Complex Fermatean fuzzy sets (CFFSs) integrate the ideas of complex fuzzy sets and Fermatean fuzzy sets, where the membership, non-membership, and hesitancy degrees are all complex numbers, allowing the express uncertain information more flexibly and comprehensively. However, how to reasonably measure the discrepancies between CFFSs in decision-making remains an open task. This paper presents a series of new distance measures of CFFSs and their weighted versions based on Hamming, Euclidean, Hausdorff, and Hellinger distances. On this basis, we explore some outstanding properties that the proposed measures satisfy (i.e., boundedness, nondegeneracy, symmetry, and triangular inequality) and demonstrate their effectiveness through several examples. Furthermore, we design a decision-making algorithm as well as a clustering algorithm based on the proposed measures and verify the performance of the proposed measures through several applications.

Ordered functional weighted averaging (OFWA) operators are a generalization of the well-known ordered weighted averaging (OWA) operators in which functions, instead of single values, are considered as weights. This fact offers an extra level of flexibility; for example, in multi-criteria decision-making, it can be used to aggregate available information and provide recommendations. This paper furthers the analysis of these general operators, studying how they can be combined to obtain conservative and aggressive perspectives from experts and studying the algebraic structure of the whole set of these operators.

Knowledge graphs are recognized as a valuable format for representing data and information. Their ability to represent semantics using different types of relations between the concepts and denoting information at different levels of abstraction creates a demand for algorithms taking advantage of such data format. In this paper, we propose a method for determining the similarity between concepts in weighted knowledge graphs. The method uses a hierarchical approach to determine the degree of similarity at different levels of 'distance' from the considered graph concepts. The proposed technique employs the T-norm and OWA operator. Similarities between concepts account for edge weights, while OWA aggregates similarities between nodes at different levels of distance from the compared nodes. The method is explained, and its merits are discussed.

Non-cooperative behavior exhibited by DMs when they must make excessive interest compromises the achievement of group consensus. This study develops an inter-subgroup compensation mechanism the Nash bargaining game under the minimum cost consensus model (MCCM) framework to managing cooperative behavior. First, a cooperative acceptability index (CAI) based on compromise limit costs is to objectively identify non-cooperative behavior. By quantifying the acceptable compromise limit costs, ensures that consensus adjustments remain within acceptable bounds. Then, an inter-subgroup compensation mechanism is designed using the Nash bargaining game from the perspective of Kaldor-Hicks improvement. This mechanism enables cooperative DMs to incentivize non-cooperative peers via resource transfers, dual optimization by minimizing collective costs and ensuring individual acceptability. Finally, a community renewal application example and comparison analysis are provided to illustrate the efficacy of the approach.

There are numerous uses for picture fuzzy sets (PFS) in the context of engineering and scientific issues. Dombi operators can flexibly work on parameter assessment. To address this, we propose the weighted versions of the ordered, hybrid, and picture fuzzy Dombi averaging operators, which we refer to as PFDWA, PFDOWA, and PFDHWA, respectively. Additionally, we created the weighted form of the PFDWG, PFDOWG, and PFDHWG operators, which stand for picture fuzzy Dombi weighted geometric, order geometric, and hybrid geometric operators, respectively, and show their features. To build the model for a multiple attribute decision-making (MADM) method, we also use PFDWA and PFDWG operators. Finally, an example has been provided to illustrate the application of the suggested MADM technique.

The Basic Uncertain Information (BUI) is a recently introduced type of uncertain data that has rapidly undergone development and practical application. The existing aggregation operators designed for BUI solely encompass the weighted mean and Choquet integral. The present study puts forth a set of general information fusion frameworks and methodologies aimed at gathering BUI granules. The first mode yields BUI granules as its output, whereas the subsequent two modes generate outputs in the form of interval values. The paper includes numerical examples and applications that correspond to the presented findings. The present study conducts an analysis of various mathematical properties pertaining to the three BUI fusion modes that have been proposed. These properties include idempotency, monotonicities, certainty derived inclusion, certainty monotonicity, homogeneities, non-symmetricity, comonotone additivities, and continuities. The proposals and analyses presented in this work are of a general nature and have the potential to inspire various practical specifications.

We propose in this paper an improved probability adjustment method to combine multiple single-label classifiers using the Dempster–Shafer (DS) combination rule. Our method employs all types of classifier outputs namely the class label, its score (also called evidence or support score), and its rank. The idea of our method is to adjust the score value assigned by a given classifier using its rank. Our method aims to assign high support to highly ranked labels and low support to lowly ranked labels. In contrast with other DS-based combination methods, our method does not require a complicated process to adjust the pieces of evidence. Through a numerical example, we proved that the proposed method is efficient and generates better results than those achieved by other DS-based combination methods. Moreover, we applied our method in combining multiple classifiers using many real-world datasets. The comparison of our method with other DS-based combination methods, non-learning combination methods, and ensemble classifiers shows that our method performs better using most of the datasets.

