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Fuzzy Number Addition with the Application of Horizontal Membership Functions.

Piegat A, Pluciński M - ScientificWorldJournal (2015)

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.

No MeSH data available.


Related in: MedlinePlus

Three different membership functions of “results” of the fuzzy formula C = A − A2 written in three equivalent forms C1 = A − A2,  C2 = A(1 − A),  C3 = (A − 1)+(1 − A)(1 + A) obtained with μ-cuts' fuzzy arithmetic, where A = [μ, 2 − μ].
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fig4: Three different membership functions of “results” of the fuzzy formula C = A − A2 written in three equivalent forms C1 = A − A2,  C2 = A(1 − A),  C3 = (A − 1)+(1 − A)(1 + A) obtained with μ-cuts' fuzzy arithmetic, where A = [μ, 2 − μ].

Mentions: Because of this feature of μ-cuts' FA, transformations of formulas are not allowable. Why? It will be shown further on. Next important paradox of μ-cuts' FA is the observation that, during calculation of results of nonlinear formulas, for example, C = A − A2, we obtain different, nonunique solutions depending on which form of the formula is used: C1 = A − A2,  C2 = A(1 − A), or C3 = (A − 1)+(1 − A)(1 + A). For A = [0,1, 2], three different solutions are obtained: C1 = [−4,0, 2],  C2 = [−2,0, 2], and C3 = [−4,0, 4]; see Figure 4. Which solution is correct? The above phenomenon means that each transformation of an equation form, in the case of μ-cuts' FA, can change its solution and that solutions are not unique. Further on, it will be shown that in the case of the multidimensional RDM FA such paradoxes do not occur.


Fuzzy Number Addition with the Application of Horizontal Membership Functions.

Piegat A, Pluciński M - ScientificWorldJournal (2015)

Three different membership functions of “results” of the fuzzy formula C = A − A2 written in three equivalent forms C1 = A − A2,  C2 = A(1 − A),  C3 = (A − 1)+(1 − A)(1 + A) obtained with μ-cuts' fuzzy arithmetic, where A = [μ, 2 − μ].
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4493312&req=5

fig4: Three different membership functions of “results” of the fuzzy formula C = A − A2 written in three equivalent forms C1 = A − A2,  C2 = A(1 − A),  C3 = (A − 1)+(1 − A)(1 + A) obtained with μ-cuts' fuzzy arithmetic, where A = [μ, 2 − μ].
Mentions: Because of this feature of μ-cuts' FA, transformations of formulas are not allowable. Why? It will be shown further on. Next important paradox of μ-cuts' FA is the observation that, during calculation of results of nonlinear formulas, for example, C = A − A2, we obtain different, nonunique solutions depending on which form of the formula is used: C1 = A − A2,  C2 = A(1 − A), or C3 = (A − 1)+(1 − A)(1 + A). For A = [0,1, 2], three different solutions are obtained: C1 = [−4,0, 2],  C2 = [−2,0, 2], and C3 = [−4,0, 4]; see Figure 4. Which solution is correct? The above phenomenon means that each transformation of an equation form, in the case of μ-cuts' FA, can change its solution and that solutions are not unique. Further on, it will be shown that in the case of the multidimensional RDM FA such paradoxes do not occur.

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.

No MeSH data available.


Related in: MedlinePlus