Opportunities for Software Testing using Neutrosophic Numbers

Neutrosophic number is an important tool which is used to express indeterminate evaluation information. The purpose of the paper is to propose some aggregation operators based on neutrosophic number, which are used to handle multiple attribute group decision-making problems. Firstly, we introduce the definition, the properties and the operational laws of the neutrosophic numbers, and the possibility degree function is briefly introduced. Then, some neutrosophic number operators are proposed, such as the neutrosophic number weighted arithmetic averaging operator, the neutrosophic number ordered weighted arithmetic averaging operator, the neutrosophic number hybrid weighted arithmetic averaging operator, the neutrosophic number weighted geometric averaging operator, the neutrosophic number ordered weighted geometric averaging operator, the neutrosophic number hybrid weighted geometric averaging operator, the neutrosophic number generalized weighted averaging operator, the neutrosophic number generalized ordered weighted averaging operator, the neutrosophic number generalized hybrid weighted averaging (NNGHWA) operator. Furthermore, some properties of these operators are discussed. Moreover, a multiple attribute group decision-making method based on the NNGHWA operator is proposed. Finally, an illustrative example is proposed to demonstrate the practicality and effectiveness of the method.

Real-life data introduce noise, uncertainty, and imprecision to statistical projects; it is advantageous to consider strategies to overcome these information expressions and processing problems. Neutrosophic (indeterminate) numbers can flexibly and conveniently represent the hybrid information of the partial determinacy and partial indeterminacy in an indeterminate setting, while a fuzzy multiset is a vital mathematical tool in the expression and processing of multi-valued fuzzy information with different and/or same fuzzy values. If neutrosophic numbers are introduced into fuzzy sequences in a fuzzy multiset, the introduced neutrosophic number sequences can be constructed as the neutrosophic number multiset or indeterminate fuzzy multiset. Motivated based on the idea, this study first proposes an indeterminate fuzzy multiset, where each element in a universe set can be repeated more than once with the different and/or identical indeterminate membership values. Then, we propose the parameterized correlation coefficients of indeterminate fuzzy multisets based on the de-neutrosophication of transforming indeterminate fuzzy multisets into the parameterized fuzzy multisets by a parameter (the parameterized de-neutrosophication method). Since indeterminate decision-making issues need to be handled by an indeterminate decision-making method, a group decision-making method using the weighted parameterized correlation coefficients of indeterminate fuzzy multisets is developed along with decision makers’ different decision risks (small, moderate, and large risks) so as to handle multicriteria group decision-making problems in indeterminate fuzzy multiset setting. Finally, the developed group decision-making approach is used in an example on a selection problem of slope design schemes for an open-pit mine to demonstrate its usability and flexibility in the indeterminate group decision-making problem with indeterminate fuzzy multisets.

An approach of TOPSIS technique for developing supplier selection with group decision making under type-2 neutrosophic number

Neutrosophic numbers are very suitable for expressing indeterminate evaluation information in complex decision-making problems, and then projection measure is a useful method for handling the decision-making problems. However, due to the lack of engineering applications of neutrosophic numbers and some shortcoming implied in general projection measures in some cases. Therefore, the paper proposes a bidirectional projection measure of interval numbers to overcome the shortcoming and extend it to the bidirectional projection measure of neutrosophic numbers, and then develops a bidirectional projection-based multiple attribute group decision-making method with neutrosophic numbers. Through the bidirectional projection measure between each alternative decision matrix and the ideal alternative matrix, all the alternatives can be ranked to select the best one. Finally, an illustrative example demonstrates the application of the proposed method. The effectiveness and advantages of the proposed method are shown by the comparative analysis with existing relative methods.

