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Multi-Attribute Decision Making Based on Averaging Aggregation Operators for Picture Hesitant Fuzzy Sets
Abstract
In this work, we proposed the theory of picture hesitant fuzzy set (PHFS) as a generalization of picture fuzzy set (PFS) and hesitant fuzzy set (HFS). Some basic operations on PHFSs have been developed and their results are studied. These aggregation operators included picture hesitant fuzzy weighted averaging (PHFWA) operator, picture hesitant fuzzy ordered weighted averaging (PHFOWA) operator and picture hesitant fuzzy hybrid averaging (PHFHA) operator. The basic properties of defined aggregation operators have been examined and some important results are discussed. Multi-Attribute decision making (MADM) process is discussed in the environment of PHFSs and numerical examples are provided for demonstration of proposed approach. The advantages of working in the environment of PHFSs are also discussed.
... PFS, which is an extension of traditional FS, provides a more enriched approach to dealing with uncertainties by providing positive, negative, neutral, and hesitant membership. This combination enables a more accurate evaluation of factors crucial to decision-making required in complex environments such as artificial intelligence ( [14], [15], [16]). ...
... Future studies could further explore the external validity of the PF-MARCOS method across different contexts and settings of decision-making. Further, a comparison of PF-MARCOS with other advanced MCDM methods ( [48], [49]) including a hesitant fuzzy set [16] and neutrosophic set [50] seems to be beneficial in identifying its advantages and disadvantages, including possible efficiency improvements. Extending the validity of the model to other sectors such as electric vehicles [51] and construction materials [52]. ...
In the complex landscape of financial accounting, selecting an optimal accounting system is crucial for organizations aiming to enhance compliance, financial reporting accuracy, and operational efficiency. To address these challenges, this study introduces a novel decision-making approach, integrating Measurement Alternatives and Ranking according to the Compromise Solution (MARCOS) with Picture Fuzzy Sets (PFS), known as the Picture Fuzzy MARCOS (PF-MARCOS) approach. PF-MARCOS provides a more accurate and comprehensive evaluation of financial accounting systems by incorporating vagueness and ambiguity in decision-making. Applying this approach to assess ten accounting platforms, the study evaluates key parameters, including compliance with international standards, cost, scalability, security, ease of implementation, accuracy, and user-friendliness. Decision-makers can efficiently rank alternatives using this systematic approach, determining which choices are most suited to long-term budgetary constraints. The results highlight the potential of PF-MARCOS for handling complex and uncertain financial scenarios, offering an effective tool to improve managerial financial accounting decision-making.
... The Choquet integral-based TOPSIS technique for IHFS was derived by Joshi and Kumar (2016). Moreover, in two different papers, we have obtained the theory of picture HFS (PHFS) which was proposed by Wang and Li (2018) and Ullah et al. (2018). But the theory of interval-valued PHFS (IVPHFS) was derived by Khalil et al. (2019). ...
... IVPHF comparative analysisMahmood et al. (2021b) Â Â Â Â Â Â ÂÂ Â Â Â Â Â Â ÂÂJoshi and Kumar (2016) Â Â Â Â Â Â ÂÂ Â Â Â Â Â Â ÂÂUllah et al. (2018) Â Â Â Â Â Â ÂÂ Â Â Â Â Â Â ÂÂ ...
In this article, we invent the Bonferroni mean (BM) operators using interval-valued picture hesitant fuzzy (IVPHF) technique, called IVPHF Bonferroni mean (IVPHFBM), IVPHF-weighted BM (IVPHFWBM), IVPHF geometric BM (IVPHFGBM), and IVPHF-weighted geometric BM (IVPHFWGBM) operators. These presented techniques are very beneficial and valuable because these are modified versions of many existing techniques. Moreover, we also examine three basic properties of each presented operator. In addition, we demonstrate the technique of multi-attribute decision-making (MADM) problem and try to describe it with the presence of evaluated techniques to show the capability and superiority of the invented theory. In last, we compare the prevailing techniques with presented studies to illustrate the supremacy and effectiveness of the derived approaches.
... Two classes of operators, Ordered Weighted Averaging (OWA) [20] and Ordered Weighted Geometric (OWG) [21], rank all values in a set and provide rankingsbased weights to those values. For IF data, the Weighted Averaging (WA) operator [22], and the geometric AO were introduced in [23]. [24] used Einstein operations, while [25] proposed modified interactive aggregation techniques in an environment of Intuitionistic Fuzzy Values (IFVs). ...
Martial arts teaching skills involve the systematic instruction of physical techniques, mental discipline, and tactical strategies to develop a student’s proficiency in self-defense and personal growth. It is relevant when there is a specific analysis of the diverse features involving the technique, versatility, and interaction features. In Multi-Attribute Decision-Making (MADM), conventional approaches cannot provide a framework to capture the uncertainties and vague judgments inevitably associated with global experts’ evaluations of such a broad spectrum of skills. These limitations are seen when the evaluations do not consider the dynamism, uncertainty, or subjective nature inherent in the evaluation process, thus having less reliable or less complete decision outcomes. To handle these challenges, the concept of Complex Picture Fuzzy Sets (C-PFS) is utilized, and new advanced Dombi aggregation operators (AO), including Complex Picture Fuzzy Dombi Weighted Averaging (C-PFDWA) and Complex Picture Fuzzy Dombi Weighted Geometric (C-PFDWG) operators are developed. These operators can be effectively utilized to overcome real-life evaluation issues in martial arts skills assessment, serving as intelligent algorithms that facilitate more accurate decision-making processes. To allow the collection of the judgments of different experts over the membership, abstention, and non-membership degrees, offering a comprehensive and detailed picture of an instructor's performance even in cases where there is uncertainty. This comprehensive approach ensures a more reliable and customized assessment, enhancing instructional quality and better student outcomes as instructors receive more targeted feedback on improvement areas. In addition, this study includes detailed simulations and complete results of selected parameters, which demonstrate the practical applicability and robustness of the proposed operators, thus providing strong justification for the research and validating the efficacy of the developed framework.
... Yager [12] and Abbasov formalized aggregation operators (AO) within PyFS, improving MADM applications. Ullah et al. [13]. Further contributed by developing weighted AOs for picture hesitant fuzzy sets, applying these to solve MADM challenges. ...
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.
... Xia et al. [11] explored advanced methods for fuzzy sampled-data stabilization in chaotic nonlinear systems. Jana et al. developed a range of AOs in the picture fuzzy [12] context frames to evaluate different alternative prioritization throughout the decisionmaking process [13]. Since associations between attributes are a common feature of practical decision-making problems, IF values (IFVs) are an effective means for representing the fuzzy information related to MCDM difficulties [14], [15], [16]. ...
Evaluating the teaching quality of traditional decorative patterns in colleges and universities is a critical task, especially when human judgment and subjective opinions are involved. This paper aims to analyze this assessment process using an enhanced framework of circular intuitionistic fuzzy sets (C-IFS) and the Dombi triangular norm. The aim of this research is to apply the Dombi t-norm (DTN) and t-conorm (DTCN) for aggregating the information within the circular intuitionistic fuzzy (C-IF) framework to evaluate the teaching quality of traditional decorative patterns. Another important feature of the proposed study is that prioritization is used in decision-making preferences by using a priority-based weight vector. To serve this purpose, we mainly proposed the C-IF Dombi prioritized weighted averaging (C-IFDPWA) and geometric (C-IFDPWG) operators. Moreover, the special cases of proposed aggregation operators (AOs) and their validity is discussed. An intelligent algorithm is provided involving prioritization of attributes and experts based on proposed AOs to evaluate the multi-criteria group decision-making (MCGDM) problem, particularly in assessing teaching quality. A numerical example is included to demonstrate the effectiveness and flexibility of the proposed work and to validate its application, the comparison anlysis has been conducted. The superiority of this model lies in the fact that general structure, less time complexity, and prioritization of factors which makes it real.
