RETRACTED: Multi-attribute decision making method based on the generalised spherical fuzzy Heronian operators

As a generalization of complex fuzzy set (CFS), complex hesitation fuzzy set (CHFS) is a powerful tool to express two-dimensional fuzzy information, but it ignores the randomness that exists widely in real life. Considering the randomness and ambiguity often simultaneously occur in real-world problems, we come up with an innovative fuzzy set called complex probabilistic hesitant fuzzy set (CPHFS) in this article, which is a joint extension of CFS and probabilistic hesitant fuzzy set (PHFS). Moreover, we establish four basic operations for complex probabilistic hesitant fuzzy element(CPHFE) and a series of useful operational laws are also developed. Secondly, several aggregation operators including complex probabilistic hesitant fuzzy weighted averaging operator(CPHFWA), complex probabilistic hesitant fuzzy weighted geometric operator(CPHFWG), complex probabilistic hesitant fuzzy Heronian Mean operator(CPHFHM) and complex probabilistic hesitant fuzzy Geometric Heronian Mean operator(CPHFGHM) as well as their weighted forms that can reflect the interrelationship among all decision attributes are proposed, which is of great use in multi-attribute decision making(MADM) or in multi-attribute group decision making(MAGDM) issues when we want to synthesize multiple evaluation values into one. Besides, we also present a number of valuable properties of these provided aggregation operators. Thirdly, a MADM process is built in the CPHF environment step by step based on the aggregation operators we have just established. Furthermore, we provide an example about purchasing a most suitable car to demonstrate the effectiveness and practicability of the proposed decision process. Finally, we point out the superiorities of MADM in working under the CPHF environment through comparing it with the existing method and also offer some possible prospects for further research. The method displayed in this work can effectively solve the MADM problems where the decision-making information is described by CPHFEs and the attributes are interactive.

Based on the hesitant triangular fuzzy set, the probabilistic hesitant fuzzy set and the power aggregation operator, the probabilistic hesitant triangular fuzzy multi-attribute decision problem with the correlation between the aggregated data is studied. Firstly, the probabilistic hesitant triangular fuzzy set (PHTFS) is defined, and the algorithm, scoring function and distance measure of the probabilistic hesitant triangular fuzzy element (PHTFE) are proposed. Then, according to the PHTFS and power aggregation operator, the power aggregation operator of PHTFS is proposed. Finally, a multi-attribute decision making model is constructed based on the probabilistic hesitant triangular fuzzy power aggregation operator, and the validity of the model is verified by a case.

<abstract><p>The soft-max function, a well-known extension of the logistic function, has been extensively utilized in numerous stochastic classification methodologies, such as linear differential analysis, soft-max extrapolation, naive Bayes detectors, and neural networks. The focus of this study is the development of soft-max based fuzzy aggregation operators (AOs) for Pythagorean fuzzy sets (PyFS), capitalizing on the benefits provided by the soft-max function. In addition to introducing these novel AOs, we also present a comprehensive approach to multi-attribute decision-making (MADM) that employs the proposed operators. To demonstrate the efficacy and applicability of our MADM method, we applied it to a real-world problem involving Pythagorean fuzzy data. The analysis of supplier selection has been extensively examined in many academic works as a crucial component of supply chain management (SCM), recognised as a significant MADM challenge. The process of choosing healthcare suppliers is a pivotal element that has the potential to greatly influence the efficacy and calibre of healthcare provisions. In addition, we given a numerical example to rigorously evaluate the accuracy and dependability of the proposed procedures. This examination demonstrates the effectiveness and potential of our proposed soft-max based AOs and their applicability in Pythagorean fuzzy environments.</p></abstract>

<abstract><p>The fractional orthotriple fuzzy sets (FOFSs) are a generalized fuzzy set model that is more accurate, practical, and realistic. It is a more advanced version of the present fuzzy set models that can be used to identify false data in real-world scenarios. Compared to the picture fuzzy set and Spherical fuzzy set, the fractional orthotriple fuzzy set (FOFS) is a powerful tool. Additionally, aggregation operators are effective mathematical tools for condensing a set of finite values into one value that assist us in decision making (DM) challenges. Due to the generality of FOFS and the benefits of aggregation operators, we established two new aggregation operators in this article using the Frank t-norm and conorm operation, which we have renamed the fractional orthotriple fuzzy Choquet-Frank averaging (FOFCFA) and fractional orthotriple fuzzy Choquet-Frank geometric (FOFCFG) operators. A few of these aggregation operators' characteristics are also discussed. To demonstrate the efficacy of the introduced work, the multi-attribute decision making (MADM) algorithm is discussed along with applications. To demonstrate the validity and value of the suggested work, a comparison of the proposed work has also been provided.</p></abstract>

This paper aims to put forward a hesitant fuzzy multi-attribute group decision making (MAGDM) method based on the weighted power aggregation operators in social network. From the point of view of social network analysis, decision makers (DMs) are interconnected in the process of MAGDM. Furthermore, the dimension of the obtained hesitant fuzzy element (HFE) by original power operators will be greater with the increasing number of attributes and alternatives and DMs, which will lead to the problem of “intermediate expression swell". This paper combines the order operation laws with the power operators to redefine two novel hesitant fuzzy power aggregation operators to simplify the involved calculation and explore new operators’ properties. Meanwhile, when two given elements have different number of values, we use the strength of social ties and social influence to develop an algorithm for extending the HFEs objectively. On the other hand, the PageRank algorithm and the deviation method are used to determine DMs’ combined weights. The feasibility of the proposed hesitant fuzzy MAGDM method based on social network is illustrated by the application to the actual issue of decision making and the comparative analysis with the existing method.

