Interval-valued Pythagorean Fuzzy Research Articles

Applications of Interval-Valued Pythagorean Fuzzy Soft Graph in Bipolar Fuzzy Frame Works

The main contribution of this paper is to get extended ideas of various interval-valued bipolar Pythagorean fuzzy soft graphs (I-VBPFSGs) and to extend the concept of the Pythagorean fuzzy soft graph to the bipolar frameworks. Finally, in order to demonstrate, how to calculate an interval-valued Pythagorean bipolar fuzzy soft graph for a specific application, a numerical example using city data from the Yunnan province is presented.

An Intelligent MCGDM Model in Green Suppliers Selection Using Interactional Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Sets

An Intelligent MCGDM Model in Green Suppliers Selection Using Interactional Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Sets

A Novel Linguistic Interval-Valued Pythagorean Fuzzy Multi-Attribute Group Decision-Making for Sustainable Building Materials Selection

The linguistic interval-valued Pythagorean fuzzy (LIVPF) sets, which absorb the advantages of linguistic terms set and interval-valued Pythagorean fuzzy sets, can efficiently describe decision makers’ evaluation information in multi-attribute group decision-making (MAGDM) problems. When investigating aggregation operators of linguistic interval-valued Pythagorean fuzzy (LIVPF) information, we have to consider two important issues, viz. the operational rules of LIVPF numbers and aggregation functions. The classical Archimedean t-norm and t-conorm (ATT) are a famous t-norm and t-conorm, which can produce some special cases. Recently, ATT has been widely applied in different fuzzy decision-making information. Hence, in this paper, for the first issue, we propose some novel operational rules of LIVPF numbers based on ATT. The new operational laws are flexible and can generate some useful operations. For the second issue, we choose a powerful function, i.e., the extended power average (EPA) operator as the aggregation function. The prominent advantages of EPA are that it not only considers the relationship among input arguments, but also dynamically changes the weights of input arguments by employing a parameter. Hence, our proposed novel aggregation operators for LIVPFNs are flexible and is suitable to handle MAGDM problems in actual life. Afterward, we further present a novel MAGDM method under LIVPF conditions. The main finding of our study is a new MAGDM method, which is more powerful and flexible than existing ones. Finally, we apply the method in a sustainable building materials selection to show its effectiveness. Additionally, comparison analysis is provided to demonstrate the advantages and superiorities of the proposed method.

Yager aggregation operators based on complex interval-valued q-rung orthopair fuzzy information and their application in decision making

As a more massive feasible and prominent tool than the complex interval-valued Pythagorean fuzzy (CIVPF) set and complex interval-valued intuitionistic fuzzy (CIVIF) set, the complex interval-valued q-rung orthopair fuzzy (CIVQROF) set has been usually used to represent ambiguity and vagueness for real-life decision-making problems. In this paper, we firstly proposed some distance measures, Yager operational laws, and their comparison method. Further, we developed CIVQROF Yager weighted averaging (CIVQROFYWA), CIVQROF Yager ordered weighted averaging (CIVQROFYOWA), CIVQROF Yager weighted geometric (CIVQROFYWG), CIVQROF Yager ordered weighted geometric (CIVQROFYOWG) operators with CIVQROF information, and some certain well-known and feasible properties and outstanding results are explored in detail. Moreover, we proposed a new and valuable technique for handling multi-attribute decision-making problems with CIVQROF information. Lastly, a practical evaluation regarding the high blood pressure diseases of the patient is evaluated to illustrate the feasibility and worth of the proposed approaches.

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.

An Interval Valued Pythagorean Fuzzy AHP–TOPSIS Integrated Model for Ergonomic Assessment of Setup Process under SMED

Single-minute exchange of dies (SMED) is one of the most significant lean tools which reduces the required time for setup operations. The consideration of appropriate ergonomic interferences during the SMED implementation improves working conditions, productivity, and flexibility. The SMED literature has an opportunity for expansion with ergonomic assessment. Moreover, ergonomic assessment studies have various limitations in terms of a broad hierarchy of ergonomic factors and their weighting. Therefore, this study enriches the literature by providing the ergonomic assessment of the setup process through integrating multi-criteria decision-making (MCDM) into SMED. In this context, a wide hierarchy of ergonomic risk factors is compiled for ergonomic assessment. The Interval Valued Pythagorean Fuzzy Analytic Hierarchy Process (IVPF-AHP) method is used to determine the weights of factors. Then, Fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is used to prioritize setup tasks in terms of ergonomic factors by using the Interval Valued Pythagorean Fuzzy Sets (IVPFSs). The proposed model was validated by applying it to a real-life manufacturing system in the white goods supplier industry. The application results justify the proposed model with a setup time reduction of 58% and an ergonomic improvement of 19%.

