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This paper presents a fuzzy maximal covering location problem (FMCLP) in which travel time between any pair of nodes is considered to be a fuzzy variable.
Let ξ be a fuzzy variable.
Let ξ be a fuzzy variable with credibility distribution Φ.
Let ξ be a fuzzy variable with regular credibility distribution Φ.
Let ξ be a fuzzy variable with the inverse credibility distribution (Phi^{-1}).
Let ξ be a fuzzy variable with finite expected value e.
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Fuzzy set theory is developed for handing uncertainty, imprecision and complexity in the real world; for example, we say "driving speed is high" wherein speed is a fuzzy variable and high is a fuzzy set, which uses the membership function to indicate the degree of a element belonging to the set (words in Italics to denote fuzzy variables or fuzzy sets).
Suppose that ξ is a fuzzy variable, μ is the membership function of ξ, and r is a real number.
The variable size of gap is a fuzzy variable, and is divided into three linguistic variables: small gap, medium gap, and large gap.
Supposing ς is a fuzzy variable, the expected value of ς is as follows: mathbf{E}[varsigma] = int_{0}^{+infty} operatorname{mathbf{Cr}}{varsigmageq s}, ds- int_{-infty}^{0}operatorname{mathbf{Cr}}{varsigmaleq s}, ds, where two integrals are finite.
operatorname{Cr}bigl{ |tau|ge tbigr} lefrac{E^{mathrm{f}}[f tau)]}{f(t)}, quadforall t>0, on condition that f is a nonnegative function and it is even and increasing on ([0,infty)). Since (xi(lambda)) is a fuzzy variable, for any (lambdainLambda), it follows that operatorname{Cr}bigl{ bigl|xi(lambda bigr|ge tbigr} lefrac{E^{mathrm{f}}[f xi(lambda))]}{f(t)} for each λ.
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