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Some Picture Fuzzy Aggregation Operators and Their Applications to Multicriteria Decision-Making
Abstract
The objective of the work is to present some series of the aggregation operators for the picture fuzzy sets (PFSs). As PFSs have been an extended version of the intuitionistic fuzzy set theory which not only considers the degree of acceptance or rejection but also taken into the account the degree of refusal during the analysis. Thus, by considering all these degrees, some aggregation operators, namely picture fuzzy weighted average, picture fuzzy ordered weighted average, and picture fuzzy hybrid average aggregation operators, have been proposed along with their desirable properties. A decision-making approach based on these operators has also been presented. Finally, an illustrative example has been given for demonstrating the approach.
... Any classical decision making technique provides only a ranking of alternatives by following some specified steps, while aggregating operators based techniques provide comprehensive values of alternatives by fusing the information through some aggregation process and then ranking the alternatives as per their aggregated values. Considering aggregation operator based approaches for the PFS environment, Wei (2017) integrated picture fuzzy values (PFVs) with the weighted AM and weighted GM and developed some novel aggregation operators for multi-attribute decision making (MADM) problems, while Garg (2017) extended this work by adopting the Archimedean tnorm and t-conorm. Khan et al. (2019) proposed Einstein tnorm and t-conorm based AM operators for multi-criteria decision making (MCDM) problems. ...
... Definition 3 (Wei 2017;Garg 2017;) Let p ¼ ðl; g; mÞ be a PFN then the score (S(p)) and accuracy (H(p)) functions of p are defined as SðpÞ ¼ l À g À m, and HðpÞ ¼ l þ g þ m respectively. ...
... Remark 3 If l 1 ¼ l 2 ¼ ::: ¼ l n ¼ 1 then the CPFEHA operator converted into the picture fuzzy hybrid averaging (PFEHA) operator (Garg 2017). ...
In today’s polluted environment, air pollution is one of the major challenges that must be controlled by adopting suitable precautionary measures and installing an appropriate number of air quality monitoring stations to safeguard human health. There are several causes of air pollution, including fossil fuels, automobiles, agricultural activities, domestic sources, mining, factories, and industries. These are typically measured in terms of common air pollutants such as , , , , , CO. The main objective of this study is to assess air quality monitoring stations by evaluating some air pollution indices using multi-criteria group decision making (MCGDM). In formulating a decision-making problem, experts often face common challenges related to the availability of information, which may be insufficient, indeterminate, or vague, as well as their familiarity with the problem and the weights assigned to criteria, which may be partially or fully unknown. To address these issues, this study uses a picture fuzzy set (PFS) to quantify the insufficiency, indeterminacy, and vagueness of the available information. The confidence level is employed to reflect the expert’s familiarity with the problem, while the maximizing deviation method is applied to manage the uncertainty related to partially or fully unknown criteria weights. By integrating PFS with confidence levels, the paper introduces novel aggregation operators, including confidence picture fuzzy Einstein weighted, ordered weighted, and hybrid averaging operators. The essential properties of these operators, such as idempotency, monotonicity, and boundedness, are also verified. A MCGDM is then presented by combining the maximizing deviation method with the proposed novel aggregation operators in a PFS environment. Finally, a case study is conducted to evaluate three air quality monitoring stations. Sensitivity analysis is performed to assess the impact of varying combinations of experts’ confidence levels on the aggregated values. Additionally, a comparative analysis is carried out, contrasting the proposed aggregation operators with existing ones to demonstrate their effectiveness. The results conclude that the proposed operators are feasible, general, consistent, and can be effectively used to evaluate air quality monitoring stations.
... 1. When p = q = r = 1 and + + ≤ 1 , then the proposed AOs reduce to picture fuzzy weighted averaging and weighted geometric operators (Wei 2017;Garg 2017). 2. When p = q , = 0 and + ≤ 1 , then the proposed AOs reduce to intuitionistic fuzzy weighted averaging and weighted geometric operators (Xu 2007;Zhao et al. 2010). ...
... Example 5 Garg (2017) a multinational company in India bases its financial strategy on a group objective. F 1 (Southern Asian markets), F 2 (Eastern Asian markets), F 3 (Northern Asian markets), and F 2 (Local markets) are the four choices mentioned. ...
... To demonstrate the superiority of the proposed AOs, we compare the results with some existing approaches, namely, picture fuzzy AOs (Wei 2017;Garg 2017;Jana et al. 2019), spherical fuzzy AOs (Ashraf et al. 2020;Jin et al. 2019b;Akram et al. 2020), T-spherical fuzzy AOs (Farid et al. 2023;Munir et al. 2020) and p, q− spherical fuzzy AO (Rahim et al. 2024). The optimal score values and ranking order of the available alternatives are summarized in Table 12. ...
... 1. When p = q = r = 1 and + + ≤ 1 , then the proposed AOs reduce to picture fuzzy weighted averaging and weighted geometric operators (Wei 2017;Garg 2017). 2. When p = q , = 0 and + ≤ 1 , then the proposed AOs reduce to intuitionistic fuzzy weighted averaging and weighted geometric operators (Xu 2007;Zhao et al. 2010). ...
... Example 5 Garg (2017) a multinational company in India bases its financial strategy on a group objective. F 1 (Southern Asian markets), F 2 (Eastern Asian markets), F 3 (Northern Asian markets), and F 2 (Local markets) are the four choices mentioned. ...
... To demonstrate the superiority of the proposed AOs, we compare the results with some existing approaches, namely, picture fuzzy AOs (Wei 2017;Garg 2017;Jana et al. 2019), spherical fuzzy AOs (Ashraf et al. 2020;Jin et al. 2019b;Akram et al. 2020), T-spherical fuzzy AOs (Farid et al. 2023;Munir et al. 2020) and p, q− spherical fuzzy AO (Rahim et al. 2024). The optimal score values and ranking order of the available alternatives are summarized in Table 12. ...
Using fractional fuzzy sets ( FFS) to demonstrate the stability of cryptocurrencies is considered due to the complex and volatile nature of cryptocurrency markets, where traditional models may fall short in capturing nuances and uncertainties. FFS provides a flexible framework for modeling cryptocurrency stability by accommodating imprecise data, multidimensional analysis of various market factors, and adaptability to the unique characteristics of the cryptocurrency space, potentially offering a more comprehensive understanding of the factors influencing stability. Existing studies have explored Picture Fuzzy Sets and Spherical Fuzzy Sets, built on membership, neutrality, and non-membership grades. However, these sets can’t reach the maximum value (equal to 1) due to grade constraints. For example, when considering , these sets fall short. This is obvious when a decision-maker possesses complete confidence in an alternative, they have the option to assign a value of 1 as the assessment score for that alternative. This signifies that they harbor no doubts or uncertainties regarding the chosen option. To address this, Fractional Fuzzy Sets ( FFSs) are introduced, using new parameters p, q, and r. These parameters abide by p, and r as the least common multiple of p and q. We establish operational laws for FFSs. Based on these operational laws, we proposed a series of aggregation operators (AOs) to aggregate the information in context of fractional fuzzy numbers. Furthermore, we constructed a novel multi-criteria group decision-making (MCGDM) method to deal with real-world decision-making problems. A numerical example is provided to demonstrate the proposed approach.
... For instance, the fuzzy Bonferroni operator is used to improve the urban traffic system [30]; an admissible ordered weighted averaging (AOWA) operator is used to evaluate sustainable development policies [31]; the intuitionistic fuzzy rough Schweizer-Sklar aggregation operators are used for investment risk management [32]; the intuitionistic fuzzy Aczel-Alsina Hamy mean operators are used to assess the construction materials [33]. [24,30,32,33] × ✓ × The operators in [34,[37][38][39]43] ✓ × × The operators in [35,36,[40][41][42][44][45][46][47][48][49] ✓ ✓ × The GIVNFWG operator [50] × × ✓ The GIVIFWA and GIVIFWG operators [51] × × ✓ The GPFWIA and GPFWIG operators [14] ✓ × ✓ The proposed operators in this paper ✓ ✓ ✓ ...
... Wang [34] proposed a series of picture fuzzy weighted averaging operators and picture fuzzy weighted geometric operators based on a probability perspective in his doctoral thesis and applied them to MCDM problems. Garg [35] constructed some general forms of picture fuzzy information aggregation operators and studied their idempotence, monotonicity, boundedness, transformation invariance, and homogeneity. Considering the uncertainty of medical diagnosis and the correlation between symptoms, Zhang et al. [36] proposed the picture fuzzy point Choquetintegral aggregation operators and successfully applied them to assist the hierarchical medical system. ...
... Because it is a generalized version of many operations and the only -(co)norm that satisfies the compatibility law, it has mathematical rigor and extensibility. (2) This work can deal with high-dimensional fuzzy data, while the methods and techniques based on matrix theory in [24,[30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] do not have this feature. (3) This work can deal with picture fuzzy data, while the fuzzy tensor-based operators in [50,51] lack this capability. ...
Picture fuzzy sets with four-dimensional features are widely used in decision-making as a mathematical tool because they can capture the uncertainty of data. However, the methods and techniques based on matrix theory are difficult to solve the decision problem involving high-dimensional data in a picture fuzzy setting. Therefore, operators that can identify high-dimensional data in a picture fuzzy environment are proposed to address this challenge. In this paper, firstly, by integrating the Frank operators into the picture fuzzy tensor, the generalized picture fuzzy Frank weighted arithmetic (GPFFWA) and generalized picture fuzzy Frank weighted geometric (GPFFWG) operators are defined. Their specific expressions are discussed, and the idempotency, order-preservation, boundedness, and commutativity of the proposed operators are also given. Then, combining the GPFFWA and GPFFWG operators, an algorithm is designed to solve the multi-criteria decision-making problem with high-dimensional data features in the picture fuzzy environment. Finally, a numerical example and related analysis demonstrate the effectiveness, superiority, and flexibility of the suggested technique. This work provides new theoretical and methodological support for developing and practicing the decision-making discipline.
... In the process of applying PFNs to practical problems, ranking the PFNs is necessary for MADM problems. Garg [30] defined score and accuracy functions as follows. ...
... Definition 4 [30]. Let ...
... Definition 5 [30]. Let ...