Interval-Valued Pythagorean Fuzzy Similarity Measure-Based Complex Proportional Assessment Method for Waste-to-Energy Technology Selection

This study introduces an integrated decision-making methodology to choose the best “waste-to-energy (WTE)” technology for “municipal solid waste (MSW)” treatment under the “interval-valued Pythagorean fuzzy sets (IPFSs)”. In this line, first, a new similarity measure is developed for IPFSs. To show the utility of the developed similarity measure, a comparison is presented with some extant similarity measures. Next, a weighting procedure based on the presented similarity measures is proposed to obtain the criteria weight. Second, an integrated approach called the “interval-valued Pythagorean fuzzy-complex proportional assessment (IPF-COPRAS)” is introduced using the similarity measure, linear programming model and the “complex proportional assessment (COPRAS)” method. Furthermore, a case study of WTE technologies selection for MSW treatment is taken to illustrate the applicability and usefulness of the presented IPF-COPRAS method. The comparative study is made to show the strength and stability of the presented methodology. Based on the results, the most important criteria are “greenhouse gas (GHG)” emissions (P3), microbial inactivation efficacy (P7), air emissions avoidance (P9) and public acceptance (P10) with the weight/significance degrees of 0.200, 0.100, 0.100 and 0.100, respectively. The evaluation results show that the most appropriate WTE technology for MSW treatment is plasma arc gasification (H4) with a maximum utility degree of 0.717 followed by anaerobic digestion (H7) with a utility degree of 0.656 over various considered criteria, which will assist with reducing the amount of waste and GHG emissions and also minimize and maintain the costs of landfills.

Green Recycling Supplier Selection of Shared Bicycles: Interval-Valued Pythagorean Fuzzy Hybrid Weighted Methods Based on Self-Confidence Level.

In the face of practical problems such as the increasing demand for shared bicycles and the number of faulty vehicles which are hard to handle and repair in time, shared bicycles operators tend to outsource recycling services to suppliers. To solve the problem of recycling supplier selection, this paper constructs a novel evaluation index system involving the three traditional dimensions and introduces an interval-valued Pythagorean fuzzy (IVPF) hybrid weighted decision-making model based on the self-confidence level. Subsequently, the self-confidence IVPF hybrid weighted average geometric operator and self-confidence IVPF ordered hybrid weighted average geometric operator are proposed by integrating the self-confidence level of experts, the superiority of the weighted and geometric average rules. The significant merit of the developed operators is that they can incorporate the self-confidence level of the expert as well as effectively combine the characteristics of the weighted and geometric average mechanism. A multi-attribute decision-making (MADM) framework is then constructed by using the proposed aggregation approach. Finally, on the basis of the established evaluation index system, a case concerning the green recycling supplier selection of shared bicycles is applied to display the superiority and practicability of the presented method.

Aggregation Operators for Interval-Valued Pythagorean Fuzzy So Set with Their Application to Solve Multi-Attribute Group Decision Making Problem

Interval-valued Pythagorean fuzzy so set (IVPFSS) is a generalization of the interval-valued intuitionistic fuzzy so set (IVIFSS) and interval-valued Pythagorean fuzzy set (IVPFS). The IVPFSS handled more uncertainty comparative to IVIFSS; it is the most signicant technique for explaining fuzzy information in the decision-making process. In this work, some novel operational laws for IVPFSS have been proposed. Based on presented operational laws, two innovative aggregation operators (AOs) have been developed such as interval-valued Pythagorean fuzzy so weighted average (IVPFSWA) and interval-valued Pythagorean fuzzy so weighted geometric (IVPFSWG) operators with their fundamental properties. A multi-attribute group decision-making (MAGDM) approach has been established utilizing our developed operators. A numerical example has been presented to ensure the validity of the proposed MAGDM technique. Finally, comparative studies have been given between the proposed approach and some existing studies. The obtained results through comparative studies show that the proposed technique is more credible and reliable than existing approaches.

