[Paper Review] Neutrosophic soft sets and neutrosophic soft matrices based on decision making

İrfan Deli̇, Said Broumi|arXiv (Cornell University)|Apr 2, 2014

Fuzzy and Soft Set TheoryDecision Sciences59 references134 citations

TL;DR

This paper introduces neutrosophic soft matrices (NSM) as a computational framework for handling indeterminate and inconsistent information in decision-making. It proposes a novel NSM-decision making method using AND-product operations on neutrosophic soft matrices, enabling efficient group decision-making under uncertainty, with a case study showing u1 as the optimal car choice based on combined preferences.

ABSTRACT

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.

Motivation & Objective

• To redefine neutrosophic soft set operations (complement, union, intersection) with improved consistency compared to Maji's original definitions.
• To develop neutrosophic soft matrices (NSM) as a practical tool for storing and computing neutrosophic soft sets in computer systems.
• To propose a new decision-making methodology, NSM-decision making, based on matrix operations for solving group decision-making problems involving indeterminate and inconsistent information.
• To demonstrate the applicability of NSM through a real-world case study on car selection under conflicting partner preferences.

Proposed method

• Redefined neutrosophic soft set operations (complement, union, intersection) using truth, indeterminacy, and falsity membership functions.
• Introduced neutrosophic soft matrices (NSM) as m×n arrays of triplets (T, I, F) representing neutrosophic values for efficient computer storage and computation.
• Defined two special matrix products: AND-product and OR-product, using t-norms (min) and s-norms (max) on T, I, F components.
• Proposed the NS-max-min decision function to aggregate results across parameters, computing max-truth, min-indeterminacy, and min-falsity per object.
• Applied the NSM-decision making algorithm to a car selection case study, using the decision function to rank alternatives based on a score (μ−ν·ω).
• Defined an optimum fuzzy set on the universe by selecting the object with the highest score from the decision matrix.

Experimental results

Research questions

• RQ1 How can neutrosophic soft set operations be redefined to improve consistency and functionality compared to Maji's original definitions?
• RQ2 What is the most effective way to represent and compute neutrosophic soft sets in a computer system for real-world applications?
• RQ3 Can neutrosophic soft matrices be used to model and solve group decision-making problems with conflicting or indeterminate preferences?
• RQ4 What matrix operations (e.g., AND-product, OR-product) are most suitable for aggregating neutrosophic soft information in decision-making?
• RQ5 How can a decision function be constructed from neutrosophic soft matrices to identify the optimal alternative in uncertain environments?

Key findings

• The proposed NSM-decision making method successfully identified u1 as the optimal car choice in the case study, with a score of 0.95, outperforming u2 (0.93) and u3 (0.92).
• The NSM framework enables efficient storage and computation of neutrosophic soft sets using matrix structures, making theoretical and practical applications more feasible.
• The AND-product of two neutrosophic soft matrices was computed using t-norms (min) for truth and s-norms (max) for indeterminacy and falsity, preserving neutrosophic logic.
• The NS-max-min decision function effectively aggregated multi-parameter information, producing a one-column decision matrix that ranks alternatives.
• The method demonstrated robustness in handling indeterminate and inconsistent data, as seen in the high indeterminacy values (e.g., I=0.8) in the input matrices.
• The case study confirmed that NSM-decision making can handle real-world group decision-making problems with conflicting preferences, such as in car selection.