A Picture Fuzzy Set (PFS) is an effective tool for handling uncertainties and incomplete cognitive information, incorporating membership, hesitancy, non-membership, and refusal degrees to accommodate diverse viewpoints. The Einstein operation has demonstrated good performance in aggregating data within uncertain contexts, garnering attention in the picture fuzzy environment. However, the existing picture fuzzy Einstein aggregation operator has limitations, particularly when a Picture Fuzzy Number (PFN) possesses a non-membership degree of zero. This leads to an aggregated non-membership degree of zero, disregarding the influence of other non-zero non-membership degrees, rendering them independent. To address these issues, a new decision-making approach is proposed, employing Einstein’s operations for PFNs. The Picture Fuzzy Einstein Interactive Weighted Averaging (PFEIWA) operator are proposed and extensively discussed for its desirable properties. Furthermore, a novel approach referring to these operators in Multi-attribute Decision-Making (MADM) problems is presented. The proposed method applied to a sustainable beef supplier selection problem, and its performance is compared with existing methods. The results show consistent optimal rankings of alternatives, indicating superiority of the proposed operator in resolving MADM problems. The suggested operator is more convincing and suitable due to its ability to produce reasonable and reliable results by considering interactions between PFNs’ degrees.
... In a PFS environment, 25 proposed WA, OWA, and hybrid averaging operators. The Einstein operations on PFSs werē rst introduced by Ref. 26. Khan et al. 27 most recently used the Hamachar operators in PFSs. ...
... A noteworthy extended form of Lukasiewicz as well as probabilistic t-norm and t-conorm 28 had been emerged as Frank t-norm and t-conorm. 26 Moreover, they constitute a su±ciently°exible type of the continuous triangular norm. The Frank models, along with the process of fusion of information, became more adaptable owing to the fact that a certain parameter is used in them, and the literature is replete with numerous works [29][30][31][32] related to these models. ...
The purpose of this study is to extend the idea of complex intuitionistic fuzzy set (CIFS) with a new notion referred to as a complex picture fuzzy set (CPFS). CPFS is a generalized version of CIFS since it includes a neutral membership degree in the concept. The capacity to cover a wider range of information with the aid of neutral membership, non-membership, and membership makes this new theory distinctive. The unit disc of complex plane has been used to cover the range of values of membership degrees. Pertaining to Frank t-norm and t-conorm operations and a few aggregation methods, we establish some fundamental CPFS aggregation operators and attributes and use them to investigate multi-attribute decision-making (MADM) problems. Then, we offer various operators for the purpose of aggregating the CPF data. These are complex picture fuzzy Frank weighted averaging (CPFFWA), complex picture fuzzy Frank ordered weighted averaging (CPFFOWA), complex picture fuzzy Frank hybrid averaging (CPFFHA), and complex picture fuzzy Frank weighted geometric averaging (CPFFWGA), complex picture fuzzy Frank ordered weighted geometric averaging (CPFFOWG), and complex picture fuzzy frank hybrid geometric averaging (CPFFHGA) operators, which benefit from the basic Frank operations and averaging, geometric aggregation techniques. Furthermore, an algorithm for solving multi-attribute decision-making MADM problems has been presented under the framework of CPFSs by using CPFFWA and CPFFWG operators. Finally, in order to depict the potential applicability of our proposed technique, a numerical problem aiming at finding the best alternative has been solved and outcomes have been well compared with some existing techniques.
... The novel bivariate 2D-q Hermite polynomials can be used to define complex membership functions in spherical fuzzy matrices, enhancing multidimensional uncertainty representation in decision-making [31]. Ganie [32] proposed a new distance measure for picture fuzzy sets and demonstrated the application of the proposed distance measure in pattern recognition. Dogra and Pal [33] developed the concept of picture fuzzy matrices and explored their properties. ...
Recent advancements have demonstrated the potential to augment matrix theory with fuzzy, intuitionistic fuzzy, picture fuzzy, interval-valued picture fuzzy matrix concepts for enhanced decision-making applications. We introduce the interval valued spherical fuzzy matrix, extending the spherical fuzzy matrix, to effectively represent and manipulate uncertain and vague information with enhanced flexibility. This paper establishes definitions and theorems for Interval-Valued spherical fuzzy matrices. We develop methods for computing determinant and adjoint, and develop algorithms using composition functions to determine the greatest and least eigenvalue interval valued spherical fuzzy sets and create a flow chart to depict the procedure. In this paper, a new distance measure has been proposed and is to be proved valid by satisfying all the conditions of the distance metric. In addition, an application of interval-valued spherical fuzzy matrices to deal with decision-making problems is presented.
... Considering the framework proposed by Garg [4] for Picture Fuzzy Numbers (PFNs), the score and accuracy functions are defined as follows: ...
This paper proposes a novel framework for sustainable solid waste management (SWM) by synergistically integrating machine learning (ML) with S-PFS, which stands for spherical picture fuzzy sets, are an extension of picture fuzzy sets that allow membership, abstention, and non-membership degrees to dwell within a spherical space. This allows for increased flexibility when modelling ambiguous information. In this study, a unique Multi-Criteria Decision Making (MCDM) method that is based on S-PFS is presented. This method is specifically adapted for the issues that are associated with solid waste management. With the primary goal of addressing the inherent difficulties that are present in decisions regarding solid waste management, the major purpose is to develop new formulations for S-PFS. These formulations will include radius calculations and a novel defuzzifi-cation function. Within the S-PFS framework, we establish the identification of optimistic and pessimistic points, which enables the development of a novel score and accuracy function that takes into account the attitude of the decision-maker, denoted by the symbol λ. When the value of λ approaches 1, the S-PFS becomes more defuzzified, bringing it closer to its optimistic point. This indicates a more risk-seeking perspective, but when λ approaches 0, it indicates a risk-averse perspective. Traditional SWM methods often struggle with complex and uncertain data. This research addresses this challenge by leveraging the power of ML for pattern recognition and prediction within waste generation and composition data. This approach is tailored to address the challenges related to solid waste management. The main objective is to tackle the challenges associated with decisions in solid 1 waste management, focussing on the development of innovative formulations for S-PFS. These formulations will encompass radius calculations and an innovative Furthermore, the framework incorporates PFS to effectively represent the inherent vagueness and uncertainty associated with SWM data, going beyond traditional fuzzy sets. This integrated approach aims to improve decision-making in SWM, leading to more efficient waste collection, processing, and resource recovery, ultimately contributing to a more sustainable and environmentally sound waste management system.
... Based on T-BSS, some algebraic structures are defined in 42 . Moreover, Khan et al. 43 , Wang and Li 44 , Kumar and Chen 45 , and Asif et al. 46 illustrated fundamental aggregation operators and applied them to decision-making problems in Pythagorean fuzzy environment, Hadi et al. 47 , Shit and Ghorai 48 , and Mateen et al. 49 in Fermatean fuzzy environment, Ashraf and Abdullah 50 , Khan et al. 51 , Mahnaz et al. 52 , Debnath and Roy 53 , Hussain and Ullah 54 with spherical fuzzy information, Garg 55 , Jana et al. 56 , Senapati 57 , and Ullah et al. 58 under picture fuzzy environment, Khan et al. 59 , Garg and Chen 60 , and Gayen et al. 61 with q-ROF information, among others. Additionally, interval-valued aggregation operators were suggested in the relevant literature. ...
In the context of telecommunications, AI enhances network efficiency by predicting and managing traffic. In many decision-making scenarios, decision-makers choose the more flexible structure that can handle all kinds of information. Bipolarity is the only case in which we can discuss the positive and negative aspects of certain scenarios. On one side, AI enhances network efficiency, proactive maintenance, and personalized customer experience but on the other hand, it has also some negative aspects (1) implementing AI infrastructure can be costly (2) Uses of AI in telecommunication may raise data security concerns and user privacy (3) AI can lead to potential issues if system fail or misused. To cover these issues, the idea of an interval-valued bipolar fuzzy soft set (IVBFSS) has been developed that can deal with both positive and negative aspects of AI. Some basic operational laws for IVBPFS numbers are developed. Several fundamental aggregation operators have been introduced like arithmetic average and geometric average aggregation operators, indicating our main contribution. An algorithm is developed to discuss the application perspective of the initiated approaches. We have utilized these developed notions to classify AI-driven techniques in the telecommunications sector to discuss the applicability of the initiated notions. A comparative analysis of the developed approaches shows the advantages and superiority of the introduced work.
... MCDM describes decision-making problems that we deal with daily. [21], [22], [23], [24]. There has been a big interest in MCDM theories and methodologies, and several notable developments have been published [25]. ...
The development of educational technology (EdTech) and artificial intelligence (AI) brings about a revolution in English learning by providing flexible, effective, and customized solutions. The purpose of this study is to assess the impact of AI and EdTec on education. In this article, we defined the multi-criteria decision-making (MCDM) procedure to manage ambiguity and awkward information by integrating the Technique for Order of Preference by Similarity to the Ideal Solution (Topsis) method with Circular q-Rung orthopair fuzzy set (Cq-ROFS), and Bonferroni mean (BM) operators to evaluate and prioritize AI-driven EdTech tools. The methodology incorporates multiple attributes, such as adaptability, learner engagement, cost-effectiveness, and scalability, within an MCDM framework. These results highlight the huge potential of intelligent teaching programs, flexible learning environments, and AI-powered language models to improve English ability. This research demonstrates how AI has advanced in education from simple computer-assisted language learning to complex AI-driven platforms like chatbots and intelligent systems for teaching. These developments, such as automated grading and feedback, have given teachers the ability to improve administrative effectiveness and instructional quality. Furthermore, customized and interactive learning experiences that are adapted to the needs and preferences of each student have been made possible by AI-based EdTech solutions. The study used the TOPSIS technique to rank important criteria for optimizing various solutions, highlighting their contribution to preservation, interaction, and overall efficacy in English language learning.
... In PFSs, the membership value μ : P −→ [0, 1], neutral value ν : P −→ [0, 1] and non-membership value ξ : P −→ [0, 1] were considered under the condition 0 ≤ μ + ν + ξ ≤ 1. PFSs were widely used in the theory of decision making, fuzzy inference, networking etc. Zeng et al. [11] provided application of PFSs in multi-criteria group decision making. Several picture fuzzy aggregation operators were introduced by Garg [12]. Some weighted distance measures in the frame of PFSs along with their applications were discussed in [13]. ...
In decision-making theory, after evaluating the information about data, the decision maker typically provides their opinion in the form of yes, no, refusal and abstain. To address such situations, a model based on picture fuzzy sets is often used. However, spherical fuzzy sets offers more extended domain compared to picture fuzzy sets and hence it is more useful in handling uncertain information. Furthermore, their prominent characteristic of an extensive domain makes them better suited for dealing with triplets, i.e., membership, non-membership and neutral values. Fuzzy graph models are power mathematical tools for managing uncertain data. Therefore, to tackle many real-world problems containing uncertainties, spherical fuzzy graphs (SFGs) model offers greater flexibility. In this article, we introduce the concepts of dominations in spherical fuzzy graphs (SFGs) utilizing perfect strong matching (PSM) using strong arcs (SAs), along with its application in decision making. Initially, various types of strong dominating sets (SDSs) like strong paired dominating set (SPDS), strong total dominating set (STDS) etc are introduced within the framework of SFGs. Additionally, characterizations of complete spherical fuzzy graphs (complete SFGs) and complete bipartite spherical fuzzy graphs (complete bipartite SFGs) are provided, alongside the introduction of novel terms related to domination in SFGs. Moreover, several useful terms related to domination in SFGs are introduced, and various characteristics of strong paired domination are examined within the framework of SFGs. Finally, we investigate the practical application of our newly established concepts in the textile industry to identify the partner with the maximum collaboration potential with the stakeholder.
