Pythagorean fuzzy soft set (PFSS) is the most influential and operative extension of the Pythagorean fuzzy set (PFS), which contracts with the parametrized standards of the substitutes. It is also a generalized form of the intuitionistic fuzzy soft set (IFSS) and delivers a well and accurate estimation in the decision-making (DM) procedure. The primary purpose is to prolong and propose ideas related to Einstein’s ordered weighted aggregation operator from fuzzy to PFSS, comforting the condition that the sum of the degrees of membership function and nonmembership function is less than one and the sum of the squares of the degree of membership function and nonmembership function is less than one. We present a novel Pythagorean fuzzy soft Einstein ordered weighted averaging (PFSEOWA) operator based on operational laws for Pythagorean fuzzy soft numbers. Furthermore, some essential properties such as idempotency, boundedness, and homogeneity for the proposed operator have been presented in detail. The choice of a sustainable supplier is also examined as an essential part of sustainable supply chain management (SSCM) and is considered a crucial multiattribute group decision-making (MAGDM) issue. In some MAGDM problems, the relationship between alternatives and uncertain environments will be the main reason for deficient consequences. We have presented a novel aggregation operator for PFSS information to choose sustainable suppliers to cope with those complex issues. The Pythagorean fuzzy soft number (PFSN) helps to represent the obscure information in such real-world perspectives. The priority relationship of PFSS details is beneficial in coping with SSCM. The proposed method’s effectiveness is proved by comparing advantages, effectiveness, and flexibility among the existing studies.

The framework of complex T-spherical fuzzy set (CTSFS) deals with unclear and imprecise information with the help of membership degree (MD), abstinence degree (AD), nonmembership degree (NMD), and refusal degree (RD). Due to this characteristic, the CTSFSs can be applied to any phenomenon having the involvement of human opinions. This article aims to familiarize some Hamacher aggregation operators (HAOs) grounded on CTSFSs. To do so, we define some Hamacher operational laws in the environment of CTSFS by using Hamacher t-norm (HTNM) and Hamacher t-conorm (HTCNM). A few numbers of AOs are developed with the help of defined operational laws based on HTNM and HTCNM including the complex T-spherical fuzzy (CTSF), Hamacher weighted averaging (HWA) (CTSFHWA), CTSF Hamacher ordered weighted averaging (CTSFHOWA) operator, CTSF Hamacher hybrid weighted averaging (CTSFHHWA) operator, CTSF Hamacher weighted geometric (CTSFHWG) operator, CTSF Hamacher ordered weighted geometric (CTSFHOWG) operator, and CTSF Hamacher hybrid weighted geometric (CTSFHHWG) operator. Some interesting properties of developed HAOs are investigated and then these HAOs are applied to the multi-attribute decision making (MADM) problem. For the significance of these HAOs, the results obtained from these HAOs are compared with existing aggregation operators (AOs).

Vendor management systems (VMSs) are web-based software packages that can be used to manage businesses. The performance of the VMSs can be assessed using multi-attribute decision-making (MADM) techniques under uncertain situations. This article aims to analyze and assess the performance of VMSs using MADM techniques, especially when the uncertainty is of complex nature. To achieve the goals, we aim to explore Hany mean (HM) operators in the environment of complex picture fuzzy (CPF) sets (CPFSs). We introduce CPF Hamy mean (CPFHM) and CPF weighted HM (CPFWHM) operators. Moreover, the reliability of the newly proposed HM operators is examined by taking into account the properties of idempotency, monotonicity, and boundedness. A case study of VMS is briefly discussed, and a comprehensive numerical example is carried out to assess VMSs using the MADM technique based on CPFHM operators. The sensitivity analysis and comprehensive comparative analysis of the proposed work are discussed to point out the significance of the newly established results.

Rough set (RS) and fuzzy set (FS) theories were developed to account for ambiguity in the data processing. The most persuasive and modernist abstraction of an FS is the linear Diophantine FS (LD-FS). This paper introduces a resilient hybrid linear Diophantine fuzzy RS model (LDF-RS) on paired universes based on a linear Diophantine fuzzy relation (LDF-R). This is a typical method of fuzzy RS (F-RS) and bipolar FRS (BF-RS) on two universes that are more appropriate and customizable. By using an LDF-level cut relation, the notions of lower approximation (L-A) and upper approximation (U-A) are defined. While this is going on, certain fundamental structural aspects of LD-FAs are thoroughly investigated, with some instances to back them up. This cutting-edge LDF-RS technique is crucial from both a theoretical and practical perspective in the field of medical assessment.