Some Normal Intuitionistic Fuzzy Heronian Mean Operators Using Hamacher Operation and Their Application

Abstract

Hamacher operation is a generalization of the algebraic and Einstein operation and expresses a family of binary operation in the unit interval [0,1]. Heronian mean can deal with correlations of different criteria or input arguments and does not bring out repeated calculation. The normal intuitionistic fuzzy numbers (NIFNs) can depict normal distribution information in practical decision making. A decision-making problem was researched under the NIFN environment in this study, and a new multi-criteria group decision-making (MCGDM) approach is herein introduced on the basis of Hamacher operation. Firstly, according to Hamacher operation, some operational laws of NIFNs are presented. Secondly, it is noted that Heronian mean not only takes into account mutuality between the attribute values once, but also considers the correlation between input argument and itself. Therefore, in order to aggregate NIFN information, we developed some operators and studied their properties. These operators include Hamacher Heronian mean (NIFHHM), Hamacher weighted Heronian mean (NIFHWHM), Hamacher geometric Heronian mean (NIFHGHM), and Hamacher weighted geometric Heronian mean (NIFHWGHM). Furthermore, we applied the proposed operators to the MCGDM problem and developed a new MCGDM approach. The characteristics of this new approach are that: (1) it is suitable for making a decision under the NIFN environment and it is more reasonable for aggregating the normal distribution data; (2) it utilizes Hamacher operation to provide an effective and powerful MCGDM algorithm and to make more reliable and more flexible decisions under the NIFN circumstance; (3) it uses the Heronian mean operator to deal with interrelations between the attributes or input arguments, and it does not bring about repeated calculation. Therefore, the proposed method can describe the interaction of the different criteria or input arguments and offer some reasonable and reliable MCGDM aggregation operators, which can open avenues for decision making and broaden perspectives of the decision experts. Lastly, an application is given for showing the effectiveness and feasibility of the approach presented in this paper.

Decision Support Algorithm for Selecting an Antivirus Mask over COVID-19 Pandemic under Spherical Normal Fuzzy Environment

“…Conversely, the NOMBD of INFN signifies the degree to which an element cannot be portrayed by NFN. Some scholars have further developed the basic theories of INFN [45,46] and presented some aggregation's operators with their applications [47,48]. Since INFN and its relevant extensions are based on the combination of IFN and NFN, some of the drawbacks of IFN remain in INFN.…”

“…Definition 2 [48]. Let Z 1 = (α 1 , σ 1 ), Z 2 = (α 2 , σ 2 ) Z 1 , Z 2 ∈ F, and λ be a nonnegative real number, we can get the operational rule of Z 1 and Z 2 as follows:…”

“…Compared with existing methods based on PtFS, INFS. As shown in Table 4, the methods proposed by Wang et al [44] and Yang et al [47] combined the IFS with NFN, and the method by Zhang et al [48] considered the correlations of different variables under INFS, but those methods only rely on IFS. As mentioned above, the IFS only use two indices (MBD and NOMBD) to portray the structure of evaluation information that cannot obtain any rational decision result in some practical decision problems.…”

Dual Hesitant Pythagorean Fuzzy Heronian Mean Operators in Multiple Attribute Decision Making

“…Thus, the intuitionistic fuzzy set (IFS) was constructed; in the IFS each intuitionistic fuzzy set is characterized by the functions of membership degree and non-membership degree between 0 and 1, and the sum of these are limited to 1. Subsequent to these studies, more and more scholars have studied the IFS in relation to many multiple attribute decision making (MADM) problems [3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Xu [17] has defined some intuitionistic fuzzy weighted average operators.…”

Some q‐rung interval‐valued orthopair fuzzy Maclaurin symmetric mean operators and their applications to multiple attribute group decision making

“…In the study of fuzzy sets (FSs) fields, finding the suitable way to express evaluation information is a hot topic. Although the intuitionistic FSs (IFSs) and Pythagorean FSs (PFSs) have emerged as useful tools to deal with actual decision‐making problems, however, the scope of their assessment information remains limited. Thus, to overcome this limitation, the q‐rung orthopair fuzzy set (q‐ROFS), which is characterized by the sum of q‐ th power of membership and nonmembership is less or equal to 1, was proposed for MADM problems.…”