Abstract
As a crucial concept of characterizing uncertainty, entropy has been widely used in fuzzy programming problems, while involving complicated calculations. To simplify the operations so as to broaden its applicable areas, this paper investigates the entropy within the framework of credibility theory and derives the formulas for calculating the entropy of regular LR fuzzy numbers by virtue of the inverse credibility distribution. By verifying the favorable property of this operator, a calculation formula of a linear function’s entropy is also proposed. Furthermore, considering the strength of semi-entropy in measuring one-side uncertainty, the lower and upper semi-entropies, as well as the corresponding formulas are suggested to handle return-oriented and cost-oriented problems, respectively. Finally, utilizing entropy and semi-entropies as risk measures, two types of entropy optimization models and their equivalent formulations derived from the proposed formulas are given according to different decision criteria, providing an effective modeling method for fuzzy programming from the perspective of entropy. The numerical examples demonstrate the high efficiency and good performance of the proposed methods in decision making.
Highlights
Fuzzy programming (FP), a technique incorporating the concept of fuzziness in programming models, has extensive applications in the field of operations research due to its advantage in handling problems with ambiguity and vagueness, which are commonly met in practice
Some necessary concepts and operational laws of fuzzy set theory, which lay the foundation for the following sections, are reviewed successively, including the LR fuzzy numbers, credibility function (CF), credibility distribution (CD), inverse credibility distribution (ICD), and expected value
Zhou et al [32] further verified that the LR fuzzy number δ ∼ (σ, α, β) LR is regular when the shape functions are continuous and strictly decreasing, and its inverse credibility distributions (ICD) is derived from Equation (8) as: Φ −1 ( γ ) =
Summary
Fuzzy programming (FP), a technique incorporating the concept of fuzziness in programming models, has extensive applications in the field of operations research (i.e., project planning [1,2], manufacturing [3,4], investment problems [5], etc.) due to its advantage in handling problems with ambiguity and vagueness, which are commonly met in practice. Considering that, Li and Liu [21] introduced a novel definition of fuzzy entropy in view of the self-dual credibility measure raised by Liu and Liu [22] to depict the uncertainty induced by information deficiency. Inspired by this new concept, theoretical research [20,23,24], as well as real applications [25,26,27,28,29,30,31] on credibility-based entropy have increased rapidly.
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