Dimensional Analysis under Pythagorean Fuzzy Approach for Supplier Selection

PurposeThe purpose of this study is to establish a synthetic group decision framework based on the Pythagorean fuzzy (PF) set to select the optimal medicine cold chain logistics provider (MCCLP). Fierce market competition makes enterprises must constantly improve every link in the process of enterprise sustainable development. The evaluation of MCCLP in pharmaceutical enterprises is an important link to enhance the comprehensive competitiveness. Because of the fuzziness of expert cognition and the complexity of the decision procedure, PF set can effectively handle the uncertainty and ambiguity in the process of multi-criteria group decision decision-making (MCGDM).Design/methodology/approachThis paper develops an integrated group decision framework through combining the Decision-Making Trial and Evaluation Laboratory (DEMATEL) technique and combined compromise solution (CoCoSo) approach to select a satisfactory MCCLP within PF circumstances. First, the PF set is used to process the ambiguity and uncertainty of the cognition ability of experts. Second, a novel PF knowledge measure is propounded to measure the vagueness of the PF set. Third, a comprehensive criterion weight determination technique is developed through aggregating subjective weights attained utilizing the PF DEMATEL approach and objective weight deduced by knowledge measure method. Furthermore, an integrated MCGDM approach based on synthetic weight and CoCoSo method is constructed.FindingsThe outcomes of sensibility analysis and comparison investigation show that the suggested decision framework can help decision experts to choose a satisfactory MCCLP scientifically and reasonably. Accordingly, the propounded comprehensive decision framework can be recommended to enterprises and organizations to assess the MCCLP for their improvement of core competitiveness.Originality/valueMCCLP selection is not only momentous for pharmaceutical enterprises to improve transportation quality and ensure medicine safety but also provides a strong guarantee for enterprises to improve their core competitiveness. Nevertheless, enterprises face certain challenges due to the uncertainty of the assessment environment as well as human cognition in the process of choosing a satisfactory MCCLP. PF set possesses a formidable capability to address the uncertainty and imprecision information in the process of MCGDM. Therefore, pharmaceutical enterprises can implement the proposed method to evaluate the suppliers to further improve the comprehensive profit of enterprises.

New Chebyshev distance measures for Pythagorean fuzzy sets with applications to multiple criteria decision analysis using an extended ELECTRE approach

This paper aims at developing a novel generalized distance measure of Pythagorean fuzzy (PF) sets and constructing a distance-based compromise approach for multiple criteria decision analysis (MCDA) within PF environments. The theory of Pythagorean fuzziness provides a representative model of nonstandard fuzzy sets; it is valuable for representing complex vague or imprecise information in many practical applications. The distance measure for Pythagorean membership grades is important because it can effectively quantify the separation between PF information. Based on the essential characteristics of PF sets (membership, non-membership, strength, and direction), this paper proposes several distance measures, namely, new Hamming and Euclidean distances and a generalized distance measure that is based on them for Pythagorean membership grades and for PF sets. Moreover, the useful and desirable properties of the proposed PF distance measures are investigated to evaluate their advantages and form a solid theoretical basis. In addition, to evaluate the performance of the proposed distance measures in practice, this paper establishes a PF-distance-based compromise approach for addressing MCDA problems that involve PF information. The effectiveness and practicability of the developed approach are further evaluated through a case study on bridge-superstructure construction methods. According to the application results and comparative analysis, the proposed PF distance measures are accurate and outperform other methods in handling the inherent uncertainties of evaluation information. Furthermore, the PF-distance-based compromise approach can accommodate the much higher degrees of uncertainty in real-life decision scenarios and effectively determine the priority ranking among candidate alternatives for managing complicated MCDA problems.

