Archimedean T-conorm Research Articles

Multi-attribute decision-making using Archimedean aggregation operator in T-spherical fuzzy environment

As an extension of several fuzzy structures such as fuzzy sets, intuitionistic fuzzy sets, picture fuzzy sets, q-rung ortho-pair fuzzy sets, and T-spherical fuzzy (TSF) sets (TSFSs), are an effective tool for controlling the vagueness of data. Archimedean t-conorm (ATCN) and t-norm (ATN), which consists of the t-conorm (TCN) and t-norm (TN) families, is a crucial approach for fuzzy sets to produce extensive operational rules. In this manuscript, for TSF numbers some core operational laws are initiated based on ATCN and ATN, also some basic characteristics of these operational laws are investigated. Secondly, based on these operational laws TSF Archimedean weighted averaging (TSPFAWA) and TSF Archimedean weighted geometric (TSPFAWG) operators are initiated. Thirdly, we investigated special cases of these aggregation operators and some basic properties. On the behalf of the TSPFAWA and TSPFAWG operators, a novel method for solving multiple attribute decision-making (MADM) problems using TSF information is also devised. Lastly, a numerical example is provided to demonstrate the applicability of the suggested technique, and a comparison analysis is done to demonstrate its advantages.

Q-Rung Orthopair Fuzzy Archimedean Aggregation Operators: Application in the Site Selection for Software Operating Units

The q-rung orthopair fuzzy (q-ROF) set is an efficient tool for dealing with uncertain and inaccurate data in real-world multi-attribute decision-making (MADM). In MADM, aggregation operators play a significant role. The majority of well-known aggregation operators are formed using algebraic, Einstein, Hamacher, Frank, and Yager t-conorms and t-norms. These existing t-conorms and t-norms are some special cases of Archimedean t-conorms (ATCNs) and Archimedean t-norms (ATNs). Therefore, this article aims to extend the ATCN and ATN operations under the q-ROF environment. In this paper, firstly, we present some new operations for q-ROF sets based on ATCN and ATN. After that, we explore a few desirable characteristics of the suggested operational laws. Then, using these operational laws, we develop q-ROF Archimedean weighted averaging (geometric) operators, q-ROF Archimedean order weighted averaging (geometric) operators, and q-ROF Archimedean hybrid averaging (geometric) operators. Next, we develop a model based on the proposed aggregation operators to handle MADM issues. Finally, we elaborate on a numerical problem about site selection for software operating units to highlight the adaptability and dependability of the developed model.

Interval-valued dual hesitant fuzzy prioritized aggregation operators based on Archimedean t-conorm and t-norm and their applications to multi-criteria decision making

Interval-valued dual hesitant fuzzy prioritized aggregation operators based on Archimedean t-conorm and t-norm and their applications to multi-criteria decision making

Probabilistic linguistic multi-attribute decision making approach based upon novel GMSM operators

<abstract><p>Probabilistic linguistic terms set (PLTS), a new tool for expressing uncertain decision information, is composed of all possible linguistic terms (LTs) and their related probabilities. It also increases the corresponding probability of LTs in hesitant fuzzy linguistic term set (HFLTS). On the other hand, aggregation operator is an important information fusion tool, the Maclaurin symmetric mean (MSM) operator can provide more flexibility and robustness in information fusion, and make it more suitable for solving MADM problems with independent attributes. This current study adopts the merits of PLTS and MSM operator, and then a novel probabilistic linguistic decision making approach are targeted. Firstly, the operations of two PLTSs are redefined based upon Archimedean t-norm (ATN) and Archimedean t-conorm (ATC); Secondly, the probabilistic linguistic generalized MSM operator (PLGMSM) is proposed based on ATN and ATC, some properties of PLGMSM are investigated, then some special PLGMSM operators have been studied in detail when the parameters take different values and the generator of ATN takes different functions. Thirdly, the weighted probabilistic linguistic generalized MSM operator (WPLGMSM) is studied along with some properties of PLGMSM, some special WPLGMSM operators have been also investigated in detail when the parameters take different values and the generator of ATN takes different functions. Finally, on the basis of our proposed aggregation operators, the aggregated-based decision making approach is designed and an example is supplied to manifest the effectiveness of the proposed approach. Furthermore, some comparison analyses with extant decision approaches are carried out to illustrate the validity and feasibility of the proposed approach.</p></abstract>

