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This study proposes a Modified Fuzzy Extended Pivot Pairwise Relative Criteria Importance Assessment (MF PIPRECIA-E) method for determining criteria weights under uncertainty. The proposed extension enables the upper bounds of fuzzy numbers obtained during calculation to remain below, equal to, or slightly above 1. In addition, this study also critically examines the procedure of direct and inverse assignment of the relative importance of the criteria, integrated in the PIPRECIA-E method, with the aim of pointing out some important characteristics of this method, which have not been sufficiently explored in previous research. The proposed approach also addresses consistency assessment and group decision-making. Therefore, the main contribution of this study is development of a modified fuzzy PIPRECIA-E method that provides a more consistent procedure for determining weights of the criteria under uncertainty. Accordingly, proposed approach incorporates both direct and inverse criteria importance assessment, extending the applicability of the PIPRECIA-E method for solving various multiple-criteria decision-making (MCDM) problems.

Shadowed set (SHS) theory was introduced by Witold Pedrycz in 1998 to play a vital role in handling vagueness and providing a way to approximate a given fuzzy set (FS) by a construct, which is easy to deal with in practice. Nonetheless, the application of SHSs and shadowed numbers (SHNs) is surprisingly underrepresented in fuzzy multi-attribute decision-making (FMADM) literature. To address this gap, the current research aims to introduce and demonstrate a new FMADM method called the SHAdowed $\boldsymbol{N}\boldsymbol{U}$mber based $\boldsymbol{R}\mathit{anking}\hspace{2.5pt}\mathit{and}\boldsymbol{S}\mathit{election}\hspace{2.5pt}\boldsymbol{A}\mathit{pproach}$ (SHANURSA). The methodology involves, inter alia, transforming FNs into their corresponding SHNs using Grzegorzewski’s (2013) approach, then implementing an efficient weighted Minkowski Distance Metric (MDM) for SHNs to better gauge the proximity/remoteness between them. Additionally, a new and unusual closeness coefficient is introduced and defined to enhance the ranking accuracy of alternatives. Finally, the applicability and effectiveness of this closeness coefficient and proximity-based FMADM method is demonstrated by means of three distinct case studies from literature. The findings of this research show that the suggested method is trustworthy.

Nowadays, it is agreed that fuzzy sets are suitable for capturing and representing the concept of vagueness and uncertainty, and various fuzzy reasoning systems are being developed based on them. Researchers have proposed fuzzy set extensions to improve the performance and accuracy of these systems. The research questions arise regarding how fuzzy sets have evolved and what the main trends in their evolution are. To address these questions, our research presents a chronological and bibliometric analysis of fuzzy sets based on papers extracted from the Web of Science database. The main findings and contributions have been identified, systematized and visualized in a fuzzy set keyword map of 65 fuzzy set extensions. These extensions are primarily used for decision-making, reasoning, and prediction, particularly in the context of digital transformation, by integrating digital technologies into all areas of business, transforming operations and enhancing value delivery to customers. As organisations increasingly adopt digital technologies, the need for robust frameworks to manage uncertainty becomes critical. The main trends indicating the directions of fuzzy sets development, an overview of the variety and popularity of fuzzy sets over the years, and the impact of countries engaged in fuzzy set research are also identified and reported. The results support researchers and practitioners working on fuzzy sets and their applications by providing valuable insights into the fuzzy set topic, its existing extensions, and, more generally, to any field of investigation where fuzzy sets are relevant, particularly in the realm of digital transformation.

When handling engineering problems associated with optimal alternative selection a researcher often deals with not sufficiently accurate data. The alternatives are usually assessed by applying several different criteria. A method takes advantage of the relationship between fuzzy sets and matrix game theories can be offered for multicriteria decision-making. Practical investigations have already been discussed for selecting the variants water supply systems.

In this paper, a new multi-criteria decision-making procedure is presented, which captures preferential information in the form of the threshold model. It is based on the ELECTRE-like sorting analysis restricted by the localization principle, which enables high adaptability of the decision model and reduces the cognitive load imposed on the decision-makers. It lays the foundation for the introduction of three concepts that have been previously insufficiently supported by outranking methods – semiautomatic derivation of criteria weights according to the selective effects of discordance and veto thresholds, convergent group consensus seeking, and autonomous multi-agent negotiation. The interdependent principles are justified, and the methodological solutions underlying their implementation are provided.