In this paper, we first review the existing entropy measures for hesitant fuzzy elements (HFEs) and demonstrate that the existing entropy measures for HFEs fail to effectively distinguish some apparently different HFEs in some cases. Then, we propose a new axiomatic framework of entropy measures for HFEs by taking fully into account two facets of uncertainty associated with an HFE (i.e., fuzziness and nonspecificity). We adopt a two-tuple entropy model to represent the two types of uncertainty associated with an HFE. Additionally, we discuss how to formulate each kind of uncertainty. For each of fuzziness and nonspecificity, some simple methods are provided to construct measures, which can well handle the problems in the existing entropy measures for HFEs. Several examples are given to illustrate each method, and comparisons with the existing entropy measures are also offered.
Recent developments in measurement science show that continuous measurements are no more precise numbers but more or less imprecise and are called fuzzy. Therefore, to utilize this imprecision of observations, the corresponding analysis techniques related to continuous quantities are essential to generalize fuzzy observations. This study is aimed to generalize the likelihood ratio test and Cox's F-test for fuzzy observations in such a way that they are able to integrate fuzziness of lifetime observations for the inference in addition to stochastic variation. The proposed generalized tests are best suited for lifetime analysis as these cover fuzziness of single observations in addition to random variation.
The Bonferroni mean has been extensively applied in multicriteria decision-making and support system and developed intuitionistic fuzzy set theory. Based on the second interpretation of the Bonferroni mean, in this paper, we introduce the geometric Bonferroni mean, which is a generalization of the Bonferroni mean and geometric mean and generalized geometric Bonferroni mean, and investigate their properties. To describe the uncertainty and fuzziness more objectively, we further develop the intuitionistic fuzzy geometric Bonferroni mean and the generalized intuitionistic fuzzy geometric Bonferroni mean, which describe the relationship between arguments, and the weighted intuitionistic fuzzy geometric Bonferroni mean and the generalized weighted intuitionistic fuzzy geometric Bonferroni mean, which consider the importance of each argument. Finally, we investigate their properties in detail. (C) 2016 Wiley Periodicals, Inc.
The aim of this article is to develop a novel multiple criteria decision analysis (MCDA) method using a Pearson-like correlation-based Pythagorean fuzzy (PF) compromise approach under complex uncertainty based on PF sets and interval-valued Pythagorean fuzzy (IVPF) sets. Because of the complexity and ambiguity involved in real-life decision-making situations, this article utilizes the theory of Pythagorean fuzziness, which is characterized by flexible degrees of membership, nonmembership, and indeterminacy to describe uncertain information more comprehensively. PF and IVPF sets possess exceptional abilities to accurately reflect the uncertainty, fuzziness, and vagueness inherent in the decision information. However, manipulating PF and IVPF information is a complicated and difficult task for most decision makers. In this regard, this article extends the well-known and widely used concept of correlation coefficients to develop simple and effective compromise models for solving MCDA problems in PF and IVPF contexts. This article conducts an extended analysis of Pearson-like correlation coefficients for PF and IVPF sets separately and introduces new concepts of PF and IVPF correlation coefficients to furnish a solid basis for the proposed methodology. Furthermore, this article develops useful concepts of PF and IVPF correlation-based closeness coefficients to simultaneously measure the relative closeness to the positive-ideal PF/IVPF solutions and the relative remoteness from the negative-ideal PF/IVPF solutions. On the basis of the developed concepts, this article proposes a novel Pearson-like correlation-based PF/IVPF compromise approach to address uncertain MCDA problems involving PF/IVPF information and determine the ultimate priority orders among competing alternatives. Finally, this article provides an illustrative application about a financing decision of working capital management to verify the developed approach and demonstrate its feasibility and practicality
Abstract Triangular intuitionistic fuzzy numbers (TIFNs) is one of the useful tools to manage the fuzziness and vagueness in expressing decision data and solving decision making problems. In this paper, triangular norm (t‐norm) based cuts of TIFNs are developed to synthesize the membership and nonmembership functions in describing the cut sets, then the possibility characteristics of TIFNs, i.e., the possibility mean, the possibility variance, and the possibility mean‐standard deviation ratio, are given. Thereby, on the ground of the possibility mean‐standard deviation ratio, a ranking method of TIFNs is introduced. With these elements, an approach to multiple attributes decision making (MADM) is proposed and illustrated by a numerical example. It is shown that the approach to MADM comprehensively considers both the membership and nonmembership functions and can lead to objective and reasonable results.
Pythagorean fuzzy set (PFS) whose main feature is that the square sum of the membership degree and the non-membership degree is equal to or less than one, is a powerful tool to express fuzziness and uncertainty. The aim of this paper is to investigate aggregation operators of Pythagorean fuzzy numbers (PFNs) based on Frank t-conorm and t-norm. We first extend the Frank t-conorm and t-norm to Pythagorean fuzzy environments and develop several new operational laws of PFNs, based on which we propose two new Pythagorean fuzzy aggregation operators, such as Pythagorean fuzzy Choquet-Frank averaging operator (PFCFA) and Pythagorean fuzzy Choquet-Frank geometric operator (PFCFG). Moreover, some desirable properties and special cases of the operators are also investigated and discussed. Then, a novel approach to multi-attribute decision making (MADM) in Pythagorean fuzzy context is proposed based on these operators. Finally, a practical example is provided to illustrate the validity of the proposed method. The result shows effectiveness and flexible of the new method. A comparative analysis is also presented.
Abstract The uncertainty and complexity of the decision‐making environment and the subjectivity of the decision makers will lead to the inevitable errors of the decision‐making data. A poor decision will be produced with those errors, whereas the linear programming technique for multidimensional analysis of preference (LINMAP) method can adjust such errors through constructing an optimal programming model based on the consistency of the decision‐making information, and it has been applied widely in multiple attribute group decision making (MAGDM). Moreover, Pythagorean fuzzy information is useful to simulate the ambiguous and uncertain decision‐making environment. Therefore, the LINMAP method under the Pythagorean fuzzy circumstance will be proposed in this paper to solve MAGDM problems. To measure the fuzziness and uncertainty of Pythagorean fuzzy set (PFS) and interval‐valued PFS, Pythagorean fuzzy entropy (PFE) and interval‐valued PFE (IVPFE) grounded on the similarity and hesitancy parts have been defined, respectively. Then, Pythagorean fuzzy LINMAP (PF LINMAP) methods are constructed on the basis of the PFE and IVPFE correspondingly. Under the given preference relations, the maximum consistency and the amount of knowledge can be realized by the proposed methods. After investigating the relevant indicator system, the decision‐making problem concerning railway project investment has been solved through the proposed PF LINMAP method with PFE. Finally, the practicability and effectiveness of the PF LINMAP method has been verified via the comparative analysis with the existing methods.