pp 751–828

Runkler T, Coupland S, John R (2017) Interval type-2 fuzzy decision making. Int J Approx Reason 80:217–224MathSciNetMATH

Stefanini LA (2010) Generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564–1584MathSciNetMATH

Sunaga T (1958) Theory of an interval algebra and its application to numerical analysis. RAAG Mem. 2:547–564MATH

Untiedt EA, Lodwick WA (2008) Using gradual numbers to analyze non-monotonic functions of fuzzy intervals. NAFIPS, New York

Vidhya R, Irene Hepzibah R (2017) A comparative study on interval arithmetic operations with intuitionistic fuzzy numbers for solving an intuitionistic fuzzy multi–objective linear programming problem. Int J Appl Math Comput Sci 27(3):563–573MathSciNetMATH

Wang C-Y, Chen S-M (2017) Multiple attribute decision making based on interval-valued intuitionistic fuzzy sets, linear programming methodology, and the extended TOPSIS method. Inf Sci 397:155–167

Warmus M (1956) Calculus of appoximations. Bulletin Acad Polon Science C1 III–IV:253–259MathSciNetMATH

Young RC (1931) The algebra of many-valued quantities. Math Annal 104:260–290MathSciNetMATH

Title: Gradual interval arithmetic and fuzzy interval arithmetic
Authors: Reda Boukezzoula
Laurent Foulloy
Didier Coquin
Sylvie Galichet
Publication date: 07-11-2019
Publisher: Springer International Publishing
Published in: Granular Computing / Issue 2/2021
Print ISSN: 2364-4966
Electronic ISSN: 2364-4974
DOI: https://doi.org/10.1007/s41066-019-00208-z

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