Fuzzy Number Addition with the Application of Horizontal Membership Functions.

Piegat A, Pluciński M - ScientificWorldJournal (2015)

Related In: Results  -  Collection

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fig8: Multidimensional granule of the addition result of two fuzzy numbers “about 1.0” and “about 1.1” being the set of quadruples {μ, αx1, αx2, y} determined by formula (35) and projected onto the 3D space αx1 × αx2 × Y.
Mentions: Formula (35) shows that the sum y = x1 + x2 is not 1-dimensional. It is a function defined in the 4D space because y = f(μ, αx1, αx2). Thus, it cannot be visualised but it can be shown as the projection onto the 3D space: αx1 × αx2 × Y (Figure 8). The fourth dimension μ is shown in a form of μ-cuts. For μ = 0, sum (35) takes a form: y = 1.9 + 0.2(αx1 + αx2). For μ = 1, the sum has value y = 2.1 independently of values of αx1 and αx2. Surfaces of both μ-cuts can be seen in Figure 8.

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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