The Pythagorean fuzzy hypersoft set is the most generalized form of the Pythagorean fuzzy soft set used to resolve indeterminate and inexplicit information in the decision-making procedure, considering the parameters’ multi-sub-attributes. Aggregation operators execute a dynamic role in considering the two prospect sequences and eliminating anxieties from this perception. The hybrid form of Pythagorean fuzzy sets with hypersoft sets has appeared as a supportive structure in fuzzy mathematics and ensued as a convenient perspective in decision-making. This paper prolongs the Einstein-weighted ordered aggregation operators for the Pythagorean fuzzy hypersoft set, which proficiently contracts with tentative and confusing data. Experts are using the Pythagorean fuzzy hypersoft set in their quest to report indefinite and specific decision-making processes. It is an effective technique for enlarging unsure facts in decision-making. Some operational laws for the Pythagorean fuzzy hypersoft set have been projected in light of Einstein’s operations. Two innovative Einstein-ordered aggregation operators were established based on operational laws: Pythagorean fuzzy hypersoft Einstein-ordered weighted average and Pythagorean fuzzy hypersoft Einstein-ordered weighted geometric operators with their essential properties. Multi-criteria decision-making is imperative in overcoming barriers to real-world complications. However, conventional methods of multi-criteria decision-making regularly provide inconsistent results. The extended model appraisals recognized score values to regulate robotic agri-farming equated to prevalent methods, which is more useful for agribusiness. A numerical illustration of decision-making complications in real-life farming is deliberated to authenticate the established method’s supremacy and applicability. Based on the anticipated aggregation operators, a robust multi-criteria decision-making method has been presented, which delivers the most appropriate outcomes compared to existing multi-criteria decision-making techniques. The consequences show that the intended approach is more effective and stable in handling rough information based on the Pythagorean fuzzy hypersoft set.

Multi-granulation rough sets (MGRSs) and decision-theoretic rough sets (DTRSs) are two important and popular generalizations of classical rough sets. The combination of two generalized rough sets have been investigated by numerous researchers in different extensions of fuzzy settings such as interval-valued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFSs), bipolar-valued fuzzy sets (BVFSs), etc. Pythagorean fuzzy (PF) set is another extension of fuzzy set, which is more capable in comparison to IFS handle vagueness in real world. However, few studies have focused on the combination of the two rough sets in PF settings. In this study, we combine the two generalized rough sets in PF settings. First, we introduce a type of PF subset (of subset of the given universe) of the PF Set (of the given universe). Then we establish two basic models of multi-granulation PF DTRS (MG-PF-DTRS) of PF subset of the PF set based on PF inclusion measure within the framework of multi-granulation PF approximation space. One model is based on a combination of PF relations (PFRs) and the construction of approximations with respect to the combined PFR. By combining PFRs through intersection and union, respectively, we construct two models. The other model is based on the construction of approximations from PFRs and a combination of the approximations. By using intersection and union to combine the approximations, respectively, we again get two models. As a result, we have total four models. Further for different constraints on parameters, we obtain three kinds of each model of the MG-PF-DTRSs. Then, their principal structure, basic properties and uncertainty measure methods are investigated as well. Second, we give a way to compute PF similarity degrees between two objects and also give a way to compute PF decision-making objects from incomplete multi-source information systems (IMSISs). Then we design an algorithm for decision-making to IMSISs using MG-PFDTRSs and their uncertainty measure methods. Finally, an example about the mutual funds investment is included to show the feasibility and potential of the theoretic results obtained.

New similarity measure of Pythagorean fuzzy sets based on the Jaccard index with its application to clustering

Multiple criteria group decision analysis using a Pythagorean fuzzy programming model for multidimensional analysis of preference based on novel distance measures

Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can describe both interval-valued Pythagorean fuzzy (IVPF) sets (IVPFSs) and Pythagorean fuzzy (PF) sets (PFSs) simultaneously for addressing information ambiguities. Since an aggregation operator (AO) is a significant mathematical approach in decision-making (DM) problems, this article presents some novel Bonferroni mean (BM) and weighted Bonferroni mean averaging operators between CPF-numbers (CPFNs) for aggregating the different preferences of the decision-makers. Then, using the proposed AOs, we develop a DM approach under the CPF environment and demonstrated it with a numerical example. Furthermore, a comparative study between the proposed and existing methods has been performed to demonstrate the practicality and efficiency of the proposed DM approach.

