Entropy, Measures of Distance and Similarity of Q-Neutrosophic Soft Sets and Some Applications

Generalized m-Polar Fuzzy Positive Implicative Ideals of BCK-Algebras

This study focuses on combining the theories of $m$ -polar fuzzy sets over $\text{BCK}$ -algebras and establishing a new framework of $m$ -polar fuzzy $\text{BCK}$ -algebras. In this paper, we define the idea of $m$ -polar fuzzy positive implicative ideals in $\text{BCK}$ -algebras and investigate some related properties. Then, we introduce the concepts of $m$ -polar $\left(\in ,\in \vee q\right)$ -fuzzy positive implicative ideals and $m$ -polar $\left(\overline{\in },\overline{\in }\vee \overline{q}\right)$ -fuzzy positive implicative ideals in $\text{BCK}$ -algebras as a generalization of $m$ -polar fuzzy positive implicative ideals. Several properties, examples, and characterization theorems of these concepts are considered.

A cosine similarity measure for multi-criteria group decision making under neutrosophic soft environment

In actual life, uncertain and inconsistent information exists widely. How to deal with the information so that it can be better applied is a problem that has to be solved. Neutrosophic soft sets can process uncertain and inconsistent information. Also, Dempster-Shafer evidence theory has the advantage of dealing with uncertain information, and it can synthesize uncertain information and deal with subjective judgments effectively. Therefore, this paper creatively combines the Dempster-Shafer evidence theory with the neutrosophic soft sets, and proposes a cosine similarity measure for multi-criteria group decision making. Different from the previous studies, the proposed similarity measure is utilized to measure the similarity between two objects in the structure of neutrosophic soft set, rather than two neutrosophic soft sets. We also propose the objective degree and credibility degree which reflect the decision makers’ subjective preference based on the similarity measure. Then parameter weights are calculated by the objective degree. Additionally, based on credibility degree and parameter weights, we propose the modified score function, modified accuracy function, and modified certainty function, which can be employed to obtain partial order relation and make decisions. Later, we construct an aggregation algorithm for multi-criteria group decision making based on Dempster’s rule of combination and apply the algorithm to a case of medical diagnosis. Finally, by testing and comparing the algorithm, the results demonstrate that the proposed algorithm can solve the multi-criteria group decision making problems effectively.

Cubic Vague Set and its Application in Decision Making

From the hybrid nature of cubic sets, we develop a new generalized hybrid structure of cubic sets known as cubic vague sets (CVSs). We also define the concept of internal cubic vague sets (ICVSs) and external cubic vague sets (ECVSs) with examples and discuss their interesting properties, including ICVSs and ECVSs under both P and R-Order. Moreover, we prove that the R and R-intersection of ICVSs (or ECVSs) need not be an ICVS (or ECVS). We also derive the different conditions for P-union (P-intersection, R and R-intersection) operations of both ICVSs (ECVSs) to become an ICVS (ECVS). Finally, we introduce a decision-making based on the proposed similarity measure of the CVSs domain and a numerical example is given to elucidate that the proposed similarity measure of CVSs is an important concept for measuring entropy in the information/data. It will be shown that the cubic vague set has the novelty to accurately represent and model two-dimensional information for real-life phenomena that are periodic in nature.

Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory

To address issues involving inconsistencies, this paper proposes a stochastic multi-criteria group decision making algorithm based on neutrosophic soft sets, which includes a pair of asymmetric functions: Truth-membership and false-membership, and an indeterminacy-membership function. For integrating an inherent stochastic, the algorithm expresses the weights of decision makers and parameter subjective weights by neutrosophic numbers instead of determinate values. Additionally, the algorithm is guided by the prospect theory, which incorporates psychological expectations of decision makers into decision making. To construct the prospect decision matrix, this research establishes a conflict degree measure of neutrosophic numbers and improves it to accommodate the stochastic multi-criteria group decision making. Moreover, we introduce the weighted average aggregation rule and weighted geometric aggregation rule of neutrosophic soft sets. Later, this study presents an algorithm for neutrosophic soft sets in the stochastic multi-criteria group decision making based on the prospect theory. Finally, we perform an illustrative example and a comparative analysis to prove the effectiveness and feasibility of the proposed algorithm.

