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Green supply chain management concerns the incorporation of globally sustainable methods into supply chain management techniques. The major goal is to decrease the environmental impact of the entire supply chain to increase the social, ecological, and economic relationships with other countries for business. In this manuscript, we aim to compute the Dombi operational laws based on the complex Pythagorean fuzzy (CPF) information. Furthermore, we construct the Dombi aggregation operators under the consideration of the CPF information, such as the CPF power Dombi averaging (CPFPDA) operator, CPF power weighted Dombi averaging (CPFPWDA) operator, CPF power Dombi geometric (CPFPDG) operator, and CPF power weighted Dombi geometric (CPFPWDG) operator. Some flexible properties for the invented operators are also discussed. Additionally, we develop the multi-attribute decision-making (MADM) method to evaluate the problem of green supply chain management based on the invented operators for CPF information. Finally, we use some numerical examples to show the supremacy and validity of the developed techniques by comparing their ranking results with the obtaining ranking results of the existing techniques. [ABSTRACT FROM AUTHOR]

In 1960, Schweizer and Sklar introduced the novel Schweizer-Sklar t-norm and t-conorm which is used in the construction of aggregation operators. Schweizer-Sklar norms are more general than algebraic norms and Einstein norms. Additionally, computing the power operators based on the Schweizer-Sklar norms for complex Atanassov intuitionistic fuzzy (CA-IF) set is very awkward and complicated. In this manuscript, firstly, we propose the Schweizer-Sklar operational laws for CA-IF values, and secondly, we develop the CA-IF Schweizer-Sklar power averaging (CA-IFSSPA) operator, CA-IF Schweizer-Sklar power ordered averaging (CA-IFSSPOA) operator, CA-IF Schweizer-Sklar power geometric (CA-IFSSPG) operator, and CA-IF Schweizer-Sklar power ordered geometric (CA-IFSSPOG) operator. Some suitable and dominant properties for the above operators are also discussed. Furthermore, to simplify the above operators, we develop the procedure of decision-making technique, called multi-attribute decision-making (MADM) methods based on the proposed operators based on CA-IF values. Finally, we compare the proposed methods with some existing methods to describe the efficiency and capability of the discovered approaches by some examples. [ABSTRACT FROM AUTHOR]

A complex intuitionistic fuzzy (CIF) set contains the membership and non-membership in the shape of a complex number whose amplitude term and phase term are covered in the unit interval. Moreover, Hamacher interaction aggregation operators are the combination of two major operators, called Hamacher aggregation operators and interaction aggregation operators, and they are used to aggregate the collection of information into one value. In this manuscript, we present the concept of Hamacher interaction operational laws for CIF sets (CIFSs). Further, we develop the CIF Hamacher interaction weighted averaging (CIFHIWA) operator, CIF Hamacher interaction ordered weighted averaging (CIFHIOWA) operator, CIF Hamacher interaction weighted geometric (CIFHIWG) operator, and CIF Hamacher interaction ordered weighted geometric (CIFHIOWG) operator. For these operators, we also discuss some basic properties, such as idempotency, monotonicity, and boundedness. Additionally, we develop a MADM method based on the developed operators and apply it to solve the green supply chain management problems, which can implement environmentally friendly practices to minimize carbon emissions, resource consumption, and waste generation while promoting long-term sustainability. Finally, we verify the superiority and effectiveness of the proposed method based on a comparative analysis between the proposed techniques and existing methods. [ABSTRACT FROM AUTHOR]

Aggregation operators are used for aggregating the collection of finite information into a singleton set. The Aczel-Alsina t-norm and t-conorm are very useful for constructing any kind of new aggregation operators, which was presented by Aczel and Alsina in 1982. Moreover, complex Pythagorean fuzzy (CPF) sets and hesitant fuzzy (HF) sets are the most generalized and very useful techniques to cope with unreliable and awkward information in genuine life problems. In this manuscript, we combine the HF set and CPF set to derive the complex Pythagorean hesitant fuzzy (CPHF) set and its fundamental laws. Furthermore, we evaluate the Aczel-Alsina operational laws based on Aczel-Alsina norms and CPHF information. Additionally, based on the Aczel-Alsina operational laws for CPHF information, we investigate the CPHF Aczel-Alsina-weighted averaging (CPHFAAWA) operator, CPHF Aczel-Alsina-ordered weighted averaging (CPHFAAOWA) operator, CPHF Aczel-Alsina-weighted geometric (CPHFAAWG) operator, and CPHF Aczel-Alsina-ordered weighted geometric (CPHFAAOWG) operator. Some remarkable properties are also examined for the invented theory. Moreover, a multi-attribute decision-making (MADM) technique is presented based on discovered operators for CPHF information. Finally, we aim to illustrate some examples for comparing the proposed techniques with some existing ones to show the worth and feasibility of the discovered approaches. [ABSTRACT FROM AUTHOR]

One of the most significant and complete approaches to accommodate greater uncertainty than current fuzzy structures is the T-Spherical Fuzzy Set (TSPFS). The primary benefit of TSPFS is that current fuzzy structures are special cases of it. Firstly, some novel TSPF power Heronian mean (TSPFPHM) operators are initiated based on Aczel–Alsina operational laws. These aggregation operators (AOs) have the capacity to eliminate the impact of uncomfortable data and can simultaneously consider the relationships between any two input arguments. Secondly, some elementary properties and core cases with respect to parameters are investigated and found that some of the existing AOs are special cases of the newly initiated aggregation operators. Thirdly, based on these AOs and Aczel–Alsina operational laws a newly advanced technique for order of preference by similarity to ideal solution (TOPSIS)-based method for dealing with multi-attribute group decision-making (MAGDM) problems in a T-Spherical fuzzy framework is established, where the weights of both the decision makers (DMs) and the criteria are completely unknowable. Finally, an illustrative example is provided to evaluate and choose the pharmaceutical firms with the capacity for high-quality, sustainable development in the TSPF environment to demonstrate the usefulness and efficacy. After that, the comparison analysis with other techniques is utilized to demonstrate the coherence and superiority of the recommended approach. [ABSTRACT FROM AUTHOR]

