Intuitionistic Fuzzy Synthetic Measure on the Basis of Survey Responses and Aggregated Ordinal Data

The paper addresses the problem of complex socio-economic phenomena assessment using questionnaire surveys. The data are represented on an ordinal scale; the object assessments may contain positive, negative, no answers, a “difficult to say” or “no opinion” answers. The general framework for Intuitionistic Fuzzy Synthetic Measure (IFSM) based on distances to the pattern object (ideal solution) is used to analyze the survey data. First, Euclidean and Hamming distances are applied in the procedure. Second, two pattern object constructions are proposed in the procedure: one based on maximum values from the survey data, and the second on maximum intuitionistic values. Third, the method for criteria comparison with the Intuitionistic Fuzzy Synthetic Measure is presented. Finally, a case study solving the problem of rank-ordering of the cities in terms of satisfaction from local public administration obtained using different variants of the proposed method is discussed. Additionally, the comparative analysis results using the Intuitionistic Fuzzy Synthetic Measure and the Intuitionistic Fuzzy TOPSIS (IFT) framework are presented.     

On the Neutrosophic,Pythagorean and Some Other Novel Fuzzy Sets Theories Used in Decision Making: Invitation to Discuss

In this short paper, a critical analysis of the Neutrosophic, Pythagorean and some other novel fuzzy sets theories foundations is provided, taking into account that they actively used for the solution of the decision-making problems. The shortcomings of these theories are exposed. It is stated that the independence hypothesis, which is a cornerstone of the Neutrosophic sets theory, is not in line with common sense and therefore leads to the paradoxical results in the asymptotic limits of this theory. It is shown that the Pythagorean sets theory possesses questionable foundations, the sense of which cannot be explained reasonably. Moreover, this theory does not completely solve the declared problem. Similarly, important methodological problems of other analyzed theories are revealed. To solve the interior problems of the Atanassov’s intuitionistic fuzzy sets and to improve upon them, this being the reason most of the criticized novel sets theories were developed, an alternative approach based on extension of the intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory is proposed. No propositions concerned with the improvement of the Cubic sets theory and Single-Valued Neutrosophic Offset theory were made, as their applicability was shown to be very dubious. In order to stimulate discussion, many statements are deliberately formulated in a hardline form.     

On Some Extension of Intuitionistic Fuzzy Synthetic Measures for Two Reference Points and Entropy Weights

In this paper, a novel Double Intuitionistic Fuzzy Synthetic Measure (DIFSM), based on intuitionistic fuzzy values for handling multi-criteria decision-making problems used to rank alternatives, is presented. In the studies, intuitionistic fuzzy sets (IFSs) represented uncertain, imprecise information or human judgment. The intuitionistic fuzzy sets can also reflect the approval, rejection, and hesitation of decision-makers. The degrees of satisfiability and non-satisfiability and uncertainty of each alternative with respect to a set of criteria are described by membership functions, non-membership functions, and hesitancy indexes, respectively. The aggregation algorithm DIFSM is inspired by Hellwig’s method based on two reference points: ideal point (pattern) and anti-ideal point (anti-pattern), measuring distances between the alternative and ideal point and distance between the ideal and anti-ideal point. The proposed methods take into consideration the entropy-based weights of criteria. An illustrative example is given to demonstrate the practicality and effectiveness of the proposed approach. Additionally, the comparative analysis results, using the DIFSM and the Intuitionistic Fuzzy TOPSIS-based framework, are presented.     

Uncertainty of Interval Type-2 Fuzzy Sets Based on Fuzzy Belief Entropy

Interval type-2 fuzzy sets (IT2 FS) play an important part in dealing with uncertain applications. However, how to measure the uncertainty of IT2 FS is still an open issue. The specific objective of this study is to present a new entropy named fuzzy belief entropy to solve the problem based on the relation among IT2 FS, belief structure, and Z-valuations. The interval of membership function can be transformed to interval BPA [Bel,Pl]. Then, Bel and Pl are put into the proposed entropy to calculate the uncertainty from the three aspects of fuzziness, discord, and nonspecificity, respectively, which makes the result more reasonable. Compared with other methods, fuzzy belief entropy is more reasonable because it can measure the uncertainty caused by multielement fuzzy subsets. Furthermore, when the membership function belongs to type-1 fuzzy sets, fuzzy belief entropy degenerates to Shannon entropy. Compared with other methods, several numerical examples are demonstrated that the proposed entropy is feasible and persuasive.     