A Generalized Approach towards Soft Expert Sets via Neutrosophic Cubic Sets with Applications in Games

Games are considered to be the most attractive and healthy event between nationsand peoples. Soft expert sets are helpful for capturing uncertain and vague information.By contrast, neutrosophic set is a tri-component logic set, thus it can deal with uncertain,indeterminate, and incompatible information where the indeterminacy is quantified explicitly andtruth membership, indeterminacy membership, and falsity membership independent of each other.Subsequently, we develop a combined approach and extend this concept further to introduce thenotion of the neutrosophic cubic soft expert sets (NCSESs) by using the concept of neutrosophiccubic soft sets, which is a powerful tool for handling uncertain information in many problems andespecially in games. Then we define and analyze the properties of internal neutrosophic cubicsoft expert sets (INCSESs) and external neutrosophic cubic soft expert sets (ENCSESs), P-order,P-union, P-intersection, P-AND, P-OR and R-order, R-union, R-intersection, R-AND, and R-OR ofNCSESs. The NCSESs satisfy the laws of commutativity, associativity, De Morgan, distributivity,idempotentency, and absorption. We derive some conditions for P-union and P-intersection of twoINCSESs to be an INCSES. It is shown that P-union and P-intersection of ENCSESs need not be anENCSES. The R-union and R-intersection of the INCSESs (resp., ENCSESs) need not be an INCSES(resp. ENCSES). Necessary conditions for the P-union, R-union and R-intersection of two ENCSESsto be an ENCSES are obtained. We also study the conditions for R-intersection and P-intersectionof two NCSESs to be an INCSES and ENCSES. Finally, for its applications in games, we use thedeveloped procedure to analyze the cricket series between Pakistan and India. It is shown that theproposed method is suitable to be used for decision-making, and as good as or better when comparedto existing models.

An Approach toward a Q-Neutrosophic Soft Set and Its Application in Decision Making

A neutrosophic set was proposed as an approach to study neutral uncertain information. It is characterized through three memberships, T , I and F, such that these independent functions stand for the truth, indeterminate, and false-membership degrees of an object. The neutrosophic set presents a symmetric form since truth enrolment T is symmetric to its opposite false enrolment F with respect to indeterminacy enrolment I that acts as an axis of symmetry. The neutrosophic set was further extended to a Q-neutrosophic soft set, which is a hybrid model that keeps the features of the neutrosophic soft set in dealing with uncertainty, and the features of a Q-fuzzy soft set that handles two-dimensional information. In this study, we discuss some operations of Q-neutrosophic soft sets, such as subset, equality, complement, intersection, union, AND operation, and OR operation. We also define the necessity and possibility operations of a Q-neutrosophic soft set. Several properties and illustrative examples are discussed. Then, we define the Q-neutrosophic-set aggregation operator and use it to develop an algorithm for using a Q-neutrosophic soft set in decision-making issues that have indeterminate and uncertain data, followed by an illustrative real-life example.

Generalized Q-Neutrosophic Soft Expert Set for Decision under Uncertainty

Neutrosophic triplet structure yields a symmetric property of truth membership on the left, indeterminacy membership in the centre and false membership on the right, as do points of object, centre and image of reflection. As an extension of a neutrosophic set, the Q-neutrosophic set was introduced to handle two-dimensional uncertain and inconsistent situations. We extend the soft expert set to generalized Q-neutrosophic soft expert set by incorporating the idea of soft expert set to the concept of Q-neutrosophic set and attaching the parameter of fuzzy set while defining a Q-neutrosophic soft expert set. This pattern carries the benefits of Q-neutrosophic sets and soft sets, enabling decision makers to recognize the views of specialists with no requirement for extra lumbering tasks, thus making it exceedingly reasonable for use in decision-making issues that include imprecise, indeterminate and inconsistent two-dimensional data. Some essential operations namely subset, equal, complement, union, intersection, AND and OR operations and additionally several properties relating to the notion of generalized Q-neutrosophic soft expert set are characterized. Finally, an algorithm on generalized Q-neutrosophic soft expert set is proposed and applied to a real-life example to show the efficiency of this notion in handling such problems.

Neutrosophic Triplet Non-Associative Semihypergroups with Application

In this paper, we extended the idea of a neutrosophic triplet set to non-associative semihypergroups and define neutrosophic triplet LA-semihypergroup. We discuss some basic results and properties. At the end, we provide an application of the proposed structure in Football.

Generalized Neutrosophic Soft Expert Set for Multiple-Criteria Decision-Making

Smarandache defined a neutrosophic set to handle problems involving incompleteness, indeterminacy, and awareness of inconsistency knowledge, and have further developed it neutrosophic soft expert sets. In this paper, this concept is further expanded to generalized neutrosophic soft expert set (GNSES). We then define its basic operations of complement, union, intersection, AND, OR, and study some related properties, with supporting proofs. Subsequently, we define a GNSES-aggregation operator to construct an algorithm for a GNSES decision-making method, which allows for a more efficient decision process. Finally, we apply the algorithm to a decision-making problem, to illustrate the effectiveness and practicality of the proposed concept. A comparative analysis with existing methods is done and the result affirms the flexibility and precision of our proposed method.