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This is mainly because the restrictive condition of intuitionistic hesitant fuzzy number (IHFN) is relaxed by the membership functions of Pythagorean probabilistic hesitant fuzzy number (PyPHFN), so the range of domain value of PyPHFN is greatly expanded. The paper aims to develop a novel decision-making technique based on aggregation operators under PyPHFNs. For this, the authors propose Algebraic operational laws using algebraic norm for PyPHFNs. Furthermore, a list of aggregation operators, namely Pythagorean probabilistic hesitant fuzzy weighted average (PyPHFWA) operator, Pythagorean probabilistic hesitant fuzzy weighted geometric (PyPHFWG) operator, Pythagorean probabilistic hesitant fuzzy ordered weighted average (PyPHFOWA) operator, Pythagorean probabilistic hesitant fuzzy ordered weighted geometric (PyPHFOWG) operator, Pythagorean probabilistic hesitant fuzzy hybrid weighted average (PyPHFHWA) operator and Pythagorean probabilistic hesitant fuzzy hybrid weighted geometric (PyPHFHWG) operator, are proposed based on the defined algebraic operational laws. Also, interesting properties of these aggregation operators are discussed in detail.

PyPHFN is not only a generalization of the traditional IHFN, but also a more effective tool to deal with uncertain multi-attribute decision-making problems.

In addition, the authors design the algorithm to handle the uncertainty in emergency decision-making issues. At last, a numerical case study of coronavirus disease 2019 (COVID-19) as an emergency decision-making is introduced to show the implementation and validity of the established technique. Besides, the comparison of the existing and the proposed technique is established to show the effectiveness and validity of the established technique.

Paper is original and not submitted elsewhere.

The purpose of this article is to present the idea of a T-spherical hesitant fuzzy set associated with probability and to develop an extended multi-attributive border approximation area comparison (MABAC) method under probabilistic T-spherical hesitant fuzzy (Pt-SHF) settings.

The authors define some basic operational laws for Pt-SHF sets (Pt-SHFSs) and a comparison method of two probabilistic T-spherical hesitant fuzzy numbers (Pt-SHFNs) is proposed. Moreover, some Pt-SHF aggregation operators and the multi-attributive border approximation area comparison (MABAC) method are established under Pt-SHF scenario to solve group decision making problems.

The developed Pt-SHF MABAC method for multi-attribute group decision making (MAGDM) can overcome the drawbacks of conventional MABAC method and limitations for decision makers, which they face while providing their assessment concerning any object.

Clearly, this paper is devoted to MABAC method, MAGDM and probabilistic T-spherical hesitant fuzzy set theory.

The approach established can be used in a variety of scenarios, including decision making, engineering, and economics. An explanatory example is illustrated which shows the superiority and effectiveness of our proposed technique.

If a T-spherical fuzzy MAGDM problem under the probabilistic scenario needs to be evaluated, the involvement of probabilities in fuzzy system will be lost because of no information. This work fills a gap in literature by establishing the notion of probabilistic t-spherical hesitant fuzzy set to deal with the ambiguity, uncertainty in decision making problems.

The purpose of this article is to conduct a main path analysis of 627 articles on the theme of Pythagorean fuzzy sets (PFSs) in the Web of Science (WoS) from 2013 to 2020, to provide a conclusive and comprehensive analysis for researchers in this field, and to provide a study on preliminary understanding of PFSs.

The research topic of Pythagorean fuzzy fields, through keyword extraction and describing the changes in characteristic themes over the past eight years, are firstly examined. Main path analysis, including local and global main paths and key route paths, is then used to reveal the most influential relationships between papers and to explore the trajectory and structure of knowledge transmission.

The application of Pythagorean fuzzy theory to the field of decision-making has been popular, and combinations of the traditional Pythagorean fuzzy decision-making method with other fuzzy sets have attracted widespread attention in recent years. In addition, over the past eight years, research interest has shifted to different types of PFSs, such as interval-valued PFSs.