A Decision Support Model for Hotel Recommendation Based on the Online Consumer Reviews Using Logarithmic Spherical Hesitant Fuzzy Information

Spherical hesitant fuzzy sets have recently become more popular in various fields. It was proposed as a generalization of picture hesitant fuzzy sets and Pythagorean hesitant fuzzy sets in order to deal with uncertainty and fuzziness information. Technique of Aggregation is one of the beneficial tools to aggregate the information. It has many crucial application areas such as decision-making, data mining, medical diagnosis, and pattern recognition. Keeping in view the importance of logarithmic function and aggregation operators, we proposed a novel algorithm to tackle the multi-attribute decision-making (MADM) problems. First, novel logarithmic operational laws are developed based on the logarithmic, t-norm, and t-conorm functions. Using these operational laws, we developed a list of logarithmic spherical hesitant fuzzy weighted averaging/geometric aggregation operators to aggregate the spherical hesitant fuzzy information. Furthermore, we developed the spherical hesitant fuzzy entropy to determine the unknown attribute weight information. Finally, the design principles for the spherical hesitant fuzzy decision-making have been developed, and a practical case study of hotel recommendation based on the online consumer reviews has been taken to illustrate the validity and superiority of presented approach. Besides this, a validity test is conducted to reveal the advantages and effectiveness of developed approach. Results indicate that the proposed method is suitable and effective for the decision process to evaluate their best alternative.     

A New Model of Interval-Valued Intuitionistic Fuzzy Weighted Operators and Their Application in Dynamic Fusion Target Threat Assessment

Existing missile defense target threat assessment methods ignore the target timing and battlefield changes, leading to low assessment accuracy. In order to overcome this problem, a dynamic multi-time fusion target threat assessment method is proposed. In this method, a new interval valued intuitionistic fuzzy weighted averaging operator is proposed to effectively aggregate multi-source uncertain information; an interval-valued intuitionistic fuzzy entropy based on a cosine function (IVIFECF) is designed to determine the target attribute weight; an improved interval-valued intuitionistic fuzzy number distance measurement model is constructed to improve the discrimination of assessment results. Specifically, first of all, we define new interval-valued intuitionistic fuzzy operation rules based on algebraic operations. We use these rules to provide a new model of interval-valued intuitionistic fuzzy weighted arithmetic averaging (IVIFWAA) and geometric averaging (IVIFWGA) operators, and prove a number of algebraic properties of these operators. Then, considering the subjective and objective weights of the incoming target, a comprehensive weight model of target attributes based on IVIFECF is proposed, and the Poisson distribution method is used to solve the time series weights to process multi-time situation information. On this basis, the IVIFWAA and IVIFWGA operators are used to aggregate the decision information from multiple times and multiple decision makers. Finally, based on the improved TOPSIS method, the interval-valued intuitionistic fuzzy numbers are ordered, and the weighted multi-time fusion target threat assessment result is obtained. Simulation results of comparison show that the proposed method can effectively improve the reliability and accuracy of target threat assessment in missile defense.     

New Pythagorean Entropy Measure with Application in Multi-Criteria Decision Analysis

The purpose of this paper is to propose a new Pythagorean fuzzy entropy for Pythagorean fuzzy sets, which is a continuation of the Pythagorean fuzzy entropy of intuitionistic sets. The Pythagorean fuzzy set continues the intuitionistic fuzzy set with the additional advantage that it is well equipped to overcome its imperfections. Its entropy determines the quantity of information in the Pythagorean fuzzy set. Thus, the proposed entropy provides a new flexible tool that is particularly useful in complex multi-criteria problems where uncertain data and inaccurate information are considered. The performance of the introduced method is illustrated in a real-life case study, including a multi-criteria company selection problem. In this example, we provide a numerical illustration to distinguish the entropy measure proposed from some existing entropies used for Pythagorean fuzzy sets and intuitionistic fuzzy sets. Statistical illustrations show that the proposed entropy measures are reliable for demonstrating the degree of fuzziness of both Pythagorean fuzzy set (PFS) and intuitionistic fuzzy sets (IFS). In addition, a multi-criteria decision-making method complex proportional assessment (COPRAS) was also proposed with weights calculated based on the proposed new entropy measure. Finally, to validate the reliability of the results obtained using the proposed entropy, a comparative analysis was performed with a set of carefully selected reference methods containing other generally used entropy measurement methods. The illustrated numerical example proves that the calculation results of the proposed new method are similar to those of several other up-to-date methods.     