Interval-valued hesitant Pythagorean fuzzy Archimedean aggregation operators and their application to multicriteria decision-making

In the context of multicriteria decision making (MCDM), it is frequently observed that the representation of cognitive information may not always be sufficient using hesitant Pythagorean fuzzy set (HPFS). From this perspective, the interval-valued hesitant Pythagorean fuzzy (IVHPF) set is more flexible than HPFS to capture the cognitive versatility of decision makers. Again, Archimedean t-conorm and t-norms (At-CN&t-Ns) possess the efficient capability to generate versatile and flexible operational rules for fuzzy numbers. Utilizing the benefit of At-CN&t-Ns some basic operations are introduced to aggregate IVHPF elements so that many other kinds of t-conorm and t-norms (t-CN&t-Ns), such as algebraic, Einstein, Hamacher, Dombi, Frank and other classes of t-CN&t-Ns can be generated. Based on those concepts, several forms of aggregation operators, viz., weighted averaging, weighted geometric, ordered weighted averaging and ordered weighted geometric aggregation operators, are updated in the IVHPF environment. Conversion processes from At-CN&t-Ns based aggregation operators to other variants are also discussed. Using the proposed operators, a methodology for solving MCDM problems with IVHPF cognitive information is proposed. To explore the applicability of the proposed approach, two examples are considered and solved. Results obtained from this method are compared with the existing approaches to establish the efficiency of the proposed method.

Uncertainty evaluations through interval-valued Pythagorean hesitant fuzzy Archimedean aggregation operators in multicriteria decision making

In many cases, use of Pythagorean hesitant fuzzy sets may not be sufficient to characterize uncertain information associated with decision making problems. From that view point the concept of interval-valued Pythagorean hesitant fuzzy sets are introduced in this paper. Considering the flexibility with the general parameters, Archimedean t-conorms and t-norms are applied to develop several operational laws in interval-valued Pythagorean hesitant fuzzy environment. Some characteristics of the developed operators are presented. The newly developed operators are used to derive a methodology for solving multicriteria decision making problems with interval-valued Pythagorean hesitant fuzzy information. Finally, two illustrative examples are provided to establish the validity of the proposed approach and are compared with the existing technique to exhibit its flexibility and effectiveness.

Properties of invertible elements in complete intuitionistic fuzzy pseudo normed algebra

In this paper, we deal with the invertible elements in a complete intuitionistic fuzzy pseudo normed algebra (in short, IFPNA) with respect to Archimedean t-norm and Archimedean t-conorm. It is done by studying the continuity of algebraic operations in a complete IFPNA and investigating the condition for existence of inverse of an element in a complete IFPNA. Also some properties of invertible elements are studied. It is observed that the set of invertible elements in a complete IFPNA is an open set.

Some new entropies and divergence measures of intuitionistic fuzzy sets based on Archimedean t-conorm and application in supplier selection

In this paper, we build the divergence measure of intuitionistic fuzzy sets from the Archimedean t-conorm operators and give their properties. Next, we built the entropy and knowledge measures of intuitionistic fuzzy sets based on their divergence measures. In this paper, relationships between entropy and knowledge measures are also investigated, which gives us many interesting results. We then apply the proposed measures into building the multi-criteria decision-making model. Finally, a numerical example to illustrate the model is also given in this paper.

