New similarity measure and distance measure for Pythagorean fuzzy set

Development of Aczel-Alsina t-norm based linear Diophantine fuzzy aggregation operators and their applications in multi-criteria decision-making with unknown weight information

Aczel-Alsina t-norm and t-conorm are intrinsically flexible and endow Aczel-Alsina aggregation operators with greater versatility and robustness in the aggregation process than operators rooted in other t-norms and t-conorm families. Moreover, the linear Diophantine fuzzy set (LD-FS) is one of the resilient extensions of the fuzzy sets (FSs), intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PyFSs), and q-rung orthopair fuzzy sets (q-ROFSs), which has acquired prominence in decision analysis due to its exceptional efficacy in resolving ambiguous data. Keeping in view the advantages of both LD-FSs and Aczel-Alsina aggregation operators, this article aims to establish Aczel-Alsina operation rules for LD-FSs, such as Aczel-Alsina sum, Aczel-Alsina product, Aczel-Alsina scalar multiplication, and Aczel-Alsina exponentiation. Based on these operation rules, we expose the linear Diophantine fuzzy Aczel-Alsina weighted average (LDFAAWA) operator, and linear Diophantine fuzzy Aczel-Alsina weighted geometric (LDFAAWG) operator and scrutinize their distinctive characteristics and results. Additionally, based on these aggregation operators (AOs), a multi-criteria decision-making (MCDM) approach is designed and tested with a practical case study related to forecasting weather under an LD-FS setting. The developed model undergoes a comparative analysis with several prevailing approaches to demonstrate the superiority and accuracy of the proposed model. Besides, the influence of the parameter Λ on the ranking order is successfully highlighted.

Tuning of controller parameters using Pythagorean fuzzy similarity measure for stable and time delayed unstable plants

This paper proposes a tuning method based on the Pythagorean fuzzy similarity measure and multi-criteria decision-making to determine the most suitable controller parameters for Fractional-order Proportional Integral Derivative (FOPID) and Integer-order Proportional Integral-Proportional Derivative (PI-PD) controllers. Due to the power of the Pythagorean fuzzy approach to evaluate a phenomenon with two memberships known as membership and non-membership, a multi-objective cost function based on the Pythagorean similarity measure is defined. The transient and steady-state properties of the system output were used for the multi-objective cost function. Thus, the determination of the controller parameters was considered a multi-criteria decision-making problem. Ant colony optimization for continuous domains (ACOR) and artificial bee colony (ABC) optimization are utilized to minimize multi-objective cost functions. The proposed method in the study was applied to three different systems: a second-order non-minimum phase stable system, a first-order unstable system with time delay, and a fractional-order unstable system with time delay, to validate its effectiveness. The cost function utilized in the proposed method is compared with the performance measures widely used in the literature based on the integral of the error, such as IAE (Integral Absolute Error), ITAE (Integral Time Absolute Error), ISE (Integral Square Error), and ITSE (Integral Time Square Error). The proposed method provides a more effective control performance by improving the system response characteristics compared to other cost functions. With the proposed method, the undershoot rate could be significantly reduced in the non-minimum phase system. In the other two systems, significant improvements were achieved compared to other methods by reducing the overshoot rate and oscillation. The proposed method does not require knowing the mathematical model of the system and offers a solution that does not require complex calculations. The proposed method can be used alone. Or it can be used as a second and fine-tuning method after a tuning process.

Picture fuzzy soft Bonferroni mean aggregation operators and their applications

Due to more advanced features of the picture fuzzy soft set and valuable characteristics of aggregation operators, in this article, we present the notion of picture fuzzy soft Bonferroni mean aggregation operators and weighted picture fuzzy soft Bonferroni mean aggregation operators. Moreover, some basic properties of these introduced aggregation operators have been given. As cancer is one of the most rapidly increasing diseases globally. But due to different kinds of cancer diseases, it is very difficult to say which type of cancer disease is increasing rapidly. So to reduce this difficulty in the medical field, we have applied our work to medical diagnosis problems to ensure that fuzzy ideas can also help in the medical field. Also, an algorithm along with a descriptive example is established that conforms to the authenticity of initiated work. Furthermore, a comparative analysis of the introduced work has been given to show how this work is more efficient and dominant than other existing theories.

