Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming

Zhou, Jian; Yang, Fan; Wang, Ke

doi:10.3233/ifs-151712

Cited by 59 publications

(56 citation statements)

References 30 publications

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“…Therefore,

((), 1 - 2 α) ζ_{4} + 2 α ζ_{3} \geq x \Leftrightarrow ({}, \frac{ζ_{4} - x}{2 (ζ_{4} - ζ_{3})}) \geq α \Leftrightarrow Cr ({}, \overset{ζ}{true} \geq x) \geq α

. When

α > \frac{1}{2}

and

Cr ({}, \overset{ζ}{true} \geq x) \geq α

, then we observe that either x ≤ ζ 1 or

({}, \frac{2 ζ_{2} - x - ζ_{1}}{2 (ζ_{2} - ζ_{1})}) \geq α

. Considering, each of the observed cases, we have (2 − 2 α ) ζ 2 + (2 α − 1) ζ 1 ≥ x . Conversely, when x < ζ 1 then

Cr ({}, \overset{ζ}{true} \geq x) = 1 \geq α .

Now,

((), 2 - 2 α) ζ_{2} + ((), 2 α - 1) ζ_{1} \geq x \Leftrightarrow ({}, \frac{2 ζ_{2} - x - ζ_{1}}{2 (ζ_{2} - ζ_{1})}) \geq α,

which eventually means

Cr ({}, \overset{ζ}{true} \geq x) \geq α

. Definition (Zhou, Yang, & Wang, ): The regular credibility distribution of a GFN,

\overset{ζ}{true} = scriptN ((),, ρ, {normalδ}^{2})

with membership function defined in , is represented as

Cr ({}, \overset{ζ}{true} \leq x) = ({, \begin{matrix} \frac{1}{2} italicexp ({}, ...) \end{matrix})

…”

Section: Preliminarymentioning

confidence: 90%

“…Conversely, when x > ρ then

Cr ({}, \overset{ζ}{true} \geq x) \geq α

is satisfied with

α \leq \frac{1}{2}

. Accordingly,

normalρ + normalδ \sqrt{- ln ((), 2 α)} \geq x \Leftrightarrow ([], \frac{1}{2} italicexp ({}, - {(\frac{normalρ - x}{δ})}^{2})) \geq α .

This eventually means that

normalρ + normalδ \sqrt{- ln ((), 2 α)} \geq x \Leftrightarrow Cr ({}, \overset{ζ}{true} \geq x) \geq α

.Definition (Liu & Liu, ): Let

\overset{ζ}{true}

be a fuzzy variable on the possibility space (Θ, Γ(Θ), Pos), then the expected value of

\overset{ζ}{true}

is defined as

normalE ([], \overset{true~}{ζ}) bold-italic= \int_{0}^{bold-italic+ bold-italic\infty} Cr ({}, \overset{ζ}{true} \geq x) normald x bold-italic- \int_{bold-italic- bold-italic\infty}^{0} Cr ({}, \overset{ζ}{true} \leq x) normald x

provided that at least one of the two integrals is finite.Theorem (Zhou et al, ): Let

\overset{ζ}{true}

be a fuzzy variable with inverse credibility distribution function Ψ −1 . If the expected value of

\overset{ζ}{true}

exists, then

normalE ([], \overset{ζ}{true}) bold-italic= \int_{0}^{1}

…”

Section: Preliminarymentioning

confidence: 99%

“…Definition 2.6: (Zhou, Yang, & Wang, 2016): The regular credibility distribution of a GFN, e ζ ¼ N ρ; δ 2 with membership function defined in 2, is represented as…”

Section: Preliminarymentioning

confidence: 99%

“…Theorem 2.1: (Zhou et al, 2016): Let e ζ be a fuzzy variable with inverse credibility distribution function Ψ −1 . If the expected value of e ζ exists, then…”

Section: Preliminarymentioning

confidence: 99%

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Uncertainty based genetic algorithm with varying population for random fuzzy maximum flow problem

et al. 2018

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This paper investigates the uncertain maximum flow of a network whose capacities are random fuzzy variables. We have developed the expected value model (EVM) and the chance‐constrained model (CCM) for maximum flow problem (MFP) under random fuzzy environment and formulated their crisp equivalent models. To solve these models, we have proposed a varying population genetic algorithm with indeterminate crossover (VPGAwIC). In VPGAwIC, selection of a chromosome depends on its lifetime. An improved lifetime allocation strategy (iLAS) has also been proposed to determine the lifetime of the chromosome. The ages of the chromosomes are defined linguistically as Young, Middle, and Old, which follow some uncertainty distributions. The crossover probability is indeterminate, and it depends on the ages of the parents, which is defined by an uncertain rule base. The number of offspring, generated from a population of parents, is determined by the reproduction ratio. The population is updated in 2 ways: (i) All the chromosomes with ages greater than their lifetimes are discarded from the population, and (ii) the offspring are combined with their parents for the next generation. The proposed VPGAwIC is compared with the genetic algorithm developed by Gen, Cheng, and Lin (2008) for maximum flow problem. Wilcoxon signed‐rank test has been performed to show the superiority of the proposed VPGAwIC.