... Munir et al [20] investigated Einstein interactive AOs of TSFS. Ullah et al extended the idea of picture fuzzy set (PFS) to picture hesitant fuzzy set (PHFS) and constructed new AOs based on PHFSs [21]. Ullah et al [22] derived mathematical approaches of Frank operators to compute unknown degree of weights and decision algorithm for resolving real-life applications. ...
Multi-Attribute Decision Making (MADM) is a powerful tool for navigating complex decision problems by systematically considering multiple criteria and alternatives. This method is widely used in various fields such as business, engineering, finance, and public policy, where decisions involve considering multiple conflicting objectives. An interval-valued intuitionistic fuzzy set is a more flexible mathematical model used to aggregate vague type and redundant information into a single set. By exploring the robustness of Aczel Alsina aggregation operators, we deduce an effective mathematical approach for handling ambiguous information of human opinion. To express the relationship among input arguments or attribute information, we study a feasible theory of Hamy mean (HM) operators. This article presents some dominant mathematical approaches in the light of interval-valued intuitionistic fuzzy (IVIF) information with Aczel Alsina operations including IVIF Aczel Alsina HM (IVIFAAHM), IVIF Aczel Alsina weighted HM (IVIFAAWHM), IVIF Aczel Alsina Dual HM (IVIFAADHM) and IVIF Aczel Alsina weighted Dual HM (IVIFAAWDHM) operators. To prove the validity and effectiveness of derived approaches, some prominent characteristics are also illustrated. A decision algorithm of the MADM technique is also established to resolve complicated real-life applications and amplifications. With the help of numerical examples, we show the compatibility of diagnosed mathematical approaches. Finally, the influence study and comparison technique also verify the consistency of pioneered approaches by contrasting the aggregated outcomes of previous operators that exist in the literature.
... Picture fuzzy Dombi AOs were suggested by [21] and their use in MADM was studied. For picture-hesitant fuzzy collections, [22] presented MADM based on averaging AOs. Frank AOs and an analytical hierarchy procedure based on interval-valued FS and its applications were proposed by [23]. ...
T-spherical fuzzy set (TSFS) is the generalization of the fuzzy set (FS) which extracts the information from the real-life scenario with certainty. Aside from the remarkable advantage of being able to account for the connections among the multi-input considerations, such as multi-attributes or multi-experts in the multi-attribute group decision-making (MAGDM), the Maclaurin symmetric mean operator (MSMO) is also the generalization of several different existing operators. Moreover, one important class of T-norms (TN) and T-conorms (TC) is the Schweizer-Sklar TN (SSTN) and TC (SSTC). In this article, the operational laws for TSFS based on SSTN and SSTC are introduced first. Then the introduced operations are used to develop a class of aggregation operators (AOs) to aggregate the information in the form of the T-spherical fuzzy values (TSFVs). The introduced operators in this article are the T-spherical fuzzy Schweizer-Sklar MSMO (TSFSSMSMO) and the T-spherical fuzzy Schweizer-Sklar weighted MSMO (TSFSSWMSMO). Further, the TSFSSMSMO and TSFWMSMO are applied to a specific multi-attribute group decision-making (MAGDM) problem to show the significance of the developed operators.
... Wei [12] provided AOs for the PFS and then utilized them in MAGDM. Ullah et al. [13] created AOs for the situation of picture-hesitant FS to address the MAGDM complications. Ullah et al. [14] introduced the AOs for TSF based on the Hamacher and used them in MAGDM. ...
Multi-attribute group decision-making (MAGDM) is very significant technique for selecting an alternative from the provided list. But the major problem is to deal with the information fusion during the information. Aczel-Alsina t-norm (AATN) and Aczel-Alsina t-conorm (AATCN) are the most generalized and flexible t-norm (TN) and t-conorm (TCN) which is used for information processing. Moreover, the interval-valued T-spherical fuzzy set (IVTSFS) is the latest framework to cover the maximum information from the real-life scenarios. Hence, the major contribution of this paper is to deal the information while the MAGDM process by introducing new aggregation operators (AOs). Consequently, the interval-valued T-spherical fuzzy (IVTSF), Aczel-Alsina weighted averaging (IVTSFAAWA), IVTSF Aczel-Alsina (IVTSFAA) ordered weighted averaging (IVTSFAAOWA), IVTSFAA weighted geometric (IVTSFAAWG), IVTSFAA ordered weighted geometric (IVTSFAAOWG), and IVTSFAA hybrid weighted geometric (IVTSFAAHWG) operators are developed. It is shown that the proposed operators are the valid and the results obtained are reliable by discussing some basic properties. To justify the developed AOs, an example of the MAGDM is discussed. The sensitivity of these AOs is observed keeping in view of the variable parameter. To show the importance of the newly developed theory, a comparison of the proposed AOs is established with already existing operators.
... Yager and Abbasov [17] formalized the aggregation operators (AO) in the environment of PyFS with application in MADM. Ullah et al. [18] developed weighted AO for the environment of picture hesitant FS and applied them to solve the MADM problem. Akram et al. [19] Introduced some AO of CSFS and developed a multi-criteria group DM (MCGDM) method called CSF-VIKOR. ...
Complex T-spherical fuzzy set (CTSF) is the extended form of complex spherical fuzzy set (CSFs). CTSF can extract the maximum information from real-life phenomena with the help of a membership degree (MD), abstinence degree (AD), and non-membership degree (NMD). Due to this nature, CTSF can describe every human opinion’s phenomenon with less uncertainty. The major contribution of this article is to develop an interesting approach to aggregate the information in the form of the CTSFS. To do this, first, we develop the operational laws for CTSF values (CTSFVs) based on Frank’s operational laws. Then two types of aggregation operators (AOs), CTSF Frank weighted averaging (CTSFWA) and CTSF Frank weighted geometric (CTSFFWG) operators are introduced based on the Frank t-norm (FTNM). Further, some basic properties of the CTSFFWA and CTSFFWG operators are studied. The developed approach is also applied to the real-life problem of multi-attribute decision-making (MADM) where the soil fertility is assessed based on some factors affecting the soil fertility. The sensitivity of the developed approach is observed with the variation of the involved parameter graphically. The comparison of the developed approach is also studied and as well as graphed.
... Various researchers started working on PFSs as soon as they were developed. Numberless research on PFSs can be seen in [19][20][21][22]. From the above analysis, we noticed that every expert or decision-maker has faced the following three major issues during the decision-making process: 1) How do we collect the information on a suitable scale to state the data? ...
The Aczel-Alsina t-norm and t-conorm were derived by Aczel and Alsina in 1982. They are modified forms of the algebraic t-norm and t-conorm. Furthermore, the theory of picture fuzzy values is a very valuable and appropriate technique for describing awkward and unreliable information in a real-life scenario. In this research, we analyze the theory of averaging and geometric aggregation operators (AOs) in the presence of the Aczel-Alsina operational laws and prioritization degree based on picture fuzzy (PF) information, such as the prioritized PF Aczel-Alsina average operator and prioritized PF Aczel-Alsina geometric operator. Moreover, we examine properties such as idempotency, monotonicity and boundedness for the derived operators and also evaluated some important results. Furthermore, we use the derived operators to create a system for controlling the multi-attribute decision-making problem using PF information. To show the approach's effectiveness and the developed operators' validity, a numerical example is given. Also, a comparative analysis is presented.
... The area of DM is then called multiple-attribute DM (MADM). MADM is a highly crucial testing domain, which helps choose the correct option associated with several leading features [1][2][3]. Mainly, the DM uses crisp figures to describe the preferences concerning the choices in usual MADM problems. Nevertheless, due to the inadequacy of statistics, shortage of time, and lack of data and quality values, fuzzy values may be used to specify the preferences exclusively for the attribute values. ...
