A New Multi-Attribute Decision Making Method with Single-Valued Neutrosophic Graphs

In most realistic situations, the theory and method of multi-attribute decision-making have been widely used in different fields, such as engineering, economy, management, military, and others. Although many studies in some extended fuzzy contexts have been explored with multi-attribute decision-making, it is widely recognized that single-valued neutrosophic sets can describe incomplete, indeterminate and inconsistent information more easier. In this paper, aiming at addressing multi-attribute decision-making problems with single-valued neutrosophic information, related models and multi-attribute decision-making approaches based on the fuzzy graph theory are studied. In specific, the notion of single-valued neutrosophic sets and graphs is firstly introduced together with several common operational laws . Then a multi-attribute decision making method based on single-valued neutrosophic graphs is established. Finally, an illustrative example and a comparative analysis are conducted to verify the feasibility and efficiency of the proposed method.

Smarandache [4][5] and Wang et al. [6], which can be seen as a generalized form of intuitionistic fuzzy sets and have numerous applications in real-life applications. At present, single-valued neutrosophic multi-attribute problems have been widely studied by scholars and practitioners [7][8][9]. Rosenfeld [10] proposed the notion of fuzzy graphs. Later on, Mordeson and Peng [11] provided several common operations of fuzzy graphs. Ye [12] gave the idea of single-valued neutrosophic minimum spanning trees and introduced a corresponding clustering method. Yang et al. [13] elaborated single-valued neutrosophic relations. Dhavaseelan et al. [14] defined strong neutrosophic graphs. Akram and Shahzadi [15] introduced the notion of neutrosophic soft graphs in 2016. Recently, Akram and Shahzadi [16] studied properties of single-valued neutrosophic graphs by level graphs. They also developed some operations according to presented novel notions in Broumi et al. [17] and Shah-Hussain [18]. Akram and Sitara [19] studied single-valued neutrosophic graph structures for decision-making issues. By taking advantages of diverse neutrosophic graphs, Broumi et al. [20] explored panning tree problems with neutrosophic edge weights. Then Broumi et al. [21] further studied a shortest path problem by virtue of the Bellman algorithm in the neutrosophic context, which can lay a solid foundation on the neutrosophic graph theory. Moreover, Karaaslan and Davvaz [22] discussed several novel notions of singlevalued neutrosophic graphs along with an application in decision-making. Karaaslan [23] explored a new Gaussian single-valued neutrosophic numbers and then presented corresponding applications in multi-attribute decisionmaking.
In this paper, the concept, operators and score functions of single-valued neutrosophic sets and graphs are firstly reviewed, which are referred to single-valued neutrosophic graph structures. Then a multi-attribute decision rule by means of single-valued neutrosophic graphs is proposed. Finally, an illustrative example and a comparative analysis are conducted to verify the feasibility and efficiency of the proposed method.

Basic
for all x X  to the set A , then the following condition is true: is a single-valued neutrosophic number. A score function with regard to x is provided as the following mathematical expression:

Single-valued neutrosophic graphs
Definition 4 [19] Suppose V is a finite universe of discourse. A single-valued neutrosophic graph G is defined as the following form: For any x a and y a , the followings are true: In specific, C is single-valued neutrosophic vertices on G , and D is single-valued neutrosophic edges on G .

Multi-attribute decision making based on single-valued neutrosophic graphs
In this section, for single-valued neutrosophic multi-attribute decision making problems with correlations and priorities among attributes, a multi-attribute decision making approach based on single-valued neutrosophic graphs is established. Firstly, the basic model of the problem is described. After that, a single-valued neutrosophic multiattribute decision making algorithm with correlations and priorities is given.

The model building
In the problem of single-valued neutrosophic multi-attribute decision making with correlations and priorities among attributes at the same time, let a set of alternatives be , and a set of attribute weights be   1 2 , , , several single-valued neutrosophic information under multiple attributes, overall attribute values that correspond to alternatives 1 2 , , , m q q q  are analyzed and an optimal alternative is selected. On the whole, the above multiattribute decision making process includes representations, constructions, analysis and other stages of decision making information.

