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As the rapidly progressing applications of uncertainty theories, the need for modifications to some of their existing mathematical tools or creating new tools to deal correctly with them in various environments is also exposed. Hesitant fuzzy numbers (HFNs), as a particular case of fuzzy numbers, are not an exception to this rule. Considering the necessity of determining the distance between given HFNs in many of their practical applications, this article shows that the existing methods either do not provide correct results or are not able to meet the needs of users. This paper aims to present new methods for distance measures of hesitant fuzzy numbers. To do them, three prevalent distance measures, i.e., the generalized distance measure, the Hamming distance measure, and the Euclidean distance measure, will be optimized into three distinct trinal categories. With the approach of reducing error propagation via reducing some unnecessary mathematical computations, new distance measures on HFNs will be introduced, first. The middle is the modification of the first category, which is more suitable when the given HFNs are equal-distance by the previous formula. Also, as the third category, the weighted form of these distance measures has been proposed, to be used where the real and membership parts of HFNs are not of equal importance. As an application of these, a TOPSIS-based technique for solving multi-attribute group decision-making problems with HFNs has been proposed. A numerical example will be implemented to describe the presented method. Finally, along with the validation of the proposed method, its numerical comparison with some other existing methods will be discussed in detail. [ABSTRACT FROM AUTHOR]

The financial management capability of enterprises, as an important component of their soft power, has a decisive impact on the success or failure of their operations. In the increasingly fierce market competition, enterprises must continuously improve their financial management capabilities in order to ensure efficient operation and achieve better economic benefits. Insufficient financial management capabilities in enterprises can seriously affect the stability of production and operation, hinder the realization of profits, and hinder the long-term development of enterprises. In order to better improve the financial management level of enterprises and promote the standardization of financial management, it is necessary to use scientific techniques to evaluate the financial management ability of enterprises, so as to accurately grasp the key links in the financial management process of enterprises and implement targeted effective measures. The enterprise financial management capability evaluation is a classical multiple attribute group decision making (MAGDM). In recent years, the MAGDM problem has become an important research field in modern decision science. This paper extends the EDAS technique to the 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs). On the basis of the original EDAS technique, 2-tuple linguistic Pythagorean fuzzy number EDAS (2TLPFN-EDAS) technique based on cosine similarity measure (CSM) and Hamming distances is managed for MAGDM. Finally, a case study for enterprise financial management capability evaluation and some comparative analysis with the other techniques show that the new technique proposed in this paper is effective, reasonable and accurate. The main contribution of the paper is summarized: (1) the 2TLPFN-EDAS technique based on CSM and Hamming distances is managed for MAGDM under 2TLPFSs; (2) The entropy is employed to manage the attribute weight based on cosine similarity measure(CSM) and Hamming distances under 2TLPFSs; (3) the 2TLPFN-EDAS technique is employed for enterprise financial management capability evaluation and were compared with some existing techniques; (4) Through the comparison, it is found that 2TLPFN-EDAS technique for enterprise financial management capability evaluation proposed are effective. [ABSTRACT FROM AUTHOR]

In this paper, we present novel approaches to multi-attribute decision-making problems using q -rung normal fuzzy numbers. The q -rung normal fuzzy sets are an important way to express uncertain information, and they are superior to the intuitionistic fuzzy sets, Pythagorean fuzzy sets and normal Pythagorean fuzzy sets. A combination of the q -rung fuzzy number and the normal fuzzy number becomes q -rung normal fuzzy numbers. The new averaging and geometric operations of q -rung normal fuzzy numbers are studied using the general aggregation function. There are both commutative, associative, idempotent and boundedness compatible using q -rung normal fuzzy numbers. There are four new aggregating operators presented such as q -rung normal fuzzy interaction weighted averaging, q -rung normal fuzzy interaction weighted geometric, q -rung generalized normal fuzzy interaction weighted averaging and q -rung generalized normal fuzzy interaction weighted geometric. Usually, the aggregation functions are taken to be the Euclidean distance and Hamming distance. The existence of these operators enables the development of algorithms that solve multi-attribute decision-making problems. We demonstrate the application of enhanced Euclidean and Hamming distances to problems arising in real-world situations. An important component of a robot sensor is computer science and machine tool technology. There are four factors that can be used to evaluate robotics systems: resolution, sensitivity, error and accuracy. In order to determine the best alternative, expert opinions should be compared to the criteria. The accuracy of these models is better and they are closer to q. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods. [ABSTRACT FROM AUTHOR]

