Pythagorean cubic fuzzy aggregation operators and their application to multi-criteria decision making problems

Khan, Faisal; Khan, Muhammad Numan; Shahzad, Muhammad; Abdullah, Saleem

doi:10.3233/jifs-18943

Cited by 45 publications

(69 citation statements)

References 34 publications

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“…Definition Assuming N to be the universe of discourse, a PCFS

normalΘ

under N would mean a combination of collections 22,23 :

normalΘ = false{〈 n, ϑ false(n false), ϖ false(n false) 〉 \sim n \in N false},

ϑ false(n false) = false{n, false〈 [Δ^{L} false(n false), Δ^{R} false(n false)], [\nabla^{L} false(n false), \nabla^{R} false(n false)] false〉 ~ n \in N false}

is an IVPFS, where

0 \leq Δ^{L} \leq Δ^{R} \leq 1

and

0 \leq \nabla^{L} \leq \nabla^{R} \leq 1

;

ϖ false(n false) = false{n, false〈 \tilde{normalΔ} (n), \tilde{\nabla} (n) false〉

~ n \in N false}

is a PFS under N , where

0 \leq \overset{Δ}{true} \leq 1, 0 \leq \overset{\nabla}{true} \leq 1

and

\overset{π}{true} false(n false) = \sqrt{1 - {\tilde{normalΔ}}^{2} false(n false) - {\tilde{\nabla}}^{2} false(n false)}

represent the hesitation of the PFS. For each element, n in N , we have

0 \leq {false(Δ^{R} false(n false) false)}^{2} + {false(\nabla^{R} false(n false) false)}^{2} \leq 1

and

0 \leq false(\overset{Δ}{true} false(n...

…”

Section: Preliminariesmentioning

confidence: 99%

“…Definition For one PCFN, 23

normalΘ = false〈 ϑ \leftrightarrow ϖ false〉 = false〈 [Δ_{L} false(n false), Δ_{R} false(n false)], [\nabla_{L} false(n false), \nabla_{R} false(n false)] false〉 \leftrightarrow

false〈 \tilde{normalΔ} (n), \tilde{\nabla} (n) false〉

, the score function is,

S ()(, Θ) = {(\frac{Δ^{L} + Δ^{R} - \overset{Δ}{true}}{3})}^{2} - {(\frac{\nabla^{L} + \nabla^{R} - \overset{\nabla}{true}}{3})}^{2} .

…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, it directly selects the maximum and minimum values of the membership and the nonmembership as the positive and negative ideal reference points, respectively, which can reflect the different risk attitudes of decision‐makers facing gains and losses. The risk‐preference coefficients under the premise of decision‐makers' bounded rationality increases the gap between the decision results; especially, our method will be more effective when choosing between several similar solutions. Comparison with the Pythagorean cubic fuzzy weighted algebraic (PCFWA) 23 and Pythagorean cubic fuzzy weighted geometric (PCFWG) 23 operators.…”

Section: Applicationmentioning

confidence: 99%

“…As can be seen from Table 1, PCFS can describe fuzzy information more comprehensively compared to the above sets; it is also more practically applicable in decision‐making and has a significant research value. Furthermore, Khan et al 23 defined the score function of PCFS and certain aggregation operators. Khan et al 24 introduced the concept of confidence to PCFS and constructed a new MCDM model.…”

Section: Introductionmentioning

confidence: 99%

“…In this process, the different sensitivities of decision‐makers towards profit and losses and the nonlinear change of probability preferences were considered, which better describes the irrational psychology of decision‐makers and has been widely used in academia 41–45 . However, available literature concerning PCFS does not consider the irrational psychological behavior of decision‐makers 22–26 . Additionally, the complex internal subset structure and actual decision‐making context involved in PCFS match the real decision‐making psychology of decision‐makers, as reflected by the prospect theory.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Prospect‐theory and geometric distance measure‐based Pythagorean cubic fuzzy multicriteria decision‐making

Wang

Zhao

2021

Int J Intell Syst

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The Pythagorean cubic fuzzy set (PCFS) describes a mixture of interval Pythagorean information and Pythagorean fuzzy values. Compared to the Pythagorean and interval Pythagorean fuzzy sets, PCFS contains comprehensive information suitable for solving complex multicriteria decision‐making (MCDM) problems. However, the use of existing PCFS comparison tools, including score‐function and distance‐measurement, is impractical in some cases. To address this concern, this study transforms PCFS into the geometric form, and proposes the use of the PCFS geometric‐distance measurements. The proposed method requires the maximum and minimum values of membership and nonmembership degrees to be considered positive and negative reference points, respectively. Moreover, the gain and loss matrices are constructed. The risk‐preference coefficient is introduced to improve the original prospect‐value expression. The final ranking results of decision‐makers considering several risk preferences are presented by balancing the proportion of the positive and negative prospect values, thereby constructing an MCDM model based on the PCFS‐geometric‐distance measure and prospect theory. The feasibility and practicality of this model are validated via a case study concerning urban human settlements to ensure satisfactory housing conditions and appropriate infrastructure development. As observed, the proposed model can adequately evaluate urban human settlements, thereby demonstrating the potential to solve similar problems. A sensitivity analysis of the risk‐preference coefficient and subsequent comparison against existing methods reveal the proposed MCDM model to yield convincing results.

