Some new correlation coefficients of picture fuzzy sets with applications

An optimized multi-attribute decision-making approach to construction supply chain management by using complex picture fuzzy soft set

Supplier selection is a critical decision-making process for any organization, as it directly impacts the quality, cost, and reliability of its products and services. However, the supplier selection problem can become highly complex due to the uncertainties and vagueness associated with it. To overcome these complexities, multi-criteria decision analysis, and fuzzy logic have been used to incorporate uncertainties and vagueness into the supplier selection process. These techniques can help organizations make informed decisions and mitigate the risks associated with supplier selection. In this article, a complex picture fuzzy soft set (cpFSS), a generalized fuzzy set-like structure, is developed to deal with information-based uncertainties involved in the supplier selection process. It can maintain the expected information-based periodicity by introducing amplitude and phase terms. The amplitude term is meant for fuzzy membership, and the phase term is for managing its periodicity within the complex plane. The cpFSS also facilitates the decision-makers by allowing them the opportunity to provide their neutral grade-based opinions for objects under observation. Firstly, the essential notions and set-theoretic operations of cpFSS are investigated and illustrated with examples. Secondly, a MADM-based algorithm is proposed by describing new matrix-based aggregations of cpFSS like the core matrix, maximum and minimum decision value matrices, and score. Lastly, the proposed algorithm is implemented in real-world applications with the aim of selecting a suitable supplier for the provision of required materials for construction projects. With the sensitivity analysis of score values through Pythagorean means, it can be concluded that the results and rankings of the suppliers are consistent. Moreover, through structural comparison, the proposed structure is proven to be more flexible and reliable as compared to existing fuzzy set-like structures.

Decision Support System for Prioritization of Offshore Wind Farm Site by Utilizing Picture Fuzzy Combined Compromise Solution Group Decision Method

The selection of offshore wind farm site (OWFS) has important strategic significance for vigorously developing offshore new energy and is deemed as a complicated uncertain multicriteria decision-making (MCDM) process. To further promote offshore wind power energy planning and provide decision support, this paper proposes a hybrid picture fuzzy (PF) combined compromise solution (CoCoSo) technique for prioritization of OWFSs. To begin with, a fresh PF similarity measure is proffered to estimate the importance of experts. Next, the novel operational rules for PF numbers based upon the generalized Dombi norms are defined, and four novel generalized Dombi operators are propounded. Afterward, the PF preference selection index (PSI) method and PF stepwise weights assessment ratio analysis (SWARA) model are propounded to identify the objective and subjective weight of criteria, separately. In addition, the enhanced CoCoSo method is proffered via the similarity measure and new operators for ranking OWFSs with PF information. Lastly, the applicability and feasibility of the propounded PF-PSI-SWARA-CoCoSo method are adopted to ascertain the optimal OWFS. The comparison and sensibility investigations are also carried out to validate the robustness and superiority of our methodology. Results manifest that the developed methodology can offer powerful decision support for departments and managers to evaluate and choose the satisfying OWFSs.

On utilizing modified TOPSIS with R-norm q-rung picture fuzzy information measure green supplier selection

The present communication introduces a new discriminant measure coined as R-norm q-rung picture fuzzy discriminant information measure which is more generalized in nature and has the capability to handle more flexibility inherited in the inexact information. The notion of q-rung picture fuzzy set (q-RPFS) has an integrated advantage of picture fuzzy set and q-rung orthopair fuzzy set with flexibility of qth level relations. The proposed parametric measure is then applied in the conventional "technique for order preference by similarity to the ideal solution (TOPSIS) method" for solving a green supplier selection problem. The numerical illustration to exhibit the proposed methodology for the green supplier selection problem has been presented in an empirical form to establish the consistency of the model. Also, the advantageous features of the proposed scheme in the setup of impreciseness have been discussed.

