Pythagorean Fuzzy Numbers Research Articles

Statistics-based approach for large-scale group decision-making under incomplete Pythagorean fuzzy information with risk attitude

As loss in decision sample information occurs during large-scale group decision-making (LSGDM), this paper proposes a statistical estimation approach for handling Pythagorean fuzzy information under the risk attitude of decision-makers (DMs). The DMs are partitioned by risk attitudes (hesitancy degrees) into subgroups. A five-number summary for the subgroups from the incomplete decision information given by the DMs is obtained. The Cornish–Fisher expansion is then applied to estimate the mean, standard variance, and skewness of the decision sample information from the five-number summary. The confidence interval constructed by the skewness is used to obtain the interval-valued Pythagorean fuzzy number (IVPFN) evaluation information of the subgroups. An optimization model based on minimizing the conflicts between the subgroups and the overall group is used to derive the weights of the subgroups. A sorting function of the IVPFNs is used to rank the alternatives. A case study on green credit and a comparison analysis are applied to validate the proposed method.

A New Technique to Solve Transportation Problems Using De-Pythagorean Value in Generalized Interval Valued Pythagorean Environment

Pythagorean fuzzy set can be defined as an augmented and amplified version of intuitionistic fuzzy set designed for measuring the impreciseness of a real-life complication. This research article advocates the idea of interval valued Pythagorean number in parametric form and its implementation on a mathematical application of a transportation problem for the first time. In adherence to the particular interval valued Pythagorean parametric fuzzy number ambience, a logical score and accuracy function has been initiated in this topic. The authors also have indentured the generalization of de-Pythagorean value and anchored the operation research transportation problem with parametric interval valued Pythagorean number followed by an industrial engineering application. This competent thought will prove useful to us for the purpose of solving surplus daily life problems in the realm of parametric interval valued Pythagorean fuzzy.

Generating method of Pythagorean fuzzy sets from the negation of probability

Fuzzy sets have been widely concerned due it can better deal with more complex and uncertain problems. To expand the application of fuzzy sets, Yager proposed Pythagorean fuzzy sets (PFS) which has been used in many fields. However, in the most cases, the PFS can be generated by expert assessment, which can limit the application of PFS. Hence, how to reasonably generate Pythagorean fuzzy number from known information is a problem worthy of discussion. The paper proposed a method to generate the Pythagorean fuzzy number by considering the negation of probability, namely Pythagorean fuzzy sets based on negation (NPFS). The probability and the probability after negation can be understood as membership and non-membership in the Pythagorean fuzzy number. Besides, NPFS is also set the connection between probability and PFS. There are some numerical examples used to explain the proposed NPFS. In order to explore the effectiveness of NPFS, the paper can not only apply NPFS to artificial decision-making, but also to data-driven applications, which used to handle the multi-criterion decision making (MCDM) and classification problems. In MCDM, the paper applied the NPFS into TOPSIS and used the fault diagnosis to verify its effectiveness by comparing with the results of probability. In the application of classification, the paper proposed the classification method based on NPFS and used real world data to verify the effectiveness of the proposed method.

LR-type fully Pythagorean fuzzy linear programming problems with equality constraints

A Pythagorean fuzzy set is a powerful model for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to an intuitionistic fuzzy model. This research article presents a new model called LR-type fully Pythagorean fuzzy linear programming problem. We consider the notions of LR-type Pythagorean fuzzy number, ranking for LR-type Pythagorean fuzzy numbers and arithmetic operations for unrestricted LR-type Pythagorean fuzzy numbers. We propose a method to solve LR-type fully Pythagorean fuzzy linear programming problems with equality constraints. We describe our proposed method with numerical examples including diet problem.

An approach to decision making with interval-valued complex Pythagorean fuzzy model for evaluating personal risk of mental patients

It is prominent important for managers to assess the personal risk of mental patients. The evaluation process refers to numerous indexes, and the evaluation values are general portrayed by uncertainty information. In order to conveniently model the complicated uncertainty information in realistic decision making, interval-valued complex Pythagorean fuzzy set is proposed. Firstly, with the aid of Einstein t-norm and t-conorm, four fundamental operations for interval-valued complex Pythagorean fuzzy number (IVCPFN) are constructed along with some operational properties. Subsequently, according to these proposed operations, the weighted average and weighted geometric forms of aggregation operators are initiated for fusing IVCPFNs, and their anticipated properties are also examined. In addition, a multiple attribute decision making issue is examined under the framework of IVCPFNs when employing the novel suggested operators. Ultimately, an example regarding the assessment on personal risk of mental patients is provided to reveal the practicability of the designed approach, and the attractiveness of our results is further found through comparing with other extant approaches.The main novelty of the coined approach is that it not only can preserve the original assessment information adequately by utilizing the IVCPFNs, but also can aggregate IVCPFNs effectively.

