Pythagorean Fuzzy Environment Research Articles

Circular Pythagorean Fuzzy Sets and Applications to Multi-Criteria Decision Making

In this paper, we introduce the concept of circular Pythagorean fuzzy set (value) (C-PFS(V)) as a new generalization of both circular intuitionistic fuzzy sets (C-IFSs) proposed by Atannassov and Pythagorean fuzzy sets (PFSs) proposed by Yager. A circular Pythagorean fuzzy set is represented by a circle that represents the membership degree and the non-membership degree and whose centre consists of non-negative real numbers μ and ν with the condition ${\mu ^{2}}+{\nu ^{2}}\leqslant 1$. A C-PFS models the fuzziness of the uncertain information more properly thanks to its structure that allows modelling the information with points of a circle of a certain centre and a radius. Therefore, a C-PFS lets decision makers to evaluate objects in a larger and more flexible region and thus more sensitive decisions can be made. After defining the concept of C-PFS we define some fundamental set operations between C-PFSs and propose some algebraic operations between C-PFVs via general triangular norms and triangular conorms. By utilizing these algebraic operations, we introduce some weighted aggregation operators to transform input values represented by C-PFVs to a single output value. Then to determine the degree of similarity between C-PFVs we define a cosine similarity measure based on radius. Furthermore, we develop a method to transform a collection of Pythagorean fuzzy values to a C-PFS. Finally, a method is given to solve multi-criteria decision making problems in circular Pythagorean fuzzy environment and the proposed method is practiced to a problem about selecting the best photovoltaic cell from the literature. We also study the comparison analysis and time complexity of the proposed method.

A novel multi-criteria group decision-making approach using aggregation operators and weight determination method for supplier selection problem in hesitant Pythagorean fuzzy environment

Uncertainty is an important factor in the decision-making process. Hesitant Pythagorean fuzzy sets (HPFS), a combination of Pythagorean and hesitant fuzzy sets, proved as a significant tool to handle uncertainty. Well-defined operational laws and attribute weights play an important role in decision-making. Thus, the paper aims to develop new Trigonometric Operational Laws, a weight determination method, and a novel score function for group decision-making (GDM) problems in the HPF environment. The approach is presented in three phases. The first phase defines new operational laws with sine trigonometric function incorporating its special properties like periodicity, symmetricity, and restricted range hence compared with previously defined aggregation operators they are more likely to satisfy the decision maker preferences. Properties of trigonometric operational laws (TOL) are studied and various aggregation operators are defined. To measure the relationship between arguments, the operators are combined with the Generalized Heronian Mean operator. The flexibility of operators is increased by the use of a real parameter λ to express the risk preference of experts. The second phase defines a novel weight determination method, which separately considers the membership and non-membership degrees hence reducing the information loss and the third phase conquers the shortcomings of previously defined score functions by defining a novel score function in HPFS. To further increase the flexibility of defined operators they are extended in the environment with unknown or incomplete attribute weights. The effectiveness of the GDM model is verified with the help of a supplier selection problem. A detailed comparative analysis demonstrates the superiority, and sensitivity analysis verifies the stability of the proposed approach.

Development of an advanced application process of Lean Manufacturing approach based on a new integrated MCDM method under Pythagorean fuzzy environment