In multi-attribute group decision-making (MAGDM), the attributes can be placed into independent groups based on their properties through partitioning. First, the partitioned dual Hamy mean (PDHM) operator is introduced, along with its essential properties. This operator integrates these separate groups while preserving the relationships between the attributes within each group. Furthermore, the partitioned Hamy mean (PHM) and the PDHM operators are also constructed in the generalized orthopair fuzzy environment, namely the q-rung orthopair fuzzy PHM (q-ROFPHM), the q-rung orthopair fuzzy PDHM (q-ROFPDHM), and their weighted forms. Their essential properties are verified to ensure the validity of the proposed aggregation operators (AOs). Subsequently, a new MAGDM approach is developed, employing the proposed AOs. The MAGDM problem of selecting the best person is examined. Moreover, the research includes a sensitivity analysis in three directions and a comparative analysis of the proposed MAGDM approach with five different approaches. The findings indicate that applying attribute partitioning in the proposed approach mitigates the adverse impact of irrelevant attributes, leading to more feasible and reliable outcomes. Additionally, a practical case study focuses on selecting a suitable industry for investment among the five available options. This case study demonstrates the approach's effectiveness by considering five distinct qualities and results that make the Internet industry the best place to invest. Furthermore, a comparative analysis with four similar papers is also performed, indicating that the developed method's results are more reliable and consistent. [ABSTRACT FROM AUTHOR]

Due to the intense competition in the market today, choosing of an appropriate healthcare device vendor in long-term organ transplant networks has emerged as a key issue in raising life expectancy. A complicated multi-attribute group decision-making (MAGDM) process problem with several viable alternatives and sustainable criteria may be used for evaluating sustainable healthcare equipment vendors. The q-rung orthopair fuzzy set (QROFS) is more effective at expressing ambiguous and fuzzy information since it is a generalization of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFS). The fact that QROFSs provide a wider range of acceptable membership grades and give decision-makers more leeway to express their real thoughts is their most valued feature. The Heronian mean (HM) operator and power aggregation (PA) operator are instances of classic aggregation operators. They are preferable because they can replicate the correlations between attributes and remove the negative effects of awkward information. The Aczel-Alsina t-norms, which were put out by Aczel and Alsina in 1982, constitute a very successful and widely used method for creating any type of aggregation operators. Additionally, the parameter Φ ∈ 0 , + ∞ makes the Algebraic t-norms as a special case of the Aczel-Alsina t-norms. To take the above advantages, in this article, initially, the Aczel-Alsina (AA) operational laws are combined, with power average and Heronian mean operators to propose the QROFAA power Heronian aggregation (QROFPWHA) operator and QROFAA power geometric Heronian aggregation (QROFAAFPGHA) operator. Moreover, some core characteristics and various core cases with respect to the parameters are investigated and found that some of the existing aggregation defined on AA operational laws are special cases of the suggested aggregation operators. Secondly, the weighted forms of the suggested aggregation operators are initiated. Thirdly, based on these newly aggregation operators two novel MAGDM models with unknown weights of the decision makers and attributes are initiated. Finally, an illustrated example about evaluating sustainable healthcare equipment vendors is provided to assess the effectiveness of the suggested models, and a comparison analysis is provided to support and corroborate the suggested approaches. Additionally, assessment for key parameters in the proposed models is carried out to evaluate the implications on results. [ABSTRACT FROM AUTHOR]

Algebraic and Einstein are two different types of norms which are the special cases of the Hamacher norm. These norms are used for evaluating or constructing three different types of aggregation operators, such as averaging/geometric, Einstein averaging/geometric, and Hamacher averaging/geometric aggregation operators. Moreover, complex Atanassov intuitionistic fuzzy (CA-IF) information is a very famous and dominant technique or tool which is used for depicting unreliable and awkward information. In this manuscript, we present the Hamacher operational laws for CA-IF values. Furthermore, we derive the power aggregation operators (PAOs) for CA-IF values, called CA-IF power Hamacher averaging (CA-IFPHA), CA-IF power Hamacher ordered averaging (CA-IFPHOA), CA-IF power Hamacher geometric (CA-IFPHG), and CA-IF power Hamacher ordered geometric (CA-IFPHOG) operators. Some dominant and valuable properties are also stated. Moreover, the multi-attribute decision-making (MADM) methods are developed based on the invented operators for CA-IF information and the detailed decision steps are given. Many prevailing operators are selected as special cases of the invented theory. Finally, the derived technique will offer many choices to the expert to evaluate the best alternatives during comparative analysis. [ABSTRACT FROM AUTHOR]

The complex q-rung orthopair fuzzy (CQ-ROF) set can describe the complex uncertain information. In this manuscript, we develop the Yager operational laws based on the CQ-ROF information and Yager t-norm and t-conorm. Furthermore, in aggregating the CQ-ROF values, the power, averaging, and geometric aggregation operators have played a very essential and critical role in the environment of fuzzy set. Inspired from the discussed operators, we propose the CQ-ROF power Yager averaging (CQ-ROFPYA), CQ-ROF power Yager ordered averaging (CQ-ROFPYOA), CQ-ROF power Yager geometric (CQ-ROFPYG), and CQ-ROF power Yager ordered geometric (CQ-ROFPYOG) operators. These operators are the modified version of the Power, Yager, averaging, geometric, and the combination of these all based on fuzzy set (FS), intuitionistic FS, Pythagorean FS, q-rung orthopair FS, complex FS, complex intuitionistic FS, and complex Pythagorean FS. Moreover, we also discuss the main properties of the proposed operators. Additionally, we develop a multi-attribute decision-making (MADM) method based on the developed operators. To show the supremacy and validity of the proposed method, the comparison between the proposed method and some existing methods is done by some examples, and results show that the proposed method is better than the others in terms of generality and effectiveness. [ABSTRACT FROM AUTHOR]

