Distance measures for cubic Pythagorean fuzzy sets and its applications to multicriteria decision making

Talukdar, Pranjal; Dutta, Palash

doi:10.1007/s41066-019-00185-3

Cited by 23 publications

(26 citation statements)

References 62 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%

“…Talukdar and Dutta 22 proposed a series of distance measures on PCFS, only a part of these distance measures are discussed and explored here.…”

Section: Preliminariesmentioning

confidence: 99%

“…Definition Assuming N to be the universe of discourse, for two PCFNs, 22

Θ_{1} = (〉〈, ϑ_{1} \leftrightarrow ϖ_{1}) = (〈, [{normalΔ}_{1}^{L} (n_{i}), {normalΔ}_{1}^{R} (n_{i})]), (〉, [\nabla_{1}^{L} (n_{i}), \nabla_{1}^{R} (n_{i})]) \leftrightarrow (〉〈, {\tilde{normalΔ}}_{1} ()(, n_{i}), {\tilde{\nabla}}_{1} ()(, n_{i}))

and

Θ_{2} = false〈 ϑ_{2} \leftrightarrow ϖ_{2} false〉 =

(〈, (][, Δ_{2}^{L} ()(, n_{i}), Δ_{2}^{R} ()(, n_{i})),) (〉, [\nabla_{2}^{L} (n_{i}), \nabla_{2}^{R} (n_{i})])

\leftrightarrow false〈 {\overset{Δ}{true}}_{2} (n_{i}), {\overset{\nabla}{true}}_{2} (n_{i}) false〉

; the distance measure is presented as follows:

d_{q}^{G} ()(, Θ_{1}, Θ_{2}) = \{\frac{1}{6} \{\begin{matrix} lefttrue \\ (||, {(Δ_{1}^{L})}^{2} - (Δ_{...})) \end{matrix}\}\}

…”

Section: Preliminariesmentioning

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%

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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.

show abstract

“…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%

“…Talukdar and Dutta 22 proposed a series of distance measures on PCFS, only a part of these distance measures are discussed and explored here.…”

Section: Preliminariesmentioning

confidence: 99%

“…Definition Assuming N to be the universe of discourse, for two PCFNs, 22

Θ_{1} = (〉〈, ϑ_{1} \leftrightarrow ϖ_{1}) = (〈, [{normalΔ}_{1}^{L} (n_{i}), {normalΔ}_{1}^{R} (n_{i})]), (〉, [\nabla_{1}^{L} (n_{i}), \nabla_{1}^{R} (n_{i})]) \leftrightarrow (〉〈, {\tilde{normalΔ}}_{1} ()(, n_{i}), {\tilde{\nabla}}_{1} ()(, n_{i}))

and

Θ_{2} = false〈 ϑ_{2} \leftrightarrow ϖ_{2} false〉 =

(〈, (][, Δ_{2}^{L} ()(, n_{i}), Δ_{2}^{R} ()(, n_{i})),) (〉, [\nabla_{2}^{L} (n_{i}), \nabla_{2}^{R} (n_{i})])

\leftrightarrow false〈 {\overset{Δ}{true}}_{2} (n_{i}), {\overset{\nabla}{true}}_{2} (n_{i}) false〉

; the distance measure is presented as follows:

d_{q}^{G} ()(, Θ_{1}, Θ_{2}) = \{\frac{1}{6} \{\begin{matrix} lefttrue \\ (||, {(Δ_{1}^{L})}^{2} - (Δ_{...})) \end{matrix}\}\}

…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Wang

Zhao

2021

Int J Intell Syst

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“…e cubic Pythagorean fuzzy set (CPFS) is a well reputed structure of fuzzy sets, proposed by Abbas et al [17] in 2019 to tackle the uncertainty in decision-making problems. Talukdar and Dutta [18] presented the distance measures under CPFS information. Fahmi et al [19] proposed the decision-making technique using cubic Pythagorean linguistic fuzzy sets.…”

Section: Introductionmentioning

confidence: 99%

Redefined “Maclaurin Symmetric Mean Aggregation Operators Based on Cubic Pythagorean Linguistic Fuzzy Numbers”

Naeem

Ashraf

Abdullah

et al. 2021

Mathematical Problems in Engineering

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In this study, we highlight the errors in Sections 2.3, 2.4, 3, and 4 in the article by Fahmi et al. (J Ambient Intell Human Comput (2020). https://doi.org/10.1007/s12652-020-02272-9) by counter definitions and theorems. We find that the definition of cubic Pythagorean fuzzy set (CPFS) (Definition 2.3.1) and operational laws (Definition 2.3.2) violates the rules to consider the membership and nonmembership functions, and then, we redefined the corrected definition and their operations for CPFS. Furthermore, we redefine the concept of cubic Pythagorean linguistic fuzzy set (CPLFS) and their basic operational laws. In addition, we find that Sections 3 and 4 (consist of a list of Maclaurin symmetric mean (MSM) and dual MSM aggregation operators) are invalid, and then, we redefined the list of updated MSM and dual MSM aggregation operators in correct way. Finally, we established the numerical application of the proposed improved algorithm using cubic Pythagorean linguistic fuzzy information to show the applicability and effectiveness of the proposed technique.

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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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Distance measures for cubic Pythagorean fuzzy sets and its applications to multicriteria decision making

Cited by 23 publications

References 62 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

Redefined “Maclaurin Symmetric Mean Aggregation Operators Based on Cubic Pythagorean Linguistic Fuzzy Numbers”

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

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