A generalization of Maclaurin's inequalities and its applications

The Maclaurin symmetric mean (MSM) operator is a classical mean type aggregation operator used in modern information fusion theory, which is suitable to aggregate numerical values. The prominent characteristic of the MSM operator is that it can capture the interrelationship among the multi‐input arguments. In this paper, we extend the MSM operator and dual MSM operator to q‐rung orthopair fuzzy sets to propose the q‐rung orthopair fuzzy MSM operator, q‐rung orthopair fuzzy dual MSM operator, q‐rung orthopair fuzzy weighted MSM operator, and q‐rung orthopair fuzzy weighted dual MSM operator. Then, some desirable properties and special cases of these operators are discussed in detail. Finally, a numerical example is provided to illustrate the feasibility of the proposed methods and deliver the sensitivity analysis and comparative analysis.

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“…Lemma Let

a_{i} > 0

b_{i} > 0 true(i = 1, 2, …, n true)

, and

\sum_{i = 1}^{n} b_{i} = 1

; then

\prod_{i = 1}^{n} {true(a_{i} true)}^{b_{i}} \leq \sum_{i = 1}^{n} a_{i} b_{i}

with equality if and only if

a_{1} = a_{2} = ⋯ = a_{n}

.…”

Section: The Q‐rofmsm Operator and The Q‐rofwmsm Operatormentioning

confidence: 99%

“…Lemma (Maclaurin inequality) Let

a_{i} true(i = 1, 2, …, n true)

be a set of nonnegative real numbers, and for

k = 1, 2, …, n

; then

{italicMSM}^{(1)} true(a_{1}, a_{2}, …, a_{n} true) \geq {italicMSM}^{(2)} true(a_{1}, a_{2}, …, a_{n} true) \geq ⋯ \geq {italicMSM}^{(n)} true(a_{1}, a_{2}, …, a_{n} true)

with equality if and only if

a_{1} = a_{2} = ⋯ = a_{n}

.…”

Section: The Q‐rofmsm Operator and The Q‐rofwmsm Operatormentioning

confidence: 99%

Some q-rung orthopair fuzzy maclaurin symmetric mean operators and their applications to potential evaluation of emerging technology commercialization

et al. 2018

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“…First, we introduce two lemmas, which will be used in the following parts. Lemma (Maclaurin inquality) . Let

a_{i} false(i = 1, 2, \dots, n false)

be a set of nonnegative real numbers, and for

k = 1, 2, \dots, n

, then

\begin{matrix} true \\ left MS {normalM}^{(1)} ((), a_{1}, a_{2}, \dots, a_{n}) \\ left \geq MS {normalM}^{(2)} ((), a_{1}, a_{2}, \dots, a_{n}) \geq \dots \geq MS {normalM}^{(n)} ((), a_{1}, a_{2}, \dots, a_{n}) \end{matrix}

with equality if and only if

a_{1} = a_{2} = \dots = a_{n}

. Lemma Let

a_{i} > 0

b_{i} > 0 false(i = 1, 2, \dots, n false)

, and

0true \sum_{i = 1}^{n} b_{i} = 1

, then

0true \prod_{i = 1}^{n} {((), a_{i})}^{b_{i}} \leq \sum_{i = 1}^{n} a_{i} b_{i}

with equality if and only if

a_{1} = a_{2} = \dots = a_{n}

. Theorem For given PFNs

{\overset{a}{true}}_{i} false(i = 1, ...

…”

Section: Pythagorean Fuzzy Maclaurin Symmetric Mean Operatorsmentioning

confidence: 99%

Pythagorean Fuzzy Maclaurin Symmetric Mean Operators in Multiple Attribute Decision Making

Wei

2017

Int. J. Intell. Syst.

231

124

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The Maclaurin symmetric mean (MSM) operator is a classical mean type aggregation operator used in modern information fusion theory, which is suitable to aggregate numerical values. The prominent characteristic of the MSM operator is that it can capture the interrelationship among the multiinput arguments. In this paper, we extend MSM to Pythagorean fuzzy environment to propose the Pythagorean fuzzy Maclaurin symmetric mean and Pythagorean fuzzy weighted Maclaurin symmetric mean operators. Then, some desirable properties and special cases of these operators are discussed in detail. Finally, a numerical example is provided to illustrate the feasibility of the proposed methods and deliver a comparative analysis.

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“…From analysis above, we can further discuss the monotonic of the IFMSM operator with respect to the parameter k. First, we introduce two lemmas, which will be used in the following parts. [37] Let a i (i = 1, 2, . .…”

Section: Property 2 (Commutativity) Letmentioning

confidence: 99%

An approach to intuitionistic fuzzy multiple attribute decision making based on Maclaurin symmetric mean operators

Qin

Liu

2014

Journal of Intelligent &Amp; Fuzzy Systems

131

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The Maclaurin symmetric mean (MSM) was originally introduced by Maclaurin and then generalized by Detemple and Robertson. The prominent characteristic of the MSM is that it can capture the interrelationship among the multi-input arguments. However, the researches on MSM are very rare, especially in fuzzy decision making. In this paper, we investigate the MSM operator and extend the MSM operator to intuitionistic fuzzy environment. Some new aggregation operators based on MSM for dealing with intuitionistic fuzzy information are developed, such as the intuitionistic fuzzy Maclaurin symmetric mean (IFMSM) and the weighted intuitionistic fuzzy Maclaurin symmetric mean (WIFMSM). Some desirable properties and special cases of these operators are discussed in detail. Based on WIFMSM operator, an approach to multiple attribute decision making (MADM) problems with intuitionistic fuzzy information is developed. Finally, a practical example is provided to illustrate the practicality and effectiveness of the proposed method.Keywords: Maclaurin symmetric mean (MSM), intuitionistic fuzzy set (IFS), intuitionistic fuzzy Maclaurin symmetric mean (IFMSM), weighted intuitionistic fuzzy Maclaurin symmetric mean (WIFMSM), multiple attribute decision making * Corresponding author. Xinwang Liu,

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A generalization of Maclaurin's inequalities and its applications

Cited by 18 publications

References 8 publications

Some q-rung orthopair fuzzy maclaurin symmetric mean operators and their applications to potential evaluation of emerging technology commercialization

Some q-rung orthopair fuzzy maclaurin symmetric mean operators and their applications to potential evaluation of emerging technology commercialization

Pythagorean Fuzzy Maclaurin Symmetric Mean Operators in Multiple Attribute Decision Making

An approach to intuitionistic fuzzy multiple attribute decision making based on Maclaurin symmetric mean operators

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