Interval-valued Pythagorean Fuzzy Numbers Research Articles

Interval-valued Pythagorean fuzzy entropy and its application to multi-criterion group decision-making

<abstract><p>Entropy is an important tool of information measurement in the fuzzy set and its inference. The research on information measurement based on interval-valued Pythagorean fuzzy sets mostly involves the distance formula for interval-valued Pythagorean fuzzy numbers, but seldom involves the measurement of fuzziness. In view of this situation, we have aimed to propose new interval-valued Pythagorean fuzzy entropy and weighted exponential entropy schemes. Based on the interval-valued Pythagorean fuzzy weighted averaging operator, a strategy based on weighted exponential entropy is proposed to solve the multi-criteria group decision-making (MCGDM) problem in the interval-valued Pythagorean environment. Two examples illustrate that this paper provides a feasible new method to solve the MCGDM problem in an interval-valued Pythagorean fuzzy (IVPF) environment. Finally, by comparing with the existing methods, it is concluded that the entropy measure of IVPF schemes and the corresponding MCGDM method can select the optimal solution of the practical problem more precisely and accurately. Therefore, the comparative analysis shows that the proposed measurement method has the characteristics of flexibility and universality.</p></abstract>

Interval-valued Pythagorean fuzzy operational competitiveness rating model for assessing the metaverse integration options of sharing economy in transportation sector

The rise of the sharing economy (SE) has enabled new forms of urban mobility and had a substantial impact on the transportation sector. The applications of SE transportation lessen vehicle ownership and single-occupancy vehicles, which contributes to the sustainability of a region. Metaverse is a digital ecosystem that integrates SE applications with transportation sector and improves the sustainability of SE applications. The aim of this work is to assess and prioritize the Metaverse integration options of sharing economy from multiple criteria and uncertainty perspectives. In this regard, the proposed study develops a four stage decision-making method on interval-valued Pythagorean fuzzy (IVPF) setting. First, new score function is proposed to compare the interval-valued Pythagorean fuzzy numbers (IVPFNs) and further used to derive the weights of decision experts in the proposed method. Second, some IVPF interaction aggregation operators are proposed to aggregate the individual’s information in group decision-making. Third, an integrated weighting process is presented to obtain the objective weights of criteria using cross entropy-based formula and subjective weights by relative closeness coefficient-based model with IVPF information. For this purpose, a cross entropy is introduced for interval-valued Pythagorean fuzzy sets. Fourth, a hybrid operational competitiveness ratings assessment (OCRA) model is developed to rank the alternatives with IVPF information. Moreover, a case study of Metaverse integration options of sharing economy in transportation sector is taken to express the practicability and effectiveness of the presented method. Sensitivity analysis with respect to various values of parameter is discussed to reveal the stability of outcomes. It is also compared with some extant models to validate the superiority of developed method.

An Improved TOPSIS Method Based on a New Distance Measure and Its Application to the House Selection Problem

It is difficult to choose an appropriate house for homebuyers. This is due to the difficulty of evaluating the multitude of factors, such as price, location, size, and so on. In order to help homebuyers in choosing an appropriate house, a method integrating the interval-valued Pythagorean FAHP and FTOPSIS is proposed. In the proposed approach, the evaluation criteria were determined by the experts, and the linguistic variables of interval-valued Pythagorean fuzzy numbers were used in the evaluations of the homebuyers and experts. A new distance between two IVPFNs is proposed. The weights of the evaluation criteria were determined by the interval-valued Pythagorean FAHP method by the homebuyers, and house selections were evaluated by interval-valued Pythagorean FTOPSIS method taking into account the new distance. Finally, a case study was executed to verify the feasibility of the proposed approach. The case study results reveal that the weights of criteria obtained by FAHP are not the same according to opinions of the different homebuyers.