... Step 3 Combine all the criteria, each associated with its own unique PF preference value for each alternative, using the PFAAPWG in Equation 6 to obtain the overall PF. α i of the corresponding A i as α 1 ...
This paper develops a robust picture fuzzy (PF) decision-making framework by integrating power aggregation operators derived from Aczél-Alsina operations. The proposed power aggregation operators effectively capture intricate interrelationships among multiple criteria, thereby enhancing the precision and reliability of decision-making processes. In this study, the familiarity of decision makers with the evaluated objects is systematically incorporated into the PF framework alongside primary data, ensuring a more comprehensive assessment. Motivated by the operational principles of Aczél-Alsina functions, this research advances the theoretical foundation of PF Aczél-Alsina power-weighted and ordered-weighted geometric operators, seamlessly integrating decision makers’ expertise into the aggregation process. The structural properties and mathematical characteristics of these newly developed operators are rigorously analyzed. To validate their practical applicability, we employ the proposed operators to solve a complex multi-criteria decision-making (MCDM) problem within the food industry, a domain where uncertainty and nuanced judgments play a critical role. A comparative evaluation against existing operators highlights the superior performance of our approach in effectively managing uncertainty, refining decision accuracy, and enhancing adaptability to real-world decision-making challenges.
... The concept of Picture FS (PFS) was proposed by Cuong [23], in which he extended the concept of Atanassov's IFS by introducing an abstinence degree along with a membership degree and non-membership degree with the condition that their sum must not exceed 1 i.e. 0 ≤ + + ≤ 1. Later on, Garg et al. [24] further generalized the concept of PFS and described some basic operations of PFS. Wang et al. [25] and Wei G et al. [26] evaluated these operations of PFS in MADM techniques. ...
This paper seeks to establish the significance of efficient evaluation of physical education programs in colleges and universities in the development of human personality, emotional health and fitness and the sociable demeanor of the learners. Nevertheless, the critical assessment methods present strong problems due to the inherent randomness and individuality. To overcome these complexities, an enhanced method of Circular Interval-Valued Fermatean Fuzzy Dombi Mean (CrIVF-FDM) operators is proposed in this research work. Through the application of Fermatean fuzzy sets, the degrees of importance and interdependence of criteria are considered in the model, together with the aggregation flexibility that the Dombi mean operator provides. Thus, the algorithm contains circular interval values to capture source uncertainty and ambiguity clearly while providing a sound assessment basis. The study also shows the effectiveness of the algorithm using physical education programs of higher learning institutions, and therefore, the Algorithm can provide summarized, accurate, and flexible quality assurance assessment. The findings suggest that it may help decision-makers crack this particular nut in determining the best courses of action toward increasing the quality of physical education, cultivating leadership, and enhancing students’ all-around growth.
... Inspired by the implementation of PFS in the process of making decisions, Garg [8] put forth several operators of aggregations inside the framework of PFS and demonstrated a method of constructing decisions utilizing the suggested operators of aggregations. Wang [9] presented a geometric aggregation operator based on picture fuzzy sets and used score and accuracy functions to compare two picture fuzzy numbers (PFNs). ...
This paper presents a new picture fuzzy set approach to modeling of B´ezier curve approximation. To produce the picture fuzzy control point, the notion is used to describe the point relation of the picture fuzzy set (PFS). The new picture fuzzy control point is combined with the B´ezier function to create a model of the picture fuzzy B´ezier curve. Subsequently, the approximation curves comprising the positive, neutral, negative, and refusal membership curves are displayed. A numerical example has been used to approximate the picture fuzzy B´ezier curve model for the illustration. Consequently, in the conclusion of this paper, an approach to obtain picture fuzzy B´ezier curve is described.
... S Das el al. [6] proposed intuitionistic multi-fuzzy weighted averaging (IMFWA) operator merging number of intuitionistic multi-fuzzy numbers into single one. H. Gang [7] proposed different type of aggregation operators for picture fuzzy numbers. Moreover, a lot researchers are sincerely concentrated on this topic and developed different type aggregation operator apply in MCDM procedure to manage the different situation of decision making problem. ...
In this study we proposed a new weighted aggregation operator for ranking the picture fuzzy numbers (PFNs) which is based on neutral membership value of PFN. As the picture fuzzy set (PFS) is an extension version of intuitionistic fuzzy set theory with introducing the neutral membership value during data analysis. The neutral membership value in PFS reflecting the ambiguous nature of the subject to judgment. The ambiguity is depending on the neutral membership value of PFN. The proposed weighted aggregation operator manages the ambiguity according to neutral membership value. Then, the aggregation operator applies in a multi attribute decision making method where attribute value of the alternative are picture fuzzy numbers. In the decision making process, the weight of attributes are calculated according to neutral values and aggregate the multiple attributes into a single PFN. Then estimate the individual score value of the alternatives. Lastly, ranking the alternative according to score value. Finally, a practical example for students' performance in the multiple paper examination is highlighted for verifying the developed approach and demonstrates its practicality and effectiveness.
... PFS identifies three degrees: MD (µ), abstinence degree (AD) or neutral degree (γ), and NMD (υ), with the condition 0 ≤ µ + γ + υ ≤ 1. While PFS is widely used in decision-making [9,10], similarity measures, [11][12][13][14][15] correlation coefficients [16,17], and clustering, it becomes insufficient when µ+γ +υ > 1. To address this, Gungogdu and Kahraman introduced the Spherical fuzzy set (SFS), which extends PFS by satisfying 0 ≤ µ 2 + γ 2 + υ 2 ≤ 1. ...
A hesitant fuzzy (HF) set enhances the concept of fuzzy sets by addressing disagreements among decision-makers about the membership degree of an element. Similarly, the Cubical Fuzzy Set (CFS) is useful for managing uncertainty in decision-making problems. However, existing methods often lack integration of hesitation and cubical uncertainty, and there is limited exploration of their combined effects on aggregation processes. In this paper, we introduce the Hesitant Cubical Fuzzy Set (HCFS), which integrates the principles of HF sets and CFS to address these limitations. We define several set-theoretical operations for HCFSs and develop Dombi operations for them. Furthermore, we present a range of aggregation operators based on Dombi operations, including the Hesitant Cubical Dombi Fuzzy Weighted Arithmetic Averaging (HCDFWAA) Operator, the Hesitant Cubical Dombi Fuzzy Weighted Geometric Averaging (HCDFWGA) Operator, the Hesitant Cubical Dombi Fuzzy Ordered Weighted Arithmetic Averaging (HCDFOWAA) Operator, and the Hesitant Cubical Dombi Fuzzy Ordered Weighted Geometric Averaging (HCDFOWGA) Operator, and examine their properties. Additionally, we propose a multi-criteria group decision-making method and algorithm within the Hesitant Cubical Fuzzy framework. To address gaps in practical application, we provide an example of the selection of green suppliers in supply chain management. We also perform a comparative analysis with existing operators to highlight the advantages and effectiveness of our approach, emphasizing how the integration of hesitation and cubical uncertainty can enhance decision-making processes.
... (7) and (9) as interval-valued picture fuzzy aggregation operators. [32] if the term levels are set to p = q = r = 1. 2. The aggregation operators are specified in Eqs. (7) and (9) as interval-valued spherical fuzzy aggregation operators. ...
The p, q, r-spherical fuzzy set represents a recent advancement in fuzzy set theory, offering improved flexibility and realism for managing uncertainty in decision-making processes. Membership degrees in p, q, r-spherical fuzzy sets are typically represented as single-point real numbers. In this paper, we introduce interval-valued p, q, r-spherical fuzzy sets (IV (p,q,r) SFSs) as an extension of p, q, r-spherical fuzzy sets. IV (p,q,r) SFSs feature membership , neutral membership, and non-membership functions expressed as intervals rather than single-point real numbers. IV (p,q,r) SFSs feature three parameters (p, q, and r) that regulate the influence of membership grades in accordance with the requirements of the decision-making process. We establish operational laws and properties for these sets and propose aggregation operators, specifically interval-valued p, q, r-spherical fuzzy weighted averaging and interval-valued p, q, r-spherical fuzzy weighted geometric operators, to handle interval-valued information. The traditional TOP-SIS method is extended to address real-life multi-criteria group decision-making problems within the IV (p,q,r) SFS framework. We employ the entropy approach to compute criteria weights, while the Best-Worst method is utilized to determine expert weights. A numerical example concerning the selection of solar energy investment locations is presented to demonstrate the feasibility of our proposed method. Finally, a comparative analysis is conducted to validate the effectiveness of our approach against existing methodolo-gies.
... With a more nuanced representation of uncertainty, PFSs consider multiple membership functions, provide a comprehensive representation, model neutrality, ambiguity, and indifference, which are crucial characteristics in dealing with varying degrees of confidence or beliefs in a list of objects, for example, criteria (Khan et al., 2019). The presence of multiple factors in the evaluation of student competencies expressed as PFS requires aggregation operators, which are available in the literature, for example, picture fuzzy weighted average operators (Garg, 2017;Wei, 2017), picture fuzzy Dombi aggregation operators , picture fuzzy Aczel-Alsina aggregation operators (Hussain et al., 2024), picture fuzzy Bonferroni mean operators (Ateş & Akay, 2020), picture fuzzy Hamacher aggregation operators (Wei, 2018;, and picture fuzzy Einstein aggregation operators (Khan et al., 2019). Among these operators, Einstein aggregation operators possess interesting characteristics representative of real-life evaluation problems, including the notion of nonlinear aggregation, higher sensitivity to extreme values, and better handling of contradictory information, among others. ...
This work highlights an evaluation of blueprint reading competencies among university students, with particular attention to common and core competencies. Recognizing the ambiguity and imprecision arising from such an evaluation, Einstein aggregation operators on picture fuzzy sets were adopted to model the judgments of participants derived from a pool of mechanical technology students. Results reveal the students’ competency level for each pre-identified task in blueprint reading. Although they display above-average performances, areas requiring enhancement in both competency types are identified. Pathways from these findings involve various strategies: conducting a separate in-depth study for a deeper understanding of the subject matter, incorporating particular emphasis on blueprint reading tasks, introducing competency-based exercises within relevant courses, and facilitating industry experts’ collaboration. Comparative analysis with those of intuitionistic fuzzy sets and Dombi aggregation operators yields similar results.
... Ullah [12] formalized Maclaurin symmetric mean AOs for the PFS. Some other AOs for the PFS can also be found in [13,14]. ...
