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Fundamentals of Fuzzy Sets pp 483–581Cite as

Fuzzy Interval Analysis

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Part of the The Handbooks of Fuzzy Sets Series book series (FSHS,volume 7)

Abstract

This chapter is an overview of past and present works dealing with fuzzy intervals and their operations. A fuzzy interval is a fuzzy set in the real line whose level-cuts are intervals. Particular cases include usual real numbers and intervals. Usual operations on the real line canonically extend to operations between fuzzy quantities, thus extending the usual interval (or error) analysis to membership functions. What is obtained is a counterpart of random variable calculus, but where, contrary to the latter case, there is no compensation between variables. Many results pertaining to basic properties of fuzzy interval analysis are summed up in the chapter. Computational methods are presented, exact or approximate ones, based on parametric representations, or level-cut approximations. The generalized fuzzy variable calculus involving interactive variables is also discussed with emphasis on triangular-norm based fuzzy additions. Dual ‘optimistic’ operations on fuzzy intervals, i.e., with maximal error compensation are also presented; its interest lies in providing tools for solving fuzzy interval equations. This chapter also contains a reasoned survey of methods for comparing and ranking fuzzy intervals. The chapter includes some historical background, as well as pointers to applications in mathematics and engineering.

Keywords

  • Membership Function
  • Fuzzy Number
  • Fuzzy Variable
  • Fuzzy Relation
  • Fuzzy Random Variable

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors
  1. Didier Dubois
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  2. Etienne Kerre
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  3. Radko Mesiar
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  4. Henri Prade
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Dubois, D., Kerre, E., Mesiar, R., Prade, H. (2000). Fuzzy Interval Analysis. In: Dubois, D., Prade, H. (eds) Fundamentals of Fuzzy Sets. The Handbooks of Fuzzy Sets Series, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4429-6_11

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