Integrations of q-Rung Orthopair Fuzzy Continuous Information

Shu, Xiaoqin; Ai, Zhenghai; Zhang, Xu; Ye, Jianmei

doi:10.1109/tfuzz.2019.2893205

Cited by 51 publications

(36 citation statements)

References 30 publications

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“…In the same way, if we note that they are the inverse operations of the addition and the multiplication, respectively. Then, Shu et al gave the following operations:…”

Section: Preliminariesmentioning

confidence: 99%

“…Definition Let

α_{q} = (μ_{α}, v_{α})

and

β_{q} = (μ_{β}, v_{β})

be two q‐ ROFNs; then

β ⊖ α = true{\begin{matrix} true({(\frac{μ_{β}^{q} - μ_{α}^{q}}{1 - μ_{α}^{q}})}^{1 / q}, \frac{v_{β}}{v_{α}} true) & i f 0 \leq \frac{v_{β}^{q}}{v_{α}^{q}} \leq \frac{1 - μ_{β}^{q}}{1 - μ_{α}^{q}} \leq 1, \\ true(0, 1 true) & otherwise, \end{matrix}

β ⊘ α = true{\begin{matrix} true(\frac{μ_{β}}{μ_{α}}, {(\frac{v_{β}^{q} - v_{α}^{q}}{1 - v_{α}^{q}})}^{1 / q} true) & i f 0 \leq \frac{μ_{β}^{q}}{μ_{β}^{q}} \leq \frac{1 - v_{β}^{q}}{1 - v_{β}^{q}} \leq 1, \\ true(1, 0 true) & otherwise . \end{matrix}

…”

Section: Preliminariesunclassified

“…On the basis of Definitions and , for a given

α_{q}^{0}

, the four regions about q‐ ROFNs can be expressed as follows:

A_{α_{q}^{-1pt 0}}^{\oplus} = {α_{q}^{0} \oplus β | β \in Q}, A_{α_{q}^{0}}^{⊖} = {α_{q}^{0} ⊖ β | β \in Q}, A_{α_{q}^{0}}^{\otimes} = {α_{q}^{0} \otimes β | β \in Q}, and A_{α_{q}^{0}}^{⊘} = {α_{q}^{0} ⊘ β | β \in Q},

where

Q

is a q‐ ROFS. Due to that the subtraction and the division are inverse operations of the addition and the multiplication, respectively, Shu et al gave two partition relationships of q‐ ROFSs between the addition and subtraction regions, and between the multiplication and division regions. Figures and show intuitively those relationships.…”

Section: Preliminariesmentioning

confidence: 99%

“…Because the differentials describe the local properties of object motion, thus, the authors also studied their continuities, derivatives, and differentials on the basis of q‐ ROFFs. In addition, since the integrals describe the overall properties of object motion, then Shu et al studied the q ‐rung orthopair fuzzy definite integrals filling the vacancy of aggregating q ‐rung orthopair fuzzy continuous information. Although this type of information is described by the form of continuous membership degrees and nonmembership degrees, the above‐mentioned study only considered that the membership and nonmembership functions are multivariable functions.…”

Section: Introductionmentioning

confidence: 99%

“…Due to that the subtraction and the division are inverse operations of the addition and the multiplication, respectively, Shu et al 23 gave two partition relationships of q-ROFSs between the addition and subtraction regions, and between the multiplication and division regions. Figures 1 and 2 show intuitively those relationships.…”

mentioning

confidence: 99%

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Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Zhang

2019

Int J Intell Syst

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The q‐rung orthopair fuzzy set ( q‐ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q‐rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q‐rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q‐rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q‐rung orthopair single variable fuzzy function ( q‐ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q‐ROSVFFs, and offer the relationship between the limit of q‐ROSVFF and that of q‐ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q‐ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q‐ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q‐ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q‐ROSVFFs.

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“…In the same way, if we note that they are the inverse operations of the addition and the multiplication, respectively. Then, Shu et al gave the following operations:…”

Section: Preliminariesmentioning

confidence: 99%

“…Definition Let

α_{q} = (μ_{α}, v_{α})

and

β_{q} = (μ_{β}, v_{β})

be two q‐ ROFNs; then

β ⊖ α = true{\begin{matrix} true({(\frac{μ_{β}^{q} - μ_{α}^{q}}{1 - μ_{α}^{q}})}^{1 / q}, \frac{v_{β}}{v_{α}} true) & i f 0 \leq \frac{v_{β}^{q}}{v_{α}^{q}} \leq \frac{1 - μ_{β}^{q}}{1 - μ_{α}^{q}} \leq 1, \\ true(0, 1 true) & otherwise, \end{matrix}

