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The problem of expanding a meaningful entropic theory for fuzzy information cannot be thought of as being a mere (more or less formal) extension of Shannon theory. By using the information theory of deterministic functions, the present author had already obtained some results in this way, and he herein continues this approach. After a short background on the different entropies of deterministic functions and on membership entropy of fuzzy sets, successively mixed entropy of fuzzy sets, joint membership functions of independent fuzzy sets, and conditional entropy of fuzzy sets with respect to other fuzzy sets are considered; the problem of defining transinformation between fuzzy sets, as a generalisation of the well known Shannon concept, is then examined. One of the conclusions of the article is that it is possible to build up a meaningful information theory of fuzzy sets by using the entropy of deterministic functions.

The entropy as a measure of diversity has been used by ecologists to characterize a community by its stability process. After the introduction of the concept of a fuzzy subset by Zadeh (1965), many definitions of entropies were given emphasizing the subjectivity in evaluations. A pioneering work which relates the classical meaning of entropy (Shannon index) with the modern fuzzy theory was due to De Luca and Termini (1972); Knopfmacher (1975) formulated a generalization of the axiomatics given by De Luca and Termini; Batle and Trillas (1979) obtained a result which is essentially analogous to Knopfmacher’s by considering a finite fuzzy measure space and the Sugeno’s integrals; and Trillas and Riera (1978) introduced the concept of fuzzy algebraic entropics. Analyses the continuity properties for these fuzzy entropies and establishes conditions which guarantee the convergence E(f_n) → E(f), where (f_n) is a sequence of fuzzy sets and E is an entropy.

In the present literature on fuzzy sets and fuzzy information, there is much confusion between entropies of fuzzy sets and fuzzy sets of entropies. After a thorough critical review of this question, proposes a unified approach based on the theory of deterministic functions. One must carefully distinguish between index of fuzziness, uncertainty of fuzziness and uncertainty of randomness on the one hand; and uncertainty of fuzzy sets and uncertainty of possibility on the other hand. This new framework could provide new approaches to management of uncertainty originating from both probability and possibility distributions.

By combining the subjective probabilistic viewpoint of fuzziness with the entropy of deterministic functions, it is possible to expand an information theory of fuzzy sets which is fully compatible and consistent with the classical Shannonian information theoretic framework. A model of transinformation between fuzzy sets, which could be of help in approximate reasoning can be obtained, an interesting feature of which is that it can be duplicated in the framework of fuzzy set theory.

This paper proposes a fuzzy random multi-objective portfolio model with different entropy measures and designs a hybrid algorithm to solve the proposed model.

Because random uncertainty and fuzzy uncertainty are often combined in a real-world setting, the security returns are considered as fuzzy random numbers. In the model, the authors also consider the effects of different entropy measures, including Yager's entropy, Shannon's entropy and min-max entropy. During the process of solving the model, the authors use a ranking method to convert the expected return into a crisp number. To find the optimal solution efficiently, a fuzzy programming technique based on artificial bee colony (ABC) algorithm is also proposed.

(1) The return of optimal portfolio increases while the level of investor risk aversion increases. (2) The difference of the investment weights of the optimal portfolio obtained with Yager's entropy are much smaller than that of the min–max entropy. (3) The performance of the ABC algorithm on solving the proposed model is superior than other intelligent algorithms such as the genetic algorithm, differential evolution and particle swarm optimization.

To the best of the authors' knowledge, no effect has been made to consider a fuzzy random portfolio model with different entropy measures. Thus, the novelty of the research is constructing a fuzzy random multi-objective portfolio model with different entropy measures and designing a hybrid fuzzy programming-ABC algorithm to solve the proposed model.