The Aczel-Alsina t-norms ware proposed by Aczel and Alsina in 1982, which are a very effective and dominant technique used in the construction of any kind of aggregation operators. Moreover, the Algebraic t-norms are a special case of the Aczel-Alsina t-norms because of the parameter 0 < p < + ∞ . Additionally, a complex intuitionistic fuzzy (COIF) set is an essential and valuable part of the fuzzy set to handle vague and awkward situations in many real-life problems. Motivated by the above valuable and dominant ideas, the major contribution of this study is to propose the aggregation operators by involving the priority degree based on Aczel-Alsina t-norms for managing the COIF values, such as COIF Aczel-Alsina prioritized weighted averaging (COIFAAPWA), COIF Aczel-Alsina prioritized ordered weighted averaging (COIFAPOWA), COIF Aczel-Alsina prioritized weighted geometric (COIFAAPWG), and COIF Aczel-Alsina prioritized ordered weighted geometric (COIFAAPOWG) operators. The fundamental properties of these operators are also examined. Afterward, we develop a decision-making technique to process the multi-attribute decision-making (MADM) problem based on the COIF information and proposed operators. Lastly, we use some examples to show the supremacy and effectiveness of the developed approaches by checking the influence of the parameters and the comparison between proposed techniques with some existing techniques. [ABSTRACT FROM AUTHOR]

Complex a-rung orthopair fuzzy soft (CQROFS) set is very famous and effective tool for depicting complex uncertain information because it is the combination of the complex q-rung orthopair fuzzy set (CQROFS) and soft set (SS), in which the truth grade and falsity grade are expressed by complex numbers. Furthermore, Einstein's t-norm and t-conorm are very useful and more dominant tools in aggregating the collection of information into a singleton one, where the algebraic t-norm and t-conorm are the specific case of the Einstein t-norm and t-conorm. In this paper, we aim to propose the complex q-rung orthopair fuzzy soft (CQROFS) information which is the modification or generation of the complex Pythagorean and complex intuitionistic fuzzy soft sets. Some specific operational laws are also derived for the proposed work based on algebraic t-norm and t-conorm. Additionally, we also develop the Einstein's operational laws in the presence of the CQROFS set. Consequently, based on the Einstein operational laws and CQROFS information, we derive the CQROFS Einstein weighted geometric (CQROFSEWG), CQROFS Einstein ordered weighted geometric (CQROFSEOWG), and CQROFS Einstein hybrid geometric (CQROFSEHG) operators. Some notable properties and valuable results are also explored. Moreover, we develop a multi-attribute decision-making (MADM) method based on the derived operators for CQROFS set. Finally, we do a comparative analysis between the proposed work and some existing operators to show the advantages and supremacy of the developed method. [ABSTRACT FROM AUTHOR]

As a more massive feasible and prominent tool than the complex interval-valued Pythagorean fuzzy (CIVPF) set and complex interval-valued intuitionistic fuzzy (CIVIF) set, the complex interval-valued q-rung orthopair fuzzy (CIVQROF) set has been usually used to represent ambiguity and vagueness for real-life decision-making problems. In this paper, we firstly proposed some distance measures, Yager operational laws, and their comparison method. Further, we developed CIVQROF Yager weighted averaging (CIVQROFYWA), CIVQROF Yager ordered weighted averaging (CIVQROFYOWA), CIVQROF Yager weighted geometric (CIVQROFYWG), CIVQROF Yager ordered weighted geometric (CIVQROFYOWG) operators with CIVQROF information, and some certain well-known and feasible properties and outstanding results are explored in detail. Moreover, we proposed a new and valuable technique for handling multi-attribute decision-making problems with CIVQROF information. Lastly, a practical evaluation regarding the high blood pressure diseases of the patient is evaluated to illustrate the feasibility and worth of the proposed approaches. [ABSTRACT FROM AUTHOR]

In this manuscript, we firstly proposed a novel concept of complex interval-valued q-rung orthopair linguistic (CIVq-ROL) information by integrating the concepts of complex interval-valued q-rung orthopair fuzzy (CIVq-ROF) information and linguistic set (LS), and it is more generalized than most existing sets, and it is very feasible and valuable for depicting awkward and unreliable information in a difficult situation. Further, we developed some new operational laws of the CIVq-ROF information based on algebraic t-norm and t-conorm. It is also clear that it is a very challenging task to fuse the collection of data into a singleton set in information fusion and decision making, and the Heronian mean (HM) operator is an important aggregation operator, which can not only achieve the aggregation function from the collection of data into a singleton one, but also consider the relationship between any two data. Therefore, based on the CIVq-ROL information and HM operator, we proposed the CIVq-ROL Heronian mean (CIVq-ROLHM), CIVq-ROL weighted HM (CIVq-ROLWHM), CIVq-ROL geometric HM (CIVq-ROLGHM), and CIVq-ROL weighted geometric HM (CIVq-ROLWGHM) operators, and then, the various desirable properties and specific cases of them are also investigated. Furthermore, we developed a multi-attribute decision-making (MADM) procedure for evaluating the finest business in the country from the collection of four different types of electronics-related businesses based on the proposed operators. Finally, various examples are given to show the application of the invented techniques, and a comparative analysis is carried out for the parameters to show the advantages of the proposed approaches. [ABSTRACT FROM AUTHOR]

The diagnosed complex Pythagorean fuzzy (CPF) set is a more valuable and dominant tool than the Pythagorean and intuitionistic fuzzy sets to describe awkward and unreliable information more effectively. Further, Archimedean t-norm and t-conorm have a significant influence to depict the relation among aggregated values. To take advantage of the CPF set and Archimedean t-norm and t-conorm, and assume the relation among Archimedean norms and algebraic, Einstein, Hamacher, and frank norms at the same time, in this analysis, first, we proposed the fundamental Archimedean operational laws. Second, based on these laws, we developed confidence CPF Archimedean-weighted averaging (CCPFSAWA), confidence CPF Archimedean-ordered weighted averaging (CCPFSAOWA), confidence CPF Archimedean-weighted geometric (CCPFSAWG), confidence CPF Archimedean-ordered weighted geometric (CCPFSAOWG) operators and implemented their valuable results and properties. We know that Archimedean t-norm and t-conorm are the general form of the all-aggregation operators, so by using different values of t-norm and t-conorm, we explored the confidence CPF-weighted averaging (CCPFWA), confidence CPF-ordered weighted averaging (CCPFOWA), confidence CPF Einstein-weighted averaging (CCPFEWA), confidence CPF Einstein-ordered weighted averaging (CCPFEOWA), confidence CPF Hamacher-weighted averaging (CCPFHWA), confidence CPF Hamacher-ordered weighted averaging (CCPFHOWA), confidence CPF frank-weighted averaging (CCPFFWA), confidence CPF frank-ordered weighted averaging (CCPFFOWA), confidence CPF-weighted geometric (CCPFWG), confidence CPF-ordered weighted geometric (CCPFOWG), confidence CPF Einstein-weighted geometric (CCPFEWG), confidence CPF Einstein-ordered weighted geometric (CCPFEOWG), confidence CPF Hamacher-weighted geometric (CCPFHWG), confidence CPF Hamacher-ordered weighted geometric (CCPFHOWG), confidence CPF frank-weighted geometric (CCPFFWG), and confidence CPF frank-ordered weighted geometric (CCPFFOWG) operators. Then, we developed a multi-attribute decision-making (MADM) method based on the proposed operators. Finally, many examples are used to do comparative analysis among proposed and existing methods to show the validation of the new approaches. [ABSTRACT FROM AUTHOR]