In urban transport decision-making, enhancing public participation is crucial for creating more inclusive and effective policies. The selection of digital voting tools is vital in facilitating this participation. This study aims to develop a decision-making model for evaluating and selecting digital voting tools based on uncertain factors. A multi-attribute group decision-making (MAGDM) approach is introduced for the evaluation and assessment of various digital voting tools, considering multiple aspects according to the needs of the policies and the stakeholders. A well-known framework called picture fuzzy rough est (PFRS) models the expert’s opinion concerning the tools considered. This study proposes the MAGDM model integrating the Schweizer-Sklar t-norm (SSTNrM) and the Schweizer-Sklar t-conorm (SSTCNrM), i.e., picture fuzzy rough weighted averaging (PFRSSWA) and picture fuzzy rough weighted geometric (PFRSSWG), are introduced to aggregate the data in the form of the picture fuzzy rough values (PFRVs). The created AOs select the most appropriate digital voting tool based on the characteristics provided in a given list. The ranking of the tools is observed by altering the values of the involved parameters in SSTNrM and SSTCNrM. The findings obtained also contrast with those of other known AOs. Additionally, a graphic representation of each observation and result is provided.
... Maclaurin symmetric mean AOs for the PFS were formalized by [15]. Some other AOs for the PFS can also be found in [16], [17]. ...
In sports, athletes’ initial training is crucial for their whole career. Initial training at the relevant board athletes are professionally trained for national and international competitions. However, the selection of potential athletes for traditional training is very hectic and uncertain due to the involvement of various factors. Thus, we need an intelligent approach that can efficiently deal with the uncertain factors involved in selecting and recommending athletes for training camp. A multi-attribute group decision-making (MAGDM) approach offers a structured framework for evaluating and selecting athletes, considering various attributes that reflect stakeholder preferences and needs. In MAGDM, the human point of view is crucial. Several frameworks deal with the extraction of information in the decision-making process. The interval-valued picture fuzzy rough set (IVPFRS) is a noteworthy framework that contributes significantly to lowering the level of uncertainty in the data derived from real-world scenarios. In this study, new aggregation operators (AOs) based on the Schweizer-Sklar t-norm (SSTN) and the Schweizer-Sklar t-conorm (SSTC), i.e., interval-valued picture fuzzy rough weighted averaging (IVPFRSSWA) and interval-valued picture fuzzy rough weighted geometric (IVPFRSSWG), are developed to aggregate the data in the form of the interval-valued picture fuzzy rough values (IVPFRVs). After their fundamental qualities are examined, the created AOs are used to solve the MAGDM problem. The results can be varied by altering the values of the involved parameters in SSTN and SSTC. The findings obtained also contrast with those of other known AOs. Additionally, a graphic representation of each observation and result is provided.
... Inspired by the implementation of PFS in the process of making decisions, [11] put forth several operators of aggregations inside the framework of PFS and demonstrated a method of constructing decisions utilizing the suggested operators of aggregations. In 2017, Wang [12] presented a geometric aggregation operator based on picture fuzzy sets and used score and accuracy functions to compare two picture fuzzy numbers (PFNs). Wang [13] suggested the cross entropy of PFSs and employed PFS in a decision-making problem. ...
This paper introduced quartic Bézier curve approximation model by using picture fuzzy set
approach. Firstly, picture fuzzy control point is introduced by using basic concepts of picture fuzzy set such
as picture fuzzy number and picture fuzzy relation. Next, the picture fuzzy control point is blended with the
Bernstein function in order to construct picture fuzzy quartic Bézier curve model with degree n=4. Later
the curves is generated through approximation method which comprised of the positive membership,
neutral membership, negative membership, and refusal membership. Finally some numerical example of
picture fuzzy quartic Bézier curve model is visualized and its properties is shown.
... where ( ) 10) and the state ...
This study introduced the picture fuzzy Bézier curve by using interpolation technique. Firstly,
some basic concepts such as picture fuzzy set, picture fuzzy number and picture fuzzy relation is discussed
to introduced picture fuzzy control point relation. Next, picture fuzzy control point relation is blended with
the Bézier basis function to construct picture fuzzy Bézier curve model. Then, the introduced curve is
generated and visualized by using an interpolation method. Finally, some numerical example and an
algorithm in constructing the desired curve is shown.
... As shown in Table 10, it is obvious that the extant methods based on the PF-aggregation operators PFWA [69], PFEWA [70], PFHWA [71], PFHWG [71] and PFWIA [75] can acquire the final rating value. However, the final ranking by the methods based on those aggregation operators cannot obtain the best candidate because of the same highest values of ranking. ...
Deepening the integration of industry and education (IIE) in classroom teaching has important strategic significance for the connotative development of higher education. The classroom teaching quality assessment is an important route to improve the level of the IIE and then promote the quality of talent training. Considering that picture fuzzy sets (PFSs) is much more efficient in comparison with fuzzy sets at handling the uncertainty in decision-making problems, this paper firstly applies PFSs to depict the indeterminacy and inaccuracy information in teaching quality assessment (TQA) process. To begin with, the related definitions of PFSs are all profiled successively. Then, we attempt to propose an integrated assessment method with the combined compromise solution (CoCoSo) and Taxonomy method (TM) to handle the TQA problem with PFSs. In this approach, we define a new cumulative method based on TM to aggregate individual opinion into group opinion. We also develop a weighting strategy based on the CRiteria Importance Through Intercriteria Correlation approach (CRITIC) to evaluate the significance of experts with PFSs. For this, we define a novel generalized chordal picture fuzzy (PF) distance measure that considers the marginal impacts of degree of refusal membership, which has a strong capacity of differentiation. Then, we evaluate its superiority and stability through some experiment comparisons. Again, a final ranking method is presented with the CoCoSo approach and the Stepwise Weight Assessment Ratio Analysis (SWARA) method. Furthermore, we apply a case study of the TQA to demonstrate the implementation of the newly proposed PF-CRITIC-SWARA-TM -CoCoSo method. The results obtained from the sensitivity analysis validate that the option “Dr. Tang” consistently achieves the highest rank and is independent of variations of balancing factor and weight information of experts and criteria. Finally, a comparison is implemented to confirm the robustness and reliability of the suggested integrated framework.
... Definition 3. (Ashraf et al., 2019;Garg, 2017;Wei, 2017): Let us have PiFNs e A j ðj ¼ 1; 2; . . . :; nÞ be a group of PiFNs and the PiFWG ("picturefuzzy-weighted-geometric-average") operator and PiFWA (picturefuzzy-weighted-arithmetic-average) operator related to w ¼ ðw 1 ; w 2 ; ...
Purpose
The main goal of this research is to analyze and assess the barriers to Digital Transformation (DT) of the healthcare Supply Chain (SC) in developing countries and evaluate strategies to overcome such barriers. The DT has been related not only to the development of SC performance but also to the expansion of its resilience capabilities in a healthcare setting to overcome the results of unavoidable risk events and return to its previous or new state, which has at least the same or better conditions, after the occurrence of an unpredicted event.
Design/methodology/approach
Assessment of the barriers and determination of the importance or effectiveness of proposed strategies to handle these barriers are conducted through a hybrid Multi-Criteria-Decision-Making method consisting of an Analytical Hierarchy Process and multi-attributive border approximation area comparison (MABAC) under picture fuzzy (PiF) environment.
Findings
Based on the outcomes of the research, the most important barriers for DT in healthcare SC functions were the organizational and technological issues, including Low support from top managers (O1), Lack of digital culture (O2), and Info/data safety and privacy (T4). To handle these barriers, the prior strategy was selected as “Making breakthroughs and incentives of top managers for the DT in the industry (ST3)”.
Research limitations/implications
The study may be extended to include developed countries and or experts from developed countries. Through this, the study will be generalized to a global level. The most common qualitative criteria, which include subjectivity, are considered. The research may be enhanced by including quantitative criteria in the decision-making process. The barriers related to DT for healthcare SC were considered. Thus, the study may be detailed by looking at the problem in terms of operational failure and determining the digital technology that can overcome such operational failures. The precedence or importance of any barrier may fluctuate with the upgradation of technological tools and improvements.
Practical implications
In terms of practical contribution, this research provides real-world suggestions to facilitate digital transformation in their SCs. Specifically, this study increases the awareness of healthcare and other industries' managers about obstacles that prevents digital transformation in SC and provides managerial path to relieve the effects of barriers.
Originality/value
To the authors' best knowledge, this is the first study to examine barriers of DT required for the promotion of resilient healthcare SCs in this concept. In addition, no previous research applied PiFSs-based AHP and MABAC integration to assess barriers and develop strategies for the DT of healthcare SC. The findings may be references for governmental institutions, policymakers, decision-makers, and stakeholders to develop proper strategies for a successful DT in healthcare SC resilience and the healthcare industry.
... Recently, several decision-making models have been proposed under the PFSs (Cuong 2013a, b) environment. Garg (2017) proposed several picture fuzzy weighted averaging operators for multicriteria decision-making. Wang et al. (2017) developed some picture fuzzy geometric aggregation operators and applied them to MCDM problems. ...
In real-life decision-making, expressing uncertainty, impreciseness, and hesitancy accurately is essential. Interval-valued spherical fuzzy sets (IVSFS) offer a suitable framework as an extension of interval-valued intuitionistic fuzzy sets, interval-valued picture fuzzy sets, and spherical fuzzy sets, allowing for interval-valued membership grades rather than exact values. This enhanced expressiveness enables more effective modeling of real-life decision-making problems by introducing suitable aggregation operators. In this paper, we propose the interval-valued spherical fuzzy Frank Choquet integral (IVSFFCI) and the interval-valued spherical fuzzy Frank geometric Choquet integral (IVSFFGCI) operators. These operators effectively capture the interaction among the criteria in real-life decision-making problems, overcoming the limitations of traditional methods. The IVSFFCI and IVSFFGCI operators utilize Frank’s t-norm and t-conorm, providing flexibility and robustness during the aggregation process. By considering the interrelation among the criteria, they exceed existing operators, making them the ideal choice for real-life decision-making situations. We develop a multicriteria decision-making (MCDM) method using the proposed operators that effectively deal with correlated criteria in real-life decision-making problems. To demonstrate the efficacy of the proposed method, an illustrative example relating to a financial body’s investment partner selection from four potential alternatives, based on criteria such as financial strength, mercantile expertise, entrepreneurial competencies, and risk management, is presented. The proposed method encapsulates immense potential across industries, promoting informed and data-driven decision-making processes.
... In the new concept, n þ 1 þ f 1. The new concept subsequently found great response among researchers [12][13][14][15][16][17][18][19]. However, here, as in the case of IFS, we have an important limitation, which in turn reduces the intellectual activities of experts in their evaluations. ...
