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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

Domain of feasible tuples of weight values {x1, x2} which can take part in adding of fuzzy numbers x1 and x2.
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fig12: Domain of feasible tuples of weight values {x1, x2} which can take part in adding of fuzzy numbers x1 and x2.

Mentions: Thus, there are two stones S1 and S2 which had been weighed on spring scales with maximal error equal to 0.1 kg. Therefore, weights x1 and x2 of stones are uncertain and can be expressed in the form of fuzzy numbers: X1 is [0.9,1.0,1.1] and X2 is [1.0,1.1,1.2]. MFs of uncertain weights are shown in Figure 7. After weighting the stones on the spring scales, their weights were compared on balance scales. The scales showed that the weight x2, though uncertain, is greater than x1. Therefore, the weights order x2 > x1 is known and can be taken into account in the weights' adding. The knowledge of the order relation changes (constraints) the domain of possible weight tuples {x1, x2} in comparison with the example from Section 3. Now less tuples will be feasible (Figure 12).


Fuzzy Number Addition with the Application of Horizontal Membership Functions.

Piegat A, Pluciński M - ScientificWorldJournal (2015)

Domain of feasible tuples of weight values {x1, x2} which can take part in adding of fuzzy numbers x1 and x2.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig12: Domain of feasible tuples of weight values {x1, x2} which can take part in adding of fuzzy numbers x1 and x2.
Mentions: Thus, there are two stones S1 and S2 which had been weighed on spring scales with maximal error equal to 0.1 kg. Therefore, weights x1 and x2 of stones are uncertain and can be expressed in the form of fuzzy numbers: X1 is [0.9,1.0,1.1] and X2 is [1.0,1.1,1.2]. MFs of uncertain weights are shown in Figure 7. After weighting the stones on the spring scales, their weights were compared on balance scales. The scales showed that the weight x2, though uncertain, is greater than x1. Therefore, the weights order x2 > x1 is known and can be taken into account in the weights' adding. The knowledge of the order relation changes (constraints) the domain of possible weight tuples {x1, x2} in comparison with the example from Section 3. Now less tuples will be feasible (Figure 12).

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