A novel approach based on similarity measure for the multiple attribute group decision-making problem in selecting a sustainable cryptocurrency

Environmental impact and sustainability challenges in the cryptocurrencies has become increasingly examined in the literature. However, studies of the multiple attribute group decision making (MAGDM) method for major selection of cryptocurrencies in advancing sustainability are still at an early stage. In particular, research on the fuzzy-MAGDM method in the evaluation of sustainability in cryptocurrencies is scarce. This paper adds contributions by developing a novel MAGDM approach to evaluate the sustainability development of major cryptocurrencies. It proposes a similarity measure for interval-valued Pythagorean fuzzy numbers (IVPFNs) based on whitenisation weight function and membership function in grey systems theory for IVPFNs. It further developed a novel generalised interval-valued Pythagorean fuzzy weighted grey similarity (GIPFWGS) measure approach to provide a more rigorous evaluation in complex decision marking problem with embedding ideal solution and membership degree. It also conducts a sustainability evaluation model of major cryptocurrencies as a numerical application and performs a robustness assessment with different variations of the expert’s weight to test how different values of parameter θ can affect the ranking results of alternatives. The results suggest that Stellar is the most sustainable cryptocurrency, while Bitcoin with its intensive energy consumption, high mining cost and high computing power provides the least effective support for its sustainable development. A comparative analysis with the average value method and Euclidean distance method was performed to validate the reliability of the proposed decision-making model and provides evidence that the GIPFWGS has better fault tolerance.

Membership Score of an Interval-Valued Pythagorean Fuzzy Numbers and Its Applications

The main aim of the paper is to define a new Membership score function on the class of interval-valued Pythagorean fuzzy numbers which can overcome the drawbacks of existing familiar ranking methods. In this paper, firstly, we show the limitations of various ranking functions in ordering/ comparing any two arbitrary interval-valued Pythagorean fuzzy numbers in detail. Secondly, we define a new Membership score function on the class of interval-valued Pythagorean fuzzy numbers and study their properties. Then we compare our proposed method with many other different existing methods for showing the efficacy of the proposed method. Finally, we show the applicability of the proposed Membership score function in solving interval-valued Pythagorean fuzzy multi-criteria decision-making problems using a numerical example.

Pythagorean Fuzzy Based AHP-VIKOR Integration to Assess Rail Transportation Systems in Turkey

Rail transit systems, one of the important public transportation (PT) systems, are widespread in large cities to satisfy passengers or commuters for their daily trips. However, rail transportation (RT) investments are costly, and the budgets are minimal. Thus assessing RT systems in big cities play a crucial role in deciding on the best RT investment. In this study, for the first time, a two-stage fuzzy set is used among the multi-criteria decision-making (MCDM) studies in the area of transportation. The proposed method is a unique technique integrating the Analytic Hierarchy Process (AHP) and Vise Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) techniques with an extension of Pythagorean fuzzy sets. In this manner, datasets from the observations of three rail lines in Antalya, one of the largest cities in Turkey, are applied to the proposed method. Fuzzy Analytic Hierarchy Process (FAHP), a commonly used MCDM method, is used to weighting four main criteria and thirteen sub-factors. AHP is strengthened by interval-valued Pythagorean fuzzy numbers (IVPFNs). The Fuzzy VIKOR (FVIKOR) approach is then applied for the prioritization of three rail line alternatives. Results are achieved by analyzing not only real RT service observations, also actual RT expenditures or costs. Consequently, the uncertainty in results is thus minimized. Outcomes of the case study reveal the most serviceable RT line in the city evaluating the main criteria, namely economy, comfort, environment, and safety. Therefore, the proposed approach methodologically contributes to adopting a new insight into the integration of MCDM methods with fuzzy set.

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.

Pandemic hospital site selection: a GIS-based MCDM approach employing Pythagorean fuzzy sets.

COVID-19 poses many challenges for hospitals around the world. Each country attempts to solve the problems in its hospitals using different methods. In Turkey, two pandemic hospitals were built in İstanbul, the most crowded province. In addition, some hospitals were designated as pandemic hospitals. This study focuses on the methods used for site selection for a pandemic hospital in Atakum, a district of Samsun City, Turkey. As a solution to the problem, initially, spatial analysis was performed using GIS to produce maps based on seven criteria obtained from the insight of an expert team. Analytic hierarchy process (AHP) augmented by interval-valued Pythagorean fuzzy numbers (PFNs) was then used to determine weights for the criteria. Distance to transportation network was the most important criterion influencing the selection process and the least significant one was the distance to fire stations. Based on the criteria weights, and five rules specified by the expert team, 13 suitable locations for a pandemic hospital were determined using GIS. The technique for order preference by similarity to ideal solution (TOPSIS) method was used to determine the final ranking of 13 alternative locations (A1–A13). A10 was identified as the most appropriate site and A11 as the least appropriate site for a pandemic hospital. Finally, sensitivity analysis was performed to investigate how changes in weight values of the criteria affect the ranking of the alternatives.

