The correlation coefficient between two variables is an important aspect of statistics. The accuracy of assessments of correlation relies on information from a set of discourses. Data collected in statistical studies are often full of exceptions. Pythagorean fuzzy soft sets (PFSS) are a parametrized family of extended Pythagorean fuzzy sets (PFS). They comprise a generalization of intuitionistic fuzzy soft sets which may be used to accurately assess deficiencies and uncertainties in evaluations. PFSS can accommodate uncertainty more competently than intuitionistic fuzzy soft sets and are the most important strategy when dealing with fuzzy information in decision-making processes. Herein, the concept and characteristics of correlation coefficients and the weighted correlation coefficients in PFSS are discussed. We also introduce the Pythagorean fuzzy soft weighted average (PFSWA) and Pythagorean fuzzy soft weighted geometric (PFSWG) operators and discuss their desirable characteristics. A prioritization technique for order preference by similarity to the ideal solution (TOPSIS) under the PFSS environment based on correlation coefficients and weighted correlation coefficients will be introduced. Through the proposed methodology, a technique for decision-making is developed. Additionally, an application of the proposed TOPSIS technique is presented for green supplier selection in green supply chain management (GSCM). The practicality, efficacy, and flexibility of the proposed approach is proved through comparative analyses, drawing upon existing studies.

The correlation coefficient between the two parameters plays a significant part in statistics. Furthermore, the exactness of the assessment of correlation depends upon information from the set of discourses. The data collected for various statistical studies are full of ambiguities. The idea of interval-valued intuitionistic fuzzy soft sets is an extension of intuitionistic fuzzy soft sets that is used to express insufficient evaluation, uncertainty, and anxiety in decision-making. Intuitionistic fuzzy soft sets consider two different types of information, such as membership degree and nonmembership degree. In this paper, the concepts and properties of the correlation coefficient and the weighted correlation coefficient of interval-valued intuitionistic fuzzy soft sets are proposed. A prioritization technique for order preference by similarity to the ideal solution based on interval-valued intuitionistic fuzzy soft sets of correlation coefficients and the weighted correlation coefficient is introduced. We also proposed interval-valued intuitionistic fuzzy soft weighted average and interval-valued intuitionistic fuzzy soft weighted geometric operators and developed decision-making techniques based on the proposed operators. By using the developed techniques, a method for solving decision-making problems is proposed. To ensure the applicability of the proposed methods, an illustrative example is given. Finally, we present a comparison of some existing methods with our proposed techniques.

Intuitionistic fuzzy hypersoft set is an extension of the intuitionistic fuzzy soft set used to express insufficient evaluation, uncertainty, and anxiety in decision-making. It is a new technique to realize computational intelligence and decision-making under uncertain conditions. The intuitionistic fuzzy hypersoft set can deal with uncertain and fuzzy information more effectively. The concepts and properties of the correlation coefficient and the weighted correlation coefficient of the intuitionistic fuzzy hypersoft sets are proposed in the following research. A prioritization technique for order preference by similarity to ideal solution (TOPSIS) based on correlation coefficients and weighted correlation coefficients is introduced under the intuitionistic fuzzy hypersoft sets. We also introduced aggregation operators, such as intuitionistic fuzzy hypersoft weighted average and intuitionistic fuzzy hypersoft weighted geometric operators. Based on the established TOPSIS method and aggregation operators, the decision-making algorithm is proposed under an intuitionistic fuzzy hypersoft environment to resolve uncertain and confusing information. A case study on decision-making difficulties proves the application of the proposed algorithm. Finally, a comparative analysis with the advantages, effectiveness, flexibility, and numerous existing studies demonstrates this method's effectiveness.

The related properties of derivations in lattices are investigated. We show that the set of all isotone derivations in a distributive lattice can form a distributive lattice. Moreover, we introduce the fixed set of derivations in lattices and prove that the fixed set of a derivation is an ideal in lattices. Using the fixed sets of isotone derivations, we establish characterizations of a chain, a distributive lattice, a modular lattice and a relatively pseudo-complemented lattice, respectively. Furthermore, we discuss the relations among derivations, ideals and fixed sets in lattices.