Abstract
Multi-attribute decision-making (MADM) is a part of management decision-making and an important branch of the modern decision theory and method. MADM focuses on the decision problem of discrete and finite decision schemes. Uncertain MADM is an extension and development of classical multi-attribute decision making theory. When the attribute value of MADM is shown by neutrosophic number, that is, the attribute value is complex data and needs three values to express, it is called the MADM problem in which the attribute values are neutrosophic numbers. However, in practical MADM problems, to minimize errors in individual decision making, we need to consider the ideas of many people and synthesize their opinions. Therefore, it is of great significance to study the method of attribute information aggregation. In this paper, we proposed a new theory—non-dual multi-granulation neutrosophic rough set (MS)—to aggregate multiple attribute information and solve a multi-attribute group decision-making (MGDM) problem where the attribute values are neutrosophic numbers. First, we defined two kinds of non-dual MS models, intersection-type MS and union-type MS. Additionally, their properties are studied. Then the relationships between MS, non-dual MS, neutrosophic rough set (NRS) based on neutrosophic intersection (union) relationship, and NRS based on neutrosophic transitive closure relation of union relationship are outlined, and a figure is given to show them directly. Finally, the definition of non-dual MS on two universes is given and we use it to solve a MGDM problem with a neutrosophic number as the attribute value.
Highlights
Fuzzy sets and rough sets are widely used to solve uncertain problems [1,2,3,4]
We show the process about the non-dual multi-granulation neutrosophic rough set (MS) on two universes to solve multi-attribute group decision-making (MGDM) problems with neutrosophic numbers as attribute values
Algorithm 1 The lower approximation of a union-type multi-granulation neutrosophic rough set Define the method to acquire a complement for a matrix A: each neutrosophic number in matrix A do complement the operator according to the following Formula: ac = (Fa, 1 − Ia, Ta )
Summary
Fuzzy sets and rough sets are widely used to solve uncertain problems [1,2,3,4]. all these theories have their own deficiency, such as in a voting, you may support, not support, be neutral, or abstain from voting, so Smarandache present the definition of the neutrosophic set (NS) [5]. Qian et al think that, in decision analysis problems, the relationship between the multiple decision makers may be independent of each other, so multiple binary relations are needed to approximate the target They put forward the concept of a multi-granularity rough set (MRS) model [20], and define the optimistic MRS model and pessimistic MRS model, respectively. We propose non-dual MS models on two universes and use it to solve MGDM problems with neutrosophic numbers as the attribute values.
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