In this article, the expert evaluations transformed in the multi-attribute decision-making (MADM) model are presented in the discrimination q-rung picture linguistic fuzzy numbers (q-RPLFNs). In the construction of a second-order additive fuzzy measure (TOAFM) the attributes’ interaction indexes and Shapley values are taken into account. The Shapley entropy maximum principle for identification of associated probabilities class (APC) of a TOAFM is constructed. Based on the APC of the TOAFM, a new aggregation operators’ class is constructed which represents some hybrid extensions of ordered weighted averaging (OWA), geometric (OWG), the Choquet integral averaging (CA) and geometric (CG) operators under discrimination q-rung orthopair fuzzy (q-ROF) and q-rung picture linguistic fuzzy (q-RPLF) information. These operators, constructed for the q-RPLF and q-ROF environments, take into account the overall pair interactions among attributes. Main properties on the correctness of extensions are proved: for the lower and upper capacities of order 2, all constructed operators consequently coincide with q-ROF and q-RPLF Choquet averaging and geometric operators, respectively. Constructed operators in the evaluation of prediction of fuzzy Collaborative Filtering Recommender Systems (CFRS) are used. New symmetric discrimination measures as some extensions of discrimination measures for the fuzzy CFRS are proposed. Users’ profile data by the constructed operators in the new similarity measure under q-rung picture linguistic environment are aggregated. The developed new approach is schematically described in such a way that it can be “embedded” in any existing CFRS model. An example is given to illustrate the results, for which the software designed to aggregate profile data for similarity comparison provides the use of new and well-known classical aggregation operators.
... Tanım 4: ̃1 = ( 1 , 1 1 ) ve ̃2 = ( 2 , 2 2 ) iki resim bulanık sayı olmak üzere, bu iki sayının kıyaslanması ve sıralanması için skor fonksiyonlarından yararlanılır. Skor fonksiyonu formülü Denklem 9'da gösterilmiştir [21]: ...
Demiryolu emniyeti, yolcuların ve çalışanların emniyetini garanti altına almak, seyrüsefer esnasında arıza yaşanma olasılığını azaltmak ve demiryolu işletmelerinin sorunsuz bir şekilde faaliyet göstermesini sağlamak açısından kritik bir öneme sahiptir. Demiryolu ulaşım emniyet önlemlerinin belirlenmesi ve sıralanması, demiryolu sektörünün barındırdığı çeşitli emniyet risklerinin ortaya çıkartılması ve bu risklerin yönetilebilmesi için temel bir adım olacaktır. Bu çalışmanın amacı, bir bulanık çok kriterli karar verme yöntemi kullanarak demiryolu emniyet önlemlerini değerlendirmektir. Çalışma kapsamında 8 farklı demiryolu ulaşım emniyet önlemi yazın taraması ve uzman ekibin görüşleri doğrultusunda belirlenmiştir. Değerlendirme için, karar vericilerin öznel değerlendirmelerindeki belirsizliği daha iyi modelleyen ve uzman görüşlerini ağırlıklandırma sürecine doğrudan dâhil eden Resim Bulanık SWARA yöntemi kullanılmıştır. Buna ek olarak, çalışma kapsamında farklı ağırlıklar altında her bir ölçütün karar verme süreci üzerindeki etkisini ölçmek amacıyla duyarlılık analizi gerçekleştirilmiştir. Ayrıca, çalışmada uygulanan Resim Bulanık SWARA yöntemine ait hesaplamaların etkinliğini ve sağlamlığını kontrol etmek amacıyla karşılaştırma analizi gerçekleştirilmiştir. Yazın taraması sonucunda, demiryolu emniyet önlemlerini kapsamlı şekilde ele alan ve bu önlemleri Resim Bulanık SWARA yöntemiyle sıralayan bir çalışmanın bulunmaması, bu çalışmanın literatüre önemli bir katkı sağladığını göstermektedir. Çalışmanın bulguları incelendiğinde; demiryolu ulaşım emniyeti açısından demiryolu altyapı periyodik bakımının ilk sırada, üstyapı bakımının ikinci sırada ve demiryolu taşıt bakımının üçüncü sırada öneme sahip olduğu ortaya çıkarılmıştır. Bu çalışmanın sonuçları demiryolu sektöründe çalışan karar vericilere ve araştırmacılara bilgi sağlamakta ve teknolojik olarak nerelere yatırım yapılması gerektiğini gözler önüne sermektedir.
... In a PFS environment, [14] proposed WA, OWA, and hybrid averaging operators. The Einstein operations on PFSs were first introduced by [23]. In [31] the most recently used the Hamachar operators in PFSs. ...
In today's data-driven landscape, to ensure continuous survival and betterment, the implementation of a robust Big Data Governance Framework (BDGF) is imperative for organizations to effectively manage and harness the potential of their vast data resources. The BDGF serves no purpose when implemented in a random manner. This article delves into the complex decision-making challenges that emerge in the context of implementation of the BDGF under uncertain conditions. Specifically, we aim to analyze and evaluate the BDGF performance using the Multi-Attribute Decision-Making (MADM) techniques aiming to address the intricacies of big data governance uncertainties. To achieve our objectives, we explore the application of Frank operators within the framework of complex picture fuzzy (CPF) sets (CPFs). We introduce complex picture fuzzy Frank weighted averaging (CPF-FWA) and complex picture fuzzy Frank ordered weighted averaging (CPFFOWA) operators to enable more accurate implementation of the BDGF. Additionally, we rigorously examine the reliability of these newly proposed fuzzy Frank (FF) operators (FFAOs), taking into consideration essential properties such as idempotency, monotonicity, and boundedness. To illustrate the practical applicability of our approach, we present a case study that highlights the decision-making challenges encountered in the implementation of the BDGF. Subsequently, we conduct a comprehensive numerical example to assess various BDGF implementation options using the MADM technique based on complex picture fuzzy Frank aggregation (CPFFA) operators. Furthermore, we perform a comprehensive comparative assessment of our proposed methodology, emphasizing the significance of the novel insights and results derived. In conclusion, this research article offers a unique and innovative perspective on decision-making within the realm of the BDGF. By employing the CPFFWA and the CPFFOWA operators, organizations can make well-informed decisions to optimize their BDGF implementations, mitigate uncertainties, and harness the full potential of their data assets in an ever-evolving data landscape. This work contributes to the advancement of decision support systems for big data governance (BDG), providing valuable insights for practitioners and scholars alike.
This paper presents a unique multi-Q valued bipolar picture fuzzy set (MQVBPFS) methodology to tackle issues in cybersecurity risk assessment under conditions of ambiguity and contradicting data. The MQVBPFS framework enhances classical fuzzy theory through three key innovations: (1) multi-granular Q-valued membership, (2) integrated bipolarity for representing conflicting evidence, and (3) refined algebraic operations, encompassing union, intersection, and complement. Contemporary fuzzy set methodologies, such as intuitionistic and image fuzzy sets, inadequately encapsulate positive, negative, and neutral membership degrees while maintaining bipolar information. Conversely, our MQVBPFS architecture effectively resolves this restriction. Utilizing this framework for threat assessment and risk ranking, we create a tailored cybersecurity algorithm that exhibits 91.7% accuracy (in contrast to 78.2–83.5% for baseline methods) and attains 94.6% contradiction tolerance in empirical evaluations, alongside an 18% decrease in false negatives relative to conventional approaches. This study offers theoretical progress in fuzzy set algebra and practical enhancements in security analytics, improving the handling of ambiguous and conflicting threat data while facilitating new research avenues in uncertainty-aware cybersecurity systems.
Picture fuzzy sets (PFSs) are an advanced extension of fuzzy sets that effectively address uncertainty and fuzzy information in the Multi-attribute Group Decision-Making (MAGDM) process. In this paper, we propose an extended TODIM model based on CRiteria Importance Through Inter criteria Correlation (CRITIC) method and Regret theory (RT) in picture fuzzy environment. First, an improved CRITIC method with PFSs is developed to deal with the decision problems of unknown criteria weights. Then, a modified TODIM method based on picture fuzzy distance is introduced to calculate the overall dominance degrees of alternatives. After that, considering that decision-makers have limited rationality, with the psychology of loss aversion and regret avoidance in the evaluation process, the regret theory is used to select the best scheme by calculating the perceived utility values. Finally, the proposed PF–CRITIC–TODIM–RT method is applied to green suppliers selection, and comparative analyses are also conducted to demonstrate its superiority in capturing nuanced differences, handling unknown criterion weights, and incorporating psychological behavior. Additionally, Spearman’s correlation tests validate the method’s reliability and stability, showing it outperforms other MADM methods in terms of consistency and adaptability.
The objective of this paper is to develop picture fuzzy aggregation operators by utilizing the concept of power aggregation operators through Schweizer-Sklar operations. The Schweizer-Sklar t-norm and t-conorm enhance the flexibility of the data integration process and the power aggregation operator, by capturing the interrelationships between various criteria during decision-making. Motivated by Schweizer-Sklar t-norm and t-conorm, this paper aims to develop the theory of the picture fuzzy Schweizer-Sklar power weighted geometric operator and the picture fuzzy Schweizer-Sklar power ordered weighted geometric operator. The paper also explores the properties and characteristics of these proposed operators. Criteria weights play a crucial role in aggregating different criteria in multiple criteria decision-making processes. This work adopts the simple multi-attribute rating technique to compute criteria weights for solving multi-criteria group decision-making problems in a picture fuzzy environment. Finally, an illustrative example of the road construction company is provided to demonstrate the applicability of the proposed operators. A comparison with existing operators validates the effectiveness of the proposed operators.
Purpose
Since conducting agile strategies provides sustainable passenger satisfaction and revenue by replacing applied policies with more profitable ones rapidly, the focus of this study is to evaluate agile attributes for managing low-cost carriers (LCCs) operations by means of resources and competences based on dynamic capabilities built on resource-based view (RBV) theory and to achieve sustainable competitive advantage in a volatile and dynamic air transport environment. LCCs in Turkey are also evaluated in this study since the competition among LCCs is high to gain market share and they can adapt quickly to all kinds of circumstances.
Design/methodology/approach
Two well-known Multi-Criteria Decision-Making Methods (MCDM) named as the Stepwise Weight Assessment Ratio Analysis (SWARA) and multi-attributive border approximation area comparison (MABAC) methods by employing Picture fuzzy sets (PiFS) are employed to determine weight of agile attributes and superiority of LCCs based on agile attributes in the market, respectively. To check the consistency and robustness of the results for the proposed approach, comparative and sensitivity analysis are performed at the end of the study.
Findings
While the ranking orders of agile attributes are Strategic Responsiveness (AG1), Financial Management (AG4), Quality (AG2), Digital integration (AG3) and Reliability (AG5), respectively, LCC2 is selected as the best agile airline company in Turkey with respect to agile attributes. SWARA and MABAC method based on PiFS is appropriate and effective method to evaluate agile attributes that has important reference value for the airline companies in aviation industry.
