A method to multi-attribute decision making problems by using heronian mean operators based on linear diophantine uncertain linguistic settings

In the real decision process, an important problem is how to express the attribute value more efficiently and accurately. In the real world, because of the complexity of decision-making problems and the fuzziness of decision-making environments, it is not enough to express attribute values of alternatives by exact values. For this managing with such sorts of issues, the principle of Linear Diophantine uncertain linguistic set is a valuable and capable technique to manage awkward and inconsistent information in everyday life problems. In this manuscript, we propose the original idea of Linear Diophantine uncertain linguistic set and elaborated their essential laws. Additionally, to determine the association among any numbers of attributes, we elaborated the Linear Diophantine uncertain linguistic arithmetic Heronian mean operator, Linear Diophantine uncertain linguistic weighted arithmetic Heronian mean operator, Linear Diophantine uncertain linguistic geometric Heronian mean operator, Linear Diophantine uncertain linguistic weighted geometric Heronian mean operator, and their properties are also discovered. By using these operators, we utilize the multi-attribute decision-making procedure by using elaborated operators. To determine the consistency and validity of the elaborated operators, we illustrate some examples by using explored operators. Finally, the superiority and comparative analysis of the elaborated operators with some existing operators are also determined and justified with the help of a graphical point of view.

T-Spherical Fuzzy Rough Interactive Power Heronian Mean Aggregation Operators for Multiple Attribute Group Decision-Making

In this article, to synthesize the merits of interaction operational laws (IOLs), rough numbers (RNs), power average (PA) and Heronian mean (HM), a new notion of T-spherical fuzzy rough numbers (T-SFRNs) is first introduced to describe the intention of group experts accurately and take the interaction between individual experts into account with complete and symmetric information. The distance measure and ordering rules of T-SFRNs are proposed, and the IOLs of T-SFRNs are extended. Next, the PA and HM are combined based on the IOLs of T-SFRNs, and the T-Spherical fuzzy rough interaction power Heronian mean operator and its weighted form are proposed. These aggregation operators can accurately express both individual and group uncertainty using T-SFRNs, capture the interaction among membership degree, abstinence degree and non-membership degree of T-SFRNs by employing IOLs, ensure the overall balance of variable values by the PA in the process of information fusion, and realize the interrelationship between attribute variables by the HM. Several properties and special cases of these aggregation operators are further presented and discussed. Subsequently, a new approach for dealing with T-spherical fuzzy multiple attribute group decision-making problems based on proposed aggregation operator is developed. Lastly, in order to validate the feasibility and reasonableness of the proposed approach, a numerical example is presented, and the superiorities of the proposed method are illustrated by describing a sensitivity analysis and a comparative analysis.

Analyzing and controlling computer security threats based on complex q-rung orthopair fuzzy heronian mean operators

Certain intellectuals have generalized the principle of the fuzzy set (FS), but the theory of complex q-rung orthopair fuzzy set (Cq-ROFS) has received massive attraction from different scholars. The goal of this study is to combine the principle of Heronian mean (HM) operator with Cq-ROFS is to initiate the complex q-rung orthopair fuzzy HM (Cq-ROFHM) operator, complex q-rung orthopair fuzzy weighted HM (Cq-ROFWHM) operator, complex q-rung orthopair fuzzy geometric HM (Cq-ROFGHM) operator, complex q-rung orthopair fuzzy weighted geometric HM (Cq-ROFWGHM) operator, and their flexible and dominant properties. These operators can help to aggregate any number of attributes to determine the reliability and consistency of the investigated operators. Moreover, there are physical and non-physical threats. Physical threats cause damage to computer systems hardware and infrastructure. Examples include theft, vandalism through to natural disasters. Non-physical threats target the software and data on the computer systems. To manage such sort of troubles, we determine the analyzing and controlling computer security threats based on presented operators under the Cq-ROFS. Finally, to show the reliability and proficiency of the presented approaches, we resolved some numerical examples by using the explored operators. The comparative analysis, advantages, and graphical interpretations of the presented works are also discovered.

Interval-Valued Hesitant Fuzzy Linguistic Multiattribute Decision-Making Method Based on Three-Parameter Heronian Mean Operators

Numerous variants have been proposed for sets of linguistic terms and the interval-valued hesitant fuzzy set (IVHFS). In particular, the interval-valued hesitant fuzzy linguistic set (IVHFLS) is more suitable for defining the hesitancy and inconsistency inherent in the human cognitive processes of decision making. A key aggregation operator is Heronian mean (HM), based on which the correlation among aggregated arguments can be captured. However, the existing HM operators partially overlook the correlation among more than two arguments and lack the properties of idempotency and reducibility. In this work, the limitations of HM operators are first analyzed. Then, two new HM variants are introduced: three-parameter weighted Heronian mean (TPWHM) and three-parameter weighted geometric Heronian mean (TPWGHM). Thus, the reducibility, idempotency, monotonicity, and boundedness properties are proven for the two computational procedures, and unique situations are mentioned. Furthermore, two more elaborate operators are also introduced which are called the interval-valued hesitant fuzzy linguistic TPWHM (IVHFLTPWHM) and the interval-valued hesitant fuzzy linguistic TPWGHM (IVHFLTPWGHM). The main properties, as well as unique situations of these two computational procedures, are discussed. Finally, the introduced methods are clarified by illustrative examples. In addition, the parameter effects on the decision-making outcomes are discussed and comparisons with other reference methods are made.

