Gradual interval arithmetic and fuzzy interval arithmetic

  • Reda BoukezzoulaEmail author
  • Laurent Foulloy
  • Didier Coquin
  • Sylvie Galichet
Original Paper


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.


Standard interval arithmetic (SIA) Instantiated interval arithmetic (IIA) Gradual intervals Gradual interval arithmetic (GIA) Fuzzy interval arithmetic (FIA) 



  1. Allahviranloo T, Ghanbari M, Hosseinzadeh AA, Haghi E, Nuraei R (2011) A note on fuzzy linear systems. Fuzzy Sets Syst 177:87–92zbMATHCrossRefGoogle Scholar
  2. Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349MathSciNetzbMATHCrossRefGoogle Scholar
  3. Barros LC, Pedro FS (2017) Fuzzy differential equations with interactive derivative. Fuzzy Sets Syst 309:64–80MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bodjanova S (2003) Alpha-bounds of fuzzy numbers. Inf Sci 152:237–266MathSciNetzbMATHCrossRefGoogle Scholar
  5. 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, GermanyGoogle Scholar
  6. Boukezzoula R, Foulloy L, Galichet S (2012) model inversion using extended gradual intervals arithmetic. IEEE Trans Fuzzy Syst 1:82–95zbMATHCrossRefGoogle Scholar
  7. Boukezzoula R, Galichet S, Foulloy L, Elmasry M (2014) Extended gradual interval (EGI) arithmetic and its application to gradual weighted averages. Fuzzy Sets Syst 257:67–84MathSciNetzbMATHCrossRefGoogle Scholar
  8. 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 85:691–712CrossRefGoogle Scholar
  9. Burkill JC (1924) Functions of intervals. Proc London Math Soc 22:275–336MathSciNetzbMATHCrossRefGoogle Scholar
  10. Cabral VM, Barros LC (2015) Fuzzy differential equations with completely correlated parameters. Fuzzy Sets Syst 265:86–98MathSciNetzbMATHCrossRefGoogle Scholar
  11. Carlsson C, Fuller R (2005) On additions of interactive fuzzy numbers. Acta Polytech Hung 2:59–73Google Scholar
  12. Chalco-Cano Y, Lodwick WA, Bede B (2014) Single level constraint interval arithmetic. Fuzzy Sets Syst 257:146–168MathSciNetzbMATHCrossRefGoogle Scholar
  13. 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):12085–12091CrossRefGoogle Scholar
  14. 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):10343–10351CrossRefGoogle Scholar
  15. Costa TM, Chalco-Cano Y, Lodwick WA, Silva GN (2015) Generalized interval vector spaces and interval optimization. Inf Sci 311:74–85MathSciNetzbMATHCrossRefGoogle Scholar
  16. Cuso I, Dubois D (2014) Statistical reasoning with set-valued information: ontic vs. epistemic views. Int J Approx Reason 55(7):1502–1518MathSciNetzbMATHCrossRefGoogle Scholar
  17. Dimitrova N, Hayes N, Markov S (2010) Motion 12.03: inner addition/subtraction over intervals (IEEE interval arithmetic standard WG). Accessed 2010
  18. Dong WM, Wong FS (1987) Fuzzy weighted averages and implementation of the extension principle. Fuzzy Sets Syst 1:183–199MathSciNetzbMATHCrossRefGoogle Scholar
  19. Dubois D (2011) Ontic vs. epistemic fuzzy sets in modeling and data processing tasks. Keynote Lecturer, Int. Joint Conference on Computational Intelligence-IJCCI, FranceGoogle Scholar
  20. Dubois D (2014) On various ways of tackling incomplete information in statistics. Int J Approx Reason 55(7):1570–1574zbMATHCrossRefGoogle Scholar
  21. Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New YorkzbMATHGoogle Scholar
  22. Dubois D, Prade H (1988) Possibility theory. An approach to computerized processing of uncertainty. Plenum Press, New YorkzbMATHGoogle Scholar
  23. Dubois D, Prade H (2008) Gradual elements in a fuzzy set. Soft Comput 12:165–175zbMATHCrossRefGoogle Scholar
  24. 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 483–581zbMATHCrossRefGoogle Scholar