Mehar Approach for Finding Shortest Path in Supply Chain Network

The aim of each company/industry is to provide a final product to customers at the minimum possible cost, as well as to protect the environment from degradation. Ensuring the shortest travel distance between involved locations plays an important role in achieving the company’s/industry’s objective as (i) the cost of a final product can be minimized by minimizing the total distance travelled (ii) finding the shortest distance between involved locations will require less fuel than the longest distance between involved locations. This will eventually result in lesser degradation of the environment. Hence, in the last few years, various algorithms have been proposed to solve different types of shortest path problems. A recently proposed algorithm for solving interval-valued Pythagorean fuzzy shortest path problems requires excessive computational efforts. Hence, to reduce the computational efforts, in this paper, firstly, an alternative lexicographic method is proposed for comparing interval-valued Pythagorean fuzzy numbers. Then, using the proposed lexicographic comparing method, a new approach (named as Mehar approach) is proposed to solve interval-valued Pythagorean fuzzy shortest path problems. Furthermore, the superiority of the proposed lexicographic comparing method, as well as the proposed Mehar approach, is discussed.

A chance-constraint programming model with interval-valued pythagorean fuzzy constraints

Pythagorean fuzzy sets and interval-valued Pythagorean fuzzy sets are more proficient in handling uncertain and imprecise information than intuitionistic fuzzy sets and fuzzy sets. In this article, we put forward a chance-constraint programming method to solve linear programming network problems with interval-valued Pythagorean fuzzy constraints. This practice is developed using score function and upper and lower membership functions of interval-valued Pythagorean fuzzy numbers. The feasibility of the anticipated approach is illustrated by solving an airway network application and shown to be used to solve different types of network problems with objective function having interval-valued Pythagorean fuzzy numbers by employing it on shortest path problem and minimum spanning tree problem. Furthermore, a comparative examination was performed to validate the effectiveness and usefulness of the projected methodology.

Novel approach to multi-attribute group decision-making based on interval-valued Pythagorean fuzzy power Maclaurin symmetric mean operator

To distinguish any two interval-valued Pythagorean fuzzy numbers (IVPFNs), two new comparison functions, namely, the membership degree uncertainty (MU) function and the hesitation degree uncertainty (HU) function are proposed in this paper. By combining the existing score and accuracy functions with the proposed MU and HU functions, a new comparison rule for IVPFNs is obtained, whereby two different IVPFNs may be distinguished. Then, the interval-valued intuitionistic Pythagorean fuzzy power Maclaurin symmetric mean aggregation (IVPFPMSM) operator and the weighted IVIPFMSM aggregation operator are introduced to address complex multi-attribute group decision-making (MAGDM) problems involving unreasonable evaluation values and interaction among the input arguments. Moreover, a series of properties of the proposed operators are studied. Further, based on the proposed comparison method for IVPFNs and the weighted IVPFPMSM operator, we develop a new method for interval-valued Pythagorean fuzzy MAGDM problems. Finally, two illustrative examples and a comparative analysis are provided to demonstrate the effectiveness and superiority of the proposed method. Specifically, the advantage of the proposed operators in MAGDM problems is that they can eliminate bad influences of extreme evaluation values from biased decision makers and capture the interaction between attributes.

Some induced generalized geometric aggregation operators based on interval-valued Pythagorean fuzzy numbers

Induced aggregation operators are more suitable for aggregating the individual preference relations into a collective fuzzy preference relation. Therefore, in this paper, we introduce the notion of some new types induced aggregation operators, namely induced interval-valued Pythagorean fuzzy ordered weighted geometric aggregation operator, induced interval-valued Pythagorean fuzzy hybrid geometric aggregation operator, induced generalized interval-valued Pythagorean fuzzy ordered weighted geometric aggregation operator and induced generalized interval-valued Pythagorean fuzzy hybrid geometric aggregation operator. Furthermore, these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean fuzzy environment to show the validity, practicality and effectiveness of the new approach. We also study the applicability in a decision-making problem concerning strategic selection of the best information system and give an illustrative example to show the effectiveness of the developed methods and operators.

