Fuzzy Number Addition with the Application of Horizontal Membership Functions.

Piegat A, Pluciński M - ScientificWorldJournal (2015)

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fig19: 3D projection of the full 4D solution of (64) from the space A × C × X × μ onto the space A × X × μ.
Mentions: It can be easily noticed that formula (64) is four-dimensional: x = f(μ, αa, αc). It cannot be expressed exactly in the 2D space, which is suggested by the extension principle and various versions of the fuzzy arithmetic. However, it can be visualised in a form of the 3D projection on subspaces, for example, on the subspace A × X × μ (variable αa is equivalent to variable a because of the transformation a = 4 + 2αa) (Figure 19).

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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