This paper implicates to investigate the growth in certain trends in the literature and to explore the main paths of knowledge dissemination in the domain of PFSs in recent years.

This paper aims to identify the topics in which researchers are currently interested, to help scholars to keep abreast of the latest research on PFSs.

Platform-based enterprises, as micro-entities in the platform economy, have the potential to effectively promote the low-carbon development of both supply and demand sides in the supply chain. Therefore, this paper aims to provide a multi-criteria decision-making method in a probabilistic hesitant fuzzy environment to assist platform-type companies in selecting cooperative suppliers for carbon reduction in green supply chains.

This paper combines the advantages of probabilistic hesitant fuzzy sets (PHFS) to address uncertainty issues and proposes an improved multi-criteria decision-making method called PHFS-DNMEREC-MABAC for aiding platform-based enterprises in selecting carbon emission reduction collaboration suppliers in green supply chains. Within this decision-making method, we enhance the standardization process of both the DNMEREC and MABAC methods by directly standardizing probabilistic hesitant fuzzy elements. Additionally, a probability splitting algorithm is introduced to handle probabilistic hesitant fuzzy elements of varying lengths, mitigating information bias that traditional approaches tend to introduce when adding values based on risk preferences.

In this paper, we apply the proposed method to a case study involving the selection of carbon emission reduction collaboration suppliers for Tmall Mart and compare it with the latest existing decision-making methods. The results demonstrate the applicability of the proposed method and the effectiveness of the introduced probability splitting algorithm in avoiding information bias.

Firstly, this paper proposes a new multi-criteria decision making method for aiding platform-based enterprises in selecting carbon emission reduction collaboration suppliers in green supply chains. Secondly, in this method, we provided a new standard method to process probability hesitant fuzzy decision making information. Finally, the probability splitting algorithm was introduced to avoid information bias in the process of dealing with inconsistent lengths of probabilistic hesitant fuzzy elements.

The authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined. Further, based on these operators, a decision-making approach is presented under the Pythagorean fuzzy set environment. An illustrative example related to the selection of the best alternative is considered to demonstrate the efficiency of the proposed approach.

In real-world situations, Pythagorean fuzzy numbers are exceptionally useful for representing ambiguous data. The authors look at multi-criteria decision-making issues in which the parameters have a prioritization relationship. The idea of a priority degree is introduced. The aggregation operators are formed by awarding non-negative real numbers known as priority degrees among strict priority levels. Consequently, the authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees.

The authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined. Further, based on these operators, a decision-making approach is presented under the Pythagorean fuzzy set environment. An illustrative example related to the selection of the best alternative is considered to demonstrate the efficiency of the proposed approach.

The aggregation operators are formed by awarding non-negative real numbers known as priority degrees among strict priority levels. Consequently, the authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined.

The purpose of this paper is to develop a multi-attribute group decision-making (MAGDM) method under the q-rung orthopair trapezoidal fuzzy environment, which calculates the interaction between the criteria depending on the proposed q-rung orthopair trapezoidal fuzzy aggregation Choquet integral (q-ROTrFACI) and employ TODIM (an acronym in Portuguese of Interactive and Multi-criteria Decision Making) to consider the risk psychology of decision-makers, to determine the optimal ranking of alternatives.

In MAGDM, q-rung orthopair trapezoidal fuzzy numbers (q-ROTrFNs) are efficient to indicate the quantitative vagueness of decision-makers. The q-ROTrFACI operator is defined and some properties are proved. Then, a novel similarity measure is developed by fusing the area and coordinates of the q-rung orthopair trapezoidal fuzzy function. Based on the above, a Choquet integral-based TODIM (CI-TODIM) method to consider the risk psychology of decision-makers is proposed and two cases are provided to prove superiority of the method.

The paper investigates q-ROTrFACI operator to productively solve problems with interdependent criteria. Then, an approach is proposed to determine the center point of q--ROTrFNs and a q-rung orthopair trapezoidal fuzzy similarity is constructed. Furthermore, CI-TODIM method is devised based on the proposed q-ROTrFACI operator and similarity in q-rung orthopair trapezoidal fuzzy context. The illustration example of business models' solutions and hypertension health management are given to demonstrate the effectiveness and superiority of proposed method.