The recent boom of decision-making under uncertain information has attracted many researchers to the field of integrating various types of sets with decision-making methods. In this paper, a combined decision-making trial and evaluation laboratory (DEMATEL) with single-valued neutrosophic sets is proposed to solve the decision problem. This new model combines the advantages of multiplicative inverse of decision matrix in DEMATEL and neutrosophic numbers in linguistic variables, which can find the interrelationship among factors of decision problem. Differently from the typical multiplicative inverse of DEMATEL, which directly used inverse of matrix using real numbers, this method introduces the concept of inverse of matrix using the proposed left–right neutrosophic numbers. This step will enhance the validity of multiplicative inverse of decision matrix in the DEMATEL with neutrosophic numbers. The proposed neutrosophic DEMATEL is also be compared with the DEMATEL and fuzzy DEMATEL. This paper includes a case study that demonstrates the applicability of the neutrosophic DEMATEL in establishing the relationship among influential factors of coastal erosion. Extensive empirical studies using 12 factors of coastal erosion were presented to study the benefits of the proposed method. The result unveils the cause-and-effect relationships among the factors, where seven factors are identified as cause factors and five factors are grouped as effect factors. It is discovered that the factor ‘imbalance sediment supply’ gives a significant influence to coastal erosion. It is also shown that the degree of importance of the factors is almost consistent with the other two methods despite differences in type of numbers used in defining linguistic variables.

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.

As a neutrosophic number, which consists of a determinate part and an indeterminate part, can more easily and better express incomplete and indeterminate information that exists commonly in real situations, the main purposes of this paper are to provide a neutrosophic number tool for group decision-making problems with indeterminate information under a neutrosophic number environment and to develop a de-neutrosophication method and a possibility degree ranking method for neutrosophic numbers from the probability viewpoint as a methodological support for group decision-making problems. In group decision-making problems with neutrosophic numbers, through the de-neutrosophication and possibility degree ranking order of neutrosophic numbers, the ranking order of alternatives is performed well as the possibility degree ranking method has the intuitive meaning from the probability viewpoint, and then the best one(s) can be determined as well. Finally, two illustrative examples show the applications and effectiveness of the proposed method.

The concept of a single valued neutrosophic number (SVN-number) is of importance for quantifying an ill-known quantity and the ranking of SVN-numbers is a very difficult problem in multi-attribute decision making problems. The aim of this paper is to present a methodology for solving multi-attribute decision making problems with SVN-numbers. Therefore, we firstly defined the concepts of cut sets of SVN-numbers and then applied to single valued trapezoidal neutrosophic numbers (SVTN-numbers) and triangular neutrosophic numbers (SVTrN-numbers). Then, we proposed the values and ambiguities of the truth-membership function, indeterminacy-membership function and falsity-membership function for a SVN-numbers and studied some desired properties. Also, we developed a ranking method by using the concept of values and ambiguities, and applied to multi-attribute decision making problems in which the ratings of alternatives on attributes are expressed with SVTN-numbers.

Due to growing risk and complexity of modern decision-making problems, the graded information associated with linguistic terms is suitable to characterize the uncertainty during cognition process. By combining 2-dimensional linguistic variables and interval neutrosophic numbers, we put forward a new concept of 2-dimensional interval neutrosophic linguistic numbers and establish pertinent operational rules. Moreover, we develop an operator for fusing 2-dimensional interval neutrosophic linguistic numbers by means of Choquet integral, by which the relationships among the attributes can be reflected. The established aggregation operator’s properties are also studied. In the sequel, a new group PROMETHEE (preference ranking organization method for enrichment evaluations) II method is created to deal with the group decision problems with 2-dimensional interval neutrosophic linguistic terms.

Neutrosophic set can deal with the uncertainties related to the information of any decision making problem in real life scenarios, where fuzzy set may fail to handle those uncertainties properly. In this study, we present the perception of trapezoidal bipolar neutrosophic numbers and its classification in different frame. We introduce the idea of disjunctive structures of trapezoidal bipolar neutrosophic numbers namely type-1 trapezoidal bipolar neutrosophic number, type-2 trapezoidal bipolar neutrosophic numbers, and type-3 trapezoidal bipolar neutrosophic number based on the perception of dependency among membership functions in neutrosophic set. In any neutrosophic decision-making problem, the decision maker uses the comparison of neutrosophic numbers to choose among alternatives solutions. Here, we introduce a ranking method, i.e., De-bipolarization scheme for trapezoidal bipolar neutrosophic number (TrBNN) using removal area technique. We also describe the utility of trapezoidal bipolar neutrosophic number and its appliance in a multi criteria group decision making problem (MCGDM) for distinct users in trapezoidal bipolar arena which is more ethical, precise and reliable in neutrosophic field.