Practical implications
The findings of this study will support managers in the airline industry to conduct airline operations more flexibly and effectively to take sustainable competitive advantage in unexpected and dynamic environment.
Originality/value
To the author' best knowledge, this study is the first developed to identify the attributes necessary to increase agility in LCCs. Thus, as a systematic tool, a framework is developed for the implementation of agile attributes to achieve sustainable competitive advantage in the airline industry and presented a roadmap for airline managers to deal with crises and challenging situations by satisfying customer and increasing competitiveness.
The study explores the utilization of Hamacher aggregation operators (HAOs) in inscription selection for medical health projects. We employ interval-valued complex T-spherical fuzzy (IVCTSF) information to address the inherent uncertainties in healthcare data. In this paper, we develop the multiple-attribute decision-making (MADM) problems with IVCTSF set information. A few HAOs built on IVCTSF sets are presented in this work. We practice the Hamacher t-norm (HTNM) and Hamacher t-conorm (HTCNM) to characterize certain operational Hamacher operational rules within the context of the IVCTSF sets. We utilize averaging and geometric operations to develop a family of operators for aggregating IVCTSF information, namely IVCTSF Hamacher weighted averaging (IVCTFHWA), IVCTSF Hamacher order weighted averaging (IVCTSFHOWA), IVCTSF Hamacher hybrid weighted averaging (IVCTSFHHWA), IVCTSF Hamacher weighted geometric (IVCTSFHWG), IVCTSF Hamacher ordered weighted geometric (IVCTSFHOWG), and IVCTSF Hamacher hybrid weighted geometric (IVCTSFHHWG) operators. Several noteworthy properties of the developed operators are examined. Besides, an approach to the MADM algorithm is formulated using the proposed operators and is applied to a detailed case study. The case study measures the effectiveness of the proposed algorithm, analyzes the effect of variable parameters on the decision-making procedure, and ensures the stability of ranking results. A comparative analysis is conducted against existing studies to underscore the significance and advantages. Our findings demonstrate the effectiveness of this approach in improving decision-making for healthcare management in complex scenarios.
For certified public accountant (CPA) firms to maintain high levels of competent assistance, achievement evaluations are essential. However, because aspects like customer fulfilment, productivity, economic success, and legality include personal and interconnected assessments, standard approaches to assessment frequently fall short of addressing the inherent complexity and uncertainty in decision-making. To overcome these issues, this paper suggests a unique multiple-criteria decision-making (MCDM) model based on complex T-Spherical fuzzy Frank (CTSFF) operations. The approach uses complex T-spherical fuzzy Frank Weighted Averaging (CTSFFWA) and complex T-spherical fuzzy Frank Weighted Geometric (CTSFFWG) operators to manage multi-attribute assessment under unclear and uncertain circumstances. The ability to handle subjective and unclear data while integrating interconnections between several assessment requirements is lacking in the practical assessment methods currently used by CPA companies. This restriction makes obtaining trustworthy and useful information for enhancing company operations more difficult. The intricacy of contemporary CPA operations, such as the increasing requirement to consider qualitative assessments and a range of client viewpoints, calls for creating a strong and flexible structure to support decision-making. The inability of conventional fuzzy frameworks to accurately represent these subtleties necessitates using sophisticated techniques. To retain mathematical accuracy while accounting for ambiguity via membership, non-membership, and abstention degrees, the suggested model uses CTSFF logic. A thorough assessment of CPA company achievement through various factors is made possible by the improved aggregation possibilities introduced using CTSFFWA and CTSFFWG operators. Flexibility evaluation is additionally built into the framework to guarantee the findings’ reliability and pinpoint ideal factor setups. This research gives decision-makers a versatile instrument to modify the framework for various expert business sectors. It offers a methodical and comprehensive way to assess the efficiency of CPA firms. This investigation makes an essential contribution to the discipline of decision-making under uncertainty by filling in the deficiencies in the assessment methods that are currently in use.
This article introduces the concept of m-polar picture fuzzy sets to address the inherent ambiguity in sustainability-related decision-making in Industry 4.0. We present a set of novel aggregation operators specifically developed for m-PPFSs, including the m-polar picture fuzzy weighted averaging operator, which assigns different weights to each element, and the ordered weighted averaging operator, which aggregates elements based on their ranked values and assigned weights. Additionally, the weighted geometric operator applies a geometric mean approach with weighted inputs, and the ordered weighted geometric operator combines the properties of ordered values and geometric aggregation. These operators provide versatile tools for handling m-PPFS data in various decision-making and computational contexts, playing a critical role in enhancing the multi-criteria decision-making process. Our framework further enables the application of blockchain technology for tracking raw material provenance, optimizing resource utilization, automating waste recycling through smart contracts, and supporting circular supply chain practices. In addition, Industry 4.0 technologies such as the Internet of Things and advanced analytics enable real-time monitoring and predictive maintenance, strengthening sustainability efforts. To evaluate the performance of the proposed aggregation operators, we compare them with existing methods in fuzzy set theory. The results demonstrate that our approach offers superior precision, robustness, and adaptability, outperforming traditional methods in addressing sustainability challenges within the manufacturing sector. In this study, we proposed novel aggregation operators that significantly improve efficiency, accuracy, and flexibility over existing methods, offering robust solutions to sustainability-related decision-making challenges.
Multi-attribute decision-making problems can be solved using a Fermatean vague set. Fermatean vague sets are extension of vague sets. We initiated generalized Fermatean vague weighted averaging, generalized Fermatean vague weighted geometric, power generalized Fermatean vague weighted averaging and power generalized Fermatean vague weighted geometric. The algebraic structures such as associative, distributive, idempotent, bounded, commutativity and monotonicity properties are satisfied by generalized Fermatean vague numbers. We discussed some mathematical properties of these sets, as well as the Hamming distance and Euclidean distance. An algorithm that uses aggregation operators to solves multi-attribute decision-making problems. The fields of computer science and medicine are essential for medical diagnosis research. Choosing cancer treatment is a complex process. Patients and health care professionals must communicate effectively to make the right decision about their cancer treatment. There are a number of factors that influence patients treatment decisions. As far as cancer patients are concerned, there are five types of cancer patients. Many different treatments are currently available and they are often combined as part of an overall treatment plan involving various treatment options. Patients with which cancer can be treated in various ways, such as cystoscopy, biopsy, blood tests and CT scan of the urogram. We intended to choose the best treatment based on our comparisons and options. Thus, it is evident that natural number �has a significant influence on the models. Additionally, flowchart based multi-criteria decision-making is offered and applied to a numerical example to show the efficiency of the recommended approach. The results are evaluated for different values of the parameter. Moreover, a comparative analysis has been performed to illustrate the superior outcomes of the suggested approach in comparison with existing methodologies.
We introduce the concept of Diophantine spherical vague set approach to multiple-attribute decision-making. The Spherical vague set is a novel expansion of the vague set and interval valued spherical fuzzy set. Three new concepts have been introduce such as Diophantine spherical vague weighted averaging operator, Diophantine spherical vague weighted geometric operator, generalized Diophantine spherical vague weighted averaging operator and generalized Diophantine spherical vague weighted geometric operator. We provide a numerical example to show how Euclidean distance and Hamming distance interact. Applications of the Diophantine spherical vague number include idempotency, boundedness, commutativity and monotonicity in algebraic operations. They can determine the optimal option and are more well-known and reasonable. Our goal was to identify the optimal choice by comparing expert opinions with the criteria. As a result, the model’s output was more accurate as well as in the range of the natural number ⅁. The weighted averaging distance and weighted geometric distance operators are distance measure that is based on aggregating model. By comparing the models under discussion with those suggested in the literature, we hoped to show their worth and reliability. It is possible to find a better solution more quickly, simply, and practically. Our objective was to compare the expert evaluations with the criteria and determine which option was the most suitable. Because they yield more precise solutions, these models are more accurate and more related to models with ⅁. To show the superiority and the validity of the proposed aggregation operations, we compared it with the existing method and concluded from the comparison and sensitivity analysis that our proposed technique is more effective and reliable. This investigation yielded some intriguing results.
The concept of the picture fuzzy set (PiFS) significantly enhances the multi-criteria decision-making (MCDM) process by incorporating membership value (MV), non-membership value (NMV), and a neutral component. PiFS extends the capabilities of traditional fuzzy sets (FSs), intuitionistic fuzzy sets (IFSs), and other fuzzy models. This paper introduces a novel MCDM approach, the picture fuzzy SWARA-CRITIC-COPRAS (PiF-SCC) method, specifically designed to assist decision-makers (DMs) in evaluating and selecting dynamic digital marketing (DDM) technologies within PiFS settings. The proposed method integrates the strengths of PiFS with step-wise weight assessment ratio analysis (SWARA), criteria importance through intercriteria correlation (CRITIC), and complex proportional assessment (COPRAS), aiming to improve the precision and effectiveness of technology evaluations. To validate the approach, a case study is conducted on DDM technology assessment within a specific business context. The PiF-SCC technique is applied to rank technological options using linguistic terms (LTs), PiFS numbers, an accuracy function (AF), and a score function (SF). Additionally, a comprehensive sensitivity analysis is performed to evaluate the robustness of the proposed method under different input scenarios and uncertainties. A thorough comparison with existing techniques is also provided, demonstrating the superior decision-making capability of the new approach, which leads to more accurate and dependable technology selection results. The manuscript also discusses marginal implications and limitations, along with potential future research directions to further enhance the applicability and effectiveness of the proposed approach.
One of the most typical examples of interactive multi-criteria decision making (MCDM) is modeling of a prediction by the collaborative filtering recommender systems (CFRS), the class of methods which recommends items to users (customers) on the basis of preferences of other users on these items. Basing on the preferences over items made by users in the past, a CFRS generates a class of users similar to the targeted user, and then recommends those items which were approved by the users from the generated class. In many cases, existing similarity relations cannot provide reliable prediction of the item recommendations in CFRS applications, which is due not only to users’ interests in objects of complex structure, but also to the lack of subjective information, which is related to the users’ social status, in short with his profile data. In many cases, in CFRS models, it is necessary to consider the interactions of model criteria and their individual degrees of dominance and influence on possible predictive decisions. The idea of our approach is based on the use of the possibility theory, when criteria importance levels and criteria pair interaction indexes in the model environment are evaluated by the decision maker or the experts, people who are involved in the assessments. In our modeling scheme the generated possibility degrees of influence on the alternatives of the criteria, take into account the values of criteria pairwise interaction indexes. Based on the principle of maximum for the Shapley entropy determined on the criteria, a mathematical programming problem is formulated, the solution of which is the possibility measure’s distribution generated on the set of criteria. We use the generated possibility measure in the definition of extensions of aggregation operators such as the ordered weighted averaging (OWA) and finite Choquet averaging (CA) operators. The confidence discrimination q-rung picture linguistic fuzzy (CD-q-RPLF) environment of expert evaluations are considered as aggregation arguments. New constructed confidence q-rung picture linguistic fuzzy ordered weighted averaging (C-q-RPLFOWA) and confidence q-rung picture linguistic fuzzy Choquet averaging (C-q-RPLFCA) operators are used in the evaluation of predictions of the CFRS. We develop CD-q-RPLF CFRS methodology, where users’ profile data by the constructed operators in the new users’ similarity measure under q-RPLF environment are aggregated. New extensions of discrimination values in the constructions of users’ similarity measures are included. Classical Jaccard index is transformed into discrimination q-RPLFNs. The main goal of the results illustration was to aggregate users’ profile data by the constructed operators in the similarity measure under confidence discrimination q-RPLF environments. For the illustration of received results, the prediction problem of the constructed Collaborative Filtering Recommender Systems is considered.