The paper develops a novel CI-TODIM method that effectively solves the MAGDM problems under the premise of fully considering the priority of criteria and the risk preference of decision-makers, which provides guiding advantages for practical decision-making and enriches the application of decision-making theory.

In the era of information overload, the density of tourism information and the increasingly sophisticated information needs of consumers have created information confusion for tourists and scenic-area managers. The study aims to help scenic-area managers determine the strengths and weaknesses in the development process of scenic areas and to solve the practical problem of tourists' difficulty in quickly and accurately obtaining the destination image of a scenic area and finding a scenic area that meets their needs.

The study uses a variety of machine learning methods, namely, the latent Dirichlet allocation (LDA) theme extraction model, term frequency-inverse document frequency (TF-IDF) weighting method and sentiment analysis. This work also incorporates probabilistic hesitant fuzzy algorithm (PHFA) in multi-attribute decision-making to form an enhanced tourism destination image mining and analysis model based on visitor expression information. The model is intended to help managers and visitors identify the strengths and weaknesses in the development of scenic areas. Jiuzhaigou is used as an example for empirical analysis.

In the study, a complete model for the mining analysis of tourism destination image was constructed, and 24,222 online reviews on Jiuzhaigou, China were analyzed in text. The results revealed a total of 10 attributes and 100 attribute elements. From the identified attributes, three negative attributes were identified, namely, crowdedness, tourism cost and accommodation environment. The study provides suggestions for tourists to select attractions and offers recommendations and improvement measures for Jiuzhaigou in terms of crowd control and post-disaster reconstruction.

Previous research in this area has used small sample data for qualitative analysis. Thus, the current study fills this gap in the literature by proposing a machine learning method that incorporates PHFA through the combination of the ideas of management and multi-attribute decision theory. In addition, the study considers visitors' emotions and thematic preferences from the perspective of their expressed information, based on which the tourism destination image is analyzed. Optimization strategies are provided to help managers of scenic spots in their decision-making.

Prefabricated components sustainable supplier (PCSS) selection is critical to the success of prefabricated projects. However, limited studies have addressed the uncertainty and complexities during the selection process, particularly in multi-criterion group decision-making (MCGDM) circumstances. Hence, the research aims to develop a group decision-making model using a modified fuzzy MCGDM approach for PCSS selection under uncertain situation.

The proposed study develops a framework for sorting decisions in PCSS selection by using the hesitant fuzzy technique for order preference by similarity to ideal solution (HF-TOPSIS) method. The maximum consistency (MC) model is used to calculate the weights of decision makers (DMs) based on the cardinality and sequence of decision data.

The proposed framework has been successfully applied and illustrated in the case example of CB01 contract section in Hong Kong-Zhuhai-Macao Bridge (HZMB) megaproject. The results show various complicated decision-making scenarios can be addressed through the proposed approach. The MC model is able to calculate the weights of DMs based on the cardinality and sequence of decision data.

The research contributes to improving accuracy and reliability decision-making processes for PCSS selection, especially under hesitant and fuzzy situations in prefabricated megaprojects.

The purpose of this paper is to develop a methodology based on TODIM (an acronym in Portuguese for interactive and multicriteria decision-making) approach for the selection of the best alternative in the context of multi criteria group decision-making (MCGDM) problems under possibilistic uncertainty in interval-valued Pythagorean fuzzy (IVPF) environment.

In this paper, IVPF-TODIM method is proposed. Some new point operator-based similarity measures (POSMs) for IVPF sets (IVPFSs) are introduced which have the capability to reduce the degree of uncertainty of the elements in the universe of discourse corresponding to IVPFS. Then the newly defined POSMs are used to compute the measure of relative dominance of each alternative over other alternatives in the IVPF-TODIM context. Finally, generalized mean aggregation operator is used to find the best alternative.