In inconsistent and indeterminate settings, as a usual tool, the neutrosophic cubic set (NCS) containing single-valued neutrosophic numbers and interval neutrosophic numbers can be applied in decision-making to present its partial indeterminate and partial determinate information. However, a few researchers have studied neutrosophic cubic decision-making problems, where the similarity measure of NCSs is one of the useful measure methods. For this work, we propose the Dice, cotangent, and Jaccard measures between NCSs, and indicate their properties. Then, under an NCS environment, the similarity measures-based decision-making method of multiple attributes is developed. In the decision-making process, all the alternatives are ranked by the similarity measure of each alternative and the ideal solution to obtain the best one. Finally, two practical examples are applied to indicate the feasibility and effectiveness of the developed method.

The neutrosophic cubic set can describe complex decision-making problems with its single-valued neutrosophic numbers and interval neutrosophic numbers simultaneously. The Dombi operations have the advantage of good flexibility with the operational parameter. In order to solve decision-making problems with flexible operational parameter under neutrosophic cubic environments, the paper extends the Dombi operations to neutrosophic cubic sets and proposes a neutrosophic cubic Dombi weighted arithmetic average (NCDWAA) operator and a neutrosophic cubic Dombi weighted geometric average (NCDWGA) operator. Then, we propose a multiple attribute decision-making (MADM) method based on the NCDWAA and NCDWGA operators. Finally, we provide two illustrative examples of MADM to demonstrate the application and effectiveness of the established method.

The neutrosophic cubic set can contain much more information to express its interval neutrosophic numbers and single-valued neutrosophic numbers simultaneously in indeterminate environments. Hence, it is a usual tool for expressing much more information in complex decision-making problems. Unfortunately, there has been no research on similarity measures of neutrosophic cubic sets so far. Since the similarity measure is an important mathematical tool in decision-making problems, this paper proposes three cosine measures between neutrosophic cubic sets based on the included angle cosine of two vectors, distance, and cosine functions, and investigates their properties. Then, we develop a cosine measures-based multiple attribute decision-making method under a neutrosophic cubic environment in which, from the cosine measure between each alternative (each evaluated neutrosophic cubic set) and the ideal alternative (the ideal neutrosophic cubic set), the ranking order of alternatives and the best option can be obtained, corresponding to the cosine measure values in the decision-making process. Finally, an illustrative example about the selection problem of investment alternatives is provided to illustrate the application and feasibility of the developed decision-making method.

The paper presents the potentials for the development of software using agile methodologies. Special consideration is devoted to the potentials and advantages of use of the Scrum methodology in the development of software and the relationship between the implementation of agile methodologies and the software development projects.

PurposeThe complex network structure causes several disruptions in the supply chain that make risk management essential for supply chain management including halal supply chain (HSM). During risk management, several challenges are associated with the risk assessment phase, such as incomplete and uncertain information about the system. To cater this, the authors propose a risk assessment framework that addresses the issues of uncertainty using neutrosophic theory and demonstrated the applicability of the proposed framework through the case of halal supply chain management (HSCM).Design/methodology/approachThe proposed framework is using the capabilities of the neutrosophic number which can handle uncertain, vague and incomplete information. Initially, the risk related to the HSC is identified through a literature review and expert’s input. Further, the probability and impact of each HSM-related risk are assessed using experts’ input through linguistic terms. These linguistic values are transformed into single-value trapezoidal neutrosophic numbers (SVTNNs). Finally, the severity of each HSM-related risk is determined through the multiplication of the probability and impact of each risk and prioritised the risks based on their severity.FindingsA comprehensive risk assessment framework is developed that could be used under uncertainty. Initially, 16 risks are identified related to the HSM. Further, the identified risks are prioritised using the severity of the risks. The high-priority risk is “raw material status”, “raw material wholesomeness” and “origin of raw material” while “information integrity” and “people integrity” are low-priority risks.Practical implicationsHSM risk can be effectively assessed through the proposed framework. The proposed framework applied neutrosophic numbers to represent real-life situations, and it could be used for other supply chains as well.Originality/valueThe proposed method is effectively addressing the issue of linguistic subjectivity, inconsistent information and uncertainty in the expert’s opinion. A case study of the HSC is adopted to illustrate the efficiency and applicability of the proposed risk framework.