The 2-dimensional uncertain linguistic variable (2DULV) can depict decision-makers' subjective assessments on the reliability of given evaluation results, which is a valid and practical tool to express decision information. In this study, we develop an improved MABAC method with 2DULVs to handle multiattribute group decision-making (MAGDM) problems where the weight information of attributes is unknown. First, some related theories of 2DULVs and the basic procedure of the MABAC method are briefly reviewed. Then, the maximum comprehensive evaluation value method is extended to 2DULVs to obtain combination weights of attributes, in which the subjective weights are determined according to the best–worst method (BWM) and the objective weights are calculated by the maximum deviation method. Besides, the generalized weighted average operator for 2DULVs (2DULGWA) is utilized to aggregate the evaluation information given by all experts. Finally, an improved MABAC for 2DULVs (2DUL-MABAC) is proposed, and an example is carried out to explain the validity of the proposed approach. [ABSTRACT FROM AUTHOR]

Massive open online courses (MOOCs) are very popular in China, and it is very important to evaluate and improve them. In this paper, a new evaluation method for MOOCs based on multi‐attribute group decision‐making is proposed. First, an evaluation index system of MOOCs is constructed, which contains six elements and 16 indicators, and multigranular unbalanced hesitant fuzzy linguistic term set (MGUHFLTS) is adopted to describe these indicators. Then based on MGUHFLTS, the aggregation operators are developed, including the multigranularity unbalanced hesitant fuzzy linguistic weighted averaging operator and the multigranularity unbalanced hesitant fuzzy linguistic order weighted averaging operator, moreover, a novel multi‐attributive border approximation area comparison model based on MGUHFLTS is proposed. This model is testified validity and superiority by comparison with other three methods and is applied in evaluation of MOOCs. After ranking five MOOCs, each indicator is analyzed to show how they influenced each element and suggestions are given. [ABSTRACT FROM AUTHOR]

The use of incomplete probabilistic linguistic term sets (InPLTSs) can enrich the flexibility of qualitative decision-making information expression, especially in decision-making situations with high time pressure and insufficient knowledge. In this article, we develop a method for group decision-making (GDM) with incomplete probabilistic linguistic preference relations (InPLPRs), considering consistency and consensus simultaneously. First, to fully explore the ability of InPLTSs to express uncertain information, InPLTSs are specifically classified. Then, an expected multiplicative consistency of InPLPRs is introduced, which is conducive to estimating the missing information more accurately and effectively. Subsequently, considering the consensus of GDM problems, a consensus index, which considers the principle of majority and minority, is developed to measure the agreement degree among multiple individuals. Because individual InPLPRs may not all meet acceptable consistency after reaching consensus, a consistency- and consensus-improving mathematical programming model considering information distortion is presented. Then, to aggregate all individual preference relations into a collective one, a reliability-induced ordered weighted geometric operator is introduced, whose induced variable reliability is determined by the confidence degree and consistency index of individual preference relations. Furthermore, a multiphase algorithm with InPLPRs is developed to solve GDM problems. Finally, a numerical example of fire emergency decisions is presented to illustrate the applicability of the proposed method, and a detailed validity test and comparative analysis are conducted to highlight the advantages of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In today's education industry, online teaching is increasingly becoming an important teaching way, and it is necessary to evaluate the quality of online teaching so as to improve the overall level of the education industry. The online teaching quality evaluation is a typical multi-attribute group decision-making (MAGDM) problem, and its evaluation index can be expressed by linguistic term sets (LTSs) by decision makers (DMs). Especially, multi-granularity probabilistic linguistic term sets (MGPLTSs) produced from many DMs are more suitable to express complex fuzzy evaluation information, and they can not only provide different linguistic term set for different DMs the give their preferences, but also reflect the importance of each linguistic term. Based on the advantages of MGPLTSs, in this paper, we propose a transformation function of MGPLTSs based on proportional 2-tuple fuzzy linguistic representation model. On this basis, the operational laws and comparison rules of MGPLTSs are given. Then, we develop a new Choquet integral operator for MGPLTSs, which considers the relationship among attributes and does not need to consider the process of normalizing the probabilistic linguistic term sets (PLTSs), and can effectively avoid the loss of evaluation information. At the same time, the properties of the proposed operator are also proved. Furthermore, we propose a new MAGDM method based on the new operator, and analyze the effectiveness of the proposed method by online teaching quality evaluation. Finally, by comparing with some existing methods, the advantages of the proposed method are shown. [ABSTRACT FROM AUTHOR]

