The Convexity of Fuzzy Sets and Augmented Extension Principle

Gadjiev, Djavanshir; Tokmazov, Georgy; Abdurazakov, Magomed

doi:10.1007/978-981-16-3246-4_19

Cited by 1 publication

(3 citation statements)

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“…Then, according to the definition of the 𝛼-cuts we can represent the fuzzy sets as stated above: The fuzzy set is convex if all its cuts are convex in terms of the convexity. We can re-define the cuts of the fuzzy sets by the pair of the fuzzy set with reference to their lowest upper bound and greatest lower bound defined as it is [13,14,15,16,17,18,19].…”

Section: 𝜶 -Cut Decomposition Methodsmentioning

confidence: 99%

“…One of the methods of the defuzzification is the 𝛂-cut method. Here for the given fuzzy set 𝐹(𝑥) we are going to define the crisp set 𝐹 𝛼 (𝑥) = {𝑥:𝜙 𝐹 𝛼 (𝑥) ≥ 𝛼, 0 < 𝛼 < 1 [13,14,15].…”

Section: Defuzzification To the Crisp Sets And Methods To Find The De...mentioning

confidence: 99%

“…The membership function of the fuzzy set 𝐹 ⊆ 𝑋 is denoted as 𝜙 𝐹 (𝑥) and defined as it is in [13,14,15] which can be interpreted by the following: nearer the value of the grade of membership to 1 then 𝑥 is more affiliated to 𝐹 :…”

Section: Fuzzy Membership Function and The Defuzzification To The Cri...mentioning

confidence: 99%

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The advanced defuzzification methods of the convex α – cut fuzzy sets

Gadjiev

Rustanov²,

Kochetkov³

2023

E3S Web of Conf.

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The main features of the fuzzy sets and their corresponding membership functions were presented in terms of the fuzzification process and further by the de-fuzzification operation. The convexity of the α -(alfa) cuts of the fuzzy sets is used in the decomposition of the fuzzy sets. The alfa cuts of the fuzzy sets were defined precisely in terms of the pair of functions and their lowest upper and greatest lower bounds. The convex combination of the intervals of the sub-regions of the fuzzy sets and their membership function were considered as the points of the defuzzified values of the fuzzy sets. The methods of the de-fuzzification to the crisp sets were presented by the formulas to find the defuzzification regions and de-fuzzified values. The compositional concepts of the inference as the expansion of the extension principle were introduced to formalize further the fuzzy reasoning by the set of fuzzy rules based on the approximate reasoning.

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Section: 𝜶 -Cut Decomposition Methodsmentioning

confidence: 99%

Section: Defuzzification To the Crisp Sets And Methods To Find The De...mentioning

confidence: 99%