Interval Neutrosophic Sets and Their Application in Multicriteria Decision Making Problems

Abstract

As a generalization of fuzzy sets and intuitionistic fuzzy sets, neutrosophic sets have been developed to represent uncertain, imprecise, incomplete, and inconsistent information existing in the real world. And interval neutrosophic sets (INSs) have been proposed exactly to address issues with a set of numbers in the real unit interval, not just a specific number. However, there are fewer reliable operations for INSs, as well as the INS aggregation operators and decision making method. For this purpose, the operations for INSs are defined and a comparison approach is put forward based on the related research of interval valued intuitionistic fuzzy sets (IVIFSs) in this paper. On the basis of the operations and comparison approach, two interval neutrosophic number aggregation operators are developed. Then, a method for multicriteria decision making problems is explored applying the aggregation operators. In addition, an example is provided to illustrate the application of the proposed method.

Some Single-Valued Neutrosophic Dombi Weighted Aggregation Operators for Multiple Attribute Decision-Making

“…For any SVNN s = <t, u, v>, its score and accuracy functions [29] can be introduced, respectively, as follows:…”

“…According to the two functions E(s) and H(s), the comparison and ranking of two SVNNs are introduced by the following definition [29]. …”

“…Recently, SNSs (INSs, and SVNSs) have been widely applied in many areas [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], such as decision-making, image processing, medical diagnosis, fault diagnosis, and clustering analysis. Especially, many researchers [7,[29][30][31][32][33][34][35][36] have developed various aggregation operators, like simplified neutrosophic weighted aggregation operators, simplified neutrosophic prioritized aggregation operators, single-valued neutrosophic normalized weighted Bonferroni mean operators, generalized neutrosophic Hamacher aggregation operators, generalized weighted aggregation operators, interval neutrosophic prioritized ordered weighted average operators, interval neutrosophic Choquet integral operators, interval neutrosophic exponential weighted aggregation operators, and so on, and applied them to decision-making problems with SNS/SVNS/INS information. Obviously, the aggregation operators give us powerful tools to deal with the aggregation of simplified (single-valued and interval) neutrosophic information in the decision making process.…”

Subtraction and Division Operations of Simplified Neutrosophic Sets

“…|x ∈ X , there are the following relations [6]: Whereas, the operational laws of the INSs N 1 and N 2 are introduced as follows [8]:…”

“…Further, Ye [7] introduced a simplified neutrosophic set (SNS), which is a subclass of the neutrosophic set and contains a SVNS and an INS, and defined some basic operational laws over SNSs, such as addition and multiplication. Then, Zhang et al [8] indicated some unreasonable phenomena of the basic operational laws over SNSs in [7] and improved some basic operational laws of INSs. Recently, Smarandache [9] defined the subtraction and division operators in the basic operational laws of neutrosophic single-valued numbers and presented some restrictions for these operations of neutrosophic single-valued numbers and neutrosophic single-valued overnumbers/undernumbers/offnumbers.…”