BIM Platform Construction Method Selection with the Interval-Valued Pythagorean Fuzzy Set Group Decision-Making Method

BIM Platform Construction Method Selection with the Interval-Valued Pythagorean Fuzzy Set Group Decision-Making Method

A new divergence measure of interval-valued pythagorean fuzzy sets and its application

As the extension of the Fuzzy sets (FSs) theory, the Interval-valued Pythagorean Fuzzy Sets (IVPFS) was introduced which play an important role in handling the uncertainty. The Pythagorean fuzzy sets (PFSs) proposed by Yager in 2013 can deal with more uncertain situations than intuitionistic fuzzy sets because of its larger range of describing the membership grades. How to measure the distance of Interval-valued Pythagorean fuzzy sets is still an open issue. Jensen–Shannon divergence is a useful distance measure in the probability distribution space. In order to efficiently deal with uncertainty in practical applications, this paper proposes a new divergence measure of Interval-valued Pythagorean fuzzy sets,which is based on the belief function in Dempster–Shafer evidence theory, and is called IVPFSDM distance. It describes the Interval-Valued Pythagorean fuzzy sets in the form of basic probability assignments (BPAs) and calculates the divergence of BPAs to get the divergence of IVPFSs, which is the step in establishing a link between the IVPFSs and BPAs. Since the proposed method combines the characters of belief function and divergence, it has a more powerful resolution than other existing methods.

Extension of interval-valued Pythagorean FDOSM for evaluating and benchmarking real-time SLRSs based on multidimensional criteria of hand gesture recognition and sensor glove perspectives

Several DataGlove wearable electronic devices based on real-time Sign Language Recognition Systems (SLRSs) have been developed recently to assist the deaf and dumb community in translating hand gestures to their spoken language equivalents. Multidimensional evaluation and benchmarking of these systems are critical for determining the most desirable approach for meeting all essential requirements. However, this process is considered a Multicriteria Decision-Making (MCDM) problem due to the presence of several issues, including multiple evaluation criteria, criteria importance and criteria confliction. Hence, the MCDM approach is required to solve these issues. In this study, a new extension of the Fuzzy Decision by Opinion Score Method (FDOSM) for evaluating and benchmarking SLRSs is developed under an Interval-Valued Pythagorean Fuzzy Set (IVPFS) named IVP-FDOSM. Fundamentally, the methodology includes two phases. In the first phase, a decision matrix is formulated on the basis of identified ‘multidimensional criteria of hand gesture recognition and sensor glove perspectives’ and ‘real-time SLRSs’. The second phase introduces the development of IVP-FDOSM in two stages. The decision matrix is transformed into an opinion matrix in the first stage (data transformation unit). Meanwhile, in the second stage (data-processing unit), the opinion matrix is converted into fuzzy opinion decision matrices with the assistance of three experts by transforming the opinion matrix’s linguistic terms to Interval-Value Pythagorean Fuzzy Numbers (IVPFNs). Results indicate the following: (1) individual benchmarking results of real-time SLRS showed high variation (90%) based on the preference of each Decision Maker (DM), with only 10% of preferences being identical. (2) The results of group benchmarking reveal that the 10th real-time SLRS was the optimal one and the 6th was the worst. In addition, the rates of ranking match between the group benchmarking and each DM (first DM, second DM and third DM) were 23%, 37% and 43%, respectively. (3) For the results evaluation, the statistical test-based objective assessment shows that the first group received the lowest mean value (2.80333), while the third group received the highest mean (3.8). These values indicate that the group benchmarked systems resulting from IVP-FDOSM are undergoing a systematic ranking. Furthermore, the comparative analysis reveals that IVP-FDOSM is superior to IVP-TOPSIS and IVP-AHP in terms of ranking and weighting.

Developing a new framework for conceptualizing the emerging sustainable community-based tourism using an extended interval-valued Pythagorean fuzzy SWARA-MULTIMOORA

During these years, the sustainable Community-Based Tourism (CBT) notion has received more attention in the context of the tourism industry. Sustainable CBT mainly emphasizes social, environmental, and cultural sustainability and gives power to local communities in any aspect of tourism management. This manuscript aims to propose a novel approach to examine the current status of sustainable CBT in the Indian Himalayan region context. In this regard, a comprehensive framework was developed using experts’ opinions, and the relevant literature in tourism studies was reviewed based on sustainable CBT. To this end, this study proposed an integrated decision-making method using Step-wise Weight Assessment Ratio Analysis (SWARA) and MULTIMOORA (Multiple Objective Optimization based on the Ratio Analysis plus Full Multiplicative Form) under Interval-Valued Pythagorean Fuzzy Sets (IVPFSs). Additionally, some comparisons were discussed with the outcomes obtained from the developed approach and those of extant ones to evaluate the efficiency of the proposed model. To confirm the applicability of the proposed IVPF-SWARA-MULTIMOORA approach to real-world decision-making problems, a sustainable CBT problem was considered as a case study. The final results confirmed the efficiency of the proposed approach as well as its consistency with the existing ones.