β ⊘ α = true{\begin{matrix} true(\frac{μ_{β}}{μ_{α}}, {(\frac{v_{β}^{q} - v_{α}^{q}}{1 - v_{α}^{q}})}^{1 / q} true) & i f 0 \leq \frac{μ_{β}^{q}}{μ_{β}^{q}} \leq \frac{1 - v_{β}^{q}}{1 - v_{β}^{q}} \leq 1, \\ true(1, 0 true) & otherwise . \end{matrix}

…”

Section: Preliminariesunclassified

“…On the basis of Definitions and , for a given

α_{q}^{0}

, the four regions about q‐ ROFNs can be expressed as follows:

A_{α_{q}^{-1pt 0}}^{\oplus} = {α_{q}^{0} \oplus β | β \in Q}, A_{α_{q}^{0}}^{⊖} = {α_{q}^{0} ⊖ β | β \in Q}, A_{α_{q}^{0}}^{\otimes} = {α_{q}^{0} \otimes β | β \in Q}, and A_{α_{q}^{0}}^{⊘} = {α_{q}^{0} ⊘ β | β \in Q},

where

Q

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

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2019

Int J Intell Syst

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Evaluation for hierarchical diagnosis and treatment policy proposals in China: A novel multi‐attribute group decision‐making method with multi‐parametric distance measures

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Weighted power means of q‐rung orthopair fuzzy information and their applications in multiattribute decision making

Wen

2019

Int J Intell Syst

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Weighted power means with weights and exponents serving as their parameters are generalizations of arithmetic means. Taking into account decision makers' flexibility in decision making, each attribute value is usually expressed by a q-rung orthopair fuzzy value (q-ROFV, ≥ q 1), where the former indicates the support for membership, the latter support against membership, and the sum of their qth powers is bounded by one. In this paper, we propose the weighted power means of qrung orthopair fuzzy values to enrich and flourish aggregations on q-ROFVs. First, the q-rung orthopair fuzzy weighted power mean operator is presented, and its boundedness is precisely characterized in terms of the power exponent. Then, the q-rung orthopair fuzzy ordered weighted power mean operator is introduced, and some of its fundamental properties are investigated in detail. Finally, a novel multiattribute decision making method is explored based on developed operators under the q-rung orthopair fuzzy environment. A numerical example is given to illustrate the feasibility and validity of the proposed approach, and it is shown that the power exponent is an index suggesting the degree of the optimism of decision makers. K E Y W O R D S aggregation operator, multiattribute decision making, q-rung orthopair fuzzy value, weighted power mean 2836 | DU operations. 22 Ye et al 23 studied the single variable differential calculus under q-rung orthopair fuzzy environment. Shu et al 24 developed q-rung orthopair fuzzy definite integrals to aggregate q-rung orthopair fuzzy continuous information. Peng et al 25 presented the exponential operation of q-ROFVs, in which the bases are positive real numbers and the exponents are q-ROFVs. Joshi et al 26 introduced the notion of interval-valued q-rung orthopair fuzzy sets by combining interval-valued and q-rung orthopair fuzzy sets. In the seminal paper of Yager, 12 a general framework of constructing aggregation operators of q-ROFSs is provided. Specially, Yager et al 12,17 developed the Sugeno and Choquet aggregation operators on q-ROFSs based on a pair of dual aggregation operators. Along this line of research, Liu et al introduced a large number of operators on q-ROFSs, including the weighted averaging/geometric operator, 18 (weighted, geometric) Bonferroni mean operator, 19 (weighted) Archimedean Bonferroni mean operator, 27 power (weighted) average operator, 28 power (weighted) Maclaurin symmetric mean operator, 28 and (weighted) Heronian mean operator. 29 Wei et al further put forward the generalized (weighted) Heronian mean operator, 30 (weighted) geometric Heronian mean operator 30 and (dual, weighted) Maclaurin symmetric mean operator 31 for q-ROFSs. Wang et al 32 presented the q-rung orthopair fuzzy (weighted, dual) Muirhead mean operators for fusing q-ROFSs. Peng et al 25 proposed the q-rung orthopair fuzzy weighted exponential aggregation operator over q-ROFSs based on the exponential operation of q-ROFVs.Weighted power mean (WPM, also called the generalized weighted averaging operator) is an i...

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Integrations of q-Rung Orthopair Fuzzy Continuous Information

Cited by 51 publications

References 30 publications

Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Evaluation for hierarchical diagnosis and treatment policy proposals in China: A novel multi‐attribute group decision‐making method with multi‐parametric distance measures

Weighted power means of q‐rung orthopair fuzzy information and their applications in multiattribute decision making

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