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Intuitionistic Neutrosophic Soft Set
Broumi Said, Florentin Smarandache
TL;DR
The paper addresses how to represent parameterized information involving imprecision through intuitionistic neutrosophic soft sets. It defines this soft-set structure and its operations, establishes related properties, and demonstrates a decision-making application in which blouse b4 is selected as best.
Problem
The paper studies how intuitionistic neutrosophic sets can represent parameterized information involving imprecision in soft-set settings.
Method
The authors define intuitionistic neutrosophic soft sets, represent them as parameterized families of intuitionistic neutrosophic sets, and introduce operations on them.
Results
The paper establishes propositions for intuitionistic neutrosophic soft-set operations and applies the framework to decision making, selecting blouse b4 with score 19.
Takeaways & Limitations
Intuitionistic neutrosophic soft sets provide the paper’s framework for describing approximations and supporting a decision-making choice.
Abstract
from arXiv · showhide
In this paper we study the concept of intuitionistic neutrosophic set of Bhowmik and Pal. We have introduced this concept in soft sets and defined intuitionistic neutrosophic soft set. Some definitions and operations have been introduced on intuitionistic neutrosophic soft set. Some properties of this concept have been established.
1. Introduction
The introduction frames imprecise, indeterminate, and inconsistent real-world data as a motivation for neutrosophic approaches. It also previews the paper’s preliminaries, intuitionistic neutrosophic soft sets, decision-making application, and conclusion.
- Real-world data in engineering, social, economic, computer science, and medical problems are often uncertain, imprecise, non-crisp, and non-deterministic.
- Fuzzy and intuitionistic fuzzy theories address vagueness but may be unsatisfactory for indeterminate and inconsistent information.
- The paper presents preliminaries, develops intuitionistic neutrosophic soft sets, applies them to decision making, and concludes with a final section.
2. Preliminaries
This section establishes the universal-set, parameter, neutrosophic-set, and intuitionistic-neutrosophic-set foundations used later in the paper.
- The framework uses a universe U and a parameter set E, where parameters describe attributes, characteristics, or properties of objects in U.
- A neutrosophic set assigns each element truth-membership, indeterminacy-membership, and non-membership degrees through functions T, I, and F.
- For technical applications, the membership values are restricted to [0,1], while the neutrosophic sum satisfies 0 ≤ T_A(x) + I_A(x) + F_A(x) ≤ 3.
- An intuitionistic neutrosophic set requires pairwise minima among truth, falsity, and indeterminacy memberships to be at most 0.5.
3. Intuitionistic Neutrosophic Soft Set
The paper introduces intuitionistic neutrosophic soft sets as a hybrid of intuitionistic neutrosophic sets and soft-set theory, then defines their basic relations, complements, operations, and algebraic properties.
- Definition: An intuitionistic neutrosophic soft set is a parameterized family of intuitionistic neutrosophic subsets assigned to elements of a parameter set.The mapping associates each parameter with an intuitionistic neutrosophic set over the universe, illustrated using blouse qualities.
- Basic relations: The paper defines containment and equality for intuitionistic neutrosophic soft sets through parameter-set inclusion and componentwise membership inequalities.Truth and indeterminacy memberships are ordered increasingly, while falsity membership is ordered decreasingly.
- Complement: Complementation maps an intuitionistic neutrosophic soft set to the complement parameter set and assigns the corresponding intuitionistic neutrosophic soft complement.The construction is illustrated with complements of blouse attributes such as bright, cheap, costly, and colorful.
- Operations: The paper defines union, intersection, AND, and OR operations using parameter combinations and componentwise maximum or minimum rules for truth, indeterminacy, and falsity memberships.Intersection uses shared parameters, whereas AND and OR use parameter pairs from A × B.
- Properties: The defined operations satisfy idempotency, commutativity, identity, associativity, distributivity, and complement-related laws for intuitionistic neutrosophic soft sets.The propositions include double complementation and De Morgan-style identities for AND and OR.
4. An application of intuitionistic neutrosophic soft set in a decision making problem
The paper applies intuitionistic neutrosophic soft sets to selecting the most suitable blouse from imprecise, parameterized performance data. It converts selected criteria into a comparison matrix, scores each blouse, and chooses the maximum-scoring object.
- The method represents each blouse's criterion values in a criteria matrix, where greater values indicate greater preferability.
- Comparison matrix: The comparison matrix records, for each object pair and parameter, how often membership, indeterminacy, and non-membership values support one object.
- The algorithm selects choice parameters, tabulates the corresponding INSS, computes pairwise comparisons and scores, then chooses an object with maximum score.
- Score 19 identifies blouse b4 as the best decision, followed by b5.
5. Conclusions
The paper introduces intuitionistic neutrosophic soft sets by incorporating intuitionistic neutrosophic sets into soft sets. It defines operations and proves propositions, then demonstrates an application to decision making.
- The paper introduces intuitionistic neutrosophic soft sets, defines operations on them, proves propositions, and applies them to a decision-making problem.