The intuitionistic hesitant fuzzy set (IHFS) is an enriched version of hesitant fuzzy sets (HFSs) that can cover both fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). By assigning membership and non-membership grades as subsets of [0, 1], the IHFS can model and handle situations more proficiently. Another related theory is the theory of set pair analysis (SPA), which considers both certainties and uncertainties as a cohesive system and represents them from three aspects: identity, discrepancy, and contrary. In this article, we explore the suitability of combining the IHFS and SPA theories in multi-attribute decision making (MADM) and present the hybrid model named intuitionistic hesitant fuzzy connection number set (IHCS). To facilitate the design of a novel MADM algorithm, we first develop several averaging and geometric aggregation operators on IHCS. Finally, we highlight the benefits of our proposed work, including a comparative examination of the recommended models with a few current models to demonstrate the practicality of an ideal decision in practice. Additionally, we provide a graphical interpretation of the devised attempt to exhibit the consistency and efficiency of our approach.
... The PHFSs proposed by Wang and Liu [27], another extension of HFSs, have been investigated and studied in various areas by many researchers due to their advantages in interpreting fuzzy information in positive, negative, and neutral aspects. Ullah et al. [50] presented some picture hesitant fuzzy aggregation operators for MADM methods. Yang et al. [51] introduced picture hesitant fuzzy entropy and similarity measurements and utilized them to make MCGDM method for end-of-life vehicle management. ...
The picture hesitant fuzzy sets (PHFSs), which consider neutral membership degree as well as positive and negative membership degrees, provide decision makers (DMs) a flexible attitude to evaluate criteria values in complex multi-criteria decision-making (MCDM) situations. However, existing MCDM approaches based on PHFSs still have some drawbacks in both evaluation information expression and criteria values fusion. In this paper, our aim is to overcome these shortcomings by proposing new decision-making methods. To achieve this purpose, a new fuzzy information representation tool, called probabilistic picture hesitant fuzzy sets (P-PHFSs), is first introduced by capturing the probability of each element in PHFSs. The characteristic of P-PHFSs is that they provide more freedom to DMs so that criterion values of each alternative can be adequately described. To facilitate the use of P-PHFSs, we define the basic operational rules and comparison method of P-PHFSs. Then we also propose some aggregation operators for P-PHFSs and provide information fusion process. Furthermore, some desirable properties of these operators is discussed, and the relationship between the developed operators and the existing ones is investigated. Based on the proposed operators, two MCDM methods are developed under probabilistic picture hesitant fuzzy environment. Finally, two numerical examples are given to show the application of the developed methods, and a comparison analysis is conducted to demonstrate the effectiveness of the proposed approaches.
... As an extension of IFSs, when the neutral membership function of a PFS is equal to 0, the PFS reduces to an IFS. Combining PFSs with HFSs, Ullah et al. [6] introduced the concept of picture hesitant fuzzy sets (PHFSs). ...
Based on the closed operational laws in picture fuzzy numbers and strict triangular norms, we extend the Bonferroni mean (BM) operator under the picture fuzzy environment to propose the picture fuzzy interactional Bonferroni mean (PFIBM), picture fuzzy interactional weighted Bonferroni mean (PFIWBM), and picture fuzzy interactional normalized weighted Bonferroni mean (PFINWBM) operators. We prove the monotonicity, idempotency, boundedness, and commutativity for the PFIBM and PFINWBM operators. We also establish a novel multi-criteria decision making (MCDM) method under the picture fuzzy environment by applying the PFINWBM operator. Furthermore, we apply our MCDM method to the enterprise resource planning (ERP) systems selection. The comparative results for our MCDM method induced by six classes of well-known triangular norms ensure that the best selection is always the same ERP system. Therefore, our MCDM method is effective for dealing with the picture fuzzy MCDM problems.
... Wei [14] gave AOs in the framework of PFSs and then applied it to MADM. Ullah et al. [15] developed weighted AOs for the environment of picture hesitant FSs and applied them to solve the MADM problem. Ullah et al. [16] proposed the TSF Hamacher AOs and applied them in MADM to evaluate the performance of search and rescue robots. ...
Aczel-Alsina t-norm (TN) and t-conorm (TCN) were proposed by Aczel and Alsina in 1982 are more flexible than the other TN and TCN. Since Aczel-Alsina TN and TCN have a great impact due to the variableness of involved parameters, they have good applications in multi-attribute decision making (MADM) under fuzzy sets (FSs) construction. Recently, Senapati
et al.
(2022) developed Aczel-Alsina aggregation operators (AOs) under intuitionistic FSs (IFSs) and interval-valued IFSs (IVIFSs) with their applications in solving IFS and IVIFS MADM problems. We know that T-spherical FSs (TSFSs) are a recently developed approach to uncertain information with less information loss and more reliability than IFSs and IVIFSs. In this paper, we develop these AOs on TSFSs as a new approach to solve MADM problems by using Aczel-Alsina TN and Aczel-Alsina TCN under T-spherical fuzzy (TSF) information. Furthermore, the basic operations of TSF numbers (TSFNs) are developed and exemplified. Based on these operations, two types of AOs, i.e., TSF Aczel-Alsina weighted average (TSFAAWA), and TSF Aczel-Alsina weighted geometric (TSFAAWG) operators, are introduced and investigated. The reliability and accuracy of the newly developed AOs are tested numerically and theoretically by the induction methods. To further give applications and also study the sensitivity of these TSF Aczel-Alsina operators, the problem of project evaluation using these proposed operators is comprehensively observed. The results obtained by using these TSF Aczel-Alsina operators are compared with some previously existing AOs of TSFSs. According to comparison results, we observe the reliability and efficiency of the proposed methods.
... Yet, these have different issues, when a chief confronts different sorts of assessment of an individual like indeed, forbearance, no, and refusal as gathering. For adapting such sort of issues, Ullah et al. [20] investigated the idea of PHFSs containing the grades of truth, restraint, deception, and refusal as a subset of the unit span with conditions, the amount of the limit of truth, limit of forbearance and limit of lie grades is having a place on the unit stretch. PHFS is more dependable than IHFS and HFS to adapt to questionable and troublesome data choices. ...
One of the most dominant and feasible technique is called the PHF setting is exist in the circumstances of fuzzy set theory for handling intricate and vague data in genuine life scenario. The perception of PHF setting is massive universal is compared to these assumptions, who must cope with two or three sorts of data in the shape of singleton element. Under the consideration of the PHF setting, we utilized some SM in the region of the PHF setting are to diagnose the PHFDSM, PHFWDSM, PHFJSM, PHFWJSM, PHFCSM, PHFWCSM, PHFHVSM, PHFWHVSM and demonstrated their flexible parts. Likewise, a lot of examples are exposed under the invented measures based on PHF data in the environment of medical diagnosis to demonstrate the stability and elasticity of the explored works. Finally, the sensitive analysis of the presented works is also implemented and illuminated their graphical structures.
... In modern decision science, multi-attribute decision making (MADM) is a vital investigation area on how to choose the correct option corresponding to many prominent attributes [1][2][3]. Usually, the decision-makers (DMs) utilize crisp figures to express the favorites regarding the alternative in conventional multi-attribute decision making difficulties. But, because of shortage of data, lack of time, deficiency of information and quality values, particularly, for subjective attribute values, usually may not be shown by real numbers, and few of them are simpler to be stated by fuzzy data. ...