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.

Multicriteria Decision-Making Approach for Pythagorean Fuzzy Hypersoft Sets’ Interaction Aggregation Operators

In this paper, we examine the multicriteria decision-making (MCDM) difficulties for Pythagorean fuzzy hypersoft sets (PFHSSs). The PFHSSs are a suitable extension of the Pythagorean fuzzy soft sets (PFSSs) which deliberates the parametrization of multi-subattributes of considered parameters. It is a most substantial notion for describing fuzzy information in the decision-making (DM) procedure to accommodate more vagueness comparative to existing PFSSs and intuitionistic fuzzy hypersoft sets (IFHSSs). The core objective of this study is to plan some innovative operational laws considering the interaction for Pythagorean fuzzy hypersoft numbers (PFHSNs). Also, based on settled interaction operational laws, two aggregation operators (AOs) i.e., Pythagorean fuzzy hypersoft interaction weighted average (PFHSIWA) and Pythagorean fuzzy hypersoft interaction weighted geometric (PFHSIWG) operators for PFHSSs operators have been presented with their fundamental properties. Furthermore, an MCDM technique has been established using planned interaction AOs. To ensure the strength and practicality of the developed MCDM method, a mathematical illustration has been presented. The usefulness, influence, and versatility of the developed method have been demonstrated via comparative analysis with the help of some conventional studies.

Methods for Solving L R -Type Pythagorean Fuzzy Linear Programming Problems with Mixed Constraints

A Pythagorean fuzzy set is the superset of fuzzy and intuitionistic fuzzy sets, respectively. Yager proposed the concept of Pythagorean fuzzy sets in which he relaxed the condition that sum of square of both membership degree and nonmembership degree of an element of a set must not be greater than 1. This paper introduces two new techniques to solve L R -type fully Pythagorean fuzzy linear programming problems with mixed constraints having unrestricted L R -type Pythagorean fuzzy numbers as variables and parameters by introducing unknown variables and using a ranking function. Furthermore, we show the equivalence of both the proposed methods and compare the solutions obtained by the two techniques. Besides this, we solve an already existing practical model using proposed techniques and compare the result.

A new score function based Fermatean fuzzy transportation problem

Today the supply, the demand and transportation costs in a transportation problem (TP) cannot be measured precisely due to the inconstant economic and environment conditions. The aim of this paper is to solve transportation problem where supply, demand and transportation costs are Fermatean fuzzy numbers (FFNs). In the existing literature, several methods have been proposed for solving transportation problems with fuzzy parameters (FPs) but in all the proposed methods the parameters relating to the transportation problems are either generalized fuzzy numbers or Pythagorean fuzzy numbers (PFNs). The concept of Fermatean fuzzy sets (FFSs) are very new and using Fermatean fuzzy sets one can handle uncertain information more easily in the process of decision making. Therefore, in this paper, first time we have solved transportation problem with Fermatean fuzzy parameters. For solving the transportation problem with Fermatean fuzzy parameters, we have proposed an algorithm and solved the same by existing method. Then using arithmetic operations of Fermatean fuzzy number the optimal value can be obtained. To illustrate the proposed method we have solved a numerical example and computed results are presented and compared with the existing literature. Finally, the significance of the paper and the scope of future study are mentioned.

Failure Mode and Effect Analysis using an integrated approach of clustering and MCDM under pythagorean fuzzy environment

Failure Mode and Effect Analysis (FMEA) is an effective risk analysis and failure avoidance approach in the design, process, services, and system. With all its benefits, FMEA has three limitations: failure mode risk assessment and prioritization, complex FMEA worksheets, and difficult application of FMEA tables. This paper seeks to overcome the shortcomings of FMEA using an integrated approach based on a developed Pythagorean fuzzy (PF) k-means clustering algorithm and a popular MCDM method called PF-VIKOR. In the first step, Pythagorean fuzzy numbers (PFNs) were used to collect Severity (S), Occurrence (O), and Detection (D) factors for failure modes to incorporate uncertainty and fuzziness into subjective judgments. Afterward, failure modes were clustered by developing a novel k-means clustering algorithm that accepts PFNs as input. Finally, the PF-VIKOR approach was used to analyze the ordering of cluster risks. The proposed approach was implemented in the dehydration unit of an Iranian gas refinery and the results were compared with the traditional FMEA. The findings showed the flexibility and applicability of the proposed approach in addressing real-world problems. This research provides two key contributions: (1) designing a PFN-based k-means clustering algorithm that tackles FMEA limitations and (2) using the PF-VIKOR method for prioritizing and evaluating failure mode clusters.