The model algorithm
In what follows, a single-valued neutrosophic multi-attribute decision making algorithm with correlations and priority relationships is given. The core of this algorithm lies in handling the relevance and priority relationships among attributes effectively.   At last, based on the above idea of coping with correlations and priority relationships among attributes, a single-valued neutrosophic multi-attribute decision making algorithm is eventually established that is shown as follows.
Input Output The best alternative.
Step 1 Calculate the energy coefficient ij  of single-valued neutrosophic information between two related attributes i a and j a .
Step 2 Calculate the attribute weight j w by using the priority relationship among attributes.
Step 3 Calculate the overall attribute value Step 4 Calculate the score function   k s q that corresponds to the overall attribute value of an alternative k q .
Step 5 Determine the best alternative

An illustrative example
This section intends to present detailed processes of addressing a single-valued neutrosophic multi-attribute decision making problem by utilizing the case study from Literature [24], then a comparative analysis is arranged to demonstrate the effectiveness of the proposed single-valued neutrosophic multi-attribute decision making algorithm.

Case descriptions
Indian government had been issued a global tender to select the contractor for these projects in the newspaper and considered five attributes required, i.e., Technology Expertise    Table 1. Afterwards, a single-valued neutrosophic graph  E a a a a a a a a a a a a a a a a a a a a   , and the established single-valued neutrosophic graph is presented in the following Figure 1. In light of the above expressions, the proposed single-valued neutrosophic multi-attribute decision making algorithm by virtue of single-valued neutrosophic graphs will be used to obtain the best contractor for Indian government.

Processes of single-valued neutrosophic multi-attribute decision making
According to the proposed single-valued neutrosophic multi-attribute decision making algorithm, the energy coefficient of the single-valued neutrosophic information between attributes is firstly calculated as follows.    q q q q q     , so the best contractor is 2 q .

Comparative analysis
For the sake of presenting validity and effectiveness of the constructed single-valued neutrosophic multiattribute decision making algorithm, a comparative analysis is studied by using single-valued neutrosophic multiattribute decision making approaches proposed in Literature [24], both specific calculation processes and discussions are presented below.
According to the method of logarithmic single-valued neutrosophic weighted average (L-SVNWA) operators and logarithmic single-valued neutrosophic weighted geometric (L-SVNWG) operators, a comparison analysis by using them in the above presented case study will be made. Suppose a single-valued neutrosophic making decision matrix is    (1) The L-SVNWA operator is provided as follows: The L-SVNWG operator is provided as follows: Compared with previous comparative analysis, the single-valued neutrosophic numbers of each contractor under corresponding attributes are integrated and the score of single-valued neutrosophic numbers after integrations is obtained. The contractor with larger score value is the best candidate for J.K. construction. By using L-SVNWA, the ranking of contractors is 2 4 3 5 1 q q q q q     , which is not completely consistent with the results obtained by using the method proposed in this paper. It is easy to see that there are differences in the ranking between 5 q and 1 q . However, the above ranking results do not affect the selection of the best contractor, the best contractor is still the contractor 2 q . By using L-SVNWG, the ranking of contractors is 2 q q q q q     , the results by using the method proposed in this paper are consistent.
Compared with the decision results obtained from the method proposed in Literature [20], the advantages of the constructed single-valued neutrosophic multi-attribute decision making algorithm are mainly reflected as follows: (1) The constructed single-valued neutrosophic multi-attribute decision making algorithm utilizes the framework of fuzzy graphs to address multi-attribute decision making, which makes single-valued neutrosophic graphs excel in expressing correlations between attributes via edges between vertices in single-valued neutrosophic information systems, thus the addressing of correlational single-valued neutrosophic multi-attribute decision making becomes more efficient.
(2) In light of the proposed theoretical aspects of single-valued neutrosophic graphs, the constructed singlevalued neutrosophic single-valued neutrosophic algorithm provides a two-stage problem solving approach by integrating the strategies with correlations and prioritization relationships at the same time, which is beneficial for completing a complicated multi-attribute decision making with high qualities.

Conclusions
In real world, single-valued neutrosophic sets have many advantages in dealing with uncertainties compared to fuzzy sets and intuitionistic fuzzy sets, it is easy to see that single-valued neutrosophic sets play important part in information depictions in single-valued neutrosophic, thus it is necessary to establish efficient information analysis tools for single-valued neutrosophic in single-valued neutrosophic information systems. In this paper, the concept and operation rules of single-value neutrosophic sets and graphs are firstly introduced. Then a single-valued neutrosophic algorithm is proposed based on single-valued neutrosophic graphs. Finally, a practical case study and a corresponding comparative analysis are conducted to show the applicability and effectiveness of the presented single-valued neutrosophic single-valued neutrosophic algorithm.