Power Improved Generalized Heronian Mean Operators Utilizing Hamacher Operations with Picture Fuzzy Information

In multiple attribute decision-making (MADM), to better denote complicated preference information of decision-makers (DMs), picture fuzzy set (PFS) as an expansion of intuitionistic fuzzy set (IFS) has become a powerful tool in the recent years. Meanwhile, to remove the impact of abnormal data and capture the correlations among attributes in MADM issue, we propose the power improved generalized Heronian mean (PIGHM) operators in this paper, which have the merits of both power average (PA) operator and improved generalized Heronian mean (IGHM) operator. Additionally, Hamacher operations as a generalization of Algebraic operations and Einstein operations demonstrate good smooth approximate. Motivated by these, the main purpose is to explore PIGHM operators utilizing Hamacher operations to cope with MADM issue with picture fuzzy information. First, we introduce the Hamacher operations, the normalized hamming distance, and similarity measure of picture fuzzy numbers (PHNs). Second, based on these, two new picture fuzzy aggregating operators (AOs), the picture fuzzy Hamacher weighted power improved generalized Heronian mean (PFHWPIGHM) operator and the picture fuzzy Hamacher weighted geometric power improved generalized Heronian mean (PFHWGPIGHM) operator, are put forward, and some properties and special instances of proposed AOs are also investigated. Third, a new MADM model in terms of the PIGHM AOs is developed. Eventually, a practical MADM example, together with sensitivity analysis and comparative analysis, is conducted to verify the credibility and superiority of the new MADM model.

Hamacher heronian mean operators for multi-critria decision-making under multi-valued picture fuzzy uncertain lingsuitic environment

Picture fuzzy set (PFS) and linguistic term set (LTS) are two significant notions in multi-criteria decision-making (MCDM). In practice, decision-makers sometimes need utilize the multiple probable membership degrees for an uncertain linguistic term to express evaluation information. Motivated by these, to better convey the vagueness and uncertainty of cognitive information, multi-valued picture fuzzy uncertain linguistic set combining picture hesitant fuzzy set with uncertain linguistic term set is proposed. We firstly define the concepts of multi-valued picture fuzzy uncertain linguistic set and multi-valued picture fuzzy uncertain linguistic number. Hamacher operations are more general and flexible in information fusion, thus, Hamacher operations and comparison method are developed at the same time. Improved generalized Heronian Mean operator can simultaneously reflect correlations between values and prevent the redundant calculation. Then, two operators of improved generalized weighted Heronian mean and improved generalized geometric weighted Heronian mean in view of Hamacher operations are proposed. Meanwhile, some distinguished properties and instances of two operators are explored as well. Moreover, a novel MCDM approach applying the developed operators is constructed. Ultimately, an illustrative example on vendor selection is performed, and sensitivity analysis and comparison analysis are provided to verify the powerfulness of the proposed method.

Multiple Attribute Decision Making Based on Linguistic Generalized Weighted Heronian Mean

In actual multiple attribute decision making, people often use language to evaluate attributes of the object, and sometimes there are associations between the attributes. Therefore, the study of multiple attribute decision making with language as attributes and associations between attributes is of great theoretical significance and practical value. The Heronian mean is not only an operator which reflects the associations between attributes, but also has excellent properties, including idempotency, monotonicity, boundedness, parameter symmetry, and alternate symmetry. In this paper, firstly a new linguistic generalized weighted Heronian mean (LGWHM) was provided, and its properties including idempotency, monotonicity, boundedness, and limit were studied. Then, a new three-parameter linguistic generalized weighted Heronian mean (TPLGWHM) and its idempotency, monotonicity, and boundedness properties were proposed. Finally, multi-attribute decision making methods based on the new linguistic generalized weighted Heronian mean were given, and an example was analyzed and compared with other methods.

A Novel Multi-Criteria Group Decision-Making Approach Based on Bonferroni and Heronian Mean Operators under Hesitant 2-Tuple Linguistic Environment

Ambiguous and uncertain facts can be handled using a hesitant 2-tuple linguistic set (H2TLS), an important expansion of the 2-tuple linguistic set. The vagueness and uncertainty of data can be grabbed by using aggregation operators. Therefore, aggregation operators play an important role in computational processes to merge the information provided by decision makers (DMs). Furthermore, the aggregation operator is a potential mechanism for merging multisource data which is synonymous with cooperative preference. The aggregation operators need to be studied and analyzed from various perspectives to represent complex choice situations more readily and capture the diverse experiences of DMs. In this manuscript, we propose some valuable operational laws for H2TLS. These new operational laws work through the individual aggregation of linguistic words and the collection of translation parameters. We introduced a hesitant 2-tuple linguistic weighted average (H2TLWA) operator to solve multi-criteria group decision-making (MCGDM) problems. We also define hesitant 2-tuple linguistic Bonferroni mean (H2TLBM) operator, hesitant 2-tuple linguistic geometric Bonferroni mean (H2TLGBM) operator, hesitant 2-tuple linguistic Heronian mean (H2TLHM) operator, and a hesitant 2-tuple linguistic geometric Heronian mean (H2TLGHM) operator based on the novel operational laws proposed in this paper. We define the aggregation operators for addition, subtraction, multiplication, division, scalar multiplication, power and complement with their respective properties. An application example and comparison analysis were examined to show the usefulness and practicality of the work.