Approaches to Some Induced Einstein Geometric Aggregation Operators Based on Interval-Valued Pythagorean Fuzzy Numbers and Their Application

Interval-valued Pythagorean fuzzy set is one of the successful extensions of the interval-valued intuitionistic fuzzy set for handling the uncertainties in the data. Under this environment, in this paper we introduce the notion of induced interval-valued Pythagorean fuzzy Einstein ordered weighted geometric (I-IVPFEOWG) operator and induced interval-valued Pythagorean fuzzy Einstein hybrid geometric (I-IVPFEHG) operator along with their some basic properties such as, idempotency, boundedness, commutatively, monotonicity. The main advantage of the induced aggregation operators is that, these operators are more suitable to aggregate the individual preference relations into a collective preference relation. Therefore these methods play a vigorous role in daily life problems. Furthermore, a method for multi-attribute group decision-making (MAGDM) problems based on these operators was developed, and the operational procedures were explained in detail.

Decision Support Methodology Based on Covering-Based Interval-Valued Pythagorean Fuzzy Rough Set Model and Its Application to Hospital Open-Source EHRs System Selection

The target of current work is to propose a new approach to deal with multiattribute decision-making (MADM) problems with interval-valued Pythagorean fuzzy set (IVPFS) based on the concepts of covering-based rough set (CRS) and TOPSIS and give its application in MADM problems. To begin with, we integrate the fuzzy rough set (FRS), IVPFS and CRS and define the covering-based interval-valued Pythagorean fuzzy rough set (CIVPFRS). Firstly, the relative notions of the CIVPFRS model are introduced. In addition, the distance measure of interval-valued Pythagorean fuzzy numbers (IVPFNs) is defined; based on the proposed distance, the rough and precision degrees of CIVPFRS are discussed. Thirdly, on the basis of the theoretical analysis for CIVPFRS models, an interval-valued Pythagorean fuzzy TOPSIS method is designed to deal with the MADM problems with interval-valued Pythagorean fuzzy information (IVPFI). Last of all, the validity and merits of the proposed approach are illustrated by an example, and the sensitivity analysis of the parameters and the comparison with the existing related methods are carried out.β

A Novel Multiple Attribute Group Decision-Making Approach Based on Interval-Valued Pythagorean Fuzzy Linguistic Sets

This article investigates multi-attribute group decision-making (MAGDM) problems based on interval-valued Pythagorean fuzzy linguistic sets (IVPFLSs). The IVPFLSs are regarded as an efficient tool to describe decision makers’ (DMs’) evaluation information from both quantitative and qualitative aspects. However, existing IVPFLSs based MAGDM methods are still insufficient and inadequate to deal with complicated practical situations. This article aims to propose a novel MAGDM method and the main contributions of the present work are three-fold. First, we propose new operations of interval-valued Pythagorean fuzzy linguistic numbers (IVPFLNs) based on linguistic scale function. Second, we propose new aggregation operators (AOs) of IVPFLNs based on power average operator and Muirhead mean. The proposed AOs take the interrelationship among any numbers of attributes into account and eliminate the bad influence of DMs’ unreasonable evaluation values on the final decision results. Third, based on the new operations and AOs of IVPFLNs, we introduce a novel approach to MAGDM and present its main steps. Finally, we discuss the effectiveness of the proposed approach and investigates their advantages through numerical examples.

A new decision approach for the sustainable transport investment selection based on the generalized entropy and knowledge measure under an interval-valued Pythagorean fuzzy environment

Finding the most suitable transport project is one of the most important tasks in transport planning. This task gets more complicated as the sustainable criteria get involved in the process. In this paper, a new multi-criteria group decision-making method with unknown expert and attribute weights is proposed to address the sustainable transport investment selection problem. To make the method more powerful in dealing with uncertain elements, an Interval-Valued Pythagorean Fuzzy (IVPF) set is used as an attractive and useful tool to handle uncertainty. First, a generalized entropy measure under an IVPF environment is introduced, which enables the method to determine the fuzziness of the attribute values, which are expressed by Interval-Valued Pythagorean Fuzzy Numbers (IVPFNs). To determine the fuzziness of IVPFNs with identical membership and non-membership degrees, a generalized knowledge measure of the IVPFNs is also introduced. Based on this measure and considering the deviation between attribute assessments, a new optimization model is presented to obtain unknown attribute weights. In addition, based on the extension of the VIKOR method, a new algorithm is presented to determine the unknown expert weights. Finally, a real case study is considered to show the efficiency of the proposed methods.