In the framework of his theory of relative information, derives formulae to convert probability into possibility and conversely, and shows how they can be utilized to reconsider many questions related to fuzzy sets and approximate reasoning on a meaningful information theoretic standpoint. The key of this approach is that the membership of a fuzzy set is now thought of as equivalent to the pair (probability, fuzziness coefficient). Therefore a sound new concept of informational membership entropy, which is fully consistent with Shannon theory, is evolved. In this new paradigm, the fuzzy object is simultaneously defined by the set itself and its complement. Thus obtains a new modelling for the union and the intersection of fuzzy sets, and new approaches to quantitatively modelling “If A then B”.

This paper aims to quantify atmospheric corrosion by image analyses. The corrosion extent, form and distribution of corrosion product on Q235B and T91 steels exposed to a Zhoushan marine atmosphere over one year are characterized by image analysis.

Image analysis of corrosion images were achieved using the gray value, wavelet analysis and fuzzy Kolmogorov–Sinai (K–S) entropy.

As corrosion becomes extensive, the gray value of corrosion images decreases, and the energy value of nine subimages after wavelength decomposition decreases. Fuzzy K–S entropy increases as localized corrosion propagates but decreases as uniform corrosion spreads.

The methods proposed in this work open a new way for fast corrosion evaluation of metallic materials exposed to atmospheric conditions.

The problem of evaluating potential suppliers has always been based on finding an optimal tradeoff between supplier’s performance consistently meeting firms’ needs and acceptable cost. The purpose of this paper is to propose a hybrid multi-criteria decision framework to quantify this qualitative judgment and reduce ambiguity in selection of suppliers in the era of Industry 4.0.

A hybrid intuitionistic fuzzy entropy weight-based multi-criteria decision model with TOPSIS is proposed. The authors make use of the intuitionistic fuzzy weighted approach operator for aggregating individual decision maker’s opinions regarding each alternative over every criterion. Additionally, the authors employ the concept of Shannon’s entropy to calculate the criteria weights.

Results obtained on the basis of the proposed hybrid methodology are analyzed against two more cases wherein the authors try to showcase the relevance of using IFS and entropy-based decision framework and find out the uniqueness of the proposed framework in supplier selection process.

The proposed model is apposite to solve management problem of supplier selection in two ways: aggregating individual decision maker’s opinion for each of the predefined criteria along with individual decision maker’s importance and ranking the suppliers based on both positive and negative ideal solutions using TOPSIS.

A robust framework incorporates not only suppliers’ performance but also provides weightage to key decision makers. Especially in the context of MCDMs wherein both qualitative and quantitative data is evaluated simultaneously, the proposed framework is unique in its practical implementation of reducing ambiguity in the supplier selection process.

Although the decision‐making problem at the lower level is generally well‐defined, the decision‐making problem at the higher level would not contain the detail. Much of decision‐making at the higher level might take place in a fuzzy environment, so that it is only necessary to decide roughly what actions, what states and what parameters should be considered. This paper deals with the higher level problem in which we can regard the elements‐states of nature, feasible actions and available information‐as fuzzy objects. Since the uncertainty of meaning of objects is represented by the fuzzy sets and the uncertainty of occurrence of objects is defined by the probability, a specific formulation of the higher level decision problem can be defined by the probability of fuzzy events. From the same aspect, the definitions of worth, entropy and quantity concerning fuzzy information are given in this paper, and we have tried to extend some of the statistical decision theory to the fuzzy decision problem. To explain our formulation, an investment problem is presented as an example.

To provide a simplified proof for Liu's results concerning the relationship between σ‐entropy and σ‐similarity measure.

First obtains Liu's clear and simple functional expression of σ‐similarity measures based on the structure of σ‐entropy by using expressions of equivalent functions of σ‐entropy. Then, by employing the general expression of equivalent functions of σ‐similarity measure of fuzzy sets, provides a different proof of Liu's results.

Here, the proof is much simpler than the original one provided by Liu. And, on the basis of Shannon's probability entropy as introduced by De Luca and Termini, a σ‐similarity measure is constructed, on which a No answer to an open question as proposed by Liu is provided.

Constructs a new approach to some known results and provides a definite answer to an open question.