The soft set-theoretic model acts as an essential methodology for handling the uncertainty in which parameterized issues appeared during the data analysis compared with fuzzy as well as advanced fuzzy sets (picture fuzzy set). In this proposed work, an effort is made to achieve the effect of picture fuzzy soft sets, as well as multicriteria group decision making (MCGDM) on picture fuzzy arguments where evaluation is executed on the evaluation of a group of experts. In this regard, the new averaging operators are introduced via weighted forms of a parameterized PFS setting, namely, a picture fuzzy soft average operator, a picture fuzzy soft geometric operator, and some properties of these proposed operators are stated. Finally, a model for the MCGDM technique has been introduced for these proposed operators and compared with the existing operators.

Dombi operations are introduced on two m-polar picture fuzzy (mPoPF) numbers in this chapter. The mPoPF Dombi weighted averaging (mPoPFDWA), mPoPF Dombi ordered weighted averaging (mPoPFDOWA), mPoPF Dombi hybrid weighted averaging (mPoPFDHWA), mPoPF Dombi weighted geometric (mPoPFDWG), and mPoPF Dombi hybrid weighted geometric (mPoPFDHWGA) operators have been proposed. Additionally, some qualities are established, including idempotency, boundedness, monotonicity, and commutativity. Next, using the mPoPFDWA and mPoPFDWG operators, we created the multiattribute decision-making (MADM) technique for the mPoPF environment. In order to choose the ideal location for the construction of a petrol station, we have provided an application of the MADM approach. Based on the operating parameter for the outcomes of the decision-making process, a sensitivity analysis of the current strategy is established.

The intuitionistic fuzzy sets (IFSs) are extended in several ways by incorporating different concepts. One such extension is the picture fuzzy set (PFS). This set measured three quantities, namely, positive membership and nonmembership like IFSs, and the third one is neutral membership of an element simultaneously. Using the PFS, one can handle uncertainties in real-life decision-making problems. Also, the power-averaging operator (PAO) reduces the significance of biased decision makers during the evaluation of extreme data. The Dombi operator has some advantages, which have flexibility when the parameters are evaluated. For this reason, the PA and Dombi operators in picture fuzzy environments are introduced. Also, the Dombi power weighted averaging, ordered weighted averaging operators, and hybrid weighted averaging operators in picture fuzzy environment are introduced. Again, some more operators in a picture fuzzy setup, namely, Dombi power weighted geometric, order weighted geometric, and hybrid weighted geometric are defined and studied for their attractive properties. Some of these newly defined operators are used to develop a multiattribute decision-making (MADM) process. Lastly, this new MADM method is illustrated by considering an application.

The concepts of cognitive interval information and cognitive uncertain information, which are two recently proposed types of uncertain information, have been extended in this work to the typical hesitant fuzzy environment. We introduce the notions of typical hesitant monopolar cognitive interval information and typical hesitant cognitive uncertain information. To facilitate their analysis, we define uncertainty degree functions and score functions for these concepts using extended aggregation operators. Furthermore, we reanalyze some decision models discussed in earlier literature using these newly proposed concepts to demonstrate their advantages and potential applications.

Goal programming is a very powerful tool to solve single- as well as multiobjective programming problems. This method assigns a goal to each objective function with a certain aspiration level. For a crisp goal programming problem, a goal is a specific number, while in fuzzy goal programming, the goal need not be a fixed number; it may be an uncertain number, or sometimes a linguistic number. Methods for fuzzy and intuitionistic fuzzy goal programming are being developed by many authors, but research is not being carried out on picture fuzzy environments. This chapter considers the goal programming problem in a picture fuzzy environment. The linear, exponential, and hyperbolic picture fuzzy numbers are defined and used in goal programming.

In this chapter, using picture fuzzy numbers in multiple attribute group decision making (MAGDM), we employ the multiple attribute border approximation area comparisons (MABAC) method with picture fuzzy numbers. We analyze the concepts of PFN and suggest their scoring, accuracy, and operating principles. Additionally, two aggregation operators—picture fuzzy weighted averaging (PFWA) and picture fuzzy weighted geometric (PFWG) operators—are presented and used to create a MAGDM strategy based on the initial MABAC method with PFNs. The suggested method is accurate and valid for considering the competing qualities. To evaluate the usefulness of the established method, we examine a numerical example for the choice of a renewable energy power generation project. Finally, we demonstrate the effectiveness of the new technique by comparing it to certain already-in-use operators.

By taking into account membership, nonmembership, and neutral degrees separately for each element, the theory of picture fuzzy sets is helpful for managing uncertainty in multiattribute decision-making issues. The picture fuzzy linear assignment technique, which is proposed in this study to solve multicriteria group decision-making issues, is an extension of the traditional linear assignment method. The priority order of various alternatives is determined by a ranking process that combines a binary mathematical model, aggregation functions, score functions, accuracy functions, and weighted rank frequency. The choice of the location for a sponge iron factory demonstrates the applicability and validity of the proposed strategy. The suggested approach aids managers in determining the ideal place to build the sponge iron factory based on the set criteria.

