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In this research article, the notions of bipolar fuzzy numbers, trapezoidal bipolar fuzzy numbers, triangular bipolar fuzzy numbers and bipolar fuzzy linguistic variables are introduced. Ranking function on the set of all bipolar fuzzy numbers, and the expressions for the ranking of trapezoidal and triangular bipolar fuzzy numbers are derived. A group decision making method based on trapezoidal bipolar fuzzy TOPSIS method is proposed, and the implementation of the proposed method in the selection of best project proposal is presented. Finally, a theoretical comparison of the proposed trapezoidal bipolar fuzzy TOPSIS method with other multi-criteria decision making methods such as TOPSIS, bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE I is discussed.

The problem of the piecewise linear approximation of fuzzy numbers giving outputs nearest to the inputs with respect to the Euclidean metric is discussed. The results given in Coroianu et al. (Fuzzy Sets Syst 233:26–51, 2013 ) for the 1-knot fuzzy numbers are generalized for arbitrary n -knot ( n ≥ 2 ) piecewise linear fuzzy numbers. Some results on the existence and properties of the approximation operator are proved. Then, the stability of some fuzzy number characteristics under approximation as the number of knots tends to infinity is considered. Finally, a simulation study concerning the computer implementations of arithmetic operations on fuzzy numbers is provided. Suggested concepts are illustrated by examples and algorithms ready for the practical use. This way, we throw a bridge between theory and applications as the latter ones are so desired in real-world problems.

Pythagorean fuzzy set, initially extended by Yager from intuitionistic fuzzy set, is capable of modeling information with more uncertainties in the process of multi-criteria decision making (MCDM), thus can be used on wider range of conditions. The fuzzy decision analysis of this paper is mainly built upon two expressions in Pythagorean fuzzy environment, named Pythagorean fuzzy number (PFN) and interval-valued Pythagorean fuzzy number (IVPFN), respectively. We initiate a novel axiomatic definition of Pythagorean fuzzy distance measurement, including PFNs and IVPFNs. After that, corresponding theorems are put forward and then proved. Based on the defined distance measurements, the closeness indexes are developed for both expressions, inspired by the idea of technique for order preference by similarity to ideal solution (TOPSIS) approach. After these basic definitions have been established, the hierarchical decision approach is presented to handle MCDM problems under Pythagorean fuzzy environment. To address hierarchical decision issues, the closeness index-based score function is defined to calculate the score of each permutation for the optimal alternative. To determine criterion weights, a new method based on the proposed similarity measure and aggregation operator of PFNs and IVPFNs is presented according to Pythagorean fuzzy information from decision matrix, rather than being provided in advance by decision makers, which can effectively reduce human subjectivity. An experimental case is then conducted to demonstrate the applicability and flexibility of the proposed decision approach. Finally, extension forms of Pythagorean fuzzy decision approach for heterogeneous information are briefly introduced to show its potentials on further applications in other processing fields with information uncertainties.

Interval type-2 fuzzy numbers (IT2FNs) are a particular kind of type-2 fuzzy numbers (T2FNs). In most scientific works, since arithmetic operations required IT2FNs are simpler than those of T2FNs, mathematical calculations on IT2FNs are used more frequently rather than the T2FNs. Hence, in recent decades, the study on IT2FNs have been intensified significantly. These numbers can be explained by trapezoidal and triangular forms. In this article, first, the concept of general interval type-2 trapezoidal fuzzy numbers (GIT2TrFNs) and then arithmetic operations among them are introduced. Next, three new ranking methods are suggested for GIT2TrFN. Finally, several examples are used to illustrate and compare new ranking methods with others.

In this paper, the problem of expressing and fusion processing of multi-channel uncertain digital information based on the method of constructing high-dimensional fuzzy numbers is studied. A new concept of multi-step platform high-dimensional fuzzy cell number is defined by providing a theorem on high-dimensional fuzzy number with special structure, and a characterization theorem of such fuzzy number is obtained. More importantly, a method for constructing such high-dimensional fuzzy numbers to represent multi-channel uncertain digital information is established. Then, a specific calculation formula of such high-dimensional fuzzy numbers is derived for ease of calculation in applications. Finally, based on the obtained results, we provided methods to solve the identification results for two types of identification problems proposed, and provided a numerical application example. And it is pointed out that the results we obtained can also be applied to other fusion processing for objects represented by multi-channel uncertain digital information, such as ranking and classifying.

