Fuzzy Number Addition with the Application of Horizontal Membership Functions.

Piegat A, Pluciński M - ScientificWorldJournal (2015)

Related In: Results  -  Collection

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fig9: Projection of the full 4D addition result granule onto the 3D space X1 × X2 × μ with slant contour lines corresponding to particular addition results y = x1 + x2 = const.
Mentions: The multidimensional result granule (Figure 8) contains an infinitive number of μ-cuts corresponding to particular fractional values of μ ∈ [0,1]. Figure 8 shows only two border cuts for μ = 0 and μ = 1. Apart from the 3D projection shown in Figure 8, other projections of the full 4D granule are also possible. Figure 9 shows the projection onto the 3D space X1 × X2 × μ. In this figure, addition results y = x1 + x2 are shown in a form of contour lines y = x1 + x2 = const.

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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