The growth of manufacturing industries and the huge competitive environment forced manufacturing organizations to develop advanced improvement strategies and enhance their sustainability performance. The integration of sustainable Manufacturing in industrial operations leads to enhanced process performances through the reduction of wastes, cost, and environmental impacts and satisfies ergonomic conditions. For this reason, various firms have adopted sustainable manufacturing concepts to enhance their performances and hold a prestigious competitive position. The purpose of this research is to develop an integrated Pythagorean Fuzzy MCDM model to enhance the application process of the conventional Lean Manufacturing approach (LM). Firstly, an extended Value Steam Mapping is proposed to assess the sustainability of the manufacturing process and identify the causes of waste from a sustainability viewpoint. Secondly, Pythagorean Fuzzy Decision-Making Trial And Evaluation Laboratory (PF-DEMATEL) is employed to analyze the interrelationship among the identified. Thirdly, Pythagorean Fuzzy Technique for Order Preference by Similarity to Ideal Solution (PF-TOPSIS) is introduced to prioritize a set of solutions in order to overcome the investigated causes and improve the durability of the manufacturing operations. Finally, sensitivity analysis is conduced to assess the effectiveness of the obtained results. The proposed method has several attractive features. It can address the drawbacks of the conventional LM and enhance its analysis and improvement tasks. However, the proposed approach offers an advanced application process for Lean Manufacturing in a sustainability context. Additionally, the suggested strategy facilitates the leaders to assess the current state of the manufacturing processes and select the appropriate solutions for successful sustainability implementation. The validity of the proposed approach was investigated in a real case study. The results confirm its effectiveness and indicate that using MCDM approaches in LM application process offers a consistent and flexible demarche for sustainable manufacturing implementation.

An Application of the Pythagorean Fuzzy Sets in the Fault Diagnosis

Abstract In this paper, a comprehensive review and critical analyses of methods based on the ordinary fuzzy set, Atanassov’s intuitionistic fuzzy set, and its extensions have been conducted to show their limitations and defects. Then, a novel similarity measure based on the generalized score function has been introduced that incorporates the significance (importance) of information, making it more intuitive to compare them. The proposed method is employed for the fault diagnosis of steam turbine generator unit under Pythagorean fuzzy environment. Ten fault types of rotating machines are established as failure patterns in nine different vibration frequency ranges, expressed in terms of Pythagorean fuzzy numbers. The superiority of the proposed method in dealing with uncertain and vague information is shown by comparing it with some existing measures in numerical examples.

Multicriteria decision-making method under the complex Pythagorean fuzzy environment

Multicriteria decision-making method under the complex Pythagorean fuzzy environment

Stakeholder requirement evaluation of smart industrial service ecosystem under Pythagorean fuzzy environment for complex industrial contexts: A case study of renewable energy park

This study focuses on ways to systematically evaluate stakeholder requirements when developing a smart industrial service ecosystem (SISE) in a complex industrial context. The SISE development requires considering the service requirement from both the complex industrial context and service ecosystem manners. This study proposes a systematic framework for stakeholder requirement evaluation in SISE. The first part of the framework is the industrial context-viable system model with ecological thinking (IC-VESM) to elicit the service requirements for the SISE, which facilitates a systematic analysis of the service value proposition and service requirement elicitation in the operational lifecycle of an entire industrial context. This second part of the framework proposes a method for evaluating service requirements that is both feasible and systematic. This is achieved by combining the Fuzzy Kano and AHP methods in a Pythagorean fuzzy (PF) environment. The PF Kano computes the categories and determines the weights of service requirements from a consumer perspective, while the PF AHP hierarchically analyzes the service requirements and provides pairwise comparison paths for design experts. Finally, an illustrative case study in a renewable energy context was used to demonstrate the feasibility and effectiveness of the methodology. The proposed theoretical model provides more reliable and systematic outcomes than traditional methods when eliciting service requirements and evaluating complex smart industrial service solutions. The study has practical implications by providing useful insights for companies to recognize key smart service requirements in complex industrial contexts and to improve sustainable development.

TOPSIS Method Based on Hesitant Factor and Priority Weighted Operator in Pythagorean Fuzzy Environment

TOPSIS Method Based on Hesitant Factor and Priority Weighted Operator in Pythagorean Fuzzy Environment

Pythagorean Fuzzy Bonferroni Mean with Weighted Interaction Operator and Its Application in Fusion of Online Multidimensional Ratings