  25. Dutta P, Doley D (2019) Fuzzy decision making for medical diagnosis using arithmetic of generalised parabolic fuzzy numbers. Granul Comput. CrossRefGoogle Scholar
  26. Dutta P, Saikia B (2019) Arithmetic operations on normal semi elliptic intuitionistic fuzzy numbers and their application in decision-making. Granul Comput. CrossRefGoogle Scholar
  27. 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 246–257Google Scholar
  28. Fahmi A, Abdullah S, Amin F (2019) Aggregation operators on cubic linguistic hesitant fuzzy numbers and their application in group decision-making. Granul Comput. CrossRefzbMATHGoogle Scholar
  29. Fortin J, Dubois D, Fargier H (2008) Gradual numbers and their application to fuzzy interval analysis. IEEE Trans Fuzzy Syst 16(2):388–402CrossRefGoogle Scholar
  30. Fuller R, Majlender P (2004) On interactive fuzzy numbers. Fuzzy Sets Syst 143(3):355–369MathSciNetzbMATHCrossRefGoogle Scholar
  31. Gardenes E, Trepat A (1980) Fundamentals of SIGLA, an interval computing system over the completed set of intervals. Computing 24:161–179MathSciNetzbMATHCrossRefGoogle Scholar
  32. 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 27–35zbMATHGoogle Scholar
  33. Giachetti RE, Young RE (1997) A parametric representation of fuzzy numbers and their arithmetic operators. Fuzzy Sets Syst 91(2):185–202MathSciNetzbMATHCrossRefGoogle Scholar
  34. Gomes LT, Barros LC (2015) A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst 280:142–145MathSciNetzbMATHCrossRefGoogle Scholar
  35. Guerra ML, Stefanini L (2005) Approximate fuzzy arithmetic operations using monotonic interpolations. Fuzzy Sets Syst 150:5–33MathSciNetzbMATHCrossRefGoogle Scholar
  36. Hanss M (2005) Applied fuzzy arithmetic. Springer, BerlinzbMATHGoogle Scholar
  37. Hukuhara M (1967) Intégration des applications mesurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj 10:205–223MathSciNetzbMATHGoogle Scholar
  38. Kaucher E (1973) Über metrische und algebraische Eigenschaften einiger beim numerischen Rechnen auftretender Räume. Dissertation, Universität KarlsruheGoogle Scholar
  39. Kaucher E (1980) Interval analysis in the extended interval space IR. Comput Suppl 2:33–49MathSciNetzbMATHCrossRefGoogle Scholar
  40. Kaufmann A, Gupta MM (1991) Introduction to fuzzy arithmetic: theory and applications. Van Nostrand Reinhold Company Inc., New YorkzbMATHGoogle Scholar
  41. Klir GJ (1997) Fuzzy arithmetic with requisite constraints. Fuzzy Sets Syst 91(2):165–175MathSciNetzbMATHCrossRefGoogle Scholar
  42. Klir GJ, Pan Y (1998) Constrained fuzzy arithmetic: basic questions and some answers. Soft Comput 2:100–108CrossRefGoogle Scholar
  43. Kulpa Z (2001) Diagrammatic representation for interval arithmetic. Linear Algebr Appl 324(1–3):55–80MathSciNetzbMATHCrossRefGoogle Scholar
  44. Liu X, Mendel JM, Wu D (2012) Analytical solution methods for the fuzzy weighted average. Inf Sci 187:151–170MathSciNetzbMATHCrossRefGoogle Scholar
  45. Lodwick WA (1999) Constrained interval arithmetic. CCM Report 138Google Scholar
  46. Lodwick WA (2007) Interval and fuzzy analysis: an unified approach. Adv Imaging Electron Phys 148:75–192CrossRefGoogle Scholar
  47. Lodwick WA, Dubois D (2015) Interval linear systems as a necessary step in fuzzy linear systems. Fuzzy Sets Syst 281:227–251MathSciNetzbMATHCrossRefGoogle Scholar
  48. Lodwick WA, Jenkins OA (2013) Constrained intervals and interval spaces. Soft Comput 17:1393–1402zbMATHCrossRefGoogle Scholar
  49. Markov SM (1977) Extended interval arithmetic. Compt Rend Acad Bulg Sci 30(9):1239–1242MathSciNetzbMATHGoogle Scholar
  50. Markov SM (1979) Calculus for interval functions of a real variable. Computing 22:325–337MathSciNetzbMATHCrossRefGoogle Scholar