Correlation coefficients for T-spherical fuzzy sets and their applications in pattern analysis and multi-attribute decision-making

T-spherical fuzzy set is an effective tool to deal with vagueness and uncertainty in real-life problems, especially in a situation when there are more than two circumstances, like in casting a ballot. The correlation coefficient of T-spherical fuzzy sets is a tool to calculate the association of two T-spherical fuzzy sets. It has several applications in various disciplines like science, management, and engineering. The noticeable applications incorporate pattern analysis, decision-making, medical diagnosis, and clustering. The aim of this article is to introduce some correlation coefficients for T-spherical fuzzy sets with their applications in pattern recognition and decision-making. The under discussion correlation coefficients are far more advantageous than the existing and many other tools used for T-spherical fuzzy sets, because they are used completely and demonstrate the nature as well as the limit of association between two T-spherical fuzzy sets. Further, an application of proposed correlation coefficients in pattern analysis is discussed here. In addition to it, the proposed correlation coefficients are applied to a multi-attribute decision-making problem, in which the selection of a suitable COVID-19 mask is presented. A comparative analysis has also been made to check the effectiveness of the proposed work with the existing correlation coefficients.

A Hybrid Picture Fuzzy Similarity Measure and Improved VIKOR Method

Picture fuzzy set (PFS) can intuitively express the answers of “yes”, “neutral”, “no” and “reject”, which have strong advantages in solving uncertain information. The similarity measure is an effective tool to determine the relationship between two picture fuzzy sets (PFSs). In this paper, we propose a hybrid picture fuzzy (PF) similarity measure which combines the Hamming distance and the transformed tetrahedral centroid distance and verifies that it satisfies the four properties of the similarity measure. The proposed and existing picture fuzzy similarity measures are compared and investigated through numerical examples and some applications of pattern recognition. The results show that the proposed similarity measure not only produces no unreasonable results, but also overcomes the shortcomings of the existing similarity measures. Furthermore, we investigate an improved VIKOR method based on the proposed similarity measure of PFS. Finally, through an example, several multi-attribute decision-making (MADM) methods are compared and analyzed to illustrate the effectiveness and practicability of the improved VIKOR method.

An Innovative Hybrid Multi-Criteria Decision-Making Approach under Picture Fuzzy Information

These days, multi-criteria decision-making (MCDM) approaches play a vital role in making decisions considering multiple criteria. Among these approaches, the picture fuzzy soft set model is emerging as a powerful mathematical tool for handling various kinds of uncertainties in complex real-life MCDM situations because it is a combination of two efficient mathematical tools, namely, picture fuzzy sets and soft sets. However, the picture fuzzy soft set model is deficient; that is, it fails to tackle information symmetrically in a bipolar soft environment. To overcome this difficulty, in this paper, a model named picture fuzzy bipolar soft sets (PRFBSSs, for short) is proposed, which is a natural hybridization of two models, namely, picture fuzzy sets and bipolar soft sets. An example discussing the selection of students for a scholarship is added to illustrate the initiated model. Some novel properties of PRFBSSs such as sub-set, super-set, equality, complement, relative null and absolute PRFBSSs, extended intersection and union, and restricted intersection and union are investigated. Moreover, two fundamental operations of PRFBSSs, namely, the AND and OR operations, are studied. Thereafter, some new results (De Morgan’s law, commutativity, associativity, and distributivity) related to these proposed notions are investigated and explained through corresponding numerical examples. An algorithm is developed to deal with uncertain information in the PRFBSS environment. To show the efficacy and applicability of the initiated technique, a descriptive numerical example regarding the selection of the best graphic designer is explored under PRFBSSs. In the end, concerning both qualitative and quantitative perspectives, a detailed comparative analysis of the initiated model with certain existing models is provided.

