Mathematical Problems in Engineering

Volume 2019, Article ID 3258018, 12 pages

https://doi.org/10.1155/2019/3258018

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

School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan, Shandong 250014, China

Correspondence should be addressed to Peide Liu; moc.liamg@uil.ediep

Received 28 April 2019; Revised 7 August 2019; Accepted 31 August 2019; Published 3 October 2019

Academic Editor: Giuseppe D'Aniello

Copyright © 2019 Peide Liu and Hongyu Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

Multiattribute decision making (MADM) is more and more momentous for modern decision science. Its essence is to use the existing decision information to sort and optimize a limited number of alternatives in a certain way. Due to the complexity and unpredictability of the external environment, the ambiguity of the object itself, the limitations of human cognition, and the subjectivity of the decision maker, decision makers usually need to provide preference information through various types of attribute values. Since Zadeh [1] introduced the concepts of the fuzzy sets (FSs), the FSs have been widely studied. Atanassov [2] put forward the intuitionistic fuzzy sets (IFSs) by adding the nonmembership degree based on the traditional FSs. IFSs consist of membership degree and nonmembership degree, and they can more easily express fuzzy information and have been rapidly developed and widely used since they were introduced. However, the degree of hesitation in IFSs cannot be defined separately. Therefore, even if IFSs can effectively describe imperfect information, they are less flexible when dealing with uncertain and inconformity information. Then, the clearly quantified neutrosophic sets (NSs) can describe the value of the proposition between true and false, which was initially proposed by Smarandache [3]. NSs are made up of membership degree, hesitancy degree, and nonmembership degree. In addition, Wang et al. [4] proposed a subcategory of NS called single-valued neutrosophic sets (SVNSs) and discussed its related rules and properties. The trait of SVNS is that membership degree, hesitancy degree, and nonmembership degree are mutually independent; all three are between 0 and 1, and the sum of them is between 0 and 3. Deli and Şubaş [5] developed a sorting method and extended it into MADM problems. Wang et al. [6, 7, 8] introduced a MADM approach in view of Maclaurin symmetric mean (MSM) operator and TODIM for SVNS. Sodenkamp et al. [9] used SVNS to process independent multisource undetermined measurements, which affected the dependability of expert evaluation in MADM problems.

In many actual MADM problems, owing to the uncertainty or imperfect of information, it is difficult to adopt a method that only accepts and rejects these two decisions. Through the expansion of the two decisions, Yao [10, 11] proposed three-way decisions (3WDs) involving acceptance, rejection, and delayed decision making [12, 13, 14]. The principle of the 3WDs is derived from the probability rough set. In light of the positive, boundary, and negative domains of the probabilistic rough set, the 3WD model including acceptance decision, delayed decision, and rejection decision is established. So far, 3WDs are widely used in some areas such as influenza emergency management, granular computing, enterprise evaluation, and social networking [13, 15, 16, 17, 18]. At the same time, many theoretical results have been achieved in the study of 3WDs. For example, Zhang et al. [19] considered a new risk measurement function by utility theory and derived a 3WD model with DTRS. Sun et al. [20] introduced a decision-theoretic 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. According to the TODIM method, Hu et al. [22] constructed a new 3WD model and demonstrated its application in online diagnosis and medical selection.

In the 3WD model, the loss function (LF) is a key parameter. The scholars have studied a great deal of 3WD rules based on LFs of diverse forms, such as interval number [23], IFSs [24], intuitionistic uncertain linguistic variables [25], dual hesitant FSs [26], and Pythagorean FSs [27]. SVNS can handle uncertain, incomplete, and inconsistent information more flexibly. To this end, we use the SVNS to express the LF in this paper. Furthermore, how to determine the conditional probability is also the key to the 3WD method. The conditional probability in many references is subjectively given by the decision makers, which makes the decision results too subjective. Liang et al. [27] used the TOPSIS method to evaluate the conditional probability by calculating the distance between each alternative and the positive or negative ideal solution, which provides a new perspective for us to identify the conditional probability. In this paper, we measure the maximum grey correlation of each alternative with the relative neutrosophic positive ideal solution (RNPIS) and the minimum grey correlation with the relative neutrosophic negative ideal solution (RNNIS) and estimate the conditional probability using the grey relational analysis (GRA) method [28–30].

In conclusion, 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. Therefore, we use the GRA method to calculate conditional probability. The goal and motivation of this paper are (1) to extend 3WDs to the environment of SVNS, using SVNS to represent the LF in 3WDs; (2) to propose the SVNDTRS model and explore its properties; and (3) to use the GRA method to calculate conditional probability in 3WDs. The proposed method extends the use environment of 3WDs and provides a new idea for the determination of conditional probability in 3WDs.

The remainder of this paper is arranged as follows. In Section 2, we briefly review the basics of the NSs and SVNSs. In Section 3, we propose a method to determine the combination weight of attributes. In Section 4, we propose a single-valued neutrosophic decision theory rough set (SVNDTRS) model and its propositions. In Section 5, we estimate the conditional probability of 3WDs based on the GRA method and presented a MADM method to deal with SVNSs based on the SVNDTRS. In Section 6, we use a numerical example to demonstrate the availability, and other methods are compared and analyzed. In Section 7, we reach the conclusion.

#### 2. SVNS

In this section, we introduce some basic concepts of the NSs and the SVNSs.

*Definition 1. *[3]. Let be an object set, and the common elements of are represented by . A NSof consists of , and . can be represented as , where , , and represent the membership degree, hesitancy degree, and nonmembership degree, respectively. and satisfies .

*Definition 2. *[4]. Let be an object set, and the common elements of are represented by . A SVNS of consists of , and . can be represented as , where , , and represent the membership degree, hesitancy degree, and nonmembership degree, respectively. and satisfies . Let be the single-valued neutrosophic number (SVNN) and abbreviated as .

*Definition 3. *[31]. For any two SVNNs and , the related operations are defined as follows:

*Definition 4. *[32]. The complement set of a SVNS is , which is defined by

*Definition 5. *[31]. Let be a SVNN; the cosine similarity of is described as follows:

*Definition 6. *[31]. For any two SVNNs and , if , then .

*Definition 7. *[32]. Let and be any two SVNNs; the Hamming distance between two SVNNs and is described as follows:The normalized Hamming distance between two SVNNs and is described as follows:where , .

#### 3. Basic Model of SVNDTRS

In this section, we introduce a model of SVNDTRS based on 3WDs. At first, we use SVNN to build a LF matrix. The loss of different decision schemes under different state variables is illustrated in Table 1.