The n-uninorms with continuous underlying t-norms and t-conorms

ABSTRACT The n-uninorms with continuous underlying t-norms and t-conorms are studied. We show that each 2-uninorm with continuous underlying functions can be expressed as an ordinal sum of a uninorm with continuous underlying functions (possibly also of a countable number of semigroups related to representable uninorms, continuous Archimedean t-norms, continuous Archimedean t-conorms and internal uninorms) and a 2-uninorm with continuous underlying functions such that (possibly restricted to open or half-open unit square ). Similar results are shown for n-uninorms with continuous underlying functions, where

Linguistic Interval-Valued Intuitionistic Fuzzy Copula Heronian Mean Operators for Multiattribute Group Decision-Making

As a generalization of the intuitionistic fuzzy number (IFN), the linguistic interval-valued intuitionistic fuzzy number (LIVIFN) is a flexible and superior tool to describe complex fuzzy uncertainty information. Heronian mean (HM) operator has the characteristic of considering the relationship between attributes. Extended copulas (ECs) and extended cocopulas (ECCs) are the promotion form of Archimedean t-norm and t-conorm (ATT). ECs and ECCs can generate versatile operational rules and can provide more choice for decision makers (DMs). Therefore, it is very necessary to take advantages of them. In this paper, ECs and ECCs, some specifics of ECs and ECCs, and score and accuracy functions of IVILFNs are gained first. Then, we propose the linguistic interval-valued intuitionistic fuzzy weighted copula Heronian mean (LIVIFWCHM) operator; also, some properties and five specific expressions of the LIVIFWCHM operator are discussed. Moreover, we also propose a new MAGDM approach based on the proposed LIVIFWCHM operator. Finally, a set of examples are used to demonstrate the effectiveness, generality, and flexibility of the proposed method.

Generalized Maclaurin symmetric mean aggregation operators based on Archimedean t-norm of the intuitionistic fuzzy soft set information

Intuitionistic fuzzy soft set (IFSS) accommodates more uncertainties within the information by considering the parameterization feature than the intuitionistic fuzzy sets and hence its applications are more extensive. Archimedean T-conorm and T-norm (ATT), consists of T-norm and T-conorm classes, is as an essential source to make the comprehensive operational laws. Meanwhile, the Maclaurin symmetric mean (MSM) has a prominent characteristic and the advantage that it can take into account the interrelation between multi-input arguments, including different attributes or different experts. Motivated by these chief characteristics, in this article, we extend the MSM operators to the IFSS based on ATT. In this paper, a method is exploited to solve the multi-criteria decision-making (MCDM) problems under the IFSS environment. To it, firstly, some generalized intuitionistic fuzzy soft operational laws are introduced based on ATT. Secondly, we reveal some averaging and geometric aggregation operators based on MSM operator. Further, some desirable features and particular cases of it are tested and build up with a new technique for illustrating MCDM problems. Finally, an illustration is given to exhibit the methodology and approach’s supremacy is shown through a comparative study with prevailing techniques.

Extensions and distortions of λ-fuzzy measures

We propose extensions and distorsion techniques to improve the flexibility of λ-fuzzy measures. As for extensions, we suggest to use the family of Archimedean t-conorms as generators of the fuzzy measures. As for distortions, we propose the composition or patchwork of different generators. As an example of application, we show that in option pricing these techniques substantially improve the flexibility of the model to reproduce features observed from market data that only one Archimedean generator would not be able to represent.

Q-Rung Orthopair Fuzzy Integrals in the Frame of Continuous Archimedean T-Norms and T-Conorms and Their Application

Yager's q-rung orthopair fuzzy set is a generalization of fuzzy sets, whose prominent feature is that the qth power sum of the membership and the nonmembership degrees is equal to or less than one, and we call its core, an ordered pair, q-rung orthopair fuzzy number (q-ROFN). More recently, the scholars have constructed the q-rung orthopair fuzzy calculus (q-ROFC), which can effectively deal with continuous q-rung orthopair fuzzy information. Nevertheless, the q-ROFC is only based on the basic operational laws of the q-ROFNs, in fuzzy theory, Archimedean t-norms and t-conorms (ATTs) are a significant class of continuous triangular norms and conorms, which are the generalizations of the intersection and union related to fuzzy sets. Thus, in order to extend the q-ROFC to a wider area, in this article, we systematically discuss the q-rung orthopair fuzzy double integrals (q-ROFDIs) in the frame of ATTs. First, we construct the q-ROFDI in the frame of Archimedean t-conorms in detail, and then provide its concrete value. In addition, we reveal the relationships with respect to two types of q-rung orthopair fuzzy spaces. Based on which, we can easily obtain another types of q-ROFDI. After that, we investigate their fundamental properties in detail so as to comprehend these kinds of q-ROFDIs in-depth. Finally, we point out the essences of these kinds of q-ROFDIs, on the basis of which we provide a practical application to show their effectiveness and elasticity via comparing with the existing methods.

