Fuzzy Number Addition with the Application of Horizontal Membership Functions.

Piegat A, Pluciński M - ScientificWorldJournal (2015)

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fig2: Visualization of the horizontal approach to fuzzy membership functions.
Mentions: Vertical MF realises the mapping x → μ. The present fuzzy arithmetic is based on just such MFs. The idea of horizontal MFs has been elaborated by Andrzej Piegat. In this paper, an example of a trapezium MF will be presented but horizontal MFs can be used for all types of MFs. A function μ(x) is unambiguous in the direction of the variable μ (Figure 1) and ambiguous in the direction of x. Therefore, it seems impossible to define a membership function in the x-direction. The function from Figure 1 assigns two values of x, xL(μ) and xR(μ), for one value of μ. However, let us introduce the RDM variable: αx ∈ [0,1]. This variable has meaning of the relative-distance-measure and allows for determining of any point between two borders xL(μ) and xR(μ) of the function (Figure 2). RDM variable αx takes a value of zero on the left border and a value of 1 on the right border of the function. Between the left border and the right border it takes fractional values. The idea of RDM variables was successfully used in the multidimensional IA [37, 67–70]. The multidimensional IA has shown that full and precise solutions of granular problems have form of multidimensional granules that cannot be explained and understood in terms of 1-dimensional approaches.

Bottom Line: The multidimensional approach allows for removing drawbacks and weaknesses of FA.It is possible thanks to the application of horizontal membership functions which considerably facilitate calculations because now uncertain values can be inserted directly into equations without using the extension principle.The paper shows how the addition operation can be realised on independent fuzzy numbers and on partly or fully dependent fuzzy numbers with taking into account the order relation and how to solve equations, which can be a difficult task for 1-dimensional FAs.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Computer Science and Information Systems, West Pomeranian University of Technology, Żołnierska 49, 71-210 Szczecin, Poland.

ABSTRACT
The paper presents addition of fuzzy numbers realised with the application of the multidimensional RDM arithmetic and horizontal membership functions (MFs). Fuzzy arithmetic (FA) is a very difficult task because operations should be performed here on multidimensional information granules. Instead, a lot of FA methods use α-cuts in connection with 1-dimensional classical interval arithmetic that operates not on multidimensional granules but on 1-dimensional intervals. Such approach causes difficulties in calculations and is a reason for arithmetical paradoxes. The multidimensional approach allows for removing drawbacks and weaknesses of FA. It is possible thanks to the application of horizontal membership functions which considerably facilitate calculations because now uncertain values can be inserted directly into equations without using the extension principle. The paper shows how the addition operation can be realised on independent fuzzy numbers and on partly or fully dependent fuzzy numbers with taking into account the order relation and how to solve equations, which can be a difficult task for 1-dimensional FAs.

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