Einstein Geometric Aggregation Operators using a Novel Complex Interval-valued Pythagorean Fuzzy Setting with Application in Green Supplier Chain Management

The principle of a complex interval-valued Pythagorean fuzzy set (CIVPFS) is a valuable procedure to manage inconsistent and awkward information genuine life troubles. The principle of CIVPFS is a mixture of the two separated theories such as complex fuzzy set and interval-valued Pythagorean fuzzy set which covers the truth grade (TG) and falsity grade (FG) in the form of the complex number whose real and unreal parts are the sub-interval of the unit interval. The superiority of the CIVPFS is that the sum of the square of the upper grade of the real part (also for an unreal part) of the duplet is restricted to the unit interval. The goal of this article is to explore the new principle of CIVPFS and its algebraic operational laws. By using the CIVPFSs, certain Einstein operational laws by using the t-norm and t-conorm are also developed. Additionally, we explore the complex interval-valued Pythagorean fuzzy Einstein weighted geometric (CIVPFEWG), complex interval-valued Pythagorean fuzzy Einstein ordered weighted geometric (CIVPFEOWG) operators and utilized their special cases. Moreover, a multicriteria decision-making (MCDM) technique is explored based on the elaborated operators by using the complex interval-valued Pythagorean fuzzy (CIVPF) information. To determine the consistency and reliability of the elaborated operators, we illustrated certain examples by using the explored principles. Finally, to determine the supremacy and dominance of the explored theories, the comparative analysis and graphical expressions of the developed principles are also discussed.

Some novel interval-valued Pythagorean fuzzy aggregation operator based on Hamacher triangular norms and their application in MADM issues

The motivation of this paper is to build up a new technique to manage multi-attribute decision-making (MADM) issues with interval-valued Pythagorean fuzzy (IVPF) set according to the concepts of Hamacher t-norm and t-conorm, and give its application in MADM problems. To begin with, we integrate the Hamacher operations on IVPF sets. Then, in view of these operations, we formulate a few IVPF aggregation operators, for example, IVPF Hamacher weighted average operator, IVPF Hamacher order weighted average operator, IVPF Hamacher hybrid average operator, IVPF Hamacher weighted geometric operator, IVPF Hamacher order weighted geometric operator, and IVPF Hamacher hybrid geometric operator. We investigate the distinguished features of these suggested operators. Moreover, we carry out the proposed aggregation operators to produce a method for solving MADM issues under IVPF data. We present an example of the emerging software system selection to elaborate on its practicality and effectiveness. We investigate the impact of the parameter for different values on decision-making outcomes. The main advantage of utilizing the suggested operator lies in the fact that this operator provides an increasingly complete perspective on the issue to the decision-makers. The technique suggested in this study gives progressively broad, improve the accuracy and exact outcomes when contrasted with the existing related methods. Therefore, this technique performs an important function in real-life MADM issues.

TODIM Method for Interval-Valued Pythagorean Fuzzy MAGDM Based on Cumulative Prospect Theory and Its Application to Green Supplier Selection

The green supplier selection could be regarded as a classical multiple attribute group decision making (MAGDM) issue. The interval-valued Pythagorean fuzzy set (IVPFSs) can fully describe the uncertain information for the risk assessment of science and technology projects. Furthermore, the classical TODIM method based on the cumulative prospect theory (CPT-TODIM) is built, which is a selectable method in reflecting the decision makers’ psychological behavior. Thus, in this paper, the interval-valued Pythagorean fuzzy CPT-TODIM (IVPF-CPT-TODIM) method is proposed for MAGDM issue. At the same time, it is enhancing rationality to get the weight information of attributes by using the entropy weight method under IVPFSs. And focusing on hot issues in contemporary society, this article applies the discussed method to green supplier selection, and demonstrates risk assessment model of science and technology projects based on the IVPF-CPT-TODIM method. Finally, through comparing the outcome of comparative analysis, we conclude that this improved approach is acceptable.

Socio-economic evaluation model for sustainable solar PV panels using a novel integrated MCDM methodology: A case in Turkey

Due to the increasing awareness of environmental, social and economic factors, solar photovoltaic (PV) system planning requires strategic decision making process for socio-economic development in many countries. The main objective of this paper is to propose a new Multi-Criteria Decision Making (MCDM) approach that is flexible and practical to the decision makers (DMs) in governments for solar PV panel manufacturer evaluation based on qualitative and quantitative factors. Accordingly, a novel two-stage MCDM model integrating Analytic Hierarchy Process (AHP) and Multiplicative Multi-Objective Ratio Analysis (MULTIMOORA) methods under Interval Valued Pythagorean Fuzzy (IVPF) environment is presented, and applied to select the most appropriate solar PV panel manufacturer for solar power plants in Southeastern Anatolia Region of Turkey. As a result, the proposed novel Integrated IVPF-AHP&MULTIMOORA method produces consistent and reasonable results to select the most appropriate solar PV panel manufacturer for the solar power plants in the cities of Southeastern Anatolia Region of Turkey considering some socio-economic sustainable indicators such as cost, environmental, efficiency and technical indices. Furthermore, sensitivity analysis and comparative analysis are also applied to prove the robustness and verification of the results of the proposed approach.