In literature, several tools have been developed to cope with ambiguity in data. Picture fuzzy set (PFS) is a very significant framework for extracting the maximum information from real-life phenomena with minimum uncertainty. Consequently, solving the muti-attribute decision-making (MADM) with the help of the PFS would be certainly useful. In this study, a short note on basic terms is provided for a better understanding of the article. A new class of operators picture fuzzy Schweizer–Sklar Maclaurin symmetric mean, picture fuzzy Schweizer–Sklar weighted Maclaurin Symmetric mean, picture fuzzy Schweizer–Sklar dual Maclaurin symmetric mean and picture fuzzy Schweizer–Sklar dual weighted Maclaurin symmetric mean is introduced. Some of the introduced AOs are applied to a real-life problem with the help of an illustrative example. For significance, the introduced AOs are compared to some existing AOs. The disparity of the results with the change in the involved parameters is also studied. The results obtained are tabulated and graphed.
This article presents a novel approach to a decision support system for handling uncertainty and impreciseness in a large amount of human opinion. Sometimes, the aggregation of real-life applications is quite complex due to incomplete and redundant information about different preferences or alternatives. To handle such type of situations, a spherical fuzzy environment is a more effective and feasible framework with four components membership, abstinence, non-membership and refusal degree. We also formulate some flexible operations of Sugeno-Weber aggregation operators. Motivated by the theory of Sugeno-Weber t-norms, we constructed a family of mathematical methodologies, including Sugeno-Weber weighted average and weighted geometric operators in the light of spherical fuzzy information. An appropriate decision-making technique of the multi-attribute decision making (MADM) problem is also demonstrated to resolve complicated real-life applications. A numerical example is used to verify the compatibility and effectiveness of discussed mathematical approaches.
This paper introduces a novel approach to enhance uncertainty representation, offering decision-makers a more comprehensive perspective for improved decision-making outcomes. We propose Generalized Orbicular (m,n,o) T-Spherical Fuzzy Set (GO-TSFS), a flexible extension of existing fuzzy set models including Globular T-spherical fuzzy sets (G-TSFSs), T-spherical fuzzy sets (T-SFSs), (p,q,r) Spherical fuzzy sets, and (p,q) Quasirung orthopair fuzzy sets (QOFSs). The framework employs three adjustable parameters m, n, and o to finely tune the influence of membership degrees, allowing for adaptable weighting of various degrees of membership. By utilizing spheres to represent membership, indeterminacy, and non-membership levels, the model enhances accuracy in depicting vague, ambiguous, and imprecise data. Building upon the foundation of GO-TSFSs, we introduce essential set operations and algebraic operations for GO-TSF Values (GO-TSFVs). Moreover, we also develop score functions, accuracy functions, and basic distance measures such as Hamming and Euclidean distances to further enhance the analytical capabilities of the framework. Additionally, we propose GO-TSF Hamacher Weighted Averaging (GO-TSFHWA) and GO-TSFH Weighted Geometric (GO-TSFHWG), aggregation operators tailored for our proposed sets. To demonstrate the practical applicability of our approach, we apply our proposed aggregation operators namely GO-TSFHWA and GO-TSFHWG to solve a Multi-Criteria Group Decision Making (MCGDM) problem, specifically for selecting the most suitable e-commerce online shopping platform from the top-rated options. Sensitivity analysis is also conducted to validate the reliability and efficacy of our results, affirming the utility and robustness of the proposed methodology in real-world decision-making scenarios.
Pythagorean fuzzy set, an extension of the intuitionistic fuzzy set which relax the condition of sum of their membership function to square sum of its membership functions is less than one. Under these environment and by incorporating the idea of the confidence levels of each Pythagorean fuzzy number, the present study investigated a new averaging and geometric operators namely confidence Pythagorean fuzzy weighted and ordered weighted operators along with their some desired properties. Based on its, a multi criteria decision-making method has been proposed and illustrated with an example for showing the validity and effectiveness of it. A computed results are compared with the aid of existing results.
In this paper, we investigate the multiple attribute decision making problems with picture fuzzy information. The advantage of picture fuzzy set is easily reflecting the ambiguous nature of subjective judgments because the picture fuzzy sets are suitable for capturing imprecise, uncertain, and inconsistent information in the multiple attribute decision making analysis. Thus, the cross entropy of picture fuzzy sets, called picture fuzzy cross entropy, is proposed as an extension of the cross entropy of fuzzy sets. Then, a multiple attribute decision making method based on the proposed picture fuzzy cross entropy is established in which attribute values for alternatives are picture fuzzy numbers. In decision making process, we utilize the picture fuzzy weighted cross entropy between the ideal alternative and an alternative to rank the alternatives corresponding to the cross entropy values and to select the most desirable one(s). Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.
In this paper, some series of new intuitionistic fuzzy averaging aggregation operators has been presented under the intuitionistic fuzzy sets environment. For this, some shortcoming of the existing operators are firstly highlighted and then new operational law, by considering the hesitation degree between the membership functions, has been proposed to overcome these. Based on these new operation laws, some new averaging aggregation operators namely, intuitionistic fuzzy Hamacher interactive weighted averaging, ordered weighted averaging and hybrid weighted averaging operators, labeled as IFHIWA, IFHIOWA and IFHIHWA respectively has been proposed. Furthermore, some desirable properties such as idempotency, boundedness, homogeneity etc. are studied. Finally, a multi-criteria decision making method has been presented based on proposed operators for selecting the best alternative. A comparative concelebration between the proposed operators and the existing operators are investigated in detail.
In this paper, group decision making methods based on intuitionistic fuzzy multiplicative preference relations has been developed. For it, firstly some new operational laws on intuitionistic multiplicative numbers have been defined and then by using these operations some new intuitionistic fuzzy multiplicative interactive weighted geometric, intuitionistic fuzzy multiplicative interactive ordered weighted geometric and intuitionistic fuzzy multiplicative interactive hybrid weighted geometric operators have been developed. Some desirable properties of these operators, such as idempotency, boundedness, monotonicity etc., are studied in the paper. The major advantage of the proposed operators as compared to existing ones are that it consider the proper interaction between the membership and non-membership functions and proposed operators are more pessimistic than existing ones. Furthermore, these operators are applied to decision making problems in which experts provide theory preference relation by intuitionistic fuzzy multiplicative intuitionistic fuzzy environment to show the validity, practicality and effectiveness of the new approach. Finally, a systematic comparison between the existing work and the proposed work has been given.
In this paper, we propose some new aggregation operators which are based on the Choquet integral and Einstein operations. The operators not only consider the importance of the elements or their ordered positions, but also consider the interactions phenomena among the decision making criteria or their ordered positions. It is shown that the proposed operators generalize several intuitionistic fuzzy Einstein aggregation operators. Moreover, some of their properties are investigated. We also study the relationship between the proposed operators and the existing intuitionistic fuzzy Choquet aggregation operators. Furthermore, an approach based on intuitionistic fuzzy Einstein Choquet integral operators is presented for multiple attribute decision-making problem. Finally, a practical decision making problem involving the water resource management is given to illustrate the multiple attribute decision making process.
Picture fuzzy sets are extension of Atanassov’s intuitionistic fuzzy sets. Picture fuzzy set based models may be adequate in situations when we face human opinions involving more answers of types: yes, abstain, no, refusal. It can be considered as a powerful tool represent an uncertain information in the process of cluster analysis. In this paper, we present a geometrical interpretation of picture fuzzy sets. We propose correlation coefficients for picture fuzzy sets which considers the degree of positive membership, degree of neutral membership, degree of negative membership and the degree of refusal membership. Effectiveness of the proposed correlation coefficient has been established in a bidirectional approximate reasoning systems. We
apply the correlation coefficient to clustering analysis under picture fuzzy environments. Advantages of proposed correlation coefficients and drawbacks of existing correlation coefficients have been discussed.
A definition of the concept ‘intuitionistic fuzzy set’ (IFS) is given, the latter being a generalization of the concept ‘fuzzy set’ and an example is described. Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.
This paper proposed a method to resolve the multi‐attribute decision‐making problem using TOPSIS method based on attribute weights and attribute values are all interval vague value. Firstly, based on the operation rules of the interval Vague value, the interval Vague attribute value is made by weighted operation, and the ideal and negative ideal solutions are calculated based on the score function. Then the distance of interval Vague value is defined, as well as the distance between each project and the ideal, and negative ideal solutions. The relative adjacent degree is calculated by TOPSIS method, then the order of the projects is confirmed according to the relative adjacent degree. Finally, a case is used to show the process of the method this paper proposed and the validity of this method is proved.
Santrauka
Straipsnyje siūlomas daugiakriterinės sprendimo priėmimo problemos sprendimas TOPSIS metodu, kai kriterijų reikšmingumai ir reikšmės yra intervaliniai dydžiai. Iš pradžių, naudojantis procedūromis, nustatomos svertinės intervalinių dydžių reikšmės, paskui apskaičiuojami idealiai teigiamas ir idealiai negiamas sprendiniai. Toliau nustatomi intervalų dydžiai, apskaičiuojami atstumai tarp kiekvienos alternatyvos ir idealiai teigiamo ir idealiai neigiamo sprendinių. TOPSIS metodu apskaičiuojami santykiniai atstumai iki minėtų idealių sprendinių ir alternatyvos išrikuojamos į eilę. Galiausiai konkrečiu pavyzdžiu demonstruojamas skaičiavimo procesas ir patvirtinamas siūlomo metodo pagrįstumas.
First published online: 21 Oct 2010
Reikšminiai žodžiai: intervalinės neapibrėžtos reikšmės, funkcija, TOPSIS, daugiakriterinis sprendimų priėmimas.