  51. Markov SM (1995) On directed interval arithmetic and its applications. J Univers Comput Sci 1(7):510–521MathSciNetzbMATHGoogle Scholar
  52. Markov SM (1997) Isomorphic embeddings of abstract interval systems. Reliab Comput 3:199–207MathSciNetzbMATHCrossRefGoogle Scholar
  53. Markov SM (2001) The mystery of intervals. Reliab Comput 7(1):63–65MathSciNetzbMATHCrossRefGoogle Scholar
  54. Mendel JM, John RI, Liu F (2006) Interval type-2 fuzzy logic systems made simple. IEEE Trans Fuzzy Syst 14(6):808–821CrossRefGoogle Scholar
  55. Moore RE (1962) Interval arithmetic and automatic error analysis in digital computing. PhD Thesis, Department of Computer Science, Stanford UniversityGoogle Scholar
  56. Moore RE (1966) Interval analysis. Prentice-Hall, NJzbMATHGoogle Scholar
  57. Moore R, Lodwick WA (2003) Interval analysis and fuzzy set theory. Fuzzy Sets Syst 135(1):5–9MathSciNetzbMATHCrossRefGoogle Scholar
  58. Moore RE, Yang CT (1959) Interval analysis I. Technical report space div. Report LMSD285875Google Scholar
  59. Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, CambridgezbMATHGoogle Scholar
  60. Ortolf HJ (1969) Eine Verallgemeinerung der Intervallarithmetik. Berichte der Geselschaft für Mathematik und Datenverarbeitung, Bonn 11:1–71MathSciNetzbMATHGoogle Scholar
  61. Oussalah M, DeSchutter J (2003) Approximated fuzzy LR computation. Inf Sci 153:155–175MathSciNetzbMATHCrossRefGoogle Scholar
  62. Pedrycz W, Skowron A, Kreinovich V (2008) Handbook of granular computing. John Wiley & Sons, ChichesterCrossRefGoogle Scholar
  63. Piegat A, Landowski M (2017) Is an interval the right result of arithmetic operations on intervals? Int J Appl Math Comput Sci 27(3):575–590MathSciNetzbMATHCrossRefGoogle Scholar
  64. Popova ED (2001) Multiplication distributivity of proper and improper intervals. Reliab Comput 7:129–140MathSciNetzbMATHCrossRefGoogle Scholar
  65. Popova ED, Markov SM (1997) Towards credible implementation of inner interval operations. In: 15th IMACS World Congress on Scientific Computation, Modelling and Applied MathematicsGoogle Scholar
  66. Qin J, Liu X (2015) Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment. Inf Sci 297:293–315MathSciNetzbMATHCrossRefGoogle Scholar
  67. 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):3207–3226zbMATHCrossRefGoogle Scholar
  68. Ratschek H, Rokne J (1995) Interval methods. In: Horst, Pardalos (eds) Handbook of global optimization. Kluwer Academic Publishers, Boston, pp 751–828CrossRefGoogle Scholar
  69. Runkler T, Coupland S, John R (2017) Interval type-2 fuzzy decision making. Int J Approx Reason 80:217–224MathSciNetzbMATHCrossRefGoogle Scholar
  70. Stefanini LA (2010) Generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564–1584MathSciNetzbMATHCrossRefGoogle Scholar
  71. Sunaga T (1958) Theory of an interval algebra and its application to numerical analysis. RAAG Mem. 2:547–564zbMATHGoogle Scholar
  72. Untiedt EA, Lodwick WA (2008) Using gradual numbers to analyze non-monotonic functions of fuzzy intervals. NAFIPS, New YorkCrossRefGoogle Scholar
  73. 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):563–573MathSciNetzbMATHCrossRefGoogle Scholar
  74. 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 397:155–167CrossRefGoogle Scholar
  75. Warmus M (1956) Calculus of appoximations. Bulletin Acad Polon Science C1 III–IV:253–259MathSciNetzbMATHGoogle Scholar
  76. Young RC (1931) The algebra of many-valued quantities. Math Annal 104:260–290MathSciNetzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire d’Informatique, Traitement de l’Information et de la Connaissance-LISTIC, SystèmesUniversité Savoie Mont Blanc USMBAnnecy CedexFrance

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