T-spherical uncertain linguistic MARCOS method based on generalized distance and Heronian mean for multi-attribute group decision-making with unknown weight information

The T-spherical uncertain linguistic (TSUL) sets (TSULSs) integrated by T-spherical fuzzy sets and uncertain linguistic variables are introduced in this article. This new concept is not only a generalized form but also can integrate decision-makers’ quantitative evaluation ideas and qualitative evaluation information. The TSULSs serve as a reliable and comprehensive tool for describing complex and uncertain decision information. This paper focuses on an extended MARCOS (Measurement of Alternatives and Ranking according to the Compromise Solution) method to handle the TSUL multi-attribute group decision-making problems where the weight information is completely unknown. First, we define, respectively, the operation rules and generalized distance measure of T-spherical uncertain linguistic numbers (TSULNs). Then, we develop two kinds of aggregation operators of TSULNs, one kind of operator with independent attributes is T-spherical uncertain linguistic weighted averaging and geometric (TSULWA and TSULWG) operators, and the other is T-spherical uncertain linguistic Heronian mean aggregation operators (TSULHM and TSULWHM) considering attributes interrelationship. Their related properties are discussed and a series of reduced forms are presented. Subsequently, a new TSUL-MARCOS-based multi-attribute group decision-making model combining the proposed aggregation operators and generalized distance is constructed. Finally, a real case of investment decision for a community group-buying platform is presented for illustration. We further test the rationality and superiorities of the proposed method through sensitivity analysis and comparative study.

Picture fuzzy Z-linguistic set and its application in multiple attribute group decision-making

Picture fuzzy linguistic set is a vital solution to express complex and uncertain information, which has been applied in multi-attribute group decision-making (MAGDM). However, the credibility of decision-making information is unconsidered, which may give rise to the inaccuracy of final result. To solve this problem, the picture fuzzy Z-linguistic set (PFZLS) composed of linguistic term, picture fuzzy number, and credibility is proposed, which could express more complete decision-making information. Subsequently, operation rules, comparison methods, and distance measures of PFZLS are introduced. In addition, the weighted geometric average operator and the classical VIKOR method are extended and combined to solve the MAGDM problem under the picture fuzzy Z-linguistic environment. Finally, an illustrative example about the emergency decision-making (EDM) problem of forest fire accident is proposed, and a series of comparative analyses are presented to verify the rationality and superiority of the PFZLS.

An Innovative Decision-Making Approach Based on Correlation Coefficients of Complex Picture Fuzzy Sets and Their Applications in Cluster Analysis

In modern times, the organizational managements greatly depend on decision-making (DM). DM is considered the management's fundamental function that helps the businesses and organizations to accomplish their targets. Several techniques and processes are proposed for the efficient DM. Sometimes, the situations are unclear and several factors make the process of DM uncertain. Fuzzy set theory has numerous tools to tackle such tentative and uncertain events. The complex picture fuzzy set (CPFS) is a super powerful fuzzy-based structure to cope with the various types of uncertainties. In this article, an innovative DM algorithm is designed which runs for several types of fuzzy information. In addition, a number of new notions are defined which act as the building blocks for the proposed algorithm, such as information energy of a CPFS, correlation between CPFSs, correlation coefficient of CPFSs, matrix of correlation coefficients, and composition of these matrices. Furthermore, some useful results and properties of the novel definitions have been presented. As an illustration, the proposed algorithm is applied to a clustering problem where a company intends to classify its products on the basis of features. Moreover, some experiments are performed for the purpose of comparison. Finally, a comprehensive analysis of the experimental results has been carried out, and the proposed technique is validated.

Hesitant T-spherical Dombi fuzzy aggregation operators and their applications in multiple criteria group decision-making

A hesitant fuzzy (HF) set is an extension of the fuzzy sets and a T-spherical fuzzy set (T-SFS) is a generalization of the spherical fuzzy set (SFS). HF set has a significant role for modelling disagreements of the decision-makers over membership degree of an element. Also, T-SFS is quite effective in the modelling of the uncertainty for decision-making (DM) problems. In this paper, we define the concept of hesitant T-spherical fuzzy (HT-SF) set (HT-SFS) by combining concepts of HF set and T-SFS, and present some set-theoretical operations of HT-SFSs. We also develop the Dombi operations on HT-SFSs. We present some aggregation operators based on Dombi operators, including hesitant T-spherical Dombi fuzzy weighted arithmetic averaging operator, hesitant T-spherical Dombi fuzzy weighted geometric averaging operator, hesitant T-spherical Dombi fuzzy ordered weighted arithmetic averaging operator, and hesitant T-spherical Dombi fuzzy ordered weighted geometric averaging operator, and investigate some properties of them. In addition, we give a multi-criteria group decision-making method and algorithm of the proposed method under the hesitant T-spherical fuzzy environment. To show the process of proposed method, we present an example related to the selection of the most suitable person for the assistant professorship position in a university. Besides this, we present a comparative analysis with existing operators to reveal the advantages and authenticity of our technique.

