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.

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

As the extension of the fuzzy sets (FSs) theory, the intuitionistic fuzzy sets (IFSs) play an important role in handling the uncertainty under the uncertain environments. The Pythagoreanfuzzy 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 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 Pythagorean fuzzy sets, which is based on the belief function in Dempster–Shafer evidence theory, and is called PFSDM distance. It describes the Pythagorean fuzzy sets in the form of basic probability assignments (BPAs) and calculates the divergence of BPAs to get the divergence of PFSs, which is the step in establishing a link between the PFSs and BPAs. Since the proposed method combines the characters of belief function and divergence, it has a more powerful resolution than other existing methods. Additionally, an improved algorithm using PFSDM distance is proposed in medical diagnosis, which can avoid producing counter-intuitive results especially when a data conflict exists. The proposed method and the magnified algorithm are both demonstrated to be rational and practical in applications.

https://www.mdpi.com/2227-7390/8/1/142/pdf

A New Divergence Measure of Pythagorean Fuzzy Sets Based on Belief Function and Its Application in Medical Diagnosis

This paper introduces a simplified presentation of a new computing procedure for solving the fuzzy Pythagorean transportation problem. To design the algorithm, we have described the Pythagorean fuzzy arithmetic and numerical conditions in three different models in Pythagorean fuzzy environment. To achieve our aim, we have first extended the initial basic feasible solution. Then an existing optimality method is used to obtain the cost of transportation. To justify the proposed method, few numerical experiments are given to show the effectiveness of the new model. Finally, some conclusion and future work are discussed.

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A Pythagorean fuzzy approach to the transportation problem

The concept of Pythagorean fuzzy sets is very much applicable in decision science because of its unique nature of indeterminacy. The main feature of Pythagorean fuzzy sets is that it is characterized by three parameters, namely, membership degree, non-membership degree, and indeterminate degree, in such a way that the sum of the square of each of the parameters is one. In this paper, we present axiomatic definitions of distance and similarity measures for Pythagorean fuzzy sets, taking into account the three parameters that describe the sets. Some distance and similarity measures in intuitionistic fuzzy sets, viz, Hamming, Euclidean, normalized Hamming, and normalized Euclidean distances, and similarities are extended to Pythagorean fuzzy set setting. However, it is discovered that Hamming and Euclidean distances and similarities fail the metric conditions in Pythagorean fuzzy set setting whenever the elements of the two Pythagorean fuzzy sets, whose distance and similarity are to be measured, are not equal. Finally, numerical examples are provided to illustrate the validity and applicability of the measures. These measures are suggestible to be resourceful in multicriteria decision-making problems (MCDMP) and multiattribute decision-making problems (MADMP), respectively.

Distance and similarity measures for Pythagorean fuzzy sets

Multi-criteria weighted aggregated sum product assessment framework for fuel technology selection using q-rung orthopair fuzzy sets

Imprecision is an important factor in any decision-making process. Different tools and approaches have been introduced to handle the imprecise environment of group decision-making. One of the latest tools in dealing with imprecision is Pythagorean fuzzy sets. These sets generalize intuitionistic fuzzy sets with a wider scope of applications, and, thus, the motivation for investigating into its resourcefulness in tackling career placements problem. In this paper, we explore the concept of Pythagorean fuzzy sets and deduce some theorems in connection to score and accuracy functions. Some properties of Pythagorean fuzzy sets are outlined. The idea of relation is established in Pythagorean fuzzy set setting called Pythagorean fuzzy relation with numerical illustrations to validate the developed relation. Finally, a decision-making approach of career placements on the basis of academic performance is presented using the proposed Pythagorean fuzzy relation called max–min–max composition to ascertain the suitability of careers to applicants. The approach adopted in this paper is suggestible to solve the other multi-criteria decision-making problems or multi-attribute decision-making problems, respectively.

/pdf/pythagorean-fuzzy-set-and-its-application-in-career-fajx87625u.pdf

Pythagorean fuzzy set and its application in career placements based on academic performance using max–min–max composition

Uncertainty is an important factor in any decision-making process. Different mathematical frameworks have been introduced to cope the ambiguity of decision-making. The concept of Pythagorean fuzzy sets (PFSs) is one of the latest mathematical frameworks that deals with uncertainty. Pythagorean fuzzy sets generalize intuitionistic fuzzy sets with a wider scope of applications, and hence, the motivation for investigating into its applicability in tackling uncertainty imbedded in medical diagnosis. This paper studies the approach of max–min–max composite relation for Pythagorean fuzzy sets, improves upon the approach, and applies its to medical diagnosis problem. The validity of the improved composite relation for Pythagorean fuzzy sets is carried out in comparison to the max–min–max composite relation for Pythagorean fuzzy sets using numerical experiments. The improved composite relation for Pythagorean fuzzy sets yields a better relation with a greater relational value when compared to the aforementioned composite relation and, hence, its choice to solving medical diagnosis problem. To this end, an application of the improved composite relation for Pythagorean fuzzy sets is explored in medical diagnosis using hypothetical medical database. This improved composite relation could be used as a sustainable approach in applying Pythagorean fuzzy sets to multi-criteria decision-making (MCDM) problems, multi-attribute decision-making (MADM) problems, pattern recognition problems, among others.

Improved composite relation for pythagorean fuzzy sets and its application to medical diagnosis

The concept of distance between Pythagorean fuzzy sets (PFSs) has been proven to be relevant in the applications of PFSs as seen in the literature. The main purpose of this paper is to show that Zhang and Xu’s distance measure between PFSs fails the conditions of distance measure; hence, it is not an appropriate distance measure for PFSs. Some numerical examples are used to validate this stance. In order to remedy this shortcoming, Zhang and Xu’s distance measure for PFSs is normalised/modified to cater for the limitation by employing the technique used to normalise both Hamming and Euclidean distances between intuitionistic fuzzy sets by Szmidt and Kacprzyk. The modified Zhang and Xu’s distance measure for PFSs satisfies the conditions of the axiomatic definition of distance measure for PFSs; hence, it is an appropriate/reliable distance measure for PFSs. Finally, the modified Zhang and Xu’s distance measure for PFSs is applied to pattern recognition problems of classification of building materials and mineral fields.

Modified Zhang and Xu’s distance measure for Pythagorean fuzzy sets and its application to pattern recognition problems

Intuitionistic fuzzy set (IFS) is very useful in providing a flexible model to elaborate uncertainty and vagueness involved in decision making. In this paper, we reviewed the concept of IFS and proposed its application in career determination using normalized Euclidean distance method to measure the distance between each student and each career respectively. Solution is obtained by looking for the smallest distance between each student and each career.

https://eujournal.org/index.php/esj/article/download/3478/3241

P. A. Ejegwa

Papers

Distance and similarity measures for Pythagorean fuzzy sets

Pythagorean fuzzy set and its application in career placements based on academic performance using max–min–max composition

Improved composite relation for pythagorean fuzzy sets and its application to medical diagnosis

Modified Zhang and Xu’s distance measure for Pythagorean fuzzy sets and its application to pattern recognition problems

Intuitionistic fuzzy set and its application in career determination via normalized euclidean distance method