Some induced generalized interval-valued Pythagorean fuzzy Einstein geometric aggregation operators and their application to group decision-making

For the multiple attribute group decision-making problems where attribute values are the interval-valued Pythagorean fuzzy numbers, the group decision-making method based on generalized Einstein geometric aggregation operators is developed. First, induced generalized interval-valued Pythagorean fuzzy Einstein ordered weighted geometric (I-GIVPFEOWG) aggregation operator and induced generalized interval-valued Pythagorean fuzzy Einstein hybrid weighted geometric (I-GIVPFEHWG) aggregation operator, were proposed. Some general properties of these operators, such as idempotency, commutativity, monotonicity, and boundedness, were discussed, and some special cases in these operators were analyzed. Furthermore, the method for multiple attribute group decision-making problems based on these operators was developed, and the operational processes were illustrated in detail. The main advantage of using the proposed methods and operators is that these operators and methods give a more complete view of the problem to the decision makers. These methods provide more general, more accurate, and precise results. Therefore, these methods play a vital role in real-world problems. Finally the proposed operators have been applied to decision-making problems to show the validity, practicality, and effectiveness of the new approach.

On Accuracy Function and Distance Measures of Interval-valued Pythagorean Fuzzy Sets with Application in Decision Making

The notion of interval-valued Pythagorean fuzzy sets permits four important parameters, i.e., membership degree, non-membership degree, and a pair of values strength of commitment and direction of commitment, to a given set to have an interval value in dealing with imprecise information. In the present communication, a new accuracy function is being provided to overcome the shortcomings of the existing score and available accuracy functions for interval-valued Pythagorean fuzzy sets. The validity of the proposed accuracy function has been discussed in detail through the illustrative examples. Further, a new interval-valued Pythagorean fuzzy $p$-distance measure for interval-valued Pythagorean fuzzy numbers has been proposed and used in context with the existing weighted averaging operators. Finally, in view of the proposed accuracy function, distance measure and weighted averaging operators, a numerical example of multi-criteria decision-making problem has been solved to validate the proposed methodology.

Models for Multiple Attribute Decision Making with Interval-Valued Pythagorean Fuzzy Muirhead Mean Operators and Their Application to Green Suppliers Selection

In this paper, we extend MM operator and dual MM (DMM) operator to process the interval-valued Pythagorean fuzzy numbers (IVPFNs) and then to solve the MADM problems. Firstly, we develop some interval-valued Pythagorean fuzzy Muirhead mean operators by extending MM and DMM operators to IVPFNs. Then, we prove some properties and discuss some special cases with respect to the parameter vector. Moreover, we present some new methods to deal with MADM problems with the IVPFNs based on the proposed MM and DMM operators. Finally, we verify the validity and reliability of our methods by using an application example for green supplier selections, and analyse the advantages of our methods by comparing it with other existing methods.

Green Supplier Selection Based on DEA Model in Interval-Valued Pythagorean Fuzzy Environment

The green supplier selection has gradually become a hot topic around the world. To achieve the purpose of optimal decision-making, it is of great importance to process the imprecise and ambiguous information that may exist in the evaluation and selection of green suppliers. In the present article, first, converting the imprecise and ambiguous information into interval-valued Pythagorean fuzzy numbers (IVPFNs) through the Likert scale. Then, using the interval-valued Pythagorean fuzzy weighted average (IVPFWA) operator and multiplicative theory to introduce the IVPFNs into the non-radial SE-BCC model of Data Envelopment Analysis (DEA), we developed the DEA model in the interval-valued Pythagorean fuzzy environment. Later, ranking all decision-making units (DMUs) to be evaluated and selecting the best of them. Moreover, The proposed model can not only deal with the case where the quantitative indicators are interval-valued numbers, but also solve the problem of qualitative analysis in the evaluation process. Finally, a numerical example of green suppliers selection is provided to demonstrate the feasibility and usefulness of the model.