Analysis of medical diagnosis based on variation co-efficient similarity measures under picture hesitant fuzzy sets and their application

One of the most dominant and feasible technique is called the PHF setting is exist in the circumstances of fuzzy set theory for handling intricate and vague data in genuine life scenario. The perception of PHF setting is massive universal is compared to these assumptions, who must cope with two or three sorts of data in the shape of singleton element. Under the consideration of the PHF setting, we utilized some SM in the region of the PHF setting are to diagnose the PHFDSM, PHFWDSM, PHFJSM, PHFWJSM, PHFCSM, PHFWCSM, PHFHVSM, PHFWHVSM and demonstrated their flexible parts. Likewise, a lot of examples are exposed under the invented measures based on PHF data in the environment of medical diagnosis to demonstrate the stability and elasticity of the explored works. Finally, the sensitive analysis of the presented works is also implemented and illuminated their graphical structures.

Applications of a picture fuzzy correlation coefficient in pattern analysis and decision-making

Picture fuzzy set is an efficient tool for dealing with uncertainty and vagueness, particularly in situations that require assimilation of more dimensions of linguistic assessment such as human voting, feature selection, etc. The correlation coefficient of picture fuzzy sets is a tool to determine the association of two picture fuzzy sets. It has several applications in various disciplines like science, engineering, and management. The prominent applications include decision-making, pattern recognition, clustering analysis, medical diagnosis, etc. In this paper, we introduce a new correlation coefficient for picture fuzzy sets with the justification of its advantages. This correlation coefficient is better than the existing correlation coefficients and other such measures in the picture fuzzy theory because it considers the picture fuzzy set as a whole and also expresses the nature (positive or negative) as well as the extent of association between two PFSs. By performing some comparative analysis based on the computation of correlation degree and linguistic hedges, we establish the effectiveness of the suggested correlation measure over some available correlation measures in a picture fuzzy environment. Further, in the context of pattern recognition, we examine the performance of the proposed correlation measure over some existing picture fuzzy correlation measures. Finally, we apply the suggested picture fuzzy correlation coefficient to a decision-making problem involving the selection of an appropriate COVID-19 mask.

Complex T-spherical fuzzy Dombi aggregation operators and their applications in multiple-criteria decision-making

Complex fuzzy (CF) sets (CFSs) have a significant role in modelling the problems involving two-dimensional information. Recently, the extensions of CFSs have gained the attention of researchers studying decision-making methods. The complex T-spherical fuzzy set (CTSFS) is an extension of the CFSs introduced in the last times. In this paper, we introduce the Dombi operations on CTSFSs. Based on Dombi operators, we define some aggregation operators, including complex T-spherical Dombi fuzzy weighted arithmetic averaging (CTSDFWAA) operator, complex T-spherical Dombi fuzzy weighted geometric averaging (CTSDFWGA) operator, complex T-spherical Dombi fuzzy ordered weighted arithmetic averaging (CTSDFOWAA) operator, complex T-spherical Dombi fuzzy ordered weighted geometric averaging (CTSDFOWGA) operator, and we obtain some of their properties. In addition, we develop a multi-criteria decision-making (MCDM) method under the CTSF environment and present an algorithm for the proposed method. To show the process of the proposed method, we present an example related to diagnosing the COVID-19. Besides this, we present a sensitivity analysis to reveal the advantages and restrictions of our method.