Intuitionistic hesitant fuzzy set (IHFS) is a mixture of two separated notions called intuitionistic fuzzy set (IFS) and hesitant fuzzy set (HFS), as an important technique to cope with uncertain and awkward information in realistic decision issues. IHFS contains the grades of truth and falsity in the form of the subset of the unit interval. The notion of IHFS was defined by many scholars with different conditions, which contain several weaknesses. Here, keeping in view the problems of already defined IHFSs, we will define IHFS in another way so that it becomes compatible with other existing notions. To examine the interrelationship between any numbers of IHFSs, we combined the notions of power averaging (PA) operators and power geometric (PG) operators with IHFSs to present the idea of intuitionistic hesitant fuzzy PA (IHFPA) operators, intuitionistic hesitant fuzzy PG (IHFPG) operators, intuitionistic hesitant fuzzy power weighted average (IHFPWA) operators, intuitionistic hesitant fuzzy power ordered weighted average (IHFPOWA) operators, intuitionistic hesitant fuzzy power ordered weighted geometric (IHFPOWG) operators, intuitionistic hesitant fuzzy power hybrid average (IHFPHA) operators, intuitionistic hesitant fuzzy power hybrid geometric (IHFPHG) operators and examined as well their fundamental properties. Some special cases of the explored work are also discovered. Additionally, the similarity measures based on IHFSs are presented and their advantages are discussed along examples. Furthermore, we initiated a new approach to multiple attribute decision making (MADM) problem applying suggested operators and a mathematical model is solved to develop an approach and to establish its common sense and adequacy. Advantages, comparative analysis, and graphical representation of the presented work are elaborated to show the reliability and effectiveness of the presented works.
... In the future, the concept of complex dual type-2 hesitant fuzzy sets can be applied to group MADM problems. Moreover, the problems discussed in this manuscript can be discussed in the environment of complex q-rung orthopair fuzzy sets [32][33][34][35][36][37][38][39], T-spherical fuzzy sets [40,41], and some others [42][43][44][45]. ...
The theory of complex dual type-2 hesitant fuzzy sets (CDT-2HFSs) is a blend of two different modifications of fuzzy sets (FSs), called complex fuzzy sets (CFSs) and dual type-2 hesitant fuzzy sets (DT-2HFSs). CDT-2HFS is a proficient technique to cope with unpredictable and awkward information in realistic decision problems. CDT-2HFS is composed of the grade of truth and the grade of falsity, and the grade of truth (also for grade of falsity) contains the grade of primary and secondary parts in the form of polar coordinates with the condition that the sum of the maximum of the real part (also for the imaginary part) of the primary grade (also for the secondary grade) cannot exceed the unit interval [0, 1]. The aims of this manuscript are to discover the novel approach of CDT-2HFS and its operational laws. These operational laws are also justified with the help of an example. Additionally, based on a novel CDT-2HFS, we explored the correlation coefficient (CC) and entropy measures (EMs), and their special cases are also discussed. TOPSIS method based on CDT-2HFS is also explored. Then, we applied our explored measures based on CDT-2HFSs in the environment of the TOPSIS method, medical diagnosis, pattern recognition, and clustering algorithm to cope with the awkward and complicated information in realistic decision issues. Finally, some numerical examples are given to examine the proficiency and validity of the explored measures. Comparative analysis, advantages, and graphical interpretation of the explored measures with some other existing measures are also discussed.
... At last, an illustrative numerical example is provided to demonstrate the efficiency and effectiveness of the proposed approaches. In future, we use the GDSMs in the environment of [55][56][57][58], neutrosophic generalisations [59][60][61][62], complex q-rung orthopair FSs [63,64] and decision making [65][66][67][68][69][70]. ...
Complex neutrosophic set (CNS) is a modified version of the complex fuzzy set, to cope with complicated and inconsistent information in the environment of fuzzy set theory. The CNS is characterised by three functions expressing the degree of complex-valued membership, complex-valued abstinence and degree of complex-valued non-membership. The aim of this manuscript is to initiate the novel dice similarity measures and generalised dice similarity using CNS. The special cases of the investigated measures are discussed with the help of some remarks. Moreover, some distance measures based on CNS are also proposed in this manuscript. Then, the authors applied the generalised dice similarity measures and weighted generalised dice similarity measures using CNS to the pattern recognition model to examine the reliability and superiority of the established approaches. The advantages and comparative analysis of the proposed measures with existing measures are also discussed in detail. At last, a numerical example is provided to illustrate the validity and applicability of the presented measures.
The importance of Internet of Things (IoT) based Healthcare systems for remote patients in modern days is very significant and multifaceted, addressing various challenges and bringing numerous benefits. These systems hold significant promises for improving the quality, accessibility, and cost-effectiveness of care for remote patients. Selection of a suitable IoT based healthcare system is a crucial task which require the evaluation of each system relative to certain conflicting criteria such as patient comfort, accuracy, cost data security and connectivity. Picture Fuzzy (PF) sets, an extension of fuzzy sets, incorporate an additional degree of freedom by considering positive, neutral and negative membership degrees, thus providing a more nuanced representation of uncertainty. Combined Compromise Solution (CoCoSo) method is one of the prominent methods used to solve decision making problems. But this method has certain limitations as the integration function used in this method fail to assign appropriate ranking to the alternatives in some special situations. A modified CoCoSo framework in PF environment is described in this research to address the issue of identifying suitable IoT-based healthcare system based on several criteria. For this, a new PF distance measure is proposed and establish its superiority in finding the pattern similarity of unknown patterns through some numerical examples. A novel score function that overcomes the shortcomings of the existing score functions is provided in order to address the comparison issue of PF numbers. A new integration function is provided which overcome the aggregation biases of function used in traditional CoCoSo method.
Multi-attribute group decision-making (MAGDM) is a very significant technique for selecting an alternative from the provided list. But the major problem is dealing with the information fusion during the information. Aczel-Alsina t-norm (AATN) and Aczel-Alsina t-conorm (AATCN) are the most generalized and flexible t-norm (TN) and t-conorm (TCN), which are used for information processing. Moreover, the interval-valued T-spherical fuzzy set (IVTSFS) is the latest framework to cover the maximum information from the real-life scenarios. Hence, the major contribution of this paper is to deal with the information during the MAGDM process by introducing new aggregation operators (AOs). Consequently, the interval-valued T-spherical fuzzy (IVTSF), Aczel-Alsina weighted averaging (IVTSFAAWA), IVTSF Aczel-Alsina (IVTSFAA) ordered weighted averaging (IVTSFAAOWA), IVTSFAA weighted geometric (IVTSFAAWG), IVTSFAA ordered weighted geometric (IVTSFAAOWG), and IVTSFAA hybrid weighted geometric (IVTSFAAHWG) operators are developed. It is shown that the proposed operators are valid and the results obtained are reliable by discussing some basic properties. To justify the developed AOs, an example of the MAGDM is discussed. The sensitivity of these AOs is observed keeping in view of the variable parameter. To show the importance of the newly developed theory, a comparison of the proposed AOs is established with already existing operators.
To reduce the risk and interruptions of the supply chain, some appropriate aggregation operators are used to deal with crucial challenges and to keep the support system reliable. In this article, we explore the notion of the complex picture fuzzy (CPF) set (CPFS), a wider framework of fuzzy and picture fuzzy sets. Some robust operations of the Aczel Alsina aggregation tools are characterized based on CPF information to overcome the impact of uncertain and imprecision situations. We also explain an innovative concept of the prioritized aggregation operators (AOs) in the light of CPF information. The theory of unknown weight vectors is used to develop some new approaches of the CPF environments, including CPF Aczel Alsina prioritized average (CPFAPA) and CPF Aczel Alsina prioritized geometric (CPFAPG) operators. We also propose a class of new methodologies under considering CPF information, such as CPF Aczel Alsina prioritized weighted average (CPFAPWA) and CPF Aczel Alsina prioritized weighted geometric (CPFAPWG) operators. To illustrate the flexibility of our invented methodologies, some appropriate characteristics of the proposed approaches are also demonstrated. The multi-attribute decision-making (MADM) technique is used to evaluate many crucial real-life challenges. An algorithm for the MADM problem is also proposed to show the robustness of our invented approaches. By using proposed methodologies, we demonstrate an experimental case study to choose a sustainable supplier agnet based on CPF information. To show the practicability and consistency of our invented approaches by contrasting the results of existing approaches in the literature review with currently presented methodologies.