Multi-attribute decision making method using advanced Pythagorean fuzzy weighted geometric operator and their applications for real estate company selection

In this paper, a novel multi-attribute decision-making method using Advanced Pythagorean fuzzy weighted geometric operator in a Pythagorean fuzzy environment is developed. Pythagorean fuzzy aggregation operators have drawbacks that they give indeterminate results in some special cases when membership value or non-membership value gets 0 value or 1 value and the weight vector is of type (1,0)T or (0,1)T. The Advanced Pythagorean fuzzy geometric operator, the proposed operator can overcome the drawbacks. In some situations, for example, where the sum of squares of membership degree and non-membership degree gets unit value of a Pythagorean fuzzy number, multi-attribute decision making (MADM) methods using some existing aggregation operators give unreasonable ranking orders (ROs) of alternatives or can't discriminate the ROs of alternatives. But the present MADM method can get over the drawbacks of the existing MADM methods. The present MADM method is devoted to eliminate the drawbacks of the existing MADM methods and to select the best real estate company for investment.

A Hybrid Method for Complex Pythagorean Fuzzy Decision Making

This article takes advantage of advancements in two different fields in order to produce a novel decision-making framework. First, we contribute to the theory of aggregation operators, which are mappings that combine large amounts of data into more advantageous forms. They are extensively used in different settings from classical to fuzzy set theory alike. Secondly, we expand the literature on complex Pythagorean fuzzy model, which has an edge over other models due to its ability to handle uncertain data of periodic nature. We propose some aggregation operators for complex Pythagorean fuzzy numbers that depend on the Hamacher t-norm and t-conorm, namely, the complex Pythagorean fuzzy Hamacher weighted average operator, the complex Pythagorean fuzzy Hamacher ordered weighted average operator, and the complex Pythagorean fuzzy Hamacher hybrid average operator. We explore some properties of these operators inclusive of idempotency, monotonicity, and boundedness. Then, the operators are applied to multicriteria decision-making problems under the complex Pythagorean fuzzy environment. Furthermore, we present an algorithm along with a flow chart, and we demonstrate their applicability with the assistance of two numerical examples (selection of most favorable country for immigrants and selection of the best programming language). We investigate the adequacy of this algorithm by conducting a comparative study with the case of complex intuitionistic fuzzy aggregation operators.

Fully Pythagorean fuzzy linear programming problems with equality constraints

There are several daily life problems where we have to deal with the uncertainties and we are forced to solve the uncertain linear programming models. Certain methods have been presented for dealing with linear programming problems based on fuzzy sets and intuitionistic fuzzy sets which are characterized by membership degree, membership and non-membership degrees, respectively. In this study, we first extend the concept of crisp linear programming problem in Pythagorean fuzzy environment based on triangular Pythagorean fuzzy numbers. The profit/cost coefficients in objective function, input/output coefficients and right-hand side coefficients and decision variables of a linear programming problem are considered as triangular Pythagorean fuzzy numbers. Further, we present methods for solving fully Pythagorean fuzzy linear programming problems for non-negative and unrestricted triangular Pythagorean fuzzy numbers with equality constraints. We also apply the proposed technique to solve practical models.

Selection of solar tracking system using extended TOPSIS technique with interval type-2 pythagorean fuzzy numbers

The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is considered among the most frequently used techniques to deal with multi-criteria group decision-making (MCGDM) conflicts. In this article, we have presented an extended TOPSIS technique in the framework of interval type-2 trapezoidal Pythagorean fuzzy numbers (IT2TrPFN). We first projected a novel approach to evaluate the distance between them using ordered weighted averaging operator and (alpha ,beta )-cut. Subsequently, we widen the concept of TOPSIS method formed on the distance method with IT2TrPFNs and applied it on MCGDM dilemma by considering the attitudes and perspectives of the decision-makers. Lastly, an application of solar tracking system and numerous contrasts with the other existing techniques are presented to express the practicality and feasibility of our projected approach.

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 comprehensive obstacle analysis framework on dispersed wind power: A case of China

Dispersed wind power is undoubtedly assisting sustainable development of society but hindered by various factors. To promote the development of dispersed wind power, this paper is devoted to obstacles identification and analysis to provide effective development proposals. Firstly, a three-stage process (namely collection, screening and improvement) is developed to extract common obstacles from literature and explore unique obstacles according to the characteristics and development status of dispersed wind power. Then, DANP method is improved to analyze interaction, interrelation and relative importance among filtered obstacles, where Pythagorean fuzzy numbers and K-Means clustering algorithm are introduced to modify incomplete information collection and arbitrary threshold determination, respectively. Three reasonable thresholds are determined according to clustering results, which divide the influence degree among obstacles into four categories (namely, high, moderate, low and no influence). The analysis results indicate that lagged unified planning ranks the first of obstacles, followed by financial difficulty and restricted grid access. On that basis, some practical proposals on regional coordinated development are suggested, including building unified coordination platforms and forming dispersed wind power project group. This paper does not only enrich material database for scholars, but also provide reference and guidance for government and related enterprises.