Aiming at the aggregation problem of online multi-attribute interactive ratings, a decision-making method based on Pythagorean Fuzzy Bonferroni mean with weighted interaction (PFWIBM) operator is developed by mining the advantages of ratings driven decision-making and the quantitative advantages of Pythagorean fuzzy sets on linguistic ratings. First, the shortcomings of existing operators in criterion interactive representation are analyzed, and the improved Bonferroni mean with weighted interaction and its dual operator are defined, which are used to aggregate the membership information and non-membership information under the Pythagorean fuzzy environment, respectively, and then the PFWIBM operator is constructed. Second, a generation method of interaction coefficients embedded in the PFWIBM operator is designed by combining expert knowledge and user ratings to overcome the limitations of the method of relying on subjective setting parameter values. Third, based on the proposed operator and the transformation relationship between Pythagorean fuzzy sets and linguistic ratings, an online multi-dimensional rating aggregation decision-making approach for solving product raking problems is developed. Finally, the proposed method is applied to a passenger car ranking example to show the feasibility of the method, and through some comparative analysis, the advantages of the proposed operator are clarified, and the influence of operator parameters on the decision-making results is analyzed.

Lifecycle risk assessment on the sustainable development of upgrading energy projects using abandoned mines: An ISM-BWM method

Mining industries have left countless opencast and underground mines in Northwest China without careful disposal for the past decades. The authorities encourage rehabilitation of abandoned mines under the green development strategy, while the path for relevant enterprises remains unclear so far. This paper thus studies the risk distribution of energy alternatives for abandoned mine upgrading. 5 types of upgrading alternatives are studied, including both generation and storage ones. A lifecycle risk system for the alternatives is established for the first time. 12 risk factors extracted through literature and expert involvement are classified as decision-making, implementation, and operation stages. Then, subjective judgements are converted into hesitant fuzzy linguistic term sets under Pythagorean fuzzy environment. And the best worst method combined with the interpretive structure model is firstly introduced to calculate relative importance based on mutual influence. Results indicate that photovoltaic and heat storage stations are generally the safest alternatives, with the lowest risk level ranging from 0.7166 to 0.8073 and hesitant degree from 0.3473 to 0.4248. Moreover, risk response measures for each alternative and policy implications for local governments are first raised to put forward AMUEP development. The analytical framework could also be applied elsewhere for smooth rehabilitation of abandoned mines.

An Extension of Bonferroni Mean under Cubic Pythagorean Fuzzy Environment and Its Applications in Selection-Based Problems

Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can describe both interval-valued Pythagorean fuzzy (IVPF) sets (IVPFSs) and Pythagorean fuzzy (PF) sets (PFSs) simultaneously for addressing information ambiguities. Since an aggregation operator (AO) is a significant mathematical approach in decision-making (DM) problems, this article presents some novel Bonferroni mean (BM) and weighted Bonferroni mean averaging operators between CPF-numbers (CPFNs) for aggregating the different preferences of the decision-makers. Then, using the proposed AOs, we develop a DM approach under the CPF environment and demonstrated it with a numerical example. Furthermore, a comparative study between the proposed and existing methods has been performed to demonstrate the practicality and efficiency of the proposed DM approach.

Assessment of Conductivity-Temperature-Depth via multi-criteria approach: Regret theory based model on the pythagorean fuzzy environment

Sustainable development of marine economy requires strengthening the marine environmental monitoring and management. Conductivity-Temperature-Depth (CTD) realises multi-dimensional monitoring of marine environment by measuring the traits of seawater. To further promote real-time online marine monitoring, conducting a robust assessment of CTD equipment is a primary task for capacity building. In this paper, to evaluate several equipment options, a multiple-criteria decision-making model, based on regret theory and squared-confidence Pythagorean fuzzy hybrid weighted arithmetic and geometric aggregation (SCPFHWAGA) operator is proposed. The SCPFHWAGA operator takes into account the confidence level, which enhances the comprehensiveness of information aggregation. In addition, considering the differences in subjective confidence in equipment use, social network analysis and regret theory are introduced. The evaluation criteria are then designed with the concerns raised before, during and after using the equipment, which reflects the whole process of CTD equipment evaluation more objectively. The case of Zhejiang (China) is analysed. The robustness and validity of the proposed framework are verified by comparing the results with those obtained using other methods.