Due to more advanced features of the picture fuzzy soft set and valuable characteristics of aggregation operators, in this article, we present the notion of picture fuzzy soft Bonferroni mean aggregation operators and weighted picture fuzzy soft Bonferroni mean aggregation operators. Moreover, some basic properties of these introduced aggregation operators have been given. As cancer is one of the most rapidly increasing diseases globally. But due to different kinds of cancer diseases, it is very difficult to say which type of cancer disease is increasing rapidly. So to reduce this difficulty in the medical field, we have applied our work to medical diagnosis problems to ensure that fuzzy ideas can also help in the medical field. Also, an algorithm along with a descriptive example is established that conforms to the authenticity of initiated work. Furthermore, a comparative analysis of the introduced work has been given to show how this work is more efficient and dominant than other existing theories.
In this paper, we propose a picture type-2 hesitant fuzzy sets (PT2HFSs) and discuss various properties of this new concept like intersection and union. Then, we define a vector similar measures including dice, Jaccard, cosine and weighted dice, weighted Jaccard and weighted cosine similarity measures and prove to some axiomatic properties. Moreover, a new algorithm is proposed to indicate effective of the similarity measures by utilizing definitions of intersection and union over decision making matrix. In addition, a numerical example is put forward to show agreement of proposed similarity measures. In the next step, we apply consistency analysis over defined vector similarity measures to see which measurement is more convenience and also two comparative analyses have been defined by utilizing a conversion of fuzzy elements to type-2 fuzzy elements and interval-valued intuitionistic fuzzy sets into end section.
To manage ambiguity and uncertainty information in real life problems, certain intellectuals have initiated numerous modifications of fuzzy sets. But, in certain situations, the decision-makers have been neglected to take a feasible decision. For instance, when an intellectual provides information in the form of a two-dimension or type-1 fuzzy set, then the prevailing theories have been neglected. For this, in this study, we elaborated the new idea of type-2 picture fuzzy sets (T2-PFS) and their algebraic laws have been given. Moreover, by using the T2-PFS, Hamacher aggregation operators are proposed and discussed their important properties. The Minkowski sorts of measures under the T2-PFS are also initiated. In the current study, we aim to utilize the matrix game in a type-2 picture fuzzy environment. By using the proposed operators and measures based on the Hausdorff metric. The elaborated idea has been defined with the biogas-plant implementation dilemma to verify the pertinence and authenticity. Lastly, to discuss the sensitivity analysis and geometrical shown of the initiated works, certain examples are illustrated to determine the supremacy and consistency of the proposed approaches.
Based on the closed operational laws in picture fuzzy numbers and strict triangular norms, we extend the Bonferroni mean (BM) operator under the picture fuzzy environment to propose the picture fuzzy interactional Bonferroni mean (PFIBM), picture fuzzy interactional weighted Bonferroni mean (PFIWBM), and picture fuzzy interactional normalized weighted Bonferroni mean (PFINWBM) operators. We prove the monotonicity, idempotency, boundedness, and commutativity for the PFIBM and PFINWBM operators. We also establish a novel multi-criteria decision making (MCDM) method under the picture fuzzy environment by applying the PFINWBM operator. Furthermore, we apply our MCDM method to the enterprise resource planning (ERP) systems selection. The comparative results for our MCDM method induced by six classes of well-known triangular norms ensure that the best selection is always the same ERP system. Therefore, our MCDM method is effective for dealing with the picture fuzzy MCDM problems.
On the heels of the online shopping boom during the Covid-19 pandemic, the electronic commerce (e-commerce) surge has many businesses facing an influx in product returns. Thus, relevant companies must implement robust reverse logistics strategies to reflect the increased importance of the capability. Reverse logistics also plays a radical role in any business’s sustainable development as a process of reusing, remanufacturing, and redistributing products. Within this context, outsourcing to a third-party reverse logistics provider (3PRLP) has been identified as one of the most important management strategies for today’s organizations, especially e-commerce players. The objective of this study is to develop a decision support system to assist businesses in the selection and evaluation of different 3PRLPs by a hybrid fuzzy multicriteria decision-making (MCDM) approach. Relevant criteria concerning the economic, environmental, social, and risk factors are incorporated and taken into the models. For obtaining more scientific and accurate ranking results, linguistic terms are adopted to reduce fuzziness and uncertainties of criteria weights in the natural decision-making process. The fuzzy analytic hierarchy process (FAHP) is applied to measure the criteria’s relative significance over the evaluation process. The fuzzy technique for order preference by similarity to an ideal solution (FTOPSIS) is then used to rank the alternatives. The prescribed method was adopted for solving a case study on the 3PRLP selection for an online merchant in Vietnam. As a result, the most compatible 3PRLP was determined. The study also indicated that “lead time,” “customer’s voice,” “cost,” “delivery and service,” and “quality” are the most dominant drivers when selecting 3PLRLs. This study aims to provide a more complete and robust evaluation process to e-commerce businesses and any organization that deals with supply chain management in determining the optimized reverse logistics partners.
Certain intellectuals have generalized the principle of the fuzzy set (FS), but the theory of complex q-rung orthopair fuzzy set (Cq-ROFS) has received massive attraction from different scholars. The goal of this study is to combine the principle of Heronian mean (HM) operator with Cq-ROFS is to initiate the complex q-rung orthopair fuzzy HM (Cq-ROFHM) operator, complex q-rung orthopair fuzzy weighted HM (Cq-ROFWHM) operator, complex q-rung orthopair fuzzy geometric HM (Cq-ROFGHM) operator, complex q-rung orthopair fuzzy weighted geometric HM (Cq-ROFWGHM) operator, and their flexible and dominant properties. These operators can help to aggregate any number of attributes to determine the reliability and consistency of the investigated operators. Moreover, there are physical and non-physical threats. Physical threats cause damage to computer systems hardware and infrastructure. Examples include theft, vandalism through to natural disasters. Non-physical threats target the software and data on the computer systems. To manage such sort of troubles, we determine the analyzing and controlling computer security threats based on presented operators under the Cq-ROFS. Finally, to show the reliability and proficiency of the presented approaches, we resolved some numerical examples by using the explored operators. The comparative analysis, advantages, and graphical interpretations of the presented works are also discovered.
The Muirhead mean (MM) operators offer a flexible arrangement with its modifiable factors because of Muirhead’s general structure. On the other hand, MM aggregation operators perform a significant role in conveying the magnitude level of options and characteristics. In this manuscript, the complex spherical fuzzy uncertain linguistic set (CSFULS), covering the grade of truth, abstinence, falsity, and their uncertain linguistic terms is proposed to accomplish with awkward and intricate data in actual life dilemmas. Furthermore, by using the MM aggregation operators with the CSFULS, the complex spherical fuzzy uncertain linguistic MM (CSFULMM), complex spherical fuzzy uncertain linguistic weighted MM (CSFULWMM), complex spherical fuzzy uncertain linguistic dual MM (CSFULDMM), complex spherical fuzzy uncertain linguistic dual weighted MM (CSFULDWMM) operators, and their important results are also elaborated with the help of some remarkable cases. Additionally, multi-attribute decision-making (MADM) based on the Multi-MOORA (Multi-Objective Optimization Based on a Ratio Analysis plus full multiplicative form), and proposed operators are developed. To determine the rationality and reliability of the elaborated approach, some numerical examples are illustrated. Finally, the supremacy and comparative analysis of the elaborated approaches with the help of graphical expressions are also developed.