Multi-Attribute Decision Making with Einstein Aggregation Operators in Complex Q-Rung Orthopair Fuzzy Hypersoft Environments

The purpose of our research is to extend the formal representation of the human mind to the concept of the complex q-rung orthopair fuzzy hypersoft set (Cq-ROFHSS), a more general hybrid theory. A great deal of imprecision and ambiguity can be captured by it, which is common in human interpretations. It provides a multiparameterized mathematical tool for the order-based fuzzy modeling of contradictory two-dimensional data, which provides a more effective way of expressing time-period problems as well as two-dimensional information within a dataset. Thus, the proposed theory combines the parametric structure of complex q-rung orthopair fuzzy sets and hypersoft sets. Through the use of the parameter q, the framework captures information beyond the limited space of complex intuitionistic fuzzy hypersoft sets and complex Pythagorean fuzzy hypersoft sets. By establishing basic set-theoretic operations, we demonstrate some of the fundamental properties of the model. To expand the mathematical toolbox in this field, Einstein and other basic operations will be introduced to complex q-rung orthopair fuzzy hypersoft values. The relationship between it and existing methods demonstrates its exceptional flexibility. The Einstein aggregation operator, score function, and accuracy function are used to develop two multi-attribute decision-making algorithms, which prioritize based on the score function and accuracy function to ideal schemes under Cq-ROFHSS, which captures subtle differences in periodically inconsistent data sets. The feasibility of the approach will be demonstrated through a case study of selected distributed control systems. The rationality of these strategies has been confirmed by comparison with mainstream technologies. Additionally, we demonstrate that these results are compatible with explicit histograms and Spearman correlation analyses. The strengths of each approach are analyzed in a comparative manner. The proposed model is then examined and compared with other theories, demonstrating its strength, validity, and flexibility.     

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn’s information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but may be irregular on the other, the extended entropy-based proof technique yields the same bound as was conjectured by Kahn (2001) and proved by Sah et al. (2019).     

A Geometric-Based LSGDM Method for Tourism Project Decision Optimization with Trust–Distrust Relationships

In this paper, we discuss the decision optimization of tourism projects in Hebei Province, China. To improve the process of analyzing tourism projects, we introduce a model that includes multiple decision makers as subjects based on a standard four-dimensional evaluation system. In order to improve the effectiveness of decision-making results, we will increase the number of decision makers to 40. A novel large-scale group decision-making (LSGDM) algorithm that incorporates the trust–distrust asymmetric relationships between decision makers is proposed. This model contains three main innovations: firstly, in the evaluation of decision makers’ social network relations, the trust–distrust value is introduced as a new carrier, and a weighted directed network and data integration operator are constructed based on the evaluation between decision makers; secondly, an extended Girvan-Newman (GN) algorithm is constructed to cluster the decision makers from this weighted network; thirdly, the interval-valued intuitionistic fuzzy number (IVIFN) is used to evaluate the alternatives, studying the IVIFN’s geometric significance by placing in a rectangular coordinate system. Finally, a new LSGDM model is proposed. Using the development of a cultural tourism project in a township as an example, the effectiveness of the proposed model is illustrated. By comparing the results of our method to those of a LSGDM algorithm that does not incorporate trust relationships, we assess the performance of the new model.     

Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario

It is well-known that the law of total probability does not generally hold in quantum theory. However, recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner’s friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. A rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner’s friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems only reconfirm the invalidity of the law of total probability in quantum theory rather than invalidating the physical statements that the no-go theorems attempt to refute.