T-spherical fuzzy numbers (FNs), which add an abstinence degree based on membership and non-membership degrees, can express neutral information conveniently and have a considerable large range of information expression. The normal FNs (NFNs) are very available to characterize normal distribution phenomenon widely existing in social life. In this paper, we first define the normal T-SFNs (NT-SFNs) which can combine the advantages of T-SFNs and NFNs. Then, we define their operational laws, score value, and accuracy value. By considering the interrelationship among multi-input parameters, we propose the Maclaurin symmetric mean operator with NT-SFNs (NT-SFMSM) and its weighted form (NT-SFWMSM). Furthermore, we study some characteristics and special cases of them. Based on the NT-SFWMSM operator, we put forward a novel multi-attribute decision-making (MADM) approach. Finally, some numerical examples are conducted to prove that the proposed approach is valid and superior to some other existing methods. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy set (PyFS) is an extension of various fuzzy concepts, such as fuzzy set (FS), intuitionistic FS, and it is enhanced mathematical gizmo to pact with uncertain and vague information. In this article, some drawbacks in the Dombi operational rules for Pythagorean fuzzy numbers (PyFNs) are examined and some improved Dombi operational laws for PyFNs are developed. We also find out that the value aggregated using the existing Dombi aggregation operators (DAOs) is not a PyFN. Furthermore, we developed two new aggregations, improved existing aggregation operators (AOs) for aggregating Pythagorean fuzzy information (PyFI) and are applied to multiple-attribute decision making (MADM). To acquire full advantage of power average (PA) operators proposed by Yager, the Pythagorean fuzzy Dombi power average (PyFDPA) operator, the Pythagorean fuzzy Dombi weighted power average (PyFDWPA) operator, Pythagorean fuzzy Dombi power geometric (PyFDPG) operator, Pythagorean fuzzy Dombi weighted geometric (PyFDPWG) operator, improved the existing AOs and their desirable properties are discussed. The foremost qualities of these developed Dombi power aggregation operators is that they purge the cause of discomfited data and are more supple due to general parameter. Additionally, based on these Dombi power AOs, a novel MADM approach is instituted. Finally, a numerical example is given to show the realism and efficacy of the proposed approach and judgment with the existing approaches is also specified. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article puts forward an innovative notion of complex picture fuzzy set (CPFS) which is particularly an extension and a generalization of picture fuzzy sets (PFSs) by the addition of phase term in the description of PFSs. The uniqueness of CPFS lies in the capability to manage the uncertainty and periodicity, simultaneously, due to the presence of phase term which broadens the range of CPFS from a real plane to the complex plane of unit disk. We describe and verify the fundamental operations and properties of CPFSs. We introduce the aggregation operators, namely; complex picture fuzzy power averaging and complex picture fuzzy power geometric operators in CPFSs environment, based on weighted and ordered weighted averaging and geometric operators. We construct multi-criteria decision making (MCDM) problem, using these operators and describe a numerical example to illustrate the validity and competence of this article. Finally, we discuss the advantages of this generalized concept of aggregation technique and analyze a comparative study to demonstrate the superiority and consistency of our model. [ABSTRACT FROM AUTHOR]

The dual generalized Bonferroni mean operator is a further extension of the generalized Bonferroni mean operator which can take the interrelationship of different numbers of attributes into account by changing the embedded parameter. The 2-dimensional uncertain linguistic variable (2DULV) adds a second dimensional uncertain linguistic variable (ULV) to express the reliability of the assessment information in first dimensional information, which is more rational and accurate than the ULV. In this paper, for combining the advantages of them, we propose the dual generalized weighted Bonferroni mean operator for 2DULVs (2DULDGWBM) and the dual generalized weighted Bonferroni geometric mean operator for 2DULVs (2DULDGWBGM). In addition, we explore several particular cases and some rational characters of them. Further, a new approach is introduced to handle multi-attribute decision making problems in the environment of 2DULVs by the proposed operators. Finally, we utilize several illustrative examples to testify the validity and superiority of this new method by comparing with several other methods. [ABSTRACT FROM AUTHOR]

As the development and extension of hesitant fuzzy linguistic term sets (HFLTSs), free double hierarchy hesitant fuzzy linguistic term sets (FDHHFLTSs) can describe the evaluation information in more detail. In addition, in practical multiple attribute decision-making (MADM) problems, priority relations and interaction relations usually exist among attributes, and the prioritized interactive Choquet (PIC) operator based on generalized prioritized measure-guided aggregation (GPMGA) operator and Choquet integral (CI) operator is an effective tool for dealing with decision-making problems with priority relations and interaction relations. Under the above contexts, we improve the PIC operator and propose the FDHHFLPIC operator which can aggregate the evaluation information under free double hesitant fuzzy linguistic environment and take both priority and interaction relations into account. Furthermore, we propose an approach based on the FDHHFLPIC operators to address the real MADM problem, and the proposed method considering the different relations between attributes in decision-making process. Finally, two numerical examples are given to demonstrate the effectiveness and advantages of the proposed method which can consider priority relations and interaction relations under the FDHHFLTSs. [ABSTRACT FROM AUTHOR]

The complex q-rung orthopair fuzzy set (Cq-ROFS), an efficient generalization of complex intuitionistic fuzzy set (CIFS) and complex Pythagorean fuzzy set (CPFS), is potent tool to handle the two-dimensional information and has larger ability to translate the more uncertainty of human judgment then CPFS as it relaxes the constrains of CPFS and thus the space of allowable orthopair increases. To solve the multi-criteria decision making (MCDM) problem by considering that criteria are at the same priority level may affect the results because in realistic situations the priority level of criteria is different. In this manuscript, we propose some useful prioritized AOs under Cq-ROF environment by considering the prioritization among attributes. We develop two prioritized AOs, namely complex q-rung orthropair fuzzy prioritized weighted averaging (C-qROFPWA) operator and complex q-rung orthropair fuzzy prioritized weighted geometric (Cq-ROFPWG) operator. We also consider their desirable properties and two special cases with their detailed proofs. Moreover, we investigate a new technique to solve the MCDM problem by initiating an algorithm along with flowchart on the bases of proposed operators. Further, we solve a practical example to reveal the importance of proposed AOs. Finally, we apply the existing operators on the same data to compare our computed result to check the superiority and validity of our proposed operators. [ABSTRACT FROM AUTHOR]

To be able to describe more complex fuzzy uncertainty information effectively, the concept of q-rung orthopair fuzzy sets (q -ROFSs) was first proposed by Yager. The q-ROFSs can dynamically adjust the range of indication of decision information by changing a parameter q based on the different hesitation degree from the decision-makers, where q ≥ 1, so they outperform the traditional intuitionistic fuzzy sets and Pythagorean fuzzy sets. In real decision-making problems, there is often an interaction phenomenon between attributes. For aggregating these complex fuzzy information, the Maclaurin symmetric mean (MSM) operator is more superior by considering interrelationships among attributes. In addition, the power average (PA) operator can reduce the effects of extreme evaluating data from some experts with prejudice. In this paper, we introduce the PA operator and the MSM operator based on q-rung orthopair fuzzy numbers (q-ROFNs). Then, we put forward the q-rung orthopair fuzzy power MSM (q-ROFPMSM) operator and the q-rung orthopair fuzzy power weighed MSM (q-ROFPWMSM) operator of q-ROFNs and present some of their properties. Finally, we present a novel multiple-attribute group decision-making (MAGDM) method based on the q-ROFPWA and the q-ROFPWMSM operators. The experimental results show that the novel MAGDM method outperforms the existing MAGDM methods for dealing with MAGDM problems. [ABSTRACT FROM AUTHOR]

