Pythagorean fuzzy set, generalized by Yager, is a new tool to deal with vagueness considering the membership grade $$\mu $$
 and non-membership $$\nu $$
 satisfying the condition $$\mu ^2+\nu ^2\le 1$$
 . It can be used to characterize the uncertain information more sufficiently and accurately than intuitionistic fuzzy set. Pythagorean fuzzy set has attracted great attention of many scholars that have been extended to new types and these extensions have been used in many areas such as decision making, aggregation operators, and information measures. Because of such a growth, we present an overview on Pythagorean fuzzy set with aim of offering a clear perspective on the different concepts, tools and trends related to their extension. In particular, we provide two novel algorithms in decision making problems under Pythagorean fuzzy environment. It may be served as a foundation for developing more algorithms in decision making.

Pythagorean fuzzy set: state of the art and future directions

This paper presents a new approach for the treatment of uncertainty which is based on interval-valued fuzzy-rough numbers (IVFRN). It is shown that by integrating the rough approach with the traditional fuzzy approach, the subjectivity that exists when defining the borders of fuzzy sets is eliminated. IVFRN make decision making possible using only the internal knowledge in the operative data available to the decision makers. In this way objective uncertainties are used and there is no need to rely on models of assumptions. Instead of different external parameters in the application of IVFRN, the structure of the given data is used. On this basis an original multi-criteria model was developed based on an IVFRN approach. In this multi-criteria model the traditional steps of the BWM (Best–Worst method) and MABAC (Multi-Attributive Border Approximation area Comparison) methods are modified. The model was tested and validated on a study of the optimal selection of fire fighting helicopters. Testing demonstrated that the model based on IVFRN enabled more objective expert evaluation of the criteria in comparison with traditional fuzzy and rough approaches. A sensitivity analysis of the IVFRN BWM-MABAC model was carried out by means of 57 scenarios, the results of which showed a high degree of stability. The results of the IVFRN model were validated by comparing them with the results of the fuzzy and rough extension of the MABAC, COPRAS and VIKOR models.

Modification of the Best–Worst and MABAC methods: A novel approach based on interval-valued fuzzy-rough numbers

q‐Rung orthopair fuzzy set (q‐ROFS) is a powerful tool that attracts the attention of many scholars in dealing with uncertainty and vagueness. The aim of paper is to present a new score function of q‐rung orthopair fuzzy number (q‐ROFN) for solving the failure problems when comparing two q‐ROFNs. Then a new exponential operational law about q‐ROFNs is defined, in which the bases are positive real numbers and the exponents are q‐ROFNs. Meanwhile, some properties of the operational law are investigated. Later, we apply them to derive the q‐rung orthopair fuzzy weighted exponential aggregation operator. Additionally, an approach for multicriteria decision‐making problems under the q‐rung orthopair fuzzy data is explored by applying proposed aggregation operator. Finally, an example is investigated to illustrate the feasibility and validity of the proposed approach. The salient features of the proposed method, compared to the existing q‐rung orthopair fuzzy decision‐making methods, are (1) it can obtain the optimal alternative without counterintuitive phenomena; (2) it has a great power in distinguishing the optimal alternative.

Exponential operation and aggregation operator for q‐rung orthopair fuzzy set and their decision‐making method with a new score function

An approach of TOPSIS technique for developing supplier selection with group decision making under type-2 neutrosophic number

Picture fuzzy Dombi aggregation operators: Application to MADM process

In this paper, we initiate a new axiomatic definition of single-valued neutrosophic distance measure and similarity measure, which is expressed by single-valued neutrosophic number that will reduce the information loss and remain more original information. Meanwhile, a novel score function is proposed. Then, the objective weights of various attributes are determined via gray system theory. Moreover, we present the combined weights, which can show both the subjective information and the objective information. Later, we present three algorithms to deal with multi-attribute decision-making problem based on revised Technique for Order Preference by Similarity to an Ideal Solution, Multi-Attributive Border Approximation area Comparison and similarity measure. Finally, the effectiveness and feasibility of approaches are demonstrated by two numerical examples.

Approaches to single-valued neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function

In this paper, we initiate a new axiomatic definition of Pythagorean fuzzy distance measure, which is expressed by Pythagorean fuzzy number that will reduce the information loss and remain more original information. Then, the objective weights of various criteria are determined via grey system theory. Combining objective weights with subjective weights, we present the combined weights, which can reflect both the subjective considerations of the decision maker and the objective information. Meanwhile, a novel score function is proposed. Later, we present two algorithms to solve stochastic multicriteria decision making problem, which takes prospect preference and regret aversion of decision makers into consideration in the decision process. Finally, the effectiveness and feasibility of approach is demonstrated by a numerical example.

Approaches to Pythagorean Fuzzy Stochastic Multi‐criteria Decision Making Based on Prospect Theory and Regret Theory with New Distance Measure and Score Function

The assessment of classroom teaching quality is critically important for producing a positive incentive and guidance role to improve service and management of universities, stimulating the enthusiasm of teachers, enhancing the teacher’s teaching ability, and improving the quality of talent training. In considering the case of teaching quality evaluation, the essential question that arises concerns strong ambiguity, fuzziness, and inexactness. The q‐rung orthopair fuzzy sets ( q‐ROFSs) dealing the indeterminacy characterized by membership degrees and nonmembership degrees are a more flexible and effective way to capture indeterminacy. In this paper, firstly, the new score function for q‐rung orthopair fuzzy number is initiated for tackling the comparison problem. Subsequently, a new distance measure for q‐ROFSs with multiple parameters is studied along with their detailed proofs. The various desirable properties among the developed similarity measures and distance measures have also been derived. Then, the objective weights of various attributes are determined via antientropy weighting method. Also, we develop the combined weights, which can reveal both the subjective information and the objective information. Moreover, two algorithms to solve q‐rung orthopair fuzzy decision‐making problem by combinative distance‐based assessment and multiparametric similarity measure are presented. Later, the feasibility of approaches is demonstrated by a classroom teaching quality problem, along with the effect of the different parameters on the ordering. Finally, a comparison between the proposed and the existing decision‐making methods has been performed for showing their effectiveness. The salient features of the proposed methods, compared to the existing q‐ROFS decision‐making methods, are as follows: (a) it can obtain the optimal alternative without counterintuitive phenomena and (b) it has a lower computational complexity.

Research on the assessment of classroom teaching quality with q‐rung orthopair fuzzy information based on multiparametric similarity measure and combinative distance‐based assessment

Jingguo Dai

Papers

Exponential operation and aggregation operator for q‐rung orthopair fuzzy set and their decision‐making method with a new score function

Approaches to single-valued neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function

Approaches to Pythagorean Fuzzy Stochastic Multi‐criteria Decision Making Based on Prospect Theory and Regret Theory with New Distance Measure and Score Function

Research on the assessment of classroom teaching quality with q‐rung orthopair fuzzy information based on multiparametric similarity measure and combinative distance‐based assessment

Hesitant fuzzy soft decision making methods based on WASPAS, MABAC and COPRAS with combined weights