Pythagoreanm-Polar Fuzzy Weighted Aggregation Operators and Algorithm for the Investment Strategic Decision Making

The role of multipolar uncertain statistics cannot be unheeded while confronting daily life problems on well-founded basis. Fusion (aggregation) of a number of input values in multipolar form into a sole multipolar output value is an essential tool not merely of physics or mathematics but also of widely held problems of economics, commerce and trade, engineering, social sciences, decision-making problems, life sciences, and many more. The problem of aggregation is very wide-ranging and fascinating, in general. We use, in this article, Pythagorean fuzzy numbers (PFNs) in multipolar form to contrive imprecise information. We introduce Pythagoreanm-polar fuzzy weighted averaging (PmFWA), Pythagoreanm-polar fuzzy weighted geometric (PmFWG), symmetric Pythagoreanm-polar fuzzy weighted averaging (SPmFWA), and symmetric Pythagoreanm-polar fuzzy weighted geometric (SPmFWG) operators for aggregating uncertain data. Finally, we present a practical example to illustrate the application of the proposed operators and to demonstrate its practicality and effectiveness towards investment strategic decision making.

Two-Phase Group Decision-Aiding System using ELECTRE III Method in Pythagorean Fuzzy Environment

This paper is in the continuation of a well-established line of research in multi-criteria decision making, namely, the development of the basic ELECTRE methodology. In the present study, we make an attempt to enhance the influence of the ELECTRE III method, which was conceived in response to the deficiency that existing variants of the ELECTRE method did not provide a full ordering of the alternatives. ELECTRE III has been recently extended so as to make use of information in the form of intuitionistic fuzzy sets. Although this enhancement has been acclaimed by many scholars, it is nonetheless true that further progresses can be made in the quest of generality. In this research, we develop a two-phase Pythagorean fuzzy version of standard ELECTRE III method, called two-phase Pythagorean fuzzy ELECTRE III (PF-ELECTRE III) approach. It takes full advantage of the capabilities of Pythagorean fuzzy sets (PFS); a successful model that generalizes intuitionistic fuzzy sets in a practical and intuitive way. The group decision support system of PF-ELECTRE III first evaluates the performances of the alternatives characterized by Pythagorean fuzzy numbers. The second stage produces a complete ranking of the alternatives from their evaluations in the first phase. We formulate the approach by defining PF indifference threshold, preference threshold and veto threshold functions which provide a more reliable basis to construct outranking relations. The ranking module of the PF-ELECTRE III method is simplified by computing the concordance credibility, discordance credibility and net credibility degrees of each alternative. A diagrammatic representation of the group decision-supporting system is presented to justify the corresponding step-by-step approach for solving problems. Finally, the haze management problem is solved to verify the applicability of the approach proposed in this paper. The investigation of this model and its performance paves the way to additional generalizations by other models like q-rung orthopair fuzzy sets.

A novel aggregation method for generating Pythagorean fuzzy numbers in multiple criteria group decision making: An application to materials selection

Pythagorean Fuzzy Numbers are more capable of modeling uncertainties in real-life decision-making situations than Intuitionistic Fuzzy Numbers. Majority of research in Pythagorean Fuzzy Numbers, us...

Some improved pythagorean fuzzy Dombi power aggregation operators with application in multiple-attribute decision making

Pythagorean fuzzy set (PyFS) is an extension of various fuzzy concepts, such as fuzzy set (FS), intuitionistic FS, and it is enhanced mathematical gizmo to pact with uncertain and vague information. In this article, some drawbacks in the Dombi operational rules for Pythagorean fuzzy numbers (PyFNs) are examined and some improved Dombi operational laws for PyFNs are developed. We also find out that the value aggregated using the existing Dombi aggregation operators (DAOs) is not a PyFN. Furthermore, we developed two new aggregations, improved existing aggregation operators (AOs) for aggregating Pythagorean fuzzy information (PyFI) and are applied to multiple-attribute decision making (MADM). To acquire full advantage of power average (PA) operators proposed by Yager, the Pythagorean fuzzy Dombi power average (PyFDPA) operator, the Pythagorean fuzzy Dombi weighted power average (PyFDWPA) operator, Pythagorean fuzzy Dombi power geometric (PyFDPG) operator, Pythagorean fuzzy Dombi weighted geometric (PyFDPWG) operator, improved the existing AOs and their desirable properties are discussed. The foremost qualities of these developed Dombi power aggregation operators is that they purge the cause of discomfited data and are more supple due to general parameter. Additionally, based on these Dombi power AOs, a novel MADM approach is instituted. Finally, a numerical example is given to show the realism and efficacy of the proposed approach and judgment with the existing approaches is also specified.