The unbalanced linguistic terms set (ULTS) is a special linguistic term set which can describe the vagueness assessment that is non-uniform and non-symmetrical distributed. So, it is effective to describe the uncertainty information existed in some special real decision making problems by ULTS. As a special prioritized operator, the scaled prioritized (SP) operator has the advantage of taking the priority among different criteria into account by detailed priority labels in known case and unknown case. In this paper, we combine the merits of SP operators and ULTS for dealing with some special multi-criteria decision making (MCDM) problems where there is a priority relationship between criteria under ULTS evaluation information. We present the unbalanced 2-tuple linguistic scaled prioritized averaging operator and the unbalanced 2-tuple linguistic scaled prioritized geometric averaging operator, which can handle the issues of the detailed priority relationship among different categories of MCDM problems in knowable case. Further, we propose the unbalanced 2-tuple linguistic scaled prioritized weighted averaging operator and the unbalanced 2-tuple linguistic scaled prioritized geometric weighted averaging operator, which can deal with the case when the detailed priority relationship among different categories of different criteria is unknowable. Then, we discussed several characteristics of the proposed operators, such as boundedness, monotonicity, and idempotency. Besides, we presented an approach for the MCDM problems according to the proposed operators. In the last, we provide an example to explain the calculating steps and effectiveness of these methods. [ABSTRACT FROM AUTHOR]

Multi-attribute group decision making (MAGDM ) can be considered as the process of ranking alternatives or selecting an optimal alternative by many decision makers based on multiple criteria. By comparing any two alternatives pairwise, preference relations provide efficient ways to represent the preference degrees of decision makers. The aim of this paper is to introduce the notions of upward intuitionistic fuzzy preference relations from fuzzy information system and to study some properties of these relations. Further the series of upward intuitionistic fuzzy preference aggregation operators including the upward intuitionistic fuzzy preference weighted averaging UIFPWA operator, upward intuitionistic fuzzy preference ordered weighted averaging UIFPOWA operator and upward intuitionistic fuzzy preference hybrid averaging UIFPHA operator and their related results are also investigated. We developed a MAGDM method based on the proposed operators under the fuzzy environment and illustrated with a numerical example to study the applicability of the new approach on a candidate selection decision-making problem. [ABSTRACT FROM AUTHOR]

The intuitionistic linguistic (IL) variable (ILV) can express the vague and uncertain information in a better way, and the partitioned Heronian mean (PHM) operator can group attributes that have relationships with each other into one zone and the independent attributes are in different zones, so in this paper, we will propose some new PHM operators for IL information (ILI) and then apply them to multiple attribute group decision-making (MAGDM). Firstly, the some improved operational rules for ILVs are developed, which can provide a more accurate result and avoid the loss of information, then we extend the PHM operator to the IL environment and propose the intuitionistic linguistic partitioned Heronian mean (ILPHM) operator and the intuitionistic linguistic weighted partitioned Heronian mean (ILWPHM) operator, which can fully consider the advantages of the ILI and the PHM operator. Meanwhile, we discuss some desirable properties and special cases of the two operators. Further, we develop the MAGDM approach with ILI based on the developed operators. Lastly, a numerical instance is given to verify the feasibility and the superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Neutrosophic cubic sets can deal with the complex information by combining the neutrosophic sets and cubic sets, the power average (PA) can weaken some effects of awkward data from biased decision makers, and Heronian mean (HM) can deal with the interrelationship between the aggregated attributes or arguments. In this article, in order to consider the advantages of the PA and HM, we combined and extended them to process neutrosophic cubic information. Firstly, we defined a distance measure for neutrosophic cubic numbers, then we presented the neutrosophic cubic power Heronian aggregation operator and neutrosophic cubic power weighted Heronian aggregation operator, and some characters and special cases of these new aggregation operators were investigated. Furthermore, we gave a new approach for multi-attribute group decision making based on new proposed operators. Finally, two examples were given to explain the validity and advantages of the developed approach by comparing with the existing method. [ABSTRACT FROM AUTHOR]

A normal wiggly hesitant fuzzy set (NWHFS) is viewed as a powerful and useful tool to dig the potential uncertainty of decision makers (DMs) in the process of expressing their preferences, which can be regarded as an extended form of the traditional hesitant fuzzy set (HFS). The NWHFSs have the ability of both reserving the original hesitant fuzzy information completely and exploring potential fuzziness of DMs, which assist the DMs in advancing the decision-making efficiency and derive the reasonable ranking orders finally. To fully exert the strengths of the combined power average and Muirhead mean operators, based on the proposed distance measure of normal wiggly hesitant fuzzy elements (NWHFEs), we extend the power Muirhead mean (PMM) to the normal wiggly hesitant fuzzy environment and develop the normal wiggly hesitant fuzzy PMM (NWHFPMM) and its weighted form included. After that, several representative cases and attractive properties of the proposed normal wiggly hesitant fuzzy operators are investigated in depth. Finally, a novel MADM method for solving normal wiggly hesitant fuzzy decision-making problems is developed, then a numerical example is performed to analyze the strengths of our proposed method, which in the way of comparing with other existing studies. [ABSTRACT FROM AUTHOR]

Single-valued neutrosophic (SVN) sets can successfully describe the uncertainty problems, and Schweizer-Sklar (SS) t-norm (TN) and t-conorm (TCN) can build the information aggregation process more flexible by a parameter. To fully consider the advantages of SVNS and SS operations, in this article, we extend the SS TN and TCN to single-valued neutrosophic numbers (SVNN) and propose the SS operational laws for SVNNs. Then, we merge the prioritized aggregation (PRA) operator with SS operations, and develop the single-valued neutrosophic Schweizer-Sklar prioritized weighted averaging (SVNSSPRWA) operator, single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted averaging (SVNSSPROWA) operator, single-valued neutrosophic Schweizer-Sklar prioritized weighted geometric (SVNSSPRWG) operator, and single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted geometric (SVNSSPROWG) operator. Moreover, we study some useful characteristics of these proposed aggregation operators (AOs) and propose two decision making models to deal with multiple-attribute decision making (MADM) problems under SVN information based on the SVNSSPRWA and SVNSSPRWG operators. Lastly, an illustrative example about talent introduction is given to testify the effectiveness of the developed methods. [ABSTRACT FROM AUTHOR]