Multi-attribute decision-making based on novel Fermatean fuzzy similarity measure and entropy measure

To deal with situations involving uncertainty, Fermatean fuzzy sets are more effective than Pythagorean fuzzy sets, intuitionistic fuzzy sets, and fuzzy sets. Applications for fuzzy similarity measures can be found in a wide range of fields, including clustering analysis, classification issues, medical diagnosis, etc. The computation of the weights of the criteria in a multi-criteria decision-making problem heavily relies on fuzzy entropy measurements. In this paper, we employ t-conorms to suggest various Fermatean fuzzy similarity measures. We have also discussed all of their interesting characteristics. Using the suggested similarity measurements, we have created some new entropy measures for Fermatean fuzzy sets. By using numerical comparison and linguistic hedging, we have established the superiority of the suggested similarity metrics and entropy measures over the existing measures in the Fermatean fuzzy environment. The usefulness of the proposed Fermatean fuzzy similarity measurements is shown by pattern analysis. Last but not least, a novel multi-attribute decision-making approach is described that tackles a significant flaw in the order preference by similarity to the ideal solution, a conventional approach to decision-making, in a Fermatean fuzzy environment.

Novel Information Measures for Fermatean Fuzzy Sets and Their Applications to Pattern Recognition and Medical Diagnosis

Fermatean fuzzy sets (FFSs) have piqued the interest of researchers in a wide range of domains. The striking framework of the FFS is keen to provide the larger preference domain for the modeling of ambiguous information deploying the degrees of membership and nonmembership. Furthermore, FFSs prevail over the theories of intuitionistic fuzzy sets and Pythagorean fuzzy sets owing to their broader space, adjustable parameter, flexible structure, and influential design. The information measures, being a significant part of the literature, are crucial and beneficial tools that are widely applied in decision-making, data mining, medical diagnosis, and pattern recognition. This paper aims to expand the literature on FFSs by proposing many innovative Fermatean fuzzy sets-based information measures, namely, distance measure, similarity measure, entropy measure, and inclusion measure. We investigate the relationship between distance, similarity, entropy, and inclusion measures for FFSs. Another achievement of this research is to establish a systematic transformation of information measures (distance measure, similarity measure, entropy measure, and inclusion measure) for the FFSs. To accomplish this aim, new formulae for information measures of FFSs have been presented. To demonstrate the validity of the measures, we employ them in pattern recognition, building materials, and medical diagnosis. Additionally, a comparison between traditional and novel similarity measures is described in terms of counter-intuitive cases. The findings demonstrate that the innovative information measures do not include any absurd cases.

Nonlinear distance measures under the framework of Pythagorean fuzzy sets with applications in problems of pattern recognition, medical diagnosis, and COVID-19 medicine selection

Background

The concept of Pythagorean fuzzy sets (PFSs) is an utmost valuable mathematical framework, which handles the ambiguity generally arising in decision-making problems. Three parameters, namely membership degree, non-membership degree, and indeterminate (hesitancy) degree, characterize a PFS, where the sum of the square of each of the parameters equals one. PFSs have the unique ability to handle indeterminate or inconsistent information at ease, and which demonstrates its wider scope of applicability over intuitionistic fuzzy sets.

Results

In the present article, we opt to define two nonlinear distances, namely generalized chordal distance and non-Archimedean chordal distance for PFSs. Most of the established measures possess linearity, and we cannot incorporate them to approximate the nonlinear nature of information as it might lead to counter-intuitive results. Moreover, the concept of non-Archimedean normed space theory plays a significant role in numerous research domains. The proficiency of our proposed measures to overcome the impediments of the existing measures is demonstrated utilizing twelve different sets of fuzzy numbers, supported by a diligent comparative analysis. Numerical examples of pattern recognition and medical diagnosis have been considered where we depict the validity and applicability of our newly constructed distances. In addition, we also demonstrate a problem of suitable medicine selection for COVID-19 so that the transmission rate of the prevailing viral pandemic could be minimized and more lives could be saved.

Conclusions

Although the issues concerning the COVID-19 pandemic are very much challenging, yet it is the current need of the hour to save the human race. Furthermore, the justifiable structure of our proposed distances and also their feasible nature suggest that their applications are not only limited to some specific research domains, but decision-makers from other spheres as well shall hugely benefit from them and possibly come up with some further extensions of the ideas.

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.

