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An interval-valued Pythagorean hesitant fuzzy set (IVPHFS) not only can be regarded as the union of some interval-valued Pythagorean fuzzy sets but also represent the Pythagorean hesitant fuzzy elements in the form of interval values. So IVPHFSs are extensions of Pythagorean hesitant fuzzy sets (PHFSs) and interval-valued Pythagorean fuzzy sets (IVPFSs), which are powerful tools to represent more complicated, uncertain and vague information. This paper focuses on the four kinds of correlation coefficients for PHFSs, and extends them to the correlation coefficients and the weighted correlation coefficients for IVPHFSs. In the processing, we develop the least common multiple expansion (LCME) method to solve the problem that the cardinalities of Pythagorean hesitant fuzzy elements (PHFEs) (or interval-valued Pythagorean hesitant fuzzy elements (IVPHFEs)) are different. In addition, we propose score functions and accuracy functions of Pythagorean fuzzy elements (PFEs) (or interval-valued Pythagorean fuzzy elements (IVPFEs)) to rank all the PFEs (or IVPFEs) in a PHFE (or an IVPHFE). Especially, score functions and accuracy functions of IVPFEs are both presented as interval numbers. Then use the comparison method of interval numbers to compare two revised IVPHFEs in order to keep the original fuzzy information as far as possible. What's more, we define the local correlations and local informational energies which can depict the similarity between two IVPHFEs more meticulously and completely. At last the numerical examples to show the feasibility and applicability of the proposed methods in multiple criteria decision making (MCDM) and clustering analysis.

The Pythagorean fuzzy set (PFS), introduced by Yager, is a generalization of the intuitionistic fuzzy set. The PFS is of great significance for the decision-making problem because it promotes the domain of intuitionistic fuzzy set. This paper aims to conduct the consistency analysis and priority weights for Pythagorean fuzzy preference relations (PFPRs). First, a consistency index to measure the multiplicative consistency of Pythagorean fuzzy preference relation (PFPR) is presented. Then, for the non-consistent PFPR, an iteration algorithm is put forward to adjust it to achieve acceptable consistency. Furthermore, an absolute deviation model is built to derive the priority weighting vector of PFPR. Finally, an illustrative example is provided to test and verify the effectiveness and applicability of the proposed method.

Pythagorean fuzzy set, characterized by membership function and non-membership function, has received increasing attention in recent years. In this paper, a new approach to decision making is proposed based on Pythagorean fuzzy preference relation and its additive consistency. Firstly, the concepts of Pythagorean fuzzy preference relations and its additive consistency are introduced, and followed by a discussion of their desirable properties. Then, a linear goal programming model is proposed to determine the consistency of PFPRs. For the PFPRs that does not satisfy the consistency, the consistency index is defined to measure the degree of consistency, and a consistency adjustment algorithm is proposed. Finally, based on the additive consistency, a new algorithm for decision making is presented. An example of CIM(Computing In Memory) is provided, and in comparison with other methods, the validity and rationality of the proposed method are verified.

Since some existing uncertainty measurement of covering-based rough intuitionistic fuzzy sets (CRIFSs) are unreasonable in some cases, an extended uncertainty measure criterion of CRIFSs is proposed. The discussion of the monotonicity of the criterion is further refined and more in line with the actual problem description. Taken into account this a modified roughness method is introduced, which is an uncertainty measure of CRIFSs satisfying the extended criterion. Furthermore, the criterion is extended to covering-based rough interval-valued intuitionistic fuzzy sets (CRIVIFSs), and the modified roughness method is proposed to measure the uncertainty of CRIVIFSs. Finally, the example is presented to illustrate the application of the modified roughness to attribute reductions. These conclusions provide a theoretical basis for the rationality (or irrationality) of existing uncertainty measures and also promote the application of CRIVIFSs.

The aim of this paper is to develop some novel operational laws for a hesitant fuzzy linguistic term set (HFLTS)-based on the improved supplementary regulation and Archimedean t-norms and s-norms. The improved supplementary regulation for HFLTSs can reserve the fidelity of original information commendably, and it brings a new conception to research on information measures of HFLTS. As a hot and key research topic for information fusion, some Archimedean t-norms and s-norms based hesitant fuzzy linguistic aggregation operators are proposed to aggregate HFLTSs. Some essential properties together with their special cases of such aggregation operators are discussed in detail. The entropy and cross-entropy of HFLTSs are proposed and applied to derive the attribute weights. An approach to multiple attributes group decision making with hesitant fuzzy linguistic information is developed. Finally, a numerical example related to the assessment of health-care waste disposal methods is provided to show the utility and effectiveness of our methods, which are then compared to the existing methods.

Complementing missing information and priority vector are of significance important aspects in group decision making (GDM) with incomplete hesitant fuzzy linguistic preference relations (HFLPRs). In this paper, an integer programming model is developed based on additive consistency to estimate missing values of incomplete HFLPRs by using additive consistency. Once the missing values are complemented, a mixed 0-1 programming model is established to derive the priority vectors from complete HFLPRs, in which the underlying idea of the mixed 0-1 programming model is the probability sampling in statistics and minimum deviation between the priority vector and HFLPR. In addition, we also propose a new GDM approach for incomplete HFLPRs by integrating the integer programming model and the mixed 0-1 programming model. Finally, two case studies and comparative analysis detail the application of the proposed models.

This paper provides a new multi-attributes decision making approach on the basis of the Gini OWA ( $\text{G}^{\mathrm {p,q}}$ -OWA) operator, in which the Gini operator is the combination of the Gini mean and the OWA operator. Desired properties and several generalized forms of the $\text{G}^{\mathrm {p,q}}$ -OWA operator are investigated. In order to determine the $\text{G}^{\mathrm {p,q}}$ -OWA operator weights, an orness measure is proposed, and a generalized least squares deviation model (GLSD) is put forward. Finally, a case study of low carbon supplier selection is shown to illustrate the effectiveness and practicality of the proposed method.