Intuitionistic fuzzy geometric aggregation operators in the framework of Aczel-Alsina triangular norms and their application to multiple attribute decision making

Aggregation operators (AOs) are utilized to overcome the effects of attributes under some specific degree of weight in the decision-making (DM) process. The AOs have a large capacity to deal with uncertain and unpredictable information in multi-attribute decision-making (MADM) problems. The Hamy mean (HM) aggregation tools are well-known aggregation models, which are utilized to define correlation among different input arguments adequately. The intuitionistic fuzzy (IF) sets (IFS) can express unpredictable and vague information. The Aczel Alsina aggregation expressions are extensions of triangular norms. Recently, Aczel Alsina aggregation tools attained a lot of attentions from numerous research scholars. By inspiring the robustness and reliability of Aczel Alsina aggregation tools, we expose some appropriate operations of Aczel Alsina expressions under consideration of IF information. In this manuscript, we developed an intuitionistic fuzzy Aczel Alsina HM (IFAAHM) and an intuitionistic fuzzy Aczel Alsina weighted HM (IFAAWHM) operator. We also expressed the theory of Dual HM (DHM) tools and established a series of new approaches including intuitionistic fuzzy Aczel Alsina Dual HM (IFAADHM) and intuitionistic fuzzy Aczel Alsina weighted Dual HM (IFAAWDHM) operators. Some reliable characteristics and special cases of our derived approaches are also presented. The authors solved an application of a MADM technique under consideration of our derived approaches. To check the reliability and dependency of our derived mythologies, we gave an experimental case study to evaluate a desirable construction material based on some specific criteria of different Alternatives. To see the advantages and compatibility of our derived approaches, by comparing the results of existing approaches with the results of currently discussed AOs.

Aggregation operators are the standard mathematical tool for the combination of several inputs into one unique output. This paper presents some aggregation operators for complex intuitionistic fuzzy (CIF) information. Complex intuitionistic fuzzy set (CIFS) theory has a large ability to capture vagueness since it can represent the complete specifications of problems with both intuitionistic uncertainty and periodicity. We introduce the CIF Hamacher weighted averaging (CIFHWA) operator, CIF Hamacher ordered weighted averaging (CIFHOWA) operator, CIF Hamacher weighted geometric (CIFHWG) operator and CIF Hamacher ordered weighted geometric (CIFHOWG) operator. We bring to light some remarkable properties of these operators and explicitly state some noteworthy special cases. Moreover, we contribute to the advancement of multi-attribute decision-making (MADM) with a novel proposal of an algorithm in the CIF environment. Furthermore, we consider and solve a MADM problem that finds the best source for generation of electricity with the aid of the proposed operators, thus proving their usefulness for the purpose of decision-making. Finally, we check the effectiveness of these operators by a validity test.

With respect to the multiattribute decision-making (MADM) problem in which the attributes have interdependent or interactive phenomena under the interval-valued intuitionistic fuzzy environment, we propose a group decision-making approach based on the interval-valued intuitionistic fuzzy Einstein geometric Choquet integral operator (IVIFEGC). Firstly, the Einstein operational laws and some basic principle on interval-valued intuitionistic fuzzy sets are introduced. Then, the IVIFEGC is developed and some desirable properties of the operator are studied. Further, an approach to multiattribute group decision-making with interval-valued intuitionistic fuzzy information is developed, where the attributes have interdependent phenomena. Finally, an illustrative example is used to illustrate the developed approach.

Retraction notice to "Applications of T-spherical fuzzy aczel-alsina power muirhead mean operators in identifying the most effective water purification process for commercial purpose" [Heliyon 10 (2024) e39524

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 ord er 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.

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.

In this paper, we present a fuzzy linguistic scale, which is characterized by trapezoidal fuzzy numbers, for the comparison between two alternatives, we introduce a possibility degree formula for comparing trapezoidal fuzzy numbers. We utilize the fuzzy linguistic scale to construct a linguistic preference matrix, and propose a trapezoidal fuzzy induced ordered weighted geometric (TFIOWG) operator to aggregate linguistic preference information. A method based on the fuzzy linguistic scale and TFIOWG operator for decision-making problems is presented. Finally, an illustrative example is given to verify the developed method and demonstrate its feasibility and practicality.

Fermatean fuzzy power Bonferroni aggregation operators and their applications to multi-attribute decision-making

The sum of membership and non-membership degrees of the Pythagorean fuzzy set is greater than one with their square sum less than or equal to one. Thus, as an extension of intuitionistic fuzzy set, the Pythagorean fuzzy set is a powerful tool to describe fuzziness and uncertainty. The aim of this study is to introduce some new operators for aggregating Pythagorean fuzzy information and apply them to multi-attribute decision making. Considering the advantages of the power average operator and Muirhead mean, we introduce the power Muirhead mean operator and investigate it under Pythagorean fuzzy environment. Thus, some new Pythagorean fuzzy aggregation operators, such as the Pythagorean fuzzy power Muirhead mean and the weighted Pythagorean fuzzy power Muirhead mean are developed. The prominent advantage of these proposed operators is that they consider the relationships between fused data and the interrelationships between all aggregated values, thereby obtaining more information in the process of multi-attribute decision making. Furthermore, we introduce a novel approach to multi-attribute decision-making problems based on the proposed operators. Finally, we provide a numerical example to illustrate the validity of the proposed approach.