As the TODIM method is used to solve the MCGDM problems under uncertainty, POSMs are developed by using three parameters which can control the effect of decision-makers’ psychological perception under risk.

The decision values are used in IVPF numbers (IVPFNs) format.

The proposed method is capable to solve real-life MCGDM problems with not only IVPFNs format but also with interval-valued intuitionistic fuzzy numbers.

As per authors’ concern, no approach using TODIM with IVPFNs is found in literature to solve MCGDM problems under uncertainty. The final judgment values of alternatives using the extended TODIM methodology are highly corroborate in compare to the results of existing methods, which proves its great potentiality in solving MCGDM problems under risk.

In this paper, the four popular multiple-criteria decision-making (MCDM) methods in fuzzy environment are utilized to reflect the vagueness and uncertainty on the judgments of decision-makers (DMs), because the crisp pairwise comparison in these conventional MCDM methods seems to be insufficient and imprecise to capture the right judgments of DMs. Of these methods, as Fuzzy analytic hierarchy process (F-AHP) is used to calculate criteria weights, the other methods; Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (F-TOPSIS), Fuzzy Grey relational analysis (F-GRA) and Fuzzy Preference Ranking Organization METhod for Enrichment of Evaluations (F- PROMETHEE II) are used to rank alternatives in the three different ways for a comparative study.

The demand for green products has dramatically increased because the importance and public awareness of the preservation of natural environment was taken into consideration much more in the last two decades. As a result of this, especially manufacturing companies have been forced to design more green products, resulting in a problem of how they incorporate environmental issues into their design and evaluate concept options. The need for the practical decision-making tools to address this problem is rapidly evolving since the problem turns into an MCDM problem in the presence of a set of green concept alternatives and criteria.

The incorporation of fuzzy set theory into these methods is discussed on a real-life case study, and a comparative analysis is done by using its numerical results in which the three fuzzy-based methods reveal the same outcomes (or rankings), while F-GRA requires less computational steps. Moreover, more detailed analyses on the numerical results of the case study are completed on the normalization methods, distance metrics, aggregation functions, defuzzification methods and other issues.

The designing and manufacturing environmental-friendly products in a product design process has been a vital issue for many companies which take care of reflecting environmental issues into their product design and meeting standards of recent green guidelines. These companies have utilized these guidelines by following special procedures at the design phase. Along the design process consisting of various steps, the environmental issues have been considered an important factor in the end-of-life of products since it can reduce the impact on the nature. In the stage of developing a new product with the aim of environmental-friendly design, the green thinking should be incorporated as early as possible in the process.

The case study was inspired from the previous work of the author, which was realized in a hot runner systems manufacturer, used in injection molding systems in a Canada. In a new product development process, the back- and front-ends of development efforts mainly determine the following criteria: cost, risk, quality and green used in this paper. The case study showed that the three fuzzy MCDM methods come to the same ranking outcomes. F-GRA has a better time complexity compared to the other two methods and uses a smaller number of computational steps. Moreover, a comparative analysis of the three F-MCDM methods; F-PROMETHEE II, F-TOPSIS and F-GRA used in ranking for green concept alternatives using the numerical results of the case study. For the case study; as seen in table 20, the three F-MCDM methods produced the numerical results on the rankings of the green concept alternatives as follows; {Concept A-Concept C–Concept B–Concept D}.

Inclusion of environmental-related criteria into concept selection problem has been gaining increasing importance in the last decade. Therefore, to facilitate necessary calculations in applying each method especially with its fuzzy extension, it can be developed a knowledge-based (KB) or an expert system (ES) to help the DMs make the required calculations of each method, and interpret its results with detailed analysis.

The objective of the research was to propose a F-AHP based F-MCDM approach to green concept selection problem through F-PROMETHEE II, F-TOPSIS and F-GRA methods. As the F-AHP is used to weight evaluation criteria, the other methods are respectively used for ranking the concept alternatives and determine the best concept alternative.