The goal of the paper is to clarify the observed irrationality of decision making in conflict situations considered as one-step games of two players. To solve such situations, we consider the asymmetry in the relation of the players to their own rewards and the rewards of the opponents. Formalization of the decision-making process is based on recently developed non-commutative operators of multivalued logic algebra. The suggested method is applied to solve the well-known Prisoners’ dilemma game and the other situations of conflict, where it results in the expected strategies.

In traditional group decision making, the inconsistent experts are usually forced to make compromises toward the group opinion to increase the group consensus level. However, the strategy of reaching group consensus via an incentive mechanism encouraging adjustment of preferences is more effective than forcing, which is the aim of this article. Specifically, this article establishes a novel incentive mechanism to support group consensus under dynamic trust relationship. First, the supremum and infimum incentives-based rule driven by trust relationship is defined. Based on the assumption that if incentive conditions are met, then experts will be willing to adjust their preferences, the incentive behavior-driven minimum adjustment consensus model is developed to generate optimal incentive-based recommendation preferences. Thus, the proposed incentive mechanism can effectively reduce the preference adjustment cost and promote group consensus reaching. Third, the updated trust relationships between experts are shown to be strengthen by the proposed incentive-driven preference revision. Consequently, the optimization model based on trust interaction relationship is constructed to obtain the final group preference matrix. Finally, a supplier selection case of high-end medical equipment is provided to illustrate the proposed method and show the rationality and advantages of the proposed methodology with both a sensitivity analysis and a comparison analysis.

In uncertain information environment, bi-polar preferences can be elicited from experts and processed to be exerted over some weights determination for multiple-agents evaluation. Recently, some weighting methodologies and models in uncertain and preference involved environment with multiple opinions from multiple experts are proposed in some literature. However, in that existing method, when collecting different types of preferences from a single expert, sometimes some subtle cognitive inconsistency may occur. To eliminate such inconsistency, this work elaborately analyzes the possible reasons and proposes some amendment together with a new distinguishable set of formulations for modeling. In addition, we further consider two situations of the weighting models for the problem, with one only considering the situation of single expert with no risk of cognitive inconsistency and the other considering the case of multiple experts wherein some inconsistency might occur. Numerical example and comparison are also presented accordingly.

In this chapter, some useful operators on picture fuzzy sets are defined and some related results are established. Some basic issues related to picture fuzzy set, namely relation, graph, and arithmetic operations are introduced. The idea of a similarity measure of two picture fuzzy sets is initiated. Several picture fuzzy sets are combined together as a convex combination. Two special types of topological operators namely a topological operator of closure and a topological operator of interior on picture fuzzy sets are initiated and some corresponding results are presented. Also, an operator, namely an implication operator, is introduced here and some of its properties are studied. Different types of picture fuzzy averaging operators and a picture fuzzy Dombi weighted aggregation operator are investigated here with suitable examples and their important properties.

Three membership functions—membership, neutral, and nonmembership degrees of the element—are incorporated by default into the PFS. The picture fuzzy linear programming problem (PFLPP), in which the various parameters are represented by picture fuzzy numbers (PFNs), is discussed in this chapter. The membership, neutral, and nonmembership degrees are used to produce the PFLPP in its purest form. To solve the PFLPP, the picture fuzzy optimization model is introduced. A numerical example and case study are provided to demonstrate the suggested research's effectiveness. Finally, the conclusion and potential future study areas are explored.

Real-world systems often exhibit intricate complexity. Navigating and examining the conditions under which these systems operate present various challenges. These systems are characterized by a web of interconnected inputs and subsystems organized in hierarchical way. The status of each subsystem depends on multiple inputs and the conditions of other interconnected subsystems. Additionally, articulating precise definitions for the states of these subsystems is a complex task, usually fraught with uncertainties. Experts frequently employ information granules to encapsulate imprecise quantities, basing these granules on specialized domain knowledge. These granules may manifest as linguistic terms or intervals, resulting in approximate state definitions. To the best of our knowledge, no methodology currently accommodates multiple uncertainties—including those tied to state definitions—while also offering an evaluation of the varying states of different subsystems. In this study, we present a novel technique for identifying local and global states in hierarchical, multi-component systems under conditions of uncertainty. Utilizing principles of Evidence Theory, we incorporate a recently devised method to evaluate how well uncertain objectives are met. These uncertain objectives correspond to state definitions formulated using information granules of a specific context. By measuring the extent to which the inputs to a given subsystem align with these imprecise state definitions, we can identify the most probable state the subsystem will likely be in. Our proposed method addresses various types of uncertainty when ascertaining system states. The specific areas of imprecision tackled by our approach include: (1) the vagueness and ambiguity inherent in the measurements serving as subsystem inputs, (2) the levels of uncertainty involved in defining subsystem states based on the conditions of other interconnected subsystems, and (3) the indistinct and incomplete knowledge incorporated into the definitions describing individual subsystems’ states. In summary, this paper introduces a method for determining the most likely state of complex systems. Its novelty lies in the application of a technique for satisfying uncertain targets. We have developed a methodology suitable for hierarchical systems. We elaborate on the intricacies of our method and include a case study to demonstrate how system states can be identified when faced with ambiguous definitions of subsystem states and uncertain values of inputs.