Modified WASPAS method based on the pythagorean fuzzy frank interaction aggregation operators and its application in cloud computing product selection

Frank operations are more robust and flexible than other algebraic operations, and interaction operational laws consider interrelationship between membership functions in Pythagorean fuzzy number. Combining the strengths of both, we define some Frank interaction operational laws of Pythagorean fuzzy numbers for the first time in this article. Based on this, the Pythagorean fuzzy Frank interaction weighted averaging and geometric operators are developed. Meanwhile, we discuss their basic properties and related special cases. Furthermore, a novel multiple attribute decision-making framework is established based on the modified WASPAS method in Pythagorean fuzzy environment. The proposed method is implemented in a real-case study of cloud computing product selection to test the proposed methodology’s plausibility. A sensitivity analysis is conducted to verify our method’s reliability, and the effectiveness and superiority are illustrated by comparative study.

Interval Pythagorean Fuzzy Decision Based on GWOWA Operator ∗

For the multiattribute group decision-making problem in an interval Pythagorean fuzzy environment, the existing experts and scholars have extended the weighted average (WA), ordered weighted average (OWA), generalized ordered weighted average (GOWA), weighted ordered weighted average (WOWA), and other operators to interval fuzzy environment, while the research on the application and promotion of interval Pythagorean fuzzy with generalized weighted ordered weighted average (GWOWA) operator has not been carried out, GWOWA operator not only retains the advantages of WOWA operator but also introduces artificial variables, which increases the ability of decision-makers to control the aggregation of fuzzy information. Therefore, the GWOWA operator model based on interval Pythagorean fuzzy sets is constructed. First, it is proved that interval Pythagorean fuzzy generalized weighted average operator (IVPFGWA) and interval Pythagorean fuzzy generalized ordered weighted average operator (IVPFGOWA) are special cases of IVPFGWOWA operator, and their idempotence, monotonicity, and boundedness are proved; second, a group decision-making method based on interval Pythagorean fuzzy GWOWA operator is presented. Finally, an example is given to illustrate the effectiveness and scientificity of this method. It is found that the interval Pythagorean fuzzy decision-making method of the GWOWA operator alleviates the loss of information in the decision-making process to a great extent. At the same time, with the increase in the value of artificial variables, the gap between the best scheme and other schemes continues to increase, making the decision-making results more obvious, scientific, and accurate.

Correction: New Extension of Fuzzy-Weighted Zero-Inconsistency and Fuzzy Decision by Opinion Score Method Based on Cubic Pythagorean Fuzzy Environment: A Benchmarking Case Study of Sign Language Recognition Systems

Correction: New Extension of Fuzzy-Weighted Zero-Inconsistency and Fuzzy Decision by Opinion Score Method Based on Cubic Pythagorean Fuzzy Environment: A Benchmarking Case Study of Sign Language Recognition Systems

Interval-valued Pythagorean fuzzy multi-criteria decision-making method based on the set pair analysis theory and Choquet integral

This paper proposes a novel fuzzy multi-criteria decision-making method based on an improved score function of connection numbers and Choquet integral under interval-valued Pythagorean fuzzy environment. To do so, we first introduce a method to convert interval-valued Pythagorean fuzzy numbers into connection numbers based on the set pair analysis theory. Then an improved score function of connection numbers is proposed to make the ranking order of connection numbers more in line with reality in multi-criteria decision-making process. In addition, some properties of the proposed score function of connection numbers and some examples have been given to illustrate the advantages of conversion method proposed in the paper. Then, considering interactions among different criteria, we propose a fuzzy multi-criteria decision-making approach based on set pair analysis and Choquet integral under interval-valued Pythagorean fuzzy environment. Finally, a case of online learning satisfaction survey and a brief comparative analysis with other existing approaches are studied to show that the proposed method is simple,convenient and easy to implement. Comparing with previous studies, the method in this paper, from a new perspective, effectively deals with multi-criteria decision-making problems that the alternatives cannot be reasonably ranked in the decision-making process under interval-valued Pythagorean fuzzy environment.