Some Enhanced Distance Measuring Approaches Based on Pythagorean Fuzzy Information with Applications in Decision Making

The construct of Pythagorean fuzzy distance measure (PFDM) is a competent measuring tool to curb incomplete information often encountered in decision making. PFDM possesses a wider scope of applications than distance measure under intuitionistic fuzzy information. Some Pythagorean fuzzy distance measure approaches (PFDMAs) have been developed and applied in decision making, albeit with some setbacks in terms of accuracy and precision. In this paper, some novel PFDMAs are developed with better accuracy and reliability rates compared to the already developed PFDMAs. In an effort to validate the novel PFDMAs, some of their properties are discussed in terms of theorems with proofs. In addition, some applications of the novel PFDMAs in problems of disease diagnosis and pattern recognition are discussed. Furthermore, we present comparative studies of the novel PFDMAs in conjunction to the existing PFDMAs to buttress the merit of the novel approaches in terms of consistency and precision. To end with, some new Pythagorean fuzzy similarity measuring approaches (PFDSAs) based on the novel PFDMAs are presented and applied to solve the problems of disease diagnosis and pattern recognition as well.

Performance of exponential similarity measures in supply of commodities in containment zones during COVID-19 pandemic under Pythagorean fuzzy sets

Following the breakout of the novel coronavirus disease 2019 (COVID-19), the government of India was forced to prohibit all forms of human movement. It became important to establish and maintain a supply of commodities in hotspots and containment zones in different parts of the country. This study critically proposes new exponential similarity measures to understand the requirement and distribution of commodities to these zones during the rapid spread of novel coronavirus (COVID-19) across the globe. The primary goal is to utilize the important aspect of similarity measures based on exponential function under Pythagorean fuzzy sets, proposed by Yager. The article aims at finding the most required commodity in the affected areas and ensures its distribution in hotspots and containment zones. The projected path of grocery delivery to different residences in containment zones is determined by estimating the similarity measure between each residence and the various necessary goods. Numerical computations have been carried out to validate our proposed measures. Moreover, a comparison of the result for the proposed measures has been carried out to prove the efficacy.

On the similarity measures of N-cubic Pythagorean fuzzy sets using the overlapping ratio

The similarity measures are essential concepts to discuss the closeness between sets. Fuzzy similarity measures and intuitionistic fuzzy similarity measures dealt with the incomplete and inconsistent data more efficiently. With time in decision-making theory, a complex frame of the environment that occurs cannot be specified entirely by these sets. A generalization like the Pythagorean fuzzy set can handle such a situation more efficiently. The applicability of this set attracted the researchers to generalize it into N-Pythagorean, interval-valued N-Pythagorean, and N-cubic Pythagorean sets. For this purpose, first, we define the overlapping ratios of N-interval valued Pythagorean and N-Pythagorean fuzzy sets. In addition, we define similarity measures in these sets. We applied this proposed measure for comparison analysis of plagiarism software.

Some t-conorm-based distance measures and knowledge measures for Pythagorean fuzzy sets with their application in decision-making

The Pythagorean fuzzy sets are more robust than fuzzy sets and intuitionistic fuzzy sets in dealing with the problems involving uncertainty. To compare two Pythagorean fuzzy sets, distance measures play a crucial role. In this paper, we have proposed some novel distance measures for Pythagorean fuzzy sets using t-conorms. We have also discussed their various desirable properties. With the help of suggested distance measures, we have introduced some new knowledge measures for Pythagorean fuzzy sets. Through numerical comparison and linguistic hedges, we have established the effectiveness of the suggested distance measures and knowledge measures, respectively, over the existing measures in the Pythagorean fuzzy setting. At last, we have demonstrated the application of the suggested measures in pattern analysis and multi-attribute decision-making.

Enhancement and segmentation of medical images through pythagorean fuzzy sets-An innovative approach

Image segmentation has attracted a lot of attention due to its potential biomedical applications. Based on these, in the current research, an attempt has been made to explore object enhancement and segmentation for CT images of lungs infected with COVID-19. By implementing Pythagorean fuzzy entropy, the considered images were enhanced. Further, by constructing Pythagorean fuzzy measures and utilizing the thresholding technique, the required values of thresholds for the segmentation of the proposed scheme are assessed. The object extraction ability of the five segmentation algorithms including current sophisticated, and proposed schemes are evaluated by applying the quality measurement factors. Ultimately, the proposed scheme has the best effect on object separation as well as the quality measurement values.

