Source-linked AI summary
Bipolar Neutrosophic Sets And Their Application Based On Multi-Criteria Decision Making Problems
Irfan Deli, Mumtaz Ali, Florentin Smarandache
TL;DR
The paper addresses multi-criteria decision making with bipolar neutrosophic information. It defines bipolar neutrosophic sets and comparison functions, develops weighted aggregation operators, and applies them to rank alternatives; a numerical example selects A3 as most desirable.
Problem
The paper addresses representing alternative evaluations and making multi-criteria decisions when the information is bipolar neutrosophic.
Method
The paper defines bipolar neutrosophic sets, score, certainty, and accuracy functions, then uses weighted average or geometric operators to aggregate and rank alternatives.
Results
A numerical example demonstrates the method's application and effectiveness, with A3 identified as the most desirable alternative.
Takeaways & Limitations
The developed approach provides a bipolar neutrosophic multiple-criteria decision-making procedure for selecting desirable alternatives.
Takeaways & Limitations
The decision-making method assumes nonnegative attribute weights that sum to one, and evaluations are represented as bipolar neutrosophic numbers.
Abstract
from arXiv · showhide
In this paper, we introduce concept of bipolar neutrosophic set and its some operations. Also, we propose score, certainty and accuracy functions to compare the bipolar neutrosophic sets. Then, we develop the bipolar neutrosophic weighted average operator and bipolar neutrosophic weighted geometric operatör to aggregate the bipolar neutrosophic information. Furthermore, based on the neutrosophic weighted geometric(aritmetic) operatör and the score, certainty and accuracy functions, we develop a bipolar neutrosophic multiple criteria decision-making approach, in which the evaluation values of alternatives on the attributes take the form of bipolar neutrosophic numbers to select the most desirable one(s). Finally, a numerical example of the method was given to demonstrate the application and effectiveness of the developed method.
1. Introduction
The paper situates bipolar neutrosophic sets within extensions of fuzzy, intuitionistic fuzzy, neutrosophic, and bipolar fuzzy models, then develops operators and a decision-making method.
- Neutrosophic sets generalize fuzzy and intuitionistic fuzzy sets for representing imprecision and uncertainty.
- Bipolarity represents reasoning through positive information about desirable or acceptable options and negative information about rejected or forbidden options.
- Bipolar fuzzy sets extend fuzzy sets by representing bipolar preferences and satisfaction-related information.
- The paper introduces bipolar neutrosophic sets as an extension of fuzzy, bipolar fuzzy, intuitionistic fuzzy, and neutrosophic sets.It also proposes comparison functions, aggregation operators, and a bipolar neutrosophic multiple-criteria decision-making approach.
2. Preliminaries
The preliminaries review neutrosophic and bipolar-valued fuzzy concepts, their membership components, basic operations, and comparison functions for single-valued neutrosophic numbers.
- A neutrosophic set uses truth-membership, indeterminacy-membership, and falsity-membership functions.The single-valued form restricts these functions to [0,1], while their sum is not required to equal one.
- Single-valued neutrosophic numbers support defined operations and comparison through score, accuracy, and certainty functions.
- The comparison procedure ranks one number above another first by score, then by accuracy, and finally by certainty when earlier values are equal.
- Bipolar-valued fuzzy sets associate positive membership with satisfaction of a property and negative membership with satisfaction of an implicit counter-property.
3. Bipolar Neutrosophic Set
This section defines bipolar neutrosophic sets and numbers, extends set operations and comparison functions, and introduces weighted average and geometric aggregation operators.
- A bipolar neutrosophic set combines positive and negative truth, indeterminacy, and falsity memberships over a universe X.The positive components describe membership information, while the negative components describe an implicit counter-property.
- Bipolar neutrosophic sets are generalized from bipolar fuzzy sets by retaining additional indeterminacy and falsity components.
- The section defines equality, inclusion, union, intersection, and complement operations for bipolar neutrosophic sets.
- Bipolar neutrosophic numbers are compared using score, accuracy, and certainty functions.The comparison method uses these functions successively to determine superiority or equality.
- The bipolar neutrosophic weighted average and weighted geometric operators aggregate families of bipolar neutrosophic numbers using normalized weights.Each weight lies in [0,1], and the weights sum to one.
- Both aggregation operators satisfy idempotency and boundedness properties, and their aggregation results remain bipolar neutrosophic numbers.
4. NBN- Decision Making Method
The paper develops a bipolar neutrosophic decision-making approach that aggregates attribute evaluations and ranks alternatives using weighted operators and comparison functions.
- The approach uses the A_w or G_w operator together with a ranking method for multiple-criteria decisions under bipolar neutrosophic information.
- Alternatives are evaluated across weighted criteria, with nonnegative attribute weights summing to one.
- Each evaluation is represented as a bipolar neutrosophic number with positive and negative truth, indeterminacy, and falsity components.
- The component values satisfy bounds in [0,1], along with a stated combined constraint for each alternative and criterion.
Algorithm
The algorithm constructs a decision matrix, aggregates each alternative’s criterion evaluations, computes scores, and ranks the alternatives. In the numerical example, A3 is selected as most desirable.
- Algorithm: Step 1 constructs the decision matrix supplied by the decision maker.
- Algorithm: The algorithm calculates score values for each alternative’s collective overall bipolar neutrosophic number.
- Algorithm: The example evaluates four cars using fuel economy, aerodynamics, comfort, and safety as attributes.
- Algorithm: A3 is the most desirable alternative after ranking the alternatives by their score values.
5. Conclusions
The paper presents bipolar neutrosophic sets, comparison functions, aggregation operators, and a multiple-criteria decision-making approach. A numerical example demonstrates the method’s application and effectiveness.
- The paper introduces bipolar neutrosophic sets, operations, and score, certainty, and accuracy functions.
- The A_w and G_w operators aggregate bipolar neutrosophic information for each alternative.
- Alternatives are ranked using score, certainty, and accuracy values to select the most desirable alternative or alternatives.
- A numerical example demonstrates the application and effectiveness of the developed method.