The power average (PA) has the property that it can eliminate the influence of inconvenient data and the Muirhead mean (MM) operator takes the correlations among the input arguments, and the single valued neutrosophic (SVN) set is a better tool to deal with incomplete, inconsistent and indeterminate information than fuzzy set (FS) and intuitionistic FS (IFS). Thus the main goal of this article is to develop a few new operators for aggregating SVN information and apply them to multiple-attribute group decision making (MAGDM). To fully utilize the advantages of MM operator and PA operator, we develop the single-valued neutrosophic power MM (SVNPMM) operator, weighted single-valued neutrosophic power MM (WSVNPMM) operator, single-valued neutrosophic power dual MM (SVNPDMM) operator and weighted single-valued neutrosophic power dual MM (WSVNPDMM) operator, and discuss their essential properties, particular cases about the parameter vector. The obvious advantages of the proposed operators are that it can eliminate the influence of inconvenient data and can take the correlation among input data at the same time. Moreover, based on the developed aggregation operators, a novel technique to MAGDM problem is proposed. Lastly, a numerical example is provided to show the efficiency and realism of the proposed technique. [ABSTRACT FROM AUTHOR]

The Maclaurin symmetric mean (MSM) operator has the characteristic of capturing the interrelationship among the multi-input arguments. The two-dimensional uncertain linguistic variables (2DULVs) add a subjective evaluation on the reliability of the evaluation results given by decision makers, so they can better express fuzzy information, and the improved operational laws of 2DULVs can avoid omitting information and make the operations more accurate. In order to fully take the advantages of the MSM operator and the improved operational laws of the 2DULVs, in this paper, we extend the MSM operator to the environment of 2DULVs, and two new aggregated operators are proposed, including the MSM operator for 2DULVs (2DULMSM) and the weighted MSM operator for 2DULVs (W2DULMSM). Simultaneously, we discuss some desirable properties and special cases with respect to different parameter values in these operators. Further, based on W2DULMSM operator, an approach to multiple-attribute group decision-making problems with 2DULVs is developed. Finally, a numerical example is provided to illustrate the feasibility of the proposed methods. [ABSTRACT FROM AUTHOR]

Linguistic q‐rung orthopair fuzzy numbers (Lq‐ROFNs) are a qualitative form of q‐rung orthopair fuzzy numbers (q‐ROFNs) where the membership and nonmembership degrees are represented by linguistic variables. The Lq‐ROFNs can describe a broader range of linguistic assessment information flexibly by adjusting the parameter q based on different situations, so they are more superior to the linguistic intuitionistic fuzzy numbers and linguistic Pythagorean fuzzy numbers in real application. Based on the Lq‐ROFNs, we introduce the entropy measure which can be used to determine the indefiniteness of the assessment information. Then, based on the linguistic entropy measure, we further propose a method to obtain the attribute weights when the weight information is incomplete known. For aggregating the assessment information, the power average (PA) operator can reduce the influence of extreme data caused by the biased decision‐makers by considering the support degree of different evaluation individuals, and the Muirhead mean (MM) operator can take the interrelationship of different numbers of attributes into account by adjusting the parameter vector based on the real situations. In this paper, based on these two operators, we firstly propose the linguistic q‐rung orthopair fuzzy PA operator and linguistic q‐rung orthopair fuzzy weighted PA operator. Further, for combing the advantages of the MM operator and PA operator, we propose the linguistic q‐rung orthopair fuzzy power MM (PMM) operator and linguistic q‐rung orthopair fuzzy weighted PMM operator, and then investigate some properties of them. Finally, a new multiple‐attribute group decision‐making (MAGDM) method is proposed to process the Lq‐ROFNs, and some practical examples are given to illustrate the effectiveness and superiority of this new method in comparison with other existing MAGDM methods. [ABSTRACT FROM AUTHOR]

We propose a novel method for hesitant fuzzy linguistic decision making by utilizing Dempster–Shafer (D–S) theory of evidence. First, we propose some novel operations of hesitant fuzzy linguistic term sets (HFLTSs) on the basis of closed operations on linguistic 2-tuples and distribution linguistic average aggregation (DLAA) operators. These novel operations not only avoid information loss and operational results exceeding the boundary of linguistic term sets, but also make the aggregation results interpretable. Then, we define the hesitant fuzzy linguistic Archimedean weighted arithmetic mean (HFLAWAM) and hesitant fuzzy linguistic Archimedean weighted geometric mean (HFLAWGM) operators of HFLTSs. These two proposed operators are able to overcome the shortcomings of the existing approaches to multicriteria decision making (MCDM) with HFLTSs. Then, to take into account the novel operations of HFLTSs and the MCDM under uncertainty, we propose the belief structure-HFLAWAM (BS-HFLAWAM) and belief structure-HFLAWGM (BS-HFLAWGM) operators of HFLTSs. After that, we create an approach to handle the hesitant fuzzy linguistic MCDM problems in light of proposed operators. Finally, we apply the created approach to a MCDM problem regarding political management of a country. [ABSTRACT FROM AUTHOR]

With respect to multi-attribute group decision-making (MAGDM) problems in which the attribute values take the form of intuitionistic uncertain linguistic variables (IULVs), the group decision-making methods based on some Bonferroni mean (BM) aggregation operators were developed. Firstly, we proposed some new operational rules for the IULVs which can overcome the weaknesses of the existing operations. Then, we extended BM operator to the IULVs and developed intuitionistic uncertain linguistic arithmetic Bonferroni mean (IULABM) operator, intuitionistic uncertain linguistic arithmetic weighted Bonferroni mean (IULWABM) operator, intuitionistic uncertain linguistic geometric Bonferroni mean (IULGBM) operator, and intuitionistic uncertain linguistic weighted geometric Bonferroni mean (IULWGBM) operator. At the same time, some desirable properties of the proposed operators, such as idempotency, boundedness, and monotonicity, were studied, and some special cases of these operators were analyzed. Moreover, some approaches based on the developed operators are proposed. Finally, an illustrative example is given to show the steps of the developed methods and to discuss the influences of different parameters on the decision-making results. [ABSTRACT FROM AUTHOR]