Multicriteria decision-making based on distance measures and knowledge measures of Fermatean fuzzy sets

Fermatean fuzzy sets are more powerful than fuzzy sets, intuitionistic fuzzy sets, and Pythagorean fuzzy sets in handling various problems involving uncertainty. The distance measures in the fuzzy and non-standard fuzzy frameworks have got their applicability in various areas such as pattern analysis, clustering, medical diagnosis, etc. Also, the fuzzy and non-standard fuzzy knowledge measures have played a vital role in computing the criteria weights in the multicriteria decision-making problems. As there is no study concerning the distance and knowledge measures of Fermatean fuzzy sets, so in this paper, we propose some novel distance measures for Fermatean fuzzy sets using t-conorms. We also discuss their various desirable properties. With the help of suggested distance measures, we introduce some knowledge measures for Fermatean fuzzy sets. Through numerical comparison and linguistic hedges, we establish the effectiveness of the suggested distance measures and knowledge measures, respectively, over the existing measures in the Pythagorean/Fermatean fuzzy setting. At last, we demonstrate the application of the suggested measures in pattern analyis and multicriteria decision-making.

Novel multi-criteria decision-making methods with soft rough q-rung orthopair fuzzy sets and q-rung orthopair fuzzy soft rough sets

The aim of this paper is to introduce the concepts of soft rough q-rung orthopair fuzzy set (SRqROFS) and q-rung orthopair fuzzy soft rough set (qROPFSRS) based on soft rough set and fuzzy soft relation, respectively. We define some fundamental operations on both SRqROFS and qROPFSRS and discuss some key properties of these models by using upper and lower approximation operators. The suggested models are superior than existing soft rough sets, intuitionistic fuzzy soft rough sets and Pythagorean fuzzy soft rough sets. These models are more efficient to deal with vagueness in multi-criteria decision-making (MCDM) problems. We develop Algorithm i (i = 1, 2, 3, 4, 5) for the construction of SRqROFS, construction of qROFSRS, selection of a smart phone, ranking of beautiful public parks, and ranking of government challenges, respectively. The notions of upper reduct and lower reduct based on the upper approximations and lower approximations by variation of the decision attributes are also proposed. The applications of the proposed MCDM methods are demonstrated by respective numerical examples. The idea of core is used to find a unanimous optimal decision which is obtained by taking the intersection of all lower reducts and upper reducts.

Distance measure for Pythagorean fuzzy sets with varied applications

Distance measure is one of the research hotspot in Pythagorean fuzzy environment due to its quantitative ability of distinguishing Pythagorean fuzzy sets (PFSs). Various distance functions for PFSs are introduced in the literature and have their own pros and cons. The common thread of incompetency for these existing distance functions is their inability to distinguish highly uncertain PFSs distinctly. To tackle this point, we introduce a novel distance measure for PFSs. An added advantage of the measure is its simple mathematical form. Moreover, superiority and reasonability of the prescribed definition are demonstrated through proper numerical examples. Boundedness and nonlinear behaviour of the distance measure is established and verified via suitable illustrations. In the current scenario, selecting an antivirus face-mask as a preventive measure in the COVID-19 pandemic and choosing the best school in private sector for children are some of the burning issues of a modern society. These issues are addressed here as multi-attribute decision-making problems and feasible solutions are obtained using the introduced definition. Applicability of the distance measure is further extended in the areas of pattern recognition and medical diagnosis.

Decision support model with Pythagorean fuzzy preference relations and its application in financial early warnings

Pythagorean fuzzy sets (PFSs) retain the advantages of intuitionistic fuzzy sets (IFSs), while PFSs portray 1.57 times more information than IFSs. In addition, Pythagorean fuzzy preference relations (PFPRs), as a generalization of intuitionistic fuzzy preference relations (IFPRs), are more flexible and applicable. The objective of this paper is to propose a novel decision support model for solving group decision-making problems in a Pythagorean fuzzy environment. First, we define the concepts of ordered consistency and multiplicative consistency for PFPRs. Then, aiming at the group decision-making problem of multiple PFPRs, a consistency improving model is constructed to improve the consistency of group preference relations. Later, a consensus reaching model is developed to reach the degree of group consensus. Furthermore, a decision support model with PFPRs is established to derive the normalized weights and output the final result. Holding these features, this paper builds a decision support model with PFPRs based on multiplicative consistency and consensus. Finally, the described method is validated by an example of financial risk management, and it is concluded that the solvency of a company is an important indicator that affects the financial early warning system.