Exploring efficiency approaches to solve the problems of decision making under uncertainty is a mainstream direction. This article explores the rough approximation of the uncertainty information with Pythagorean fuzzy information on multi-granularity space over two universes combined with grey relational analysis. Based on grey relational analysis, we present a new approach to calculate the relative degree or the attribute weight with Pythagorean fuzzy set and give a new descriptions for membership degree and non-membership. Then, this paper proposes a multi-granulation rough sets combined with Pythagorean fuzzy set, including optimistic multi-granulation Pythagorean fuzzy rough set, pessimistic multi-granulation Pythagorean fuzzy rough set and variable precision Pythagorean fuzzy rough set. Several basic properties for the established models are investigated in detail. Meanwhile, we present an approach to solving the multiple-criteria group decision making problems with fuzzy information based on the proposed model. Eventually, a case study of psychological evaluation of health care workers in COVID-19 show the principle of the established model and is utilized to verify the availability. The main contributions have three aspects. The first contribution of an approach of calculating the attribute weight is presented based on Grey Relational Analysis and gives a new perspective for the Pythagorean fuzzy set. Then, this paper proposes a mutli-granulation rough set model with Pythagorean fuzzy set over two universes. Finally, we apply the proposed model to solving the psychological evaluation problems.

Pythagorean fuzzy set, an extension form of intuitionistic fuzzy set, which owns many advantages for dealing with uncertainties, and it has been developed to deal with various complex decision-making problems. Furthermore, based on lower and upper approximations induced by multiple binary relations, the multigranulation rough set has become one of the most promising directions in rough set theory. To combine the two ideas and explore the practical decision-making problems, we develop a new multigranulation rough set model, called Pythagorean fuzzy multigranulation rough set over two universes. In the framework of our study, we introduce the models of Pythagorean fuzzy rough set over two universes and Pythagorean fuzzy multigranulation rough set over two universes, respectively. Both the definition and basic properties are explored. Finally, we give a general algorithm, which is applied to a decision-making problem in merger and acquisition, and the effectiveness of the algorithm is demonstrated by a numerical example.

The diverse decision values may fail to capture an accurate perspective when multiple decision-makers are part of the process. To address this challenge, this work introduces the diamond Pythagorean fuzzy set (Dia‑PyFS), an advancement over both the Pythagorean fuzzy (PyFS) set and the interval-valued Pythagorean fuzzy set (IVPyFS). Due to the established extension of intuitionistic fuzzy sets into Pythagorean fuzzy sets, the Dia‑PyFS model, a broader form of the diamond intuitionistic fuzzy set model, demonstrates enhanced performance. We then introduce the Dia‑PyFS as an extension of PyFS. The Diamond Pythagorean fuzzy values that define the elements within Dia‑PyFS may share a common norm. For Dia‑PyFS, we define fundamental algebraic and arithmetic operations, including union, intersection, addition, multiplication, and scalar multiplication, and analyze their primary properties. Additionally, we propose some new Dia‑PyF weighted average and geometric aggregation operators as well as explore their unique properties. We also then propose several algebraic operations between Dia‑PyFVs using general triangular 𝓉-norms and 𝓉-conorms. To transform input values represented by Dia‑PyFs into a single output value, we also introduce specific weighted aggregation operators based on these algebraic methods. Additionally, the Dia‑PyFS framework builds upon the “combinative distance-based assess” (CODAS) methodology, which relies on both Euclidean and Hamming distances. To illustrate the applicability of this new approach, the feasibility and suitability of the Dia‑PyFS set approach for choosing the best options are demonstrated by the summary and comparative analysis of the produced reports.

