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Horizontal Membership Function and Examples of its Applications

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Abstract

The paper introduces horizontal membership functions (HMFs) which define a fuzzy set not in form of commonly used vertical membership functions of type μ = f 1(x) but in the horizontal form x = f 2(μ). Until now, constructing HMFs had seemed impossible because of horizontal ambiguity of this function. Now, however, it became possible thanks to the multidimensional, RDM-interval arithmetic based on relative-distance-measure variables. HMFs enable direct introducing uncertain, interval or fuzzy variable-values in usual mathematical formulas of type y = f(x 1 ,…,x 2) together with crisp values, without using Zadeh’s extension principle. Thus, a relatively easy aggregation of crisp and uncertain knowledge became possible. The paper shows application of HMFs, first on example of a classical mathematical function y = f(x 1 ,x 2) and next, on example of a computing with words challenge problem.

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Authors and Affiliations

  1. Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Zolnierska 49, 71-210, Szczecin, Poland

    Andrzej Piegat

  2. Maritime University of Szczecin, Waly Chrobrego 1-2, 70-500, Szczecin, Poland

    Marek Landowski

Authors
  1. Andrzej Piegat
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  2. Marek Landowski
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Correspondence to Marek Landowski.

Additional information

In 2014 professor Lotfi Zadeh, creator of fuzzy sets and of Computing with Words idea, celebrated his 93rd birthday. On this occasion a special conference “4th World Conference on Soft Computing” dedicated to his research heritage was organized by University of California in Berkeley. Authors of this paper dedicate it, and especially the novel notion of horizontal membership functions, to professor Zadeh.

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Piegat, A., Landowski, M. Horizontal Membership Function and Examples of its Applications. Int. J. Fuzzy Syst. 17, 22–30 (2015). https://doi.org/10.1007/s40815-015-0013-8

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  • DOI: https://doi.org/10.1007/s40815-015-0013-8

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