Generalized Geometric Aggregation Operators Based on T-Norm Operations for Complex Intuitionistic Fuzzy Sets and Their Application to Decision-making

Complex intuitionistic fuzzy set (CIFS) is a special intuitionistic fuzzy set where the membership and non-membership degrees are expressed by a complex-valued membership degree and can more easily describe the vagueness and uncertainty in the real world. Archimedean t-conorm and t-norm (ATT), as an important class of the t-norm (TN) and t-conorm (TC), have greater flexibility in the information fusion process. In this paper, we extend the ATT to CIFSs and present some generalized geometric aggregation operators, which can be used to handle the multiple criteria decision-making (MCDM) problems. For it, we firstly define some new operational laws of the CIFSs based on ATT, then some weighted geometric aggregation operators based on proposed operations are proposed. Further, some desirable properties and special cases of them are studied. Finally, a decision-making approach is developed for the MCDM problem with complex intuitionistic fuzzy information. A practical example is given to show the availability and advantages of the proposed method by comparison with some existing methods. The proposed aggregation operators are more generalized than the existing ones to utilize the uncertain and imprecise information. Several existing operators are considered as special cases of the proposed one. Finally, the proposed method will offer various choices to the decision-maker to access the finest alternatives.

Dual hesitant fuzzy linguistic power-geometric operators based on Archimedean t-conorms and t-norms and their application to group decision making

Dual hesitant fuzzy linguistic power-geometric operators based on Archimedean t-conorms and t-norms and their application to group decision making

Interval-Valued Intuitionistic Uncertain Linguistic Multi-attribute Decision-Making Method for Plant Location Selection with Partitioned Hamy Mean

Plant location selection (PLS) is a typical and complex multi-attribute group decision-making (MAGDM) problem, which requires consideration of several alternatives and multiple attributes. However, existing research results on PLS lack sufficient consideration of complex relationship patterns between attributes, which often leads to inaccurate decision results. Thus, in this paper, a new MAGDM technique is proposed to solve the problem of PLS, in which complex interrelationship structures exist between multi-inputs and attribute values are represented as interval-valued intuitionistic uncertain linguistic variables (IVIULVs). First, to solve the shortcomings of the existing operations, some new operational rules are redefined for IVIULVs based on linguistic scale functions and Archimedean T-conorms and T-norms. These new rules of operation hold the closedness, and are flexible for semantic transformation processing. Then, to more accurately reflect the partition structure and the complex interrelationship pattern among multi-input arguments, we extend the traditional Hamy mean and propose the partitioned Hamy mean (PHAM), which can eliminate the impact of unrelated attributes on the results and meet the semantic conversion needs of different decision makers. Furthermore, we develop two new fuzzy linguistic aggregation operators: interval-valued intuitionistic uncertain linguistic PHAM (IVIULPHAM) and its weighted version (IVIULWPHAM). Finally, a novel MAGDM technique (IVIUL-MAGDM) based on the IVIULWPHAM operator is proposed to solve the problem of PLS. Several examples of PLS are also illustrated to show the validity and advantages of the IVIUL-MAGDM method by comparing with the other existing MAGDM methods.

Some Novel Picture 2-Tuple Linguistic Maclaurin Symmetric Mean Operators and Their Application to Multiple Attribute Decision Making