This paper introduces a simplified presentation of a new computing procedure for solving the fuzzy Pythagorean transportation problem. To design the algorithm, we have described the Pythagorean fuzzy arithmetic and numerical conditions in three different models in Pythagorean fuzzy environment. To achieve our aim, we have first extended the initial basic feasible solution. Then an existing optimality method is used to obtain the cost of transportation. To justify the proposed method, few numerical experiments are given to show the effectiveness of the new model. Finally, some conclusion and future work are discussed.

Organizational decisions, often originating from diverse groups of managers with varying criteria, encounter challenges due to the inherent ambiguity in human judgments, particularly those involving preferences. This paper proposes a nuanced solution using a heterogeneous fuzzy group decision‐making method, accommodating diverse criteria based on real numbers, interval numbers, triangular fuzzy numbers, trapezoidal fuzzy numbers, and intuitive fuzzy numbers. Two numerical examples validate the method’s effectiveness, demonstrating noteworthy consistency. Utilizing Spearman’s rank correlation coefficient as a metric underscores its robust agreement with established methods. The subsequent analysis underscores the method’s adeptness in handling decision‐making complexities within diverse groups, thereby contributing significantly to the understanding and management of organizational relationships in today’s context.

In this paper, different measures of interval-valued pentagonal fuzzy numbers (IVPFN) associated with assorted membership functions (MF) were explored, considering significant exposure of multifarious interval-valued fuzzy numbers in neoteric studies. Also, the idea of MF is generalized somewhat to nonlinear membership functions for viewing the symmetries and asymmetries of the pentagonal fuzzy structures. Accordingly, the construction of level sets, for each case of linear and nonlinear MF was also carried out. Besides, defuzzification was undertaken using three methods and a ranking method, which were also the main features of this framework. The developed intellects were implemented in a game problem by taking the parameters as PFNs, ultimately resulting in a new direction for modeling real world problems and to comprehend the uncertainty of the parameters more precisely in the evaluation process.

In this paper, based on the weighted metric, the problem of approximating a general continuous fuzzy number $u$ with derivable $\\underline{u}(r)$ and $\\overline{u}(r)$ (left and right cut functions of level value $r$) for $r \\in (0,1]$ by using the $m-n-$step type fuzzy number is studied. Two kinds of approximations ($I-$nearest approximation and $II-$nearest approximation which satisfy different conditions) of using $m-n-$step type fuzzy numbers to approximate general continuous fuzzy numbers are defined, and the relationship between the $I-$nearest approximation and $II-$nearest approximation is obtained. Then the methods of the two kinds of approximations are respectively presented by the theorems more specifically. At last, an example is given to show the effectiveness and usability of the methods set up by us.

Generalized hesitant fuzzy numbers (GHFNs) are able to directly manage situations in which we may encounter a finite set of known values with a finite set of degrees of doubt as quantitative approximations of an uncertain situation/quantification of a linguistic expression. They are new extensions of hesitant fuzzy sets, which have been considered in this paper. In fact, in this paper, GHFNs will be utilized to model the uncertainty of the assessment values of options against criteria in multi-attribute decision making (MADM) problems. It means that all of the elements of decision matrix are GHFNs. Then, the technique for order of preference by similarity to ideal solution (TOPSIS) method, as a very successful method in solving MADM problems, will be updated to be used with GHFNs. To this end, the distance between GHFNs must be defined to obtain the distances between given alternatives from each of two subjective alternatives (positive/negative ideal solutions). Thus, three existing famous distance measures, i.e., general distance ( d g ), Hamming distance ( d h ), and Euclidean distance ( d e ) measures, have been updated for GHFNs firstly. Then, the new TOPSIS method will be proposed based on GHFNs. Finally, the numerical examples have been appointed to illustrate the proposed method, analyze comparatively and validate it.