Three-Way Decisions with Single-Valued Neutrosophic Decision Theory Rough Sets Based on Grey Relational Analysis

Peide Liu,Hongyu Yang

doi:10.1155/2019/3258018

Abstract

The single-valued neutrosophic set (SVNS) can not only depict imperfect information in the real decision system but also handle undetermined and inconformity information flexibly and effectively. Three-way decisions (3WDs) are often used as an effective method to deal with uncertainties, but the conditional probability is given by the decision maker subjectively, which makes the decision result too subjective. This paper proposes a novel model based on 3WDs to settle the multiattribute decision-making (MADM) problems, where the attribute values are described by SVNS, and the attribute weights are entirely unknown. At first, we build a single-valued neutrosophic decision theory rough set (SVNDTRS) model based on Bayesian decision process. Then, we use the analytic hierarchy process (AHP) approach to calculate the subjective weight of each attribute, the information entropy to obtain the attribute’s objective weight, and the minimum total deviation approach to determine the combined weight of the attributes. After obtaining the standard weight, the grey relational analysis (GRA) method is utilized to calculate the grey correlation closeness with the ideal solution, and the conditional probability is estimated by it. In addition, we develop a decision-making method in view of the ideal solution of 3WDs with the SVNS. This approach not only considers the lowest cost but also gives a corresponding semantic explanation for the decision result of each alternative, which can supplement the decision results of GRA. At last, we illustrate the feasibility and effectiveness of 3WDs through an example of supplier selection and compare it with other methods to verify the advantages of our approach.

Highlights

Multiattribute decision making (MADM) is more and more momentous for modern decision science
single-valued neutrosophic set (SVNS) can handle uncertain, incomplete, and inconsistent information more flexibly, and 3WDs are often used as an effective method to deal with uncertainties, but the conditional probability in many references is given by the decision maker subjectively, which makes the decision result too subjective. erefore, we use the grey relational analysis (GRA) method to calculate conditional probability. e goal and motivation of this paper are (1) to extend 3WDs to the environment of SVNS, using SVNS to represent the loss function (LF) in 3WDs; (2) to propose the single-valued neutrosophic decision theory rough set (SVNDTRS) model and explore its properties; and (3) to use the GRA method to calculate conditional probability in 3WDs. e proposed method extends the use environment of 3WDs and provides a new idea for the determination of conditional probability in 3WDs
We extended 3WDs to the environment of SVNSs and used SVNSs to express the evaluation values and LFs given by decision makers

Summary

Introduction

Multiattribute decision making (MADM) is more and more momentous for modern decision science. Sun et al [20] introduced a decisiontheoretic rough fuzzy set model with linguistic information based on 3WDs and applied it to MADMs. Zhang et al [21] proposed a dynamic 3WD model and proved the model is practicable and valid. SVNS can handle uncertain, incomplete, and inconsistent information more flexibly, and 3WDs are often used as an effective method to deal with uncertainties, but the conditional probability in many references is given by the decision maker subjectively, which makes the decision result too subjective. Y􏽥 can be represented as Y􏽥 􏼈T􏽥Y(x), I􏽥Y(x), F􏽥Y(x) | x ∈ X􏼉, where T􏽥Y(x), I􏽥Y(x), and F􏽥Y(x) represent the membership degree, hesitancy degree, and nonmembership degree, respectively.

Basic Model of SVNDTRS

Establishing the Weights

Decision Analysis for 3WDs with SVNDTRS

Decision-Making Steps

Conclusion