The picture hesitant fuzzy sets (PHFSs) are an important way to express uncertain information, and they are superior to the intuitionistic hesitant fuzzy sets and the hesitant fuzzy sets. Their eminent characteristic is that the sum of the supremum of the membership degree, the abstinence degree, and the non-membership is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we propose the picture hesitant fuzzy Bonferroni mean (PHFBM) operator, picture hesitant fuzzy weighted Bonferroni mean (PHFWBM) operator, picture hesitant fuzzy geometric Bonferroni mean (PHFGBM) operator, and picture hesitant fuzzy weighted geometric Bonferroni mean (PHFWGBM) operator to deal with the decision information, and there some properties are well proved. Further, based on these operators, we presented a new method to deal with the multi-attribute group decision-making (MAGDM) problems under the fuzzy environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing it with other existing methods.
Interval-valued T-spherical fuzzy set (IVTSFS) handles uncertain and vague information by discussing their membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). MD, AD, NMD, and RD are defined in terms of closed subintervals of [0, 1] that reduce information loss compared to the T-spherical fuzzy set (TSFS), which takes crisp values from [0, 1] intervals; hence, some information may be lost. The purpose of this manuscript is to develop some Hamacher aggregation operators (HAOs) in the environment of IVTSFSs. To do so, some Hamacher operational laws based on Hamacher t-norms (HTNs) and Hamacher t-conorms (HTCNs) are introduced. Using Hamacher operational laws, we develop some aggregation operators (AOs), including an interval-valued T-spherical fuzzy Hamacher (IVTSFH) weighted averaging (IVTSFHWA) operator, an IVTSFH-ordered weighted averaging (IVTSFHOWA) operator, an IVTSFH hybrid averaging (IVTSFHHA) operator, an IVTSFH-weighted geometric (IVTSFHWG) operator, an IVTSFH-ordered weighted geometric (IVTSFHOWG) operator, and an IVTSFH hybrid geometric (IVTSFHHG) operator. The validation of the newly developed HAOs is investigated, and their basic properties are examined. In view of some restrictions, the generalization and proposed HAOs are shown, and a multi-attribute decision-making (MADM) procedure is explored based on the HAOs, which are further exemplified. Finally, a comparative analysis of the proposed work is also discussed with previous literature to show the superiority of our work.
The main purpose of this planned manuscript is to establish an algorithm for the solution of multiattribute decision-making (MADM) issues, where the experts utilizing linguistic variables provide the information about attributes in the form of picture hesitant fuzzy numbers. So, for the solution of these kinds of issues, we develop the TOPSIS algorithm under picture hesitant fuzzy environment using linguistic variables, which plays a vital role in practical applications, notably MADM issues, where the decision information is arranged by the decision-makers (DMs) in the form of picture hesitant fuzzy numbers. Finally, a sample example is given as an application and appropriateness of the planned method. At the end, we conduct comparison analysis of the planned method with picture fuzzy TOPSIS method and intuitionistic fuzzy TOPSIS method.
Complex q-rung orthopair fuzzy set (CQROFS) is a proficient technique to describe awkward and complicated information by the truth and falsity grades with a condition that the sum of the q-powers of the real part and imaginary part is in unit interval. Further, Schweizer–Sklar (SS) operations are more flexible to aggregate the information, and the Muirhead mean (MM) operator can examine the interrelationships among the attributes, and it is more proficient and more generalized than many aggregation operators to cope with awkward and inconsistence information in realistic decision issues. The objectives of this manuscript are to explore the SS operators based on CQROFS and to study their score function, accuracy function, and their relationships. Further, based on these operators, some MM operators based on PFS, called complex q-rung orthopair fuzzy MM (CQROFMM) operator, complex q-rung orthopair fuzzy weighted MM (CQROFWMM) operator, and their special cases are presented. Additionally, the multi-criteria decision making (MCDM) approach is developed by using the explored operators based on CQROFS. Finally, the advantages and comparative analysis are also discussed.
Spherical fuzzy set (SFS) is a modified version of fuzzy set (FS) to cope with uncertainty and complicated data in real-decision theory. In this article, some similarity measures, called cosine similarity measure (CSM), weighted cosine similarity measure (WCSM), set-theoretic similarity measure (STSM), weighted set-theoretic similarity measure (WSTSM), gray similarity measure (GSM), and weighted gray similarity measure (WGSM) are utilized in the environment of SFSs. Further, the information energy, correlation co-efficient (CC) and weighted correlation co-efficient (WCC) of SFSs are also introduced in this manuscript. The established measures based on SFSs are utilized in the environment of pattern recognition and medical diagnosis to express the validity and reliability of the explored measures with the help of some numerical examples. The projected measures based on SFSs are compared with existing measures, to show that the established measures for SFSs are more generalized than existing measures. The advantages and sensitive analysis of the investigated measures are also discussed in detail.
In rule optimization, some rule characteristics were extracted to describe the uncertainty correlations of fuzzy relations, but the concrete numbers cannot express correlations with uncertainty, such as “at least 0.1 and up to 0.5.” To solve this problem, a novel definition concerning interval information content of fuzzy relation has been proposed in this manuscript to realize the fuzziness measurement of the fuzzy relation. Also, its definition and expressions have also been constructed. Meanwhile based on the interval information content, the issues of fuzzy implication ranking and clustering were analyzed. Finally, utilizing the combination of possibility’s interval comparison equations and interval value’s similarity measure, the classifications of implication operators were proved to be realizable. The achievements in the presented work will provide a reasonable index to measure the fuzzy implication operators and lay a solid foundation for further research.
The main purpose of this study is to develop a way of solving multiattribute decision‐making problems (MADM), where the value of an attribute appears as picture hesitant fuzzy number and the attribute weight information is not entirely known. So as to acquire attribute weight vector, we build up a model of optimization that relies on the basic ideal of the conventional grey relational analysis method, by which attribute weight can be resolved. We build up another model of optimization for the uncommon circumstance where the attribute weight information is totally unknown. By solving this model, we get a basic and definite equation which can be utilized to be determined attribute weight. The grey relation degree between each alternative and positive ideal solution (PIS) and negative ideal solution (NIS) are determined. At the same time, a degree of relative relational is defined to decide all alternative order of ranking by ascertaining the grey relation degree to both the PIS and NIS. To examine the produced approach and to show its effectiveness and applicability an illustrative examples are given. A comparative analysis with pre‐existing MADM methods are considered to show the predominance of the proposed strategy.
The recently proposed q-rung orthopair fuzzy set, which is characterized by a membership degree and a non-membership degree, is effective for handling uncertainty and vagueness. This paper proposes the concept of complex q-rung orthopair
fuzzy sets (Cq-ROFS) and their operational laws. A multi-attribute decision making (MADM) method with complex q-rung orthopair fuzzy information is investigated. To aggregate complex q-rung orthopair fuzzy numbers, we extend the Einstein
operations to Cq-ROFSs and propose a family of complex q-rung orthopair fuzzy Einstein averaging operators, such as the complex q-rung orthopair fuzzy Einstein weighted averaging operator, the complex q-rung orthopair fuzzy Einstein ordered
weighted averaging operator, the generalized complex q-rung orthopair fuzzy Einstein weighted averaging operator, and the generalized complex q-rung orthopair fuzzy Einstein ordered weighted averaging operator. Desirable properties and special
cases of the introduced operators are discussed. Further, we develop a novel MADM approach based on the proposed operators
in a complex q-rung orthopair fuzzy context. Numerical examples are provided to demonstrate the effectiveness and superiority
of the proposed method through a detailed comparison with existing methods
Schweizer–Sklar (SS) operations are more flexible to aggregate the information, and the Muirhead mean (MM) operator can examine the interrelationships among the family of attributes. MM operator is more proficient and more generalized than many aggregation operators to cope with awkward and inconsistence information in realistic decision issues. The objectives of this manuscript are to explore the SS operators based on Pythagorean fuzzy set (PFS) and studied their score function, accuracy function, and their relationships. Further, based on these operators, the MM operators based on PFS, called Pythagorean fuzzy MM (PFMM) operator, Pythagorean fuzzy weighted MM (PFWMM) operator, and their special cases are presented. Additionally, the multi-attribute decision making (MADM) problem is solved by using the explored operators based on PFS to observe the consistency and efficiency of the discovered approach. Finally, the advantages, comparative analysis, and their geometrical representations are also discussed.