When solving multiple attribute decision making (MADM) problems, the 2-tuple linguistic variable is an effective tool that can not only express complex cognitive information but also prevent loss of information in calculation. The picture fuzzy set (PFS) has three degrees and has more freedom to express cognitive information. In addition, Archimedean t-conorm and t-norm (ATT) can generalize most existing t-conorms and t-norms and Maclaurin symmetric mean (MSM) operators can catch the relationships among the multi-input parameters. Therefore, we investigate several novel aggregation operators, such as the picture 2-tuple linguistic MSM (2TLMSM) operator based on the ATT (ATT-P2TLMSM) and the picture 2-tuple linguistic generalized MSM (2TLGMSM) operator based on ATT (ATT-P2TLGMSM). Considering that the input parameters have different importance, we proposed picture 2-tuple linguistic weighted MSM (2TLWMSM) operators based on ATT (ATT-P2TLWMSM) and picture 2-tuple linguistic weighted generalized MSM (2TLWGMSM) operators based on ATT (ATT-P2TLWGMSM). Finally, a MADM method is introduced, and an expositive example is presented to explain the availability and applicability of the developed operators and methods.

New generalised Bonferroni mean aggregation operators of complex intuitionistic fuzzy information based on Archimedean t-norm and t-conorm

ABSTRACTThis manuscript presents aggregation operators which encapsulate the interaction among the criteria and preferences under complex intuitionistic fuzzy (CIF) conditions. The current extensions of fuzzy set theory handle the uncertain data by representing the satisfaction and dissatisfaction degrees as real values and can trade with only one-dimensional problems due to which some important knowledge may be lost in some situations. A modification to these, CIF sets are portrayed by complex-valued degrees of satisfaction and dissatisfaction and handle two-dimensional data concurrently in one set using additional terms, called phase terms, which usually give knowledge related with periodicity. Motivated by the features of the CIF model, we present some aggregation operators, namely, generalised CIF Bonferroni mean and generalised CIF weighted Bonferroni mean. Some features related to proposed operators are addressed. In light of the developed operators, a decision-making method is put forward and is drawn with the aid of an example. The reliability of the presented decision-making method is examined with the help of a validity test and by comparing the results of the example with several predominating studies.

Dual Hesitant Fuzzy Linguistic Power-Average Operators Based on Archimedean t-Conorms and t-Norms

This paper introduces some novel power-average (PA) operators of dual hesitant fuzzy linguistic (DHFL) set with the Archimedean t-conorms and t-norms (ATT) functions, such as the weighted DHFL power-average (ATT-WDHFLPA) operator based on ATT functions and the weighted generalized DHFL power-average operator on ATT functions (ATT-WGDHFLPA). Subsequently, the properties of those new operators are studied and the relationships between operators and existing ones are given. Furthermore, an approach of those operators is developed to solve the group decision-making problem with dual hesitant fuzzy linguistic information. Finally, an example is showing the advantages of those operators, and a sensitivity analysis of the main parameters has illustrated the results.

Multi-Attribute Decision-Making Method Based on Interval-Valued $q$ -Rung Orthopair Fuzzy Archimedean Muirhead Mean Operators

Interval-valued $q$ -rung orthopair fuzzy sets (IV $q$ -ROFSs), as a generalization of $q$ -rung orthopair fuzzy sets ( $q$ -ROFSs), are the powerful tool for mastering the fuzziness of information. Archimedean t-conorm and t-norm (ATT) consist of t-conorm and t-norm families, which is an important tool for fuzzy sets to generate general operational laws. Meanwhile, the Muirhead mean (MM) operator is a useful aggregation operator that considers the interdependent phenomena among the aggregated arguments. Motivated by those primary characteristics, in this paper, the MM operator to the interval-valued $q$ -rung orthopair fuzzy numbers (IV $q$ -ROFNs) based on the ATT is studied. First, some interval-valued $q$ -rung orthopair fuzzy operational rules are proposed based on ATT. Second, the interval-valued $q$ -rung orthopair fuzzy Archimedean Muirhead mean (IV $q$ -ROFAMM) operator and the interval-valued $q$ -rung orthopair fuzzy weighted Archimedean Muirhead mean (IV $q$ -ROFWAMM) operator are proposed. Third, some desirable properties of the two operators are discussed, and some special cases of the developed operators are investigated. Furthermore, a novel approach based on the IV $q$ -ROFWAMM operator is developed to solve multiple attribute decision-making problem with the interval-valued $q$ -rung orthopair fuzzy information. Finally, a numerical example is given to illustrate the validity of the proposed method and a comparative analysis is conducted to show the superiorities of the method.