Selecting a Project Delivery System for Wastewater Treatment Plants With Related-Indicators Under a Pythagorean Fuzzy Environment

Identifying appropriate project delivery systems for wastewater treatment plants (PDSWTPs) plays an important role for wastewater industry decision makers. This study aims to develop a PDSWTP selection model to deal with the related-indicators case by combining the advantages of Pythagorean fuzzy sets and Bonferroni mean operators. The works of this study are as follows: 1) the case with related indicators is innovatively considered as specific to the water industry, and Bonferroni mean operator and Pythagorean fuzzy sets are introduced to PDSWTP selection, which can handle complexity and fuzziness for the actual application. 2) Pythagorean fuzzy weighted Bonferroni mean (PFWBM) and Pythagorean fuzzy weighted geometry Bonferroni mean (PFWGBM) operators are used to aggregate all related indicators in PDSWTP selection, and taking full advantage of PFWBM and PFWGBM operators, a selection framework for PDSWTPs is constructed. 3) To show the robustness, the PDSWTP selection results were given with different parameters in the proposed operators. Finally, a practice example is created, and the results shown are effective and applicable.

Group-based Pythagorean fuzzy soft sets with medical decision-making applications

ABSTRACTThe Pythagorean fuzzy soft set is an instrument to overcome the uncertainty in the data by adding a parametrisation element. Group decision‐making is a continuum in which multiple individuals interact at the same time, resolve problems, appraise the probable existing alternatives, characterised by multiple contradictory criteria, and select an appropriate alternative solution to the problem. In this study, the generalised Pythagorean fuzzy soft set and the group-based generalised Pythagorean fuzzy soft set are defined. A group-based generalised Pythagorean fuzzy soft set is to be used in the evaluation of the object by a group of experts rather than a single expert. According to new definitions, weighted averaging and geometric aggregation operators have been given. To solve the problems in the Pythagorean fuzzy environment, the decision-making process established by considering the new soft sets and the aggregation operators obtained with these sets were presented with an algorithm. A medical example of the choice of the optimal alternative has been designed to indicate the developed decision-making process. Finally, a comparison has been made between the new method and the existing method. It is seen from the results obtained that an expert opinion does not give appropriate results at the desired rate without the generalisation parameter.

A causality analysis of risks to perishable product supply chain networks during the COVID-19 outbreak era: An extended DEMATEL method under Pythagorean fuzzy environment

In nowadays world, firms are encountered with many challenges that can jeopardize business continuity. Recently, the coronavirus has brought some problems for supply chain networks. Remarkably, perishable product supply chain networks, such as pharmaceutical, dairy, blood, and food supply chains deal with more sophisticated situations. Generally, during pandemic outbreaks, the activities of these industries can play an influential role in society. On the one hand, products of these industries are considered to be daily necessities for living. However, on the other hand, there are many new restrictions to control the coronavirus prevalence, such as closing down all official gatherings and lessening the work hours, which subsequently affect the economic growth and gross domestic product. Therefore, risk assessment can be a useful tool to forestall side-effects of the coronavirus outbreaks on supply chain networks. To that aim, the decision-making trial and evaluation laboratory approach is used to evaluate the risks to perishable product supply chain networks during the coronavirus outbreak era. Feedback from academics was received to identify the most important risks. Then, experts in pharmaceutical, food, and dairy industries were inquired to specify the interrelations among risks. Then, Pythagorean fuzzy sets are employed in order to take the uncertainty of the experts’ judgments into account. Finally, analyses demonstrated that the perishability of products, unhealthy working conditions, supply-side risks, and work-hours are highly influential risks that can easily affect other risk factors. Plus, it turned out that competitive risks are the most susceptive risk in the effect category. In other words, competition among perishable product supply chain networks has become even more fierce during the coronavirus outbreak era. The practical outcomes of this study provide a wide range of insights for managers and decision-makers in order to prevent risks to perishable product supply chain networks during the coronavirus outbreak era.

An improved $$A^*$$ search algorithm for the shortest path under interval-valued Pythagorean fuzzy environment

An improved $$A^*$$ search algorithm for the shortest path under interval-valued Pythagorean fuzzy environment

Granulation of protein–protein interaction networks in Pythagorean fuzzy soft environment

Granulation of protein–protein interaction networks in Pythagorean fuzzy soft environment