A product ranking method combining the features–opinion pairs mining and interval-valued Pythagorean fuzzy sets

Mining online reviews has become an important means of identifying consumer behavior and the innovation direction of products. However, it is difficult for both producers and consumers to effectively analyze and extract relevant opinions from a vast number of online reviews. To overcome this problem, a product ranking method that combines feature–opinion pairs mining and interval-valued Pythagorean fuzzy (IVPF) sets was proposed in this study. First, three types of important feature–opinion pairs were clearly defined based on the diversity and complexity of opinion expression forms in Chinese ecommerce reviews. Two deep learning models were then designed to automatically extract the feature–opinion terms and match them into pairs. Afterwards, sentiment analysis techniques were applied to identify sentiment orientation, and the feature–opinion pairs were clustered into groups using K-means clustering algorithm. Meanwhile, considering the confidence level based on the number of online reviews on different products, sentiment value was transformed into interval-value from, including interval membership and non-membership. As the sum of the converted interval membership and non-membership was greater than 1 and their quadratic sum was less than 1, IVPF set was introduced to represent the interval-valued sentiment. Furthermore, based on the interrelationship between product attributes, we proposed an IVPF weighted Heronian mean operator to aggregate the attribute information. Product ranking was then achieved based on the operator and operations under the IVPF information. Finally, a case study was used to verify the feasibility of the proposed method, and comparisons and sensitivity analysis were performed to demonstrate the superiority of our method.

Linguistic Interval-Valued Pythagorean Fuzzy Sets and Their Application to Multiple Attribute Group Decision-making Process

The paper’s aims are to present a novel concept of linguistic interval-valued Pythagorean fuzzy set (LIVPFS) or called a linguistic interval-valued intuitionistic type-2 fuzzy set, which is a robust and trustworthy tool, and to accomplish the imprecise information while solving the decision-making problems. The presented LIVPFS is a generalization of the linguistic Pythagorean fuzzy set, by characterizing the membership and non-membership degrees as the interval-valued linguistic terms to represent the uncertain information. To explore the study, we firstly define some basic operational rules, score and accuracy functions, and the ordering relations of LIVPFS with a brief study of the desirable properties. Based on the stated operational laws, we proposed several weighted averages and geometric aggregating operators to aggregate the linguistic interval-valued Pythagorean fuzzy information. The fundamental inequalities between the proposed operators and their properties are discussed in detail. Finally, a multiple attribute group decision-making (MAGDM) algorithm is promoted to solve the group decision-making problems with uncertain information using linguistic features and the proposed operators. The fundamental inequalities between the proposed operators and their properties are discussed in detail. Also, the illustration of the stated algorithm is given through several numerical examples and compared their performance with the results of the existing algorithms. Based on the stated MAGDM algorithm and the suitable operators, the decision-makers’ can be selected their best alternatives with their own attitude character towards optimism or pessimism choice. The presented LIVPFS is an extension of the several existing sets and is more generalized to utilize the uncertain and imprecise information with a wider range of information. Based on the presented aggregation operators, a decision-maker can select the desired one as per their choices to access the finest alternatives.

A novel Pythagorean fuzzy AHP and fuzzy TOPSIS methodology for green supplier selection in the Industry 4.0 era

Advances in information and communication technology have created innovator technologies such as cloud computing, Internet of Things, big data analysis and artificial intelligence. These technologies have penetrated production systems and converted them smart. However, this transformation did not only affect production systems, but also differentiated supplier selection processes. In the supplier selection process, the usage of new technologies along with traditional and green criteria extensively has been investigated in recent years. This paper aims to develop a new group decision-making approach based on Industry 4.0 components for selecting the best green supplier by integrating AHP and TOPSIS methods under the Pythagorean fuzzy environment. In the proposed approach, judgments of different experts are expressed by linguistic terms based on Pythagorean fuzzy numbers. The interval-valued Pythagorean Fuzzy AHP method is utilized to determine the criteria weights. The Pythagorean Fuzzy TOPSIS method based on the distances of suppliers is applied to obtain the ranking of the suppliers and determine the most suitable one. Finally, a real case study on an agricultural tools and machinery company is presented to indicate the effectiveness and accuracy of the proposed selection approach.