Entropy measure (EM) and similarity measures (SM) are important techniques in the environment of fuzzy set (FS) theory to resolve the SM between two objects. The q-rung orthopair FS (q-ROFS) and complex FS (CFS) are new extensions of FS theory and have been widely used in various fields. In this article, the EM, TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) based on the correlation coefficient (CC) is investigated. It is very important to study the SM of Cq-ROFS. Then, the established approaches and the existing drawbacks are compared by an example, and it is verified that the explored work can distinguish highly similar but inconsistent Cq-ROFS. Finally, to examine the reliability and feasibility of the new approaches we illustrate an example, using the TOPSIS method based on Cq-ROFS and is utilized to manage a case related the selection of firewall productions, and then, a situation concerning the security evaluation of computer systems is given to conduct the comparative analysis between the established TOPSIS method based on Cq-ROFS and previous decision-making methods for validating the advantages of the established work by comparing them with the other existing drawbacks.
Recently, Yager has established that the notion of q-rung orthopair fuzzy set (q-ROFS) is more accomplished than pythagorean fuzzy set (PyFS) and intuitionistic fuzzy set (IFS) to cope with awkward and complicated information in real decision theory. This notion works with yes-, no- and refusal-type fuzzy information. The constraint of q-ROFS is that the sum of n-power of the truth grade and the n-power of the falsity grade is bounded to unit interval. Generalized dice similarity measures are complimentary concepts quantifying the difference and closeness of q-ROFSs. In this paper, we suggested a number of novel dice similarity measures (DSMs) in the surroundings of the q-ROFS, and we examined some prevailing dice similarity measures and their limitations. In addition, we took the DSMs broad view to some globalized dice similarity measures (GDSMs), and we examined some of their particular cases. We employed the novel suggested GDSMs to the best selections of items on identification problems, and we analyzed their acquired consequences. There is a development of novel work in which many situations are evaluated, and from this perspective, the suggested work is changed into already prevailing work. This study also examines the merits of novel DSMs and the limitations for DSMs of IFSs and PyFSs. The comparison between established measures with existing measures is explored and their graphical interpretations are also discussed to show the reliability and effectiveness of the explored measures.
The Maclaurin symmetric mean (MSM) and dual Maclaurin symmetric mean (DMSM) operators are two aggregation operators to aggregate the q-rung orthopair fuzzy numbers (q-ROFNs) into a single element. The complex q-rung orthopair fuzzy framework is more effective to describe fuzzy information in real decision-making problems. The complex q-rung orthopair fuzzy sets (Cq-ROFSs) are more superior to the complex intuitionistic fuzzy sets (CIFSs) and complex pythagorean fuzzy sets (CPFSs) to cope with uncertain and difficult information in the environment of fuzzy set theory. Thus, this manuscript proposes some complex q-rung orthopair fuzzy Maclaurin symmetric mean (Cq-ROFMSM), complex q-rung orthopair fuzzy weighted Maclaurin symmetric mean (Cq-ROFWMSM), complex q-rung orthopair fuzzy dual Maclaurin symmetric mean (Cq-ROFDMSM) and complex q-rung orthopair fuzzy weighted dual Maclaurin symmetric mean (Cq-ROFWDMSM) operators. Moreover, some properties and special cases of our proposed methods are also introduced. Then we present a multi-attributive group decision-making based on proposed methods. Further, a numerical example is provided to illustrate the flexibility and accuracy of the proposed operators. Last, the proposed methods are compared with existing methods to examine the best emerging technology enterprises. Keywords q-Rung orthopair fuzzy sets · Complex q-rung orthopair fuzzy sets · Maclaurin symmetric mean operators · Dual Maclaurin symmetric mean operators Mathematics Subject Classification 03E72 · 03A72 · 90B50 · 94D05 Communicated by Regivan
The objective of this manuscript is to present some new, improved aggregation operators for the T-spherical fuzzy sets, which is an extension of the several existing sets, such as intuitionistic fuzzy sets, picture fuzzy sets, neutrosophic sets, and Pythagorean fuzzy sets. In it, some new, improved operational laws and their corresponding properties are studied. Further, based on these laws, we propose some geometric aggregation operators and study their various relationships. Desirable properties, as well as some special cases of the proposed operators, are studied. Then, based on these proposed operators, we present a decision-making approach to solve the multi-attribute decision-making problems. The reliability of the presented decision-making method is explored with the help of a numerical example and the proposed results are compared with several prevailing studies’ results. Finally, the superiority of the proposed approach is explained with a counter example to show the advantages of the proposed work.
In this manuscript, a novel structure of bipolar-valued hesitant fuzzy set (BPVHFS) is proposed as a generalization of fuzzy set (FS). The basic set theoretic operations of BPVHFSs are defined and their related results are studied. Based on the basic operations, some aggregation operators for BPVHFSs are developed and their fitness is verified using principle of mathematical induction. The multi-attribute decision making (MADM) is established in the framework of BPVHFSs and a numerical example is provided to illustrate the process. The article ended with some concluding remarks along with some future directions.
In this article, we give some basic definitions related to picture
fuzzy graphs and we explore several properties of these fuzzy graphs. Moreover,
the operations union, join, Cartesian product and composition of picture fuzzy
graphs and their properties are also studied.
In this manuscript, two generalizations of fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets, known as spherical fuzzy sets and T-spherical fuzzy sets, are discussed and a numerical and geometrical comparison among them is established. A T-spherical fuzzy set can model phenomena like voting using four characteristic functions denoting the degree of vote in favor, abstinence, vote in opposition, and refusal with an infinite domain, whereas an intuitionistic fuzzy set can model only phenomena of yes or no types. First, in this manuscript, some similarity measures in the frameworks of intuitionistic fuzzy sets and picture fuzzy sets are discussed. With the help of some numerical results, it is discussed that existing similarity measures have some limitations and could not be applied to problems where information is provided in T-spherical fuzzy environment. Therefore, some new similarity measures in the framework of spherical fuzzy sets and T-spherical fuzzy sets are proposed including cosine similarity measures, grey similarity measures, and set theoretic similarity measures. With the help of some results, it was proved that the proposed similarity measures are a generalization of existing similarity measures. The newly-defined similarity measures were subjected to a well-known problem of building material recognition and the results are discussed. A comparative study of new and existing similarity measures was established and some advantages of the proposed work are discussed.
Human opinion cannot be restricted to yes or no as depicted by conventional fuzzy set (FS) and intuitionistic fuzzy set (IFS) but it can be yes, abstain, no and refusal as explained by picture fuzzy set (PFS). In this article, the concept of spherical fuzzy set (SFS) and T-spherical fuzzy set (T-SFS) is introduced as a generalization of FS, IFS and PFS. The novelty of SFS and T-SFS is shown by examples and graphical comparison with early established concepts. Some operations of SFSs and T-SFSs along with spherical fuzzy relations are defined, and related results are conferred. Medical diagnostics and decision-making problem are discussed in the environment of SFSs and T-SFSs as practical applications.
his article is based on Einstein operations for bipolar-valued hesitant fuzzy sets (BVHFSs). We extend the concept of Einstein operators for BVHFSs by defining bipolar-valued hesitant fuzzy Einstein weighted averaging (BPVHFEWA) operators and weighted geometric (BPVHFEWG) operators. Similarly, we define ordered weighted averaging operators and hybrid operators i.e. BVHFEOWA operators, BVHFEOWG operators, BVHFEHA operator and BVHFEHG operator. Further, these operators are applied in decision making DM) problems.