An integral aspect of global businesses and economic activities is the supply chain networks. Importantly, the coronavirus (COVID-19) pandemic scenario has further shown that the outbreak of diseases can create a global network-scale disruption to supply chain or logistics, thereby damaging several aspects of economic activities and business life. Hence, this study aims to assess the resilient supplier selection (RSS) process in the wake of the COVID-19 outbreak. A two-stage hybrid decision model using Pythagorean fuzzy sets was proposed as a case study from the automotive industry to deal with RSS during the COVID-19 outbreak. In the first stage, significant criteria and their corresponding sub-criteria were determined through a vast review of the literature and nominal group technique, while the relative weights for RSS were obtained through the Pythagorean Fuzzy Analytic Hierarchy Process (PFAHP) method. In the second stage, nine suppliers were evaluated with Pythagorean Fuzzy VIKOR (PFVIKOR) method. The results of the hybrid approach revealed that flexibility is the most important criterion among resilience criteria that constitute the most significant dimensions for RSS. In many studies, strategic criteria such as quality, cost, and delivery are found to be the most important criteria in supplier selection, however, in the wake of the COVID-19 outbreak, the opinions of decision-makers were significantly changed as the present study reveals that flexibility is the most important criterion to improve the operations of the supply chain for RSS. Next to flexibility is process capabilities, while quality (Q), and cost (C) existed as the first and second in the category of influential criteria for strategic supplier selection criteria, respectively. The managerial and practical implication is that, in the wake of COVID-19 disruptions, suppliers need to be re-evaluated based on resilience-related indicators.

Multiple-attribute group decision-making (MAGDM) technique is often used to make decisions when several optimal options are under consideration. It can be difficult to select a reasonable optimal option for the decision maker under consideration of insufficient information. The theory of Hamy mean (HM) operators are used to express correlation among different input arguments and provide a smooth approximation during the decision-making process. Recently, Aczel Alsina aggregating expressions gained a lot of attention from numerous mathematicians under different fuzzy circumstances. This article aims to illustrate the notion of a Pythagorean fuzzy (PyF) set (PyFS) with some restricted constraints, such as a sum of the square of truth membership value and falsity membership value. We developed a series of new approaches under consideration of the HM tools, including PyF Aczel Alsina Hamy mean (PyFAAHM), and PyF Aczel Alsina weighted Hamy mean (PyFAAWHM) operators. Further, we also extend the theory of Dual Hamy mean (DHM) operators and derived a series of new methodologies such as PyF Aczel Alsina Dual Hamy mean (PyFAADHM) and PyF Aczel Alsina weighted Dual Hamy mean (PyFAAWDHM) operators. To demonstrate the flexibility of our derived approaches, we illustrate an application of a multinational company considering the MAGDM technique. An experimental case study is also illustrated to evaluate a reasonable option from a group of options. We see the advantages and compatibility of our findings by comparing the results of existing approaches with the results of currently discussed methodologies.

Pythagorean fuzzy linear programming technique for multidimensional analysis of preference using a squared-distance-based approach for multiple criteria decision analysis

Due to insufficient information in multi-criteria decision-making (MCDM) problems, the decision values given by experts are often fuzzy and uncertain. As an extension of Pythagorean fuzzy set (PFS), interval-valued Pythagorean fuzzy (IPF) set is a more effective and powerful tool to handle fuzzy information in decision problems. But, there are two key issues that needed to be solved: weights of experts in the IPF environment and IPF aggregation operators. For these issues, a two-stage MCDM method is constructed in the IPF environment. In the first stage, a novel method for determining the weights of experts is proposed by introducing IPF set (IPFS) into social networks. To do that, the concepts of trust function (TF) and trust score (TS) in the IPF environment are defined to obtain the objective weights of experts. Meanwhile, the subjective weights of experts are obtained from the number of experts. Afterward, the objective weight and subjective weight of each expert are combined to derive the weight of each expert. In the second stage, a novel weighted sum model (WSM) with novel IPF aggregation operators is constructed to rank alternatives. Considering the psychological behavior of experts, that is, self-confidence level, four IPF aggregation operators with self-confidence levels are defined, namely, the self-confidence interval-valued Pythagorean fuzzy weighted averaging (SC-IPFWA) and ordered weighted averaging (SC-IPFOWA) operator, the self-confidence interval-valued Pythagorean fuzzy weighted geometric (SC-IPFWG) and ordered weighted geometric (SC-IPFOWG) operator. Finally, a numerical case is used to verify the effectiveness of the proposed two-stage MCDM method.