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“…Therefore,

((), 1 - 2 α) ζ_{4} + 2 α ζ_{3} \geq x \Leftrightarrow ({}, \frac{ζ_{4} - x}{2 (ζ_{4} - ζ_{3})}) \geq α \Leftrightarrow Cr ({}, \overset{ζ}{true} \geq x) \geq α

. When

α > \frac{1}{2}

and

Cr ({}, \overset{ζ}{true} \geq x) \geq α

, then we observe that either x ≤ ζ 1 or

({}, \frac{2 ζ_{2} - x - ζ_{1}}{2 (ζ_{2} - ζ_{1})}) \geq α

. Considering, each of the observed cases, we have (2 − 2 α ) ζ 2 + (2 α − 1) ζ 1 ≥ x . Conversely, when x < ζ 1 then

Cr ({}, \overset{ζ}{true} \geq x) = 1 \geq α .

Now,

((), 2 - 2 α) ζ_{2} + ((), 2 α - 1) ζ_{1} \geq x \Leftrightarrow ({}, \frac{2 ζ_{2} - x - ζ_{1}}{2 (ζ_{2} - ζ_{1})}) \geq α,

which eventually means

Cr ({}, \overset{ζ}{true} \geq x) \geq α

. Definition (Zhou, Yang, & Wang, ): The regular credibility distribution of a GFN,

\overset{ζ}{true} = scriptN ((),, ρ, {normalδ}^{2})

with membership function defined in , is represented as

Cr ({}, \overset{ζ}{true} \leq x) = ({, \begin{matrix} \frac{1}{2} italicexp ({}, ...) \end{matrix})

…”

Section: Preliminarymentioning

confidence: 90%

“…Conversely, when x > ρ then

Cr ({}, \overset{ζ}{true} \geq x) \geq α

is satisfied with

α \leq \frac{1}{2}

. Accordingly,

normalρ + normalδ \sqrt{- ln ((), 2 α)} \geq x \Leftrightarrow ([], \frac{1}{2} italicexp ({}, - {(\frac{normalρ - x}{δ})}^{2})) \geq α .

This eventually means that

normalρ + normalδ \sqrt{- ln ((), 2 α)} \geq x \Leftrightarrow Cr ({}, \overset{ζ}{true} \geq x) \geq α

.Definition (Liu & Liu, ): Let

\overset{ζ}{true}

be a fuzzy variable on the possibility space (Θ, Γ(Θ), Pos), then the expected value of

\overset{ζ}{true}

is defined as

normalE ([], \overset{true~}{ζ}) bold-italic= \int_{0}^{bold-italic+ bold-italic\infty} Cr ({}, \overset{ζ}{true} \geq x) normald x bold-italic- \int_{bold-italic- bold-italic\infty}^{0} Cr ({}, \overset{ζ}{true} \leq x) normald x

provided that at least one of the two integrals is finite.Theorem (Zhou et al, ): Let

\overset{ζ}{true}

be a fuzzy variable with inverse credibility distribution function Ψ −1 . If the expected value of

\overset{ζ}{true}

exists, then

normalE ([], \overset{ζ}{true}) bold-italic= \int_{0}^{1}

…”

Section: Preliminarymentioning

confidence: 99%

“…Definition 2.6: (Zhou, Yang, & Wang, 2016): The regular credibility distribution of a GFN, e ζ ¼ N ρ; δ 2 with membership function defined in 2, is represented as…”

Section: Preliminarymentioning

confidence: 99%

“…Theorem 2.1: (Zhou et al, 2016): Let e ζ be a fuzzy variable with inverse credibility distribution function Ψ −1 . If the expected value of e ζ exists, then…”

Section: Preliminarymentioning

confidence: 99%

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Uncertainty based genetic algorithm with varying population for random fuzzy maximum flow problem

et al. 2018

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“…It is apparent that the overall AQI is also a fuzzy number. In order to achieve the concrete distribution of this fuzzy number, we introduce the following operational law put forward by Zhou et al [32]. Let ξ 1 , ξ 2 , · · · , ξ n be independent regular LRfuzzy numbers with credibility distributions Φ 1 , Φ 2 , · · · , Φ n , respectively.…”

Section: Fuzzy Air Quality Indexmentioning

confidence: 99%

A Fuzzy Expression Way for Air Quality Index with More Comprehensive Information

et al. 2017

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Abstract:The Air Quality Index (AQI) is an evaluating indicator for the atmospheric environment released by various environmental monitoring centers to communicate the present air quality status to the public, which is calculated by the aid of the monitored concentrations of six common air pollutants and relevant computational formulae. Considering that the historical data of daily overall AQI illustrated by the traditional expression way merely contain limited information about the original data, this paper puts forward a more concrete and intuitive way to express the air quality in the past day. By analyzing the data concerning individual air quality indices of pollutants gathered from five cities of China for six consecutive months and conducting the curve fitting, each sub-index is recommended to be set as a Gaussian fuzzy number. Accordingly, taking advantage of the novel operational law for fuzzy numbers, the fuzzy distribution and membership function of the daily overall AQI can be deduced immediately, which as a reference contributes to the users acquiring the information more intuitively and facilitates making plans or decisions. Subsequently, a case study taking Shanghai as a background is conducted to elaborate the application of the proposed approach. Furthermore, the line chart reflecting the overall air quality status in a past period is depicted, based on which an example of selecting a tourist destination is given to demonstrate its utilization.

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Full Fuzzy Fractional Programming Based on the Extension Principle

Stanojević

Stanojević²

2022

Sustainable Business Management and Digital Transformation: Challenges and Opportunities in the Post-Covid Era

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Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming

Cited by 59 publications

References 30 publications

Uncertainty based genetic algorithm with varying population for random fuzzy maximum flow problem

Uncertainty based genetic algorithm with varying population for random fuzzy maximum flow problem

A Fuzzy Expression Way for Air Quality Index with More Comprehensive Information

Full Fuzzy Fractional Programming Based on the Extension Principle

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