In this article, a new hesitant Pythagorean fuzzy set (HPFS) has been presented by combining the concepts of Pythagorean fuzzy set (PFS) and Hesitant fuzzy set (HFS). The proposed set is the generalization of the sets of fuzzy, intuitionistic fuzzy, hesitant fuzzy and Pythagorean fuzzy. Some of the basic operational laws and their properties have been investigated in details. Also, we have developed some new weighted averaging and geometric aggregation operators named as hesitant Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric, hybrid average and geometric aggregation operators with hesitant Pythagorean fuzzy information. Furthermore, a multi-attribute decision-making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.
Intuitionistic fuzzy sets is one of the successful extensions of the fuzzy set to deal with the data, where the information is vague, imprecise, or insufficient during the decision-making process. The objective of the manuscript is to present generalized intuitionistic fuzzy sets and their corresponding operations. For it, we first presented an idea of the generalized IFSs and their corresponding operational laws. Furthermore, some new arithmetic and geometric mean operations are defined to aggregate the different preferences of the decision makers during the process. Various relations related to these operations are investigated in detail. Finally, a decision-making approach has been presented under the GIFSs environment and illustrated with a numerical example.
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.
Dealing with uncertainty is always a challenging problem. Intuitionistic fuzzy sets was presented to manage situations in which experts have some membership and non-membership value to assess an alternative. Hesitant fuzzy sets was used to handle such situations in which experts hesitate between several possible membership values to assess an alternative. In this paper, the concept of intuitionistic hesitant fuzzy set is introduced to provide computational basis to manage the situations in which experts assess an alternative in possible membership values and non-membership values. Distance measure is defined between any two intuitionistic hesitant fuzzy elements. Fuzzy technique for order preference by similarity to ideal solution is developed for intuitionistic hesitant fuzzy set to solve multi-criteria decision making problem in group decision environment. An example is given to illustrate this technique.
Hesitant fuzzy sets, permitting the membership of an element to be a set of several possible values, can be used as an efficient mathematical tool for modelling people’s hesitancy in daily life. In this paper, we extend the hesitant fuzzy set to interval-valued intuitionistic fuzzy environments and propose the concept of interval-valued intuitionistic hesitant fuzzy set, which allows the membership of an element to be a set of several possible interval-valued intuitionistic fuzzy numbers. The aim of this paper is to develop a series of aggregation operators for interval-valued intuitionistic hesitant fuzzy information. Then, some desired properties of the developed operators are studied, and the relationships among these operators are discussed. Furthermore, we apply these aggregation operators to develop an approach to multiple attribute group decision-making with interval-valued intuitionistic hesitant fuzzy information. Finally, a numerical example is provided to illustrate the application of the developed approach.
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.
A generalized trapezoidal-valued intuitionistic fuzzy geometric aggregation operator is proposed which is then used to aggregate decision makers' opinions in group decision making process. An extension of TOPSIS, a multi-criteria trapezoidal-valued intuitionistic fuzzy decision making technique, to a group decision environment is also proposed, where inter-dependent or interactive characteristics among criteria and preference of decision makers are under consideration. Furthermore, Choquet integral-based distance between trapezoidal-valued intuitionistic fuzzy values is defined. Combining the trapezoidal-valued intuitionistic fuzzy geometric aggregation operator with Choquet integral-based distance, an extension of TOPSIS method is developed to deal with a multi-criteria trapezoidal-valued intuitionistic fuzzy group decision making problems. Finally, an illustrative example is provided to understand the proposed method.
In this paper, we investigate the decision-making problem under the Pythagorean fuzzy environment by proposing some generalised aggregation operators. For it, we improve the existing aggregation operators by adding the pairs of the hesitation between the membership functions and hence proposed some new operational laws for Pythagorean fuzzy numbers using Einstein norm operations. Based on these laws, some weighted, ordered weighted and hybrid geometric interaction aggregation operators are proposed. The prominent characteristics of these operators are also studied. Further, a group decision-making problem is illustrated and validated through a numerical example. A comparative analysis of the proposed and existing studies is performed to show the validity of the proposed operators.
In this article, a new linguistic Pythagorean fuzzy set (LPFS) is presented by combining the concepts of a Pythagorean fuzzy set and linguistic fuzzy set. LPFS is a better way to deal with the uncertain and imprecise information in decision making, which is characterized by linguistic membership and nonmembership degrees. Some of the basic operational laws, score, and accuracy functions are defined to compare the two or more linguistic Pythagorean fuzzy numbers and their properties are investigated in detail. Based on the norm operations, some series of the linguistic Pythagorean weighted averaging and geometric aggregation operators, named as linguistic Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric with linguistic Pythagorean fuzzy information are proposed. Furthermore, a multiattribute decision‐making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.
In this paper, we investigate the multiple attribute decision making (MADM) problems with picture fuzzy information. Firstly, concepts and some operational laws of picture fuzzy sets are introduced. Then, we develop some picture fuzzy geometric operators and discuss their basic properties. Next, we apply the proposed operators to deal with multiple attribute decision making problems under picture fuzzy environment. Finally, an illustrative example is given to demonstrate the practicality and effectiveness of our proposed method.
In this paper, we have investigated the multiple attribute decision-making problems with bipolar fuzzy information. Motivated by the Hamacher operations, we have proposed bipolar fuzzy Hamacher weighted average operator, bipolar fuzzy Hamacher ordered weighted average operator, bipolar fuzzy Hamacher hybrid average operator, bipolar fuzzy Hamacher weighted geometric operator, bipolar fuzzy Hamacher ordered weighted geometric operator, bipolar fuzzy Hamacher hybrid geometric operator. We investigate the characteristics and special cases of these operators. Then, we have utilized these operators to develop some approaches to solve the bipolar fuzzy multiple attribute decision-making problems. Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.
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.
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.
Interval-valued hesitant fuzzy sets (IVHFSs), as an extension of hesitant fuzzy sets, can account for the membership degrees of an element to a given set having a few different interval values, which provides an intuitionistic description on the differences among decision makers. We derive the properties and relationships of fundamental operations on IVHFSs for Algebraic t-norm and t-conorm. Furthermore, we present the operations based on Archimedean t-norm and t-conorm and investigate their properties. The results obtained using the two types of t-norms and t-conorms could be useful for applications of IVHFSs in information aggregation and decision making.
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.
We try to provide a tentative assessment of the role of fuzzy sets in decision analysis. We discuss membership functions, aggregation operations, linguistic variables, fuzzy intervals and the valued preference relations they induce. The importance of the notion of bipolarity and the potential of qualitative evaluation methods are also pointed out. We take a critical standpoint on the state-of-the-art, in order to highlight the actual achievements and question what is often considered debatable by decision scientists observing the fuzzy decision analysis literature.
Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.
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.
Multiple Attribute Decision Making M e t h o d U n d e r L i n g u i s t i c C u b i c Information
- N Jan
- L Zedam
- T Mahmood
- K Ullah
- Ali
N. Jan, L. Zedam, T. Mahmood,K. Ullah andZ.
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Information,"Journal of Intelligent & Fuzzy
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Some Aggregation Operators F o r B i p o l a r -Va l u e d H e s i t a n t F u z z y Information
- T Mahmood
- K Ullah
- Q Khan
- Smarandache
T. Mahmood, K. Ullah, Q. Khan andF.
Smarandache,"Some Aggregation Operators
F o r B i p o l a r -Va l u e d H e s i t a n t F u z z y
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Applied Sciences, vol. 10, no. 4S, pp. 240-245,
2018.
New Operations of P i c t u r e F u z z y R e l a t i o n s a n d F u z z y Comprehensive Evaluation
- B Chunxin
- X Zhang
B. Chunxin and X. Zhang, "New Operations of
P i c t u r e F u z z y R e l a t i o n s a n d F u z z y
Comprehensive Evaluation," Symmetry, vol. 9,
no.11,pp. 268, 2017.