The cloud model (CM) is an important tool to describe qualitative concept by the quantitative method, and the Maclaurin symmetric mean (MSM) can capture the interrelationship among the multi-inputs and it can generalize most of existing operators. In this paper, we firstly convert the uncertain linguistic variables (ULVs), which are easily used to express the qualitative information, to CM. Then, we combine the MSM with the CM, and propose the cloud MSM (CMSM) operator and cloud weighted MSM (CWMSM) operator. In addition, we explore some of their desirable features and develop a new approach to deal with some multi-attribute group decision-making (MAGDM) problems under the uncertain environment based on the proposed operators. Finally, by comparing with other approaches, an illustrative example is arranged to demonstrate the usability of the proposed method. [ABSTRACT FROM AUTHOR]

Schweizer-Sklar T-norm and T-conorm (SSTT), as an important class of the T-norm (TN) and T-conorm (TC), have greater flexibility in the information fusion process, and the Maclaurin symmetric mean (MSM) has a prominent advantage that it can take into account the interrelationships among the multi-input arguments, including multi-attributes or multi-experts in the multi-attribute group decision making (MAGDM), and it is also the generalization of many existing operators. In order to make full use of the advantages of both SSTT and MSM, we extend SSTT to intuitionistic fuzzy numbers (IFNs) and define Schweizer-Sklar operational rules of IFNs. Then, we combine the MSM with Schweizer-Sklar operational rules, and propose the intuitionistic fuzzy Schweizer-Sklar Maclaurin symmetric mean (IFSSMSM) operator, the intuitionistic fuzzy Schweizer-Sklar weighted Maclaurin symmetric mean (IFSSWMSM) operator, respectively. Furthermore, we study some desirable characteristics of them and develop a new method based on these operators to deal with some MAGDM problems. Finally, some examples of practical applications are presented to show the availability and advantages of the proposed method by comparing with some existing methods. [ABSTRACT FROM AUTHOR]

The trapezoid intuitionistic fuzzy linguistic (TIFL) variables (TIFLVs) can deal with uncertain information more efficiently and precisely and avoid the information losing and distortion in the decision making process. Bonferroni mean (BM) operator can process the relationship of the inputs. In order to take full advantages of them, in this paper, we extend BM to process TIFL information, and propose the BM operator for TIFLVs (TIFLBM), the weighted BM operator for TIFLVs (TIFLWBM), the geometric BM (GBM) operator for TIFLVs (TIFLGBM), and the weighted GBM operator for TIFLVs (TIFLWGBM). Then some properties and some particular cases of these proposed operators are analyzed. Furthermore, two approaches for multi-attribute group decision making (MAGDM) based on these operators are developed. Finally, by an example, the effectiveness and the advantages of the developed methods are explained. [ABSTRACT FROM AUTHOR]

The Hamy mean (HM) operator, as a convenient mathematical aggregation tool, can deal with the interrelationship among multiple input parameters, and the power average (PA) operator can relieve the influence of awkward assessment values in the decision consequences. The interval neutrosophic sets (INSs) are amore powerful mathematical tool to handle insufficient, indeterminate and vague information that exists in real life problems. Yet, in some complicated decision-making situations, we require to consider the correlation between multi-input arguments and remove the influence of awkward data at the same time. To deal with such situations, in this paper, we combine the conventional HM operator to the traditional PA operator in interval neutrosophic settings and present two novel interval neutrosophic aggregation operators, that is, the interval neutrosophic power Hamy mean (INPHM) operator and the weighted interval neutrosophic power Hamy mean (WINPHM) operators. Then, some preferable properties of the developed aggregation operators are discussed. Moreover, based on these developed aggregation operators, we propose a new method for multiple attribute group decision making (MAGDM) under the INSs. Lastly, some examples are given to show the effectiveness of the developed method by comparing it with other existing methods. [ABSTRACT FROM AUTHOR]

For a multi-attribute group decision making (MAGDM) problem where the attribute values are intuitionistic uncertain linguistic variables (IULVs), we propose a novel decision making method based on hybrid aggregation operator of IULVs. Firstly, new operational rules of IULVs are proposed based on linguistic scale functions (LSFs) to overcome the existing shortcoming in which the operations on linguistic variables (LVs) directly based on the subscripts of linguistic terms (LTs) are not closed, then the expected value and accuracy function of IULVs are introduced. Further, some aggregation operators for IULVs are proposed, including the weighted geometric average operator for IULVs (IULWGA), ordered weighted geometric operator for IULVs (IULOWG), then the hybrid geometric operator for IULVs (IULHG) are developed, which could consider the weights of attributes and their ranking positions. However, because IULHG don't meet some desirable properties, we proposed a new hybrid weighted geometric operator for IULVs (IULHWG) which can not only weight the importance of each argument and its ordered position but also maintain some desirable properties. Based on these operators, an approach to MAGDM with IULVs has been proposed. Finally, an illustrative example is provided to show the steps of the developed approach and to demonstrate its practicality and effectiveness. [ABSTRACT FROM AUTHOR]

Linguistic intuitionistic fuzzy number (LIFN) is characterized by the degrees of membership and non-membership which take the form of linguistic variables, so it can more easily describe the vague and imprecise information existing in the real decision-making problems. Density weighted averaging operator considers the density preference of information distribution, so it can produce the more reasonable decision results. In this paper, some arithmetic aggregation operators are combined with density weighted averaging operator under the linguistic intuitionistic fuzzy environment and some linguistic intuitionistic fuzzy density aggregation operators are proposed. Firstly, the related theories of LIFN have been reviewed briefly, and the method of calculating density weighted vector and the clustering method are presented. Then, some linguistic intuitionistic fuzzy density aggregation operators, such as linguistic intuitionistic fuzzy density weighted averaging operator, linguistic intuitionistic fuzzy density ordered weighted averaging operator, and linguistic intuitionistic fuzzy density hybrid weighted averaging (LIFDHWA) operator are proposed, and some properties are discussed. Moreover, a new group decision-making method based on LIFDHWA operator is proposed. Finally, an illustrative example is used to demonstrate the validity of the proposed method. [ABSTRACT FROM AUTHOR]