Co-degree and Degree of classes of Neutrosophic Hypergraphs

New setting is introduced to study types of coloring numbers, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges in neutrosophic hypergraphs. Different types of procedures including neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs are proposed in this way, some results are obtained. General classes of neutrosophic hypergraphs are used to obtain chromatic number, the representatives of the colors, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges in neutrosophic hypergraphs. Using colors to assign to the vertices of neutrosophic hypergraphs and characterizing representatives of the colors are applied in neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs. Some questions and problems are posed concerning ways to do further studies on this topic. Using different ways of study on neutrosophic hypergraphs to get new results about number, degree and co-degree in the way that some number, degree and co-degree get understandable perspective. Neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs are studied to investigate about the notions, coloring, the representatives of the colors, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges in neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs. In this way, sets of representatives of colors, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges have key points to get new results but in some cases, there are usages of sets and numbers instead of optimal ones. Simultaneously, notions chromatic number, the representatives of the colors, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges are applied into neutrosophic hypergraphs, especially, neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs to get sensible results about their structures. Basic familiarities with neutrosophic hypergraphs theory and hypergraph theory are proposed for this article.

Neutrosophic Triangular Fuzzy Travelling Salesman Problem Based on Dhouib-Matrix-TSP1 Heuristic

In this paper, the Travelling Salesman Problem is considered in neutrosophic environment which is more realistic in real-world industries. In fact, the distances between cities in the Travelling Salesman Problem are presented as neutrosophic triangular fuzzy number. This problem is solved in two steps: At first, the Yager’s ranking function is applied to convert the neutrosophic triangular fuzzy number to neutrosophic number then to generate the crisp number. At second, the heuristic Dhouib-Matrix-TSP1 is used to solve this problem. A numerical test example on neutrosophic triangular fuzzy environment shows that, by the use of Dhouib-Matrix-TSP1 heuristic, the optimal or a near optimal solution as well as the crisp and fuzzy total cost can be reached.

Neutrosophic Number Optimization Models and Their Application in the Practical Production Process

In order to simplify the complex calculation and solve the difficult solution problems of neutrosophic number optimization models (NNOMs) in the practical production process, this paper presents two methods to solve NNOMs, where Matlab built-in function “fmincon()” and neutrosophic number operations (NNOs) are used in indeterminate environments. Next, the two methods are applied to linear and nonlinear programming problems with neutrosophic number information to obtain the optimal solution of the maximum/minimum objective function under the constrained conditions of practical productions by neutrosophic number optimization programming (NNOP) examples. Finally, under indeterminate environments, the fit optimal solutions of the examples can also be achieved by using some specified indeterminate scales to fulfill some specified actual requirements. The NNOP methods can obtain the feasible and flexible optimal solutions and indicate the advantage of simple calculations in practical applications.

A Hybrid BSC-DEA Model with Indeterminate Information

Strategy is the main source of long-term growth for organizations, and if it is not successfully implemented, even if appropriate ones are adopted, the process is futile. The balanced scorecard which focuses on four aspects such as growth and learning, internal processes, customer, and financial is considered as a comprehensive framework for assessing performance and the progress of the strategy. Moreover, the data envelopment analysis is one of the best mathematical methods to compute the efficiency of organizations. The combination of these two techniques is a significant quantitative measurement with respect to the organization’s performance. However, in the real world, determinate and indeterminate information exists. Henceforth, the indeterminate issues are inescapable and must be considered in the performance evaluation. Neutrosophic number is a helpful tool for dealing with information that is indeterminate and incomplete. In this paper, we propose a new model of data envelopment analysis in the neutrosophic number environment. Furthermore, we attempt to combine the new model with the balanced scorecard to rank different decision-making units. Finally, the proposed method is illustrated by an empirical study involving 20 banking branches. The results show the effectiveness of the proposed method and indicate that the model has practical outcomes for decision-makers.

Size Effect on Recycled Concrete Strength and Its Prediction Model Using Standard Neutrosophic Number

In recent years, research on recycled aggregate concrete has become a hot issue in the field of civil engineering. This paper mainly studies the size effects on compressive and tensile strengths of the recycled aggregate concrete. Firstly, four sets of recycled concrete cube specimens with different sizes are produced in the laboratory. Secondly, the experiments on compressive and tensile strengths are carried out to obtain the rules of the strength value with the change of the specimen size. Thirdly, a standard neutrosophic number is proposed and used in modelling the size effect law more reasonably. According to the experimental results, it was found that the compressive and tensile strengths of recycled concrete both have obvious size effects. In general, the strength value decreases gradually with the increase of specimen size. Using the standard neutrosophic number, the proposed new formula on size effect law is more suitable for tackling the indeterminacy in the experimental data. It has been shown that the size effect law based on the standard neutrosophic number is more realistic than the existing size effect law. The results may be useful for the engineering application of the recycled concrete and can be extended to other types of size effect laws in the future.

Arccosine and arctangent similarity measures of refined simplified neutrosophic indeterminate sets and their multicriteria decision-making method

In indeterminate and inconsistent setting, existing simplified neutrosophic indeterminate set (SNIS) can be depicted by the neutrosophic number (NN) functions of the truth, falsity and indeterminacy. Then, the three NN functions in SNIS lack their refined expressions and then the simplified neutrosophic indeterminate DM method cannot carry out the multicriteria DM problems with both criteria and sub-criteria in the setting of SNISs. To overcome the flaws, this study first proposes a new notion of a refined simplified neutrosophic indeterminate set (RSNIS), which is described by the refined truth, falsity and indeterminate NN information regarding both elements and sub-elements in a universe set, as the extension of SNIS. Next, we propose the arccosine and arctangent similarity measures of RSNISs and their multicriteria decision making (DM) method with various indeterminate risk ranges so as to carry out multicriteria DM problems with weight values of both criteria and sub-criteria in RSNIS setting. Lastly, the proposed DM method is applied to a multicriteria DM example of slope design schemes for an open pit mine to illustrate its application in the indeterminate DM problem with RSNISs. The decision results and comparative analysis indicate the rationality and efficiency of the proposed DM method with different indeterminate risk ranges.