A new type of generalized picture fuzzy soft set and its application in decision making

 In this article, we propose a novel concept of the generalized picture fuzzy soft set by combining the picture fuzzy soft set and the fuzzy parameter set. For possible applications, we explain five kinds of operations (e.g., subset, equal, union, intersection, and complement) based on generalized picture fuzzy soft sets. Then, we establish several theoretical operations of generalized picture fuzzy soft sets. In addition, we present the new type by using the AND operation of the generalized picture fuzzy soft set for fuzzy decision-making and clarify its applicability with a numerical example. Finally, we give a comparison between the picture fuzzy soft set theory and the generalized picture fuzzy soft set theory. It is shown that our proposed (i.e., generalized picture fuzzy soft set theory) is viable and provide decision makers a more mathematical insight before making decisions on their options.

An innovative picture fuzzy distance measure and novel multi-attribute decision-making method

Picture fuzzy set (PFS) is a direct generalization of the fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). The concept of PFS is suitable to model the situations that involve more answers of the type yes, no, abstain, and refuse. In this study, we introduce a novel picture fuzzy (PF) distance measure on the basis of direct operation on the functions of membership, non-membership, neutrality, refusal, and the upper bound of the function of membership of two PFSs. We contrast the proposed PF distance measure with the existing PF distance measures and discuss the advantages in the pattern classification problems. The application of fuzzy and non-standard fuzzy models in the real data is very challenging as real data is always found in crisp form. Here, we also derive some conversion formulae to apply proposed method in the real data set. Moreover, we introduce a new multi-attribute decision-making (MADM) method using the proposed PF distance measure. In addition, we justify necessity of the newly proposed MADM method using appropriate counterintuitive examples. Finally, we contrast the performance of the proposed MADM method with the classical MADM methods in the PF environment.

Picture Fuzzy MCDM Approach for Risk Assessment of Railway Infrastructure

To increase the level of safety and prevent significant accidents, it is essential to prioritize risk factors and assess railway infrastructure. The key question is how to identify unsafe railway infrastructure so authorities can undertake safety improvement projects on time. The paper aims to introduce a picture fuzzy group multi-criteria decision-making approach for risk assessment of railway infrastructure. Firstly, picture fuzzy sets are employed for representing and handling risk-related information. Secondly, a picture fuzzy hybrid method based on the direct rating, and Tsallis–Havrda–Charvát entropy is provided to prioritize risk factors. Thirdly, a picture fuzzy measurement of alternatives and ranking according to compromise solution method is developed to rank railway infrastructures. Lastly, the formulated approach is implemented in the Czech Republic context. Two sensitivity analyses verified the high robustness of the formulated approach. The comparative analysis with five state-of-the-art picture fuzzy approaches approved its high reliability. Compared to the state-of-the-art picture fuzzy approaches, the provided three-parametric approach has superior flexibility.

Decision support framework for the risk ranking of agroforestry biomass power generation projects with picture fuzzy information

With the rapid growth of the global population and economy, energy consumption and demad are increasing sharply. As an essential renewable energy, biomass energy can promote the reform of energy production and consumption. Considering the characteristics of long investment cycle and large investment scale of agroforestry biomass power generation (AFBPG) projects, this study establishes a decision support framework for risk ranking of AFBPG project under picture fuzzy environment. The proposed framework considers not only the fuzziness and uncertainty of decision-making problems but also the decision-makers’ (DMs) psychological behavior. First, given the integrity of information representation, DMs provide risk assessment information expressed with picture fuzzy numbers, and then gives the distance of the picture fuzzy set (PFS) to maximize the PFS information. Second, the entropy weight method is used to compute the objective weight. Third, the VIKOR (Vlse Kriterijumska Optimizacija I Kompromisno Resenje) – TODIM (an acronym in Portuguese for an interactive multi-criteria decision making) method is suggested for ranking risk factors, which reflects the behavioral psychology of DMs. Moreover, the proposed evaluation model is successfully applied in a practical case. The results show that the model is valid for ranking risk factors under picture fuzzy environment. Last but not least, comparison and sensitivity analysis are implemented to verify the effectiveness and applicability of the proposed method and some suggestions for practical application are put forward.