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This paper describes a generalized Diophantine fuzzy sets, which can be seen as a generalization of both Diophantine fuzzy sets and Pythagorean fuzzy sets. We define the basic properties of generalized Diophantine fuzzy set, as well as their relationships and distances. We compare Diophantine fuzzy sets with other Diophantine Pythagorean fuzzy sets to demonstrate their importance in the literature. We introduce new operators including necessity, possibility, accuracy function and score function. Furthermore, we discuss the new distance between normalized Euclidean distance and normalized Hamming distance. For a generalized Diophantine fuzzy relation, image and inverse image functions are defined. Numerous real-world applications can be found for the prevalent ideas of intuitionistic fuzzy sets, Pythagorean fuzzy sets, Diophantine fuzzy sets and q-rung orthopair fuzzy sets. Regretfully, these theories about the membership and non-membership grades have their own limits. We provide a new idea the generalized Diophantine fuzzy set that eliminates these limitations by including reference parameters. Compared to other kinds of fuzzy sets, there are more applications for generalized Diophantine fuzzy sets. We offer practical examples that show how different enhanced distances might be used in everyday situations. Additionally, to demonstrate the effectiveness of the suggested approach, flowchart based multi-criteria decision-making is provided and used to a numerical example. The outcomes are assessed for various parameter values. Furthermore, a comparative analysis developed to demonstrate the superiority of the suggested technique over current methodologies. [ABSTRACT FROM AUTHOR]

Effective and optimal decision-making can enhance system performance, potentially leading to a positive reputation and financial gains. Multi-criteria decision-making (MCDM) is an important research topic widely applied to practical decision-making problems. Using the basic idea of symmetry to balance the arrangement where elements or features have an equality or similarity in distribution, MCDM provides robust decisions in such multi-dimensional complex issues. This study proposes MultiFuzzTOPS, a decision-making model to deal with complexity of multi-criteria decision-making. The proposed MultiFuzzTOPS leverages the fuzzy logic and soft sets such as type-2 soft sets (T2SS) and technique for order preference by similarity to ideal solution (TOPSIS) for decision-making. We validate the proposed model by implementing it to solve the pesticide selection problem in food science by considering various criteria for the selection of pesticides. Our proposed MultiFuzzTOPS recommends the best pesticide compared with its counterparts because it covers the maximum information for the selection of the best alternative. Results are ranked on the basis of the Hamming distance and similarity coefficient. We also validate the effectiveness by performing the sensitivity analysis, and the validation shows the reliability and effectiveness of our proposed model. [ABSTRACT FROM AUTHOR]

In this article, multiple attribute decision-making problems are solved using the vague normal set (VNS). It is possible to generalize the vague set (VS) and q-rung fuzzy set (FS) into the q-rung vague set (VS). A log q-rung normal vague weighted averaging (log q-rung NVWA), a log q-rung normal vague weighted geometric (log q-rung NVWG), a log generalized q-rung normal vague weighted averaging (log Gq-rung NVWA), and a log generalized q-rungnormal vagueweightedgeometric (logGq-rungNVWG) operator are discussed in this article. Adescription is provided of the scoring function, accuracy function and operational laws of the log q-rung VS. The algorithms underlying these functions are also described. A numerical example is provided to extend the Euclidean distance and the Humming distance. Additionally, idempotency, boundedness, commutativity, and monotonicity of the log q-rung VS are examined as they facilitate recognizing the optimal alternative more quickly and help clarify conceptualization. We chose five anemia patients with four types of symptoms including seizures, emotional shock or hysteria, brain cause, and high fever, who had either retrograde amnesia, anterograde amnesia, transient global amnesia, post-traumatic amnesia, or infantile amnesia. Natural numbers q are used to express the results of the models. To demonstrate the effectiveness and accuracy of the models we are investigating, we compare several existing models with those that have been developed. [ABSTRACT FROM AUTHOR]

This paper aims to support decision-makers improve their ability to accurately capture and represent their judgment in a wide range of situations. To do this, we propose a new type of fuzzy set called a p , q -cubic quasi-rung orthopair fuzzy set ( p , q -CQOFS). The p , q -CQOFS allows for a more flexible and detailed expression of incomplete information through the use of an additional parameter. The paper describes the concept of p , q -CQOFS and its relationship to other types of fuzzy sets, introduces score and accuracy functions for p , q -CQOFS and analyzes some of its mathematical properties, defines the Hamming distance measure between two p , q -CQOFSs and examines some of its important properties, investigates the basic operations of p , q -CQOFSs and extends these laws to aggregation operators, and introduces weighted averaging and geometric aggregation operators for combining p , q -cubic quasi-rung orthopair fuzzy data. [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]

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]