• A CODAS-based group decision model is designed for the T-spherical fuzzy context. • A T-spherical fuzzy Minkowski distance operator based on Heronian mean is proposed. • A novel ordered weighted Heronian Minkowski distance (TSFOWHMD) operator is advanced. • The TSFOWHMD operator is utilized to improve the CODAS method. • The applicability of the developed methodology is verified through a case study. T-spherical fuzzy sets (TSFSs) are more flexible and efficient tools to deal with ambiguous, uncertain and vague information in complex real-world decision-making problems than various extended intuitionistic fuzzy sets. This paper aims to develop a novel T-spherical fuzzy (TSF) Combinative Distance-Based ASsessment (CODAS) based on the Heronian Minkowski distance aggregation operator, this new method can capture interrelationship between input arguments. Some TSF weighted Heronian Minkowski distance (TSFWHMD) aggregation operators with generalization are developed based on Heronian mean and Minkowski-type distance, their properties are discussed as well as their families are analyzed. Furthermore, the TSF MAGDM methodology based on the improved CODAS is designed, where the Minkowski-type distance is used to define the TSF similarity for computing the expert weights and to construct the maximizing deviation method (MDM) for determining the attribute weights, respectively. The TSF ordered weighted Heronian Hamming distance (TSFOWHHD) and TSF ordered weighted Heronian Euclidean distance (TSFOWHED) operators derived from the TSFOWHMD operator are integrated into the CODAS method, which is an improved method for both measuring the deviation of the negative ideal solution from each alternative and capturing the correlation between attributes. Finally, the feasibility and practicality of developed methodology are illustrated with an example of CAE (Computer Aided Engineering) software selection for lithium-ion power battery (LiPB) design, sensitivity analysis and method comparisons are performed to elucidate the reliability and validity of the developed methodology. [ABSTRACT FROM AUTHOR]

Atanassov's intuitionistic fuzzy sets play a vital role in handling vagueness and uncertainty successfully as they associate an additional non-membership degree to each element of the universal set. Interval-valued intuitionistic fuzzy (IVIF) sets are more generalized quantities in this direction as they use intervals (⊂ [ 0 , 1 ]) to define the membership and non-membership grades of each element of the set. However, without defining basic arithmetic operations on the IVIF values/sets, the study is incomplete. This paper broadly examines the division and subtraction operations over any arbitrary IVIF values/sets using the concept of Hamming distance between them. A deterministic linear optimization model is proposed and solved in order to obtain the complete expressions for these operations. Next, various fundamental properties and relationships are examined and proved for the proposed arithmetic operations over IVIF values/sets. Furthermore, an example of a multi-criteria decision-making problem under uncertain conditions is studied and analyzed using the developed arithmetic operations on IVIF numbers. • Derived the generalized subtraction and division operations for IVIF values/sets. • The notion of Hamming distance and dual-simplex algorithm are used. • A deterministic LPP is proposed to obtain the expressions for the operations. • The fundamental properties and relationships for the basic operations are explored. • A case study of multi-criteria decision-making problems has been discussed. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy (IF) theory has become main approach to representing imprecision and vagueness. The IF divergence measure (IFDivM) based on Jensen–Shannon divergence is perhaps the most widely used measure to compare the similarity of multiple intuitionistic fuzzy sets (IFSs). In the present paper, this IFDivM is examined and applied to multiple examples. It is found that some extant IFDivMs hardly satisfy the axiomatic definition, and in a few cases even unable to show divergence of trivial IFSs. To address these inconsistencies, a new IFDivM based on Jensen–Shannon divergence is proposed, free from these problems. The effectiveness of the proposed IFDivM is tested on several critical cases, and precise analysis of its properties is performed. It is proved that the proposed IFDivM satisfies the axiomatic definition of IFDivMs. To illustrate the practical significance of the IFDivM, a novel intuitionistic fuzzy (IF) TODIM method, based on the proposed IFDivM, is developed, termed as GIF-TODIM method. Unlike the existing IF-TODIM methods, GIF-TODIM does not suffer from the revere ordering inconsistencies. The proposed GIF-TODIM method and the proposed IFDivM are applied to a real-world case study on supplier selection. A detailed comparative analysis is performed taking the TOPSIS method and other IFDivMs as baselines. The role of attitude on the final choice is analyzed in great detail. It is found that the proposed GIF-TODIM method is indeed useful, effective, and superior to the counterpart methods, when it comes to real-world situations. Concomitantly, in the present work, it is also revealed that the TOPSIS method based on the 2-D Hamming distance is a special form of the proposed GIF-TODIM method, when decision-makers have the same attitude towards losses and gains. Thus, an interesting relationship between TOPSIS and TODIM is identified under the intuitionistic fuzzy environment, which is bound to propel significant research in the area of decision making under uncertain conditions. As a whole, the article offers comprehensive analyses of IFDivMs and the TODIM method under the intuitionistic fuzzy environment. • Construct a new IF divergence measure (IFDivM) based on Jensen–Shannon divergence. • Develop a GIF-TODIM method based on the new IFDivM. • Apply the proposed GIF-TODIM method to a supplier selection problem. • Provide parameter analysis and comparative analysis with TOPSIS and other IFDivMs. • Show that TOPSIS based on 2-D Hamming distance is a special form of our GIF-TODIM. [ABSTRACT FROM AUTHOR]