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“…Definition Assuming N to be the universe of discourse, a PCFS

normalΘ

under N would mean a combination of collections 22,23 :

normalΘ = false{〈 n, ϑ false(n false), ϖ false(n false) 〉 \sim n \in N false},

ϑ false(n false) = false{n, false〈 [Δ^{L} false(n false), Δ^{R} false(n false)], [\nabla^{L} false(n false), \nabla^{R} false(n false)] false〉 ~ n \in N false}

is an IVPFS, where

0 \leq Δ^{L} \leq Δ^{R} \leq 1

and

0 \leq \nabla^{L} \leq \nabla^{R} \leq 1

;

ϖ false(n false) = false{n, false〈 \tilde{normalΔ} (n), \tilde{\nabla} (n) false〉

~ n \in N false}

is a PFS under N , where

0 \leq \overset{Δ}{true} \leq 1, 0 \leq \overset{\nabla}{true} \leq 1

and

\overset{π}{true} false(n false) = \sqrt{1 - {\tilde{normalΔ}}^{2} false(n false) - {\tilde{\nabla}}^{2} false(n false)}

represent the hesitation of the PFS. For each element, n in N , we have

0 \leq {false(Δ^{R} false(n false) false)}^{2} + {false(\nabla^{R} false(n false) false)}^{2} \leq 1

and

0 \leq false(\overset{Δ}{true} false(n...

…”

Section: Preliminariesmentioning

confidence: 99%

“…Definition For one PCFN, 23

normalΘ = false〈 ϑ \leftrightarrow ϖ false〉 = false〈 [Δ_{L} false(n false), Δ_{R} false(n false)], [\nabla_{L} false(n false), \nabla_{R} false(n false)] false〉 \leftrightarrow

false〈 \tilde{normalΔ} (n), \tilde{\nabla} (n) false〉

, the score function is,

S ()(, Θ) = {(\frac{Δ^{L} + Δ^{R} - \overset{Δ}{true}}{3})}^{2} - {(\frac{\nabla^{L} + \nabla^{R} - \overset{\nabla}{true}}{3})}^{2} .

…”

Section: Preliminariesmentioning

confidence: 99%

Section: Applicationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Prospect‐theory and geometric distance measure‐based Pythagorean cubic fuzzy multicriteria decision‐making

Wang

Zhao

2021

Int J Intell Syst

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Similarity and Pythagorean reliability measures of multivalued neutrosophic cubic set and its application to multiple‐criteria decision‐making

Wang

Zhao

2021

Int J Intell Syst

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Considering the limitations of interval neutrosophic set (INS), multivalued neutrosophic set (MVNS), and neutrosophic cubic set (NCS) in describing uncertainty, respectively, this study proposes a multivalue neutrosophic cubic set (MVNCS). Originating from the combination of INS and MVNS, MVNCS is regarded as the general form of NCS. First, some basic operations of MVNCS and the similarity measures of MVNCS are defined. Second, considering the complexity of actual decision-making problems, the Pythagorean reliability measure of MVNCS is further developed to process the evaluation indicators and attribute weights given by decision-makers more accurately. Third, a multicriteria decision-making approach under the multivalue neutrosophic cubic environment is proposed based on the above measures. Finally, an example of investment options that follow sensitivity and comparative analysis provided to justify the advantages of our method is given.

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A modified method of generating Z-number based on OWA weights and maximum entropy

Tian

Kang

2020

Soft Comput

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How to generate Z -number is an important and open issue in the uncertain information processing of Z -number. In Kang et al. (Int J Intell Syst 33(8):1745-1755, 2018), a method of generating Z -number using OWA weight and maximum entropy is investigated. However, the meaning of the method in Kang et al. (2018) is not clear enough according to the definition of Z -number. Inspired by the methodology in Kang et al. ( 2018), we modify the method of determining Z -number based on OWA weights and maximum entropy, which is more clear about the meaning of Z -number. In addition, the model of generating Z -number under the environment of group decision making is well investigated based the modified model. Some numerical examples are used to illustrate the effectiveness of the proposed methodology.

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Pythagorean cubic fuzzy aggregation operators and their application to multi-criteria decision making problems

Cited by 45 publications

References 34 publications

Prospect‐theory and geometric distance measure‐based Pythagorean cubic fuzzy multicriteria decision‐making

Prospect‐theory and geometric distance measure‐based Pythagorean cubic fuzzy multicriteria decision‐making

Similarity and Pythagorean reliability measures of multivalued neutrosophic cubic set and its application to multiple‐criteria decision‐making

A modified method of generating Z-number based on OWA weights and maximum entropy

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