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Neutrosophic soft sets and neutrosophic soft matrices based on decision making
Irfan Deli, Said Broumi
TL;DR
Neutrosophic soft sets address indeterminate and inconsistent information, while prior soft-matrix approaches motivate more functional matrix representations. The paper redefines neutrosophic soft-set operations, introduces neutrosophic soft matrices and operators, and develops an NSM decision-making methodology for group decision problems.
Problem
Neutrosophic soft-set theory requires revised operations and more functional matrix representations for theoretical study and decision-making applications.
Method
The paper redefines neutrosophic soft sets and operations, introduces neutrosophic soft matrices with products and properties, and bases a decision method on matrix operations.
Results
The paper develops an NSM decision-making methodology and applies it to a neutrosophic soft-matrix decision problem.
Takeaways & Limitations
Neutrosophic soft matrices provide a representation for storing neutrosophic soft sets in computer memory and supporting decision-making methods.
Abstract
from arXiv · showhide
Maji\cite{maj-13}, firstly proposed neutrosophic soft sets can handle the indeterminate information and inconsistent information which exists commonly in belief systems. In this paper, we have firstly redefined complement, union and compared our definitions of neutrosophic soft with the definitions given by Maji. Then, we have introduced the concept of neutrosophic soft matrix and their operators which are more functional to make theoretical studies in the neutrosophic soft set theory. The matrix is useful for storing an neutrosophic soft set in computer memory which are very useful and applicable. Finally, based on some of these matrix operations a efficient methodology named as NSM-decision making has been developed to solve neutrosophic soft set based group decision making problems.
1. Introduction
The introduction situates neutrosophic soft sets within uncertainty theories and identifies parameterization difficulties as a motivation for matrix-based decision making. The paper redefines neutrosophic soft-set operations, develops neutrosophic matrices, and outlines an associated decision method.
- Motivation: Existing uncertainty theories address uncertainty, imprecision, vagueness, and indeterminacy, but their parameterization tools have inherent difficulties.The introduction lists probability, fuzzy, intuitionistic fuzzy, vague, rough, neutrosophic, and interval neutrosophic theories.
- Related work: Prior work developed fuzzy and intuitionistic fuzzy soft matrices, including matrix products and decision-making applications.Examples include applications in real-life scenarios and medical diagnosis.
- Contributions: The paper redefines neutrosophic soft sets and compares its complement, union, and related definitions with Maji’s definitions.
- Contributions: It introduces neutrosophic matrices, their basic properties, special products, and a soft decision-making method based on an and-product.These topics are presented across Sections 4–6.
2. Preliminary
The preliminary section introduces neutrosophic, soft, neutrosophic soft, and soft-matrix concepts used later. It also defines matrix products and max-min decision functions.
- Neutrosophic sets: A neutrosophic set assigns each object truth-membership, indeterminacy-membership, and falsity-membership values represented as subsets of [0, 1].The three functions may use real standard or nonstandard subsets, and their supremum sum is bounded between 0 and 3.
- Soft sets: A soft set is a parameterized family of subsets of the universe, with an approximate function assigning each parameter a subset and assigning the empty set outside the parameter domain.
- Soft matrices: A soft set over finite universes can be represented as an m × n s-matrix whose entries are characteristic-function values.Rows correspond to universe elements and columns to parameters.
- Matrix operations and decision functions: The And-product uses minimum aggregation and the Or-product uses maximum aggregation, after which a max-min decision function produces a one-column decision matrix.The resulting matrix is used to obtain an optimum fuzzy set on the universe.
- Neutrosophic soft sets: A neutrosophic soft set is a parameterized family of neutrosophic sets represented by an approximate function over a parameter subset.Its approximations may be empty or have nonempty intersections.
3. Neutrosophic soft set and some operations redefined
The paper redefines neutrosophic soft sets and their complement, union, and intersection operations, then compares these definitions with Maji’s formulations. It also establishes basic properties using t-norm and s-norm operations.
- Redefined neutrosophic soft sets: The paper modifies the definition of neutrosophic soft sets and presents the approximate function as a parametrized family of neutrosophic sets.Each parameter’s approximation may be empty, nonempty, or overlap with others.
- Operations: The complement of a neutrosophic soft set exchanges its truth-membership and falsity-membership values while retaining indeterminacy.
- Operations: The paper defines union and intersection of neutrosophic soft sets through componentwise truth, indeterminacy, and falsity memberships.The displayed formulas use t-norm and s-norm functions for the respective membership components.
- Examples and properties: Using min as the t-norm and max as the s-norm, the paper illustrates these operations on two neutrosophic soft sets.
- Examples and properties: The paper states algebraic propositions for neutrosophic soft sets, with proofs relying on the commutativity and associativity of t-norm and s-norm functions.
- Comparison with prior definitions: The authors compare their neutrosophic soft-set, complement, union, and intersection definitions with Maji’s definitions in tabulated comparisons.
4. Neutrosophic Soft Matrices
This section represents neutrosophic soft sets as neutrosophic soft matrices and defines their principal forms and operations. The matrix representation is presented as useful for storing neutrosophic soft sets in computer memory.
- Matrix representation: Neutrosophic soft matrices represent neutrosophic soft sets and provide a computer-memory representation described as useful and applicable.
- Matrix representation: The matrix construction uses rows for elements of the universe and columns for parameters, with entries formed from neutrosophic membership information.The section defines the relation form and characteristic function that organize these truth, indeterminacy, and falsity values.
- Matrix representation: A neutrosophic soft set is uniquely characterized by an m × n neutrosophic soft matrix, allowing the two concepts to be used interchangeably.
- Matrix operations: It also introduces zero and universal matrices, submatrix and proper-submatrix relations, matrix equality, disjointness, and matrix complement, union, and intersection.
- Matrix operations: The section states that De Morgan’s laws hold for neutrosophic soft matrices.
5. Products of NS-Matrices
The paper defines two special products for neutrosophic soft matrices and states that their De Morgan-type results hold.
- Definitions: Two special products, And-product and Or-product, are defined for neutrosophic matrices to support soft decision-making methods.The products map pairs of neutrosophic matrices into matrices with n^2 columns.
- Properties: The paper states that the De Morgan’s types of results are true for the defined matrix operations.
6. Decision making problem using and-product of neutrosophic soft matrices
The decision-making method combines neutrosophic matrices through an And-product, applies max-min-max aggregation, and selects the universe element with the maximum resulting value. In the car-selection example, the method chooses u1.
- Decision function: The NS-max-min decision function groups product-matrix entries by parameter blocks and aggregates truth, indeterminacy, and falsity components.For nonempty blocks, the component rules use maxima or minima over the indexed entries; empty blocks receive zero.
- Algorithm: The algorithm selects feasible parameters, constructs neutrosophic matrices, chooses a matrix product, applies min-max-max decision processing, and derives an optimum fuzzy set.
- Decision variants: The method can also define max-min-min decision-making variants, with usefulness depending on the problem type.
- Car-selection example: In the car example, two decision-makers choose costly and fuel-efficiency parameters, construct their matrices, and combine them using the And-product.
- Car-selection result: u1 has the maximum value, so the couple may decide to buy car u1.
7. Conclusion
The conclusion presents the paper as redefining neutrosophic sets, introducing neutrosophic soft matrices, and developing matrix types, operations, and properties.
- Contributions: The paper redefines the notion of neutrosophic set and proposes neutrosophic soft matrices.
- Contributions: It introduces different matrix types, operations, and properties within neutrosophic soft theory.