The ELECTRE II and III methods enjoy a wide acceptance in solving multi-criteria decision-making (MCDM) problems. Research results in this paper reveal that there are some compelling reasons to doubt the correctness of the proposed rankings when the ELECTRE II and III methods are used. In a typical test we first used these methods to determine the best alternative for a given MCDM problem. Next, we randomly replaced a non-optimal alternative by a worse one and repeated the calculations without changing any of the other data. Our computational tests revealed that sometimes the ELECTRE II and III methods might change the indication of the best alternative. We treat such phenomena as rank reversals. Although such ranking irregularities are well known for the additive variants of the AHP method, it is the very first time that they are reported to occur when the ELECTRE methods are used. These two methods are also evaluated in terms of two other ranking tests and they failed them as well. Two real-life cases are described to demonstrate the occurrence of rank reversals with the ELECTRE II and III methods. Based on the three test criteria presented in this paper, some computational experiments on randomly generated decision problems were executed to test the performance of the ELECTRE II and III methods and an examination of some real-life case studies are also discussed. The results of these examinations show that the rates of the three types of ranking irregularities were rather significant in both the simulated decision problems and the real-life cases studied in this paper.
Keywords: Multi-criteria decision-making; Ranking irregularities; ELECTRE methods; The analytic hierarchy process (AHP); Multiplicative AHP
The basic definitions of the concept of interval-valued intuitionistic fuzzy set and of the operations, relations and operators over it are given. Some of the most important applications are described. Ideas for future development of the theory of interval-valued intuitionistic fuzzy sets are discussed.
In this paper, some series of averaging aggregation operators have been presented under the intuitionistic fuzzy environment by considering the degrees of hesitation between the membership functions. For it, firstly, shortcoming of some existing aggregation operators has been identified and then new operational laws have been proposed for overcoming these shortcoming. Based on these operations, weighted, ordered weighted and hybrid averaging aggregation operators have been proposed by using Einstein operational laws. Furthermore, some desirable properties such as idempotency, boundedness, homogeneity etc. are studied. Finally, a multi-criteria decision making (MCDM) method has been presented based on proposed operators and compare their performance with the existing operators.
The objective of this paper is to present some series of geometric-aggregated operators under Pythagorean fuzzy environment by relaxing the condition that the sum of the degree of membership functions is less than one with the square sum of the degree of membership functions is less than one. Under these environments, aggregator operators, namely, Pythagorean fuzzy Einstein weighted geometric, Pythagorean fuzzy Einstein ordered weighted geometric, generalized Pythagorean fuzzy Einstein weighted geometric, and generalized Pythagorean fuzzy Einstein ordered weighted geometric operators, are proposed in this paper. Some of its properties have also been investigated in details. Finally, an illustrative example for multicriteria decision-making problems of alternatives is taken to demonstrate the effectiveness of the approach.
The objective of this manuscript is divided into two fold. Firstly, a more generalized intuitionistic fuzzy entropy measure of order and degree has been presented for measuring the degree of fuzziness of the set with a proof of its validity. A structured linguistic variable has been taken as an illustrative example to show its validity and superiority than the existing measures. Furthermore, based on this measure, an approach to deal with multi-criteria decision making (MCDM) problem is developed. Finally, a practical example is provided to illustrate the decision making process. A computed result is compared with the help of existing results. A sensitivity analysis on the different values of the parameters will make a decision maker more choice for accessing their results.
The objective of this paper is to focus on multi-attribute decision-making for interval-valued intuitionistic fuzzy set environment based on set pair analysis (SPA). For it, the major component of the SPA known as connection number has been constructed based on the set pairs between two preference values consists of every attribute and ideal pairs of it. Based on these connection numbers, an extension of technique for order of preference by similarity to ideal solution method is developed by combining the proposed connection number for IVIFSs and hence finding the best alternative(s) using relative degree of closeness coefficient. An illustrative example has been given for demonstrating the approach and compares their performance with some existing measures.
The present paper proposes some new geometric aggregation operations on the intuitionistic fuzzy sets (IFSs) environment. Based on it, a new class of generalized geometric interaction averaging aggregation operators using Einstein norms and conorms are developed, which includes the weighted, ordered weighted and hybrid weighted averaging operators. Furthermore, desirable properties corresponding to proposed operators have been stated. Finally, a multi-criteria decision making (MCDM) problem has been illustrated to show the validity and effectiveness of the proposed operators. The computed results have been compared with the existing results.
Pythagorean fuzzy set (PFS) is one of the most successful in terms of representing comprehensively uncertain and vague information. Considering that the correlation coefficient plays an important role in statistics and engineering sciences, in this paper, after pointing out the weakness of the existing correlation coefficients between intuitionistic fuzzy sets (IFSs), we propose a novel correlation coefficient and weighted correlation coefficient formulation to measure the relationship between two PFSs. Pairs of membership, nonmembership, and hesitation degree as a vector representation with the two elements have been considered during formulation. Numerical examples of pattern recognition and medical diagnosis have been taken to demonstrate the efficiency of the proposed approach. Results computed by the proposed approach are compared with the existing indices.
Picture fuzzy set (PFS), which is a generalization of traditional fuzzy set and intuitionistic fuzzy set, shows great promises of better adaptation to many practical problems in pattern recognition, artificial life, robotic, expert and knowledge-based systems than existing types of fuzzy sets. An emerging research trend in PFS is development of clustering algorithms which can exploit and investigate hidden knowledge from a mass of datasets. Distance measure is one of the most important tools in clustering that determine the degree of relationship between two objects. In this paper, we propose a generalized picture distance measure and integrate it to a novel hierarchical picture fuzzy clustering method called Hierarchical Picture Clustering (HPC). Experimental results show that the clustering quality of the proposed algorithm is better than those of the relevant ones.
The objective of this article is to extend and present an idea related to weighted aggregated operators from fuzzy to Pythagorean fuzzy sets (PFSs). The main feature of the PFS is to relax the condition that the sum of the degree of membership functions is less than one with the square sum of the degree of membership functions is less than one. Under these environments, aggregator operators, namely, Pythagorean fuzzy Einstein weighted averaging (PFEWA), Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA), generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA), and generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA), are proposed in this article. Some desirable properties corresponding to it have also been investigated. Furthermore, these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean fuzzy environment to show the validity, practicality, and effectiveness of the new approach. Finally, a systematic comparison between the existing work and the proposed work has been given.
This paper develops a new method for solving multiple attribute group decision-making (MAGDM) problems with Atanassov’s interval-valued intuitionistic fuzzy values (AIVIFVs) and incomplete attribute weight information. Firstly, we investigate the asymptotic property of the Atanassov’s interval-valued intuitionistic fuzzy (AIVIF) matrix. It is demonstrated that after applying weights an infinite number of times, all elements in an AIVIF matrix will approach the same AIVIFV without regard to the initial values of elements. Then, the weights of each decision maker (DM) with respect to every attribute are determined by considering the similarity degree and proximity degree simultaneously. To avoid weighting an AIVIF matrix too many times, the collective decision matrix is transformed into an interval matrix using the risk coefficient of DMs. Subsequently, to derive the attribute weights objectively, we construct a multi-objective interval-programming model that is solved by transforming it into a linear program. The ranking order of alternatives is generated by the comprehensive interval values of alternatives. Finally, an example of a research and development (R & D) project selection problem is provided to illustrate the implementation process and applicability of the method developed in this paper.
With respect to multiple attribute decision making (MADM) problems in which the attribute value takes the form of intuitionistic trapezoidal fuzzy number, and the attribute weight is unknown, a new decision making analysis methods are developed. Firstly, some operational laws and expected values of intuitionistic trapezoidal fuzzy numbers, and distance between two intuitionistic trapezoidal fuzzy numbers, are introduced. Then, the maximizing deviation method is used to determine the attribute weight, and three extensions of VIKOR method based on the expected value of the intuitionistic trapezoidal fuzzy number, based on distance between two intuitionistic trapezoidal fuzzy numbers and based on interval numbers, are proposed to rank the alternatives. Finally, an illustrative example is given to verify the developed approaches and to demonstrate their practicality and effectiveness.
A generalization of the notion of intuitionistic fuzzy set is given in the spirit of ordinary interval valued fuzzy sets. The new notion is called interval valued intuitionistic fuzzy set (IVIFS). Here we present the basic preliminaries of IVIFS theory.
The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets.
Aggregation of fuzzy information is a new branch of Atanassov's intuitionistic fuzzy set (AIFS) theory, which has attracted significant interest from researchers in recent years. In this paper, we treat the intuitionistic fuzzy aggregation operators with the help of Einstein operations. We first introduce some new operations of AIFSs, such as Einstein sum, Einstein product, and Einstein scalar multiplication. Then, we develop some intuitionistic fuzzy aggregation operators, such as the intuitionistic fuzzy Einstein weighted averaging operator and the intuitionistic fuzzy Einstein ordered weighted averaging operator, which extend the weighted averaging operator and the ordered weighted averaging operator to aggregate Atanassov's intuitionistic fuzzy values, respectively. We further establish various properties of these operators and analyze the relations between these operators and the existing intuitionistic fuzzy aggregation operators. Moreover, we give some numerical examples to illustrate the developed aggregation operators. Finally, we apply the intuitionistic fuzzy Einstein weighted averaging operator to multiple attribute decision making with intuitionistic fuzzy information.
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
An intuitionistic fuzzy set, characterized by a membership function and a non-membership function, is a generalization of fuzzy set. In this paper, based on score function and accuracy function, we introduce a method for the comparison between two intuitionistic fuzzy values and then develop some aggregation operators, such as the intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, and intuitionistic fuzzy hybrid aggregation operator, for aggregating intuitionistic fuzzy values and establish various properties of these operators.
We are primarily concerned with the problem of aggregating multicriteria to form an overall decision function. We introduce a new type of operator for aggregation called an ordered weighted aggregation (OWA) operator. We investigate the properties of this operator. We particularly see that it has the property of lying between the “and,” requiring all the criteria to be satisfied, and the “or,” requiring at least one of the criteria to be satisfied. We see these new OWA operators as some new family of mean operators.
Picture fuzzy sets -first results. part 1, seminar neuro-fuzzy systems with applications
- B C Cuong
Cuong, B.C.: Picture fuzzy sets -first results. part 1, seminar
neuro-fuzzy systems with applications. Tech. rep., Instiute of Mathematics, Hanoi (2013)
On ordered weighted avergaing aggregation operators in multi-criteria decision making
- R R Yager
Yager, R.R.: On ordered weighted avergaing aggregation operators
in multi-criteria decision making. IEEE Trans. Syst. Man Cybern.
18(1), 183-190 (1988)
Picture fuzzy sets -first results. part 2, seminar neuro-fuzzy systems with applications
- B C Cuong
Cuong, B.C.: Picture fuzzy sets -first results. part 2, seminar
neuro-fuzzy systems with applications. Tech. rep., Instiute of Mathematics, Hanoi (2013)