Directional correlation coefficient measures for Pythagorean fuzzy sets: their applications to medical diagnosis and cluster analysis

Compared to the intuitionistic fuzzy sets, the Pythagorean fuzzy sets (PFSs) can provide the decision makers with more freedom to express their evaluation information. There exist some research results on the correlation coefficient between PFSs, but sometimes they fail to deal with the problems of disease diagnosis and cluster analysis. To tackle the drawbacks of the existing correlation coefficients between PFSs, some novel directional correlation coefficients are put forward to compute the relationship between two PFSs by taking four parameters of the PFSs into consideration, which are the membership degree, non-membership degree, strength of commitment, and direction of commitment. Afterwards, two practical examples are given to show the application of the proposed directional correlation coefficient in the disease diagnosis, and the application of the proposed weighted directional correlation coefficient in the cluster analysis. Finally, they are compared with the previous correlation coefficients that have been developed for PFSs.

Single-stage and two-stage total failure-based group-sampling plans for the Weibull distribution under neutrosophic statistics

If the sample or population has vague, inaccurate, unidentified, deficient, indecisive, or fuzzy data, then the available sampling plans could not be suitable to use for decision-making. In this article, an improved group-sampling plan based on time truncated life tests for Weibull distribution under neutrosophic statistics (NS) has been developed. We developed improved single and double group-sampling plans based on the NS. The proposed design neutrosophic plan parameters are obtained by satisfying both producer’s and consumer’s risks simultaneously under neutrosophic optimization solution. Tables are constructed for the selected shape parameter of Weibull distribution and various combinations of neutrosophic group size. The efficiency of the proposed group-sampling plan under the neutrosophic statistical interval method is also compared with the crisp method grouped sampling plan under classical statistics.

Some novel features of Pythagorean m-polar fuzzy sets with applications

We face many situations in day to day life where multi-polar statistics is offered. The prevailing models like Pythagorean fuzzy sets and m-polar fuzzy sets become inoperable in handling such situation efficiently e.g. if someone wishes to invest his capital in some scheme, he would for sure like to know repeated information about pros and cons of that scheme. Pythagorean m-polar fuzzy sets (PmFSs) serve as the most appropriate model to cope with such situations. The motivation behind this article is to extend the notions of PmFSs coined by Naeem et al. (J Intell Fuzzy Syst 37(6): 8441–8458, 2019) and introduce some new operations and results on PmFSs. Owing to the idea of Pythagorean m-polar fuzzy relation, we render an application in the selection of most appropriate life partner.

Enhanced synchronization-inspired clustering for high-dimensional data

The synchronization-inspired clustering algorithm (Sync) is a novel and outstanding clustering algorithm, which can accurately cluster datasets with any shape, density and distribution. However, the high-dimensional dataset with high dimensionality, high noise, and high redundancy brings some new challenges for the synchronization-inspired clustering algorithm, resulting in a significant increase in clustering time and a decrease in clustering accuracy. To address these challenges, an enhanced synchronization-inspired clustering algorithm, namely SyncHigh, is developed in this paper to quickly and accurately cluster the high-dimensional datasets. First, a PCA-based (Principal Component Analysis) dimension purification strategy is designed to find the principal components in all attributes. Second, a density-based data merge strategy is constructed to reduce the number of objects participating in the synchronization-inspired clustering algorithm, thereby speeding up clustering time. Third, the Kuramoto Model is enhanced to deal with mass differences between objects caused by the density-based data merge strategy. Finally, extensive experimental results on synthetic and real-world datasets show the effectiveness and efficiency of our SyncHigh algorithm.

Similarity Measure of Lattice Ordered Multi-Fuzzy Soft Sets Based on Set Theoretic Approach and Its Application in Decision Making

Many effective tools in fuzzy soft set theory have been proposed to handle various complicated problems in different fields of our real life, especially in decision making. Molodtsov’s soft set theory has been regarded as a newly emerging mathematical tool to deal with uncertainty and vagueness. Lattice ordered multi-fuzzy soft set (LMFSS) has been applied in forecasting process. However, similarity measure is not used in this application. In our research, similarity measure of LMFSS is proposed to calculate the similarity between two LMFSSs. Moreover, some of its properties are introduced and proved. Finally, an application of LMFSS in decision making using similarity measure is analysed.