Source-linked AI summary
Bipolar Fuzzy Soft sets and its applications in decision making problem
Muhammad Aslam, Saleem Abdullah, Kifayat ullah
TL;DR
The paper addresses uncertainty-handling limitations associated with existing mathematical tools and their parametrization. It combines bipolar fuzzy sets with soft sets, develops their properties and operations, and applies the framework to a car-selection decision problem. The resulting approach models the decision problem and selects an optimal object using an algorithm based on available information.
Problem
Existing mathematical tools for uncertainty have inherent limitations, including inadequate parametrization, motivating a combined bipolar fuzzy-set and soft-set framework.
Method
The paper defines bipolar fuzzy soft sets, studies their fundamental properties and operations, and uses comparison tables and an algorithm for decision making.
Results
The paper develops bipolar fuzzy soft-set operations, including union, intersection, and De Morgan laws, and applies them to model a car-choice problem.
Takeaways & Limitations
The framework selects car c3 as the decision-maker’s choice, with c2 or c4 as equally scoring second choices if c3 is unavailable.
Abstract
from arXiv · showhide
In this article, we combine the concept of a bipolar fuzzy set and a soft set. We introduce the notion of bipolar fuzzy soft set and study fundamental properties. We study basic operations on bipolar fuzzy soft set. We define exdended union, intersection of two bipolar fuzzy soft set. We also give an application of bipolar fuzzy soft set into decision making problem. We give a general algorithm to solve decision making problems by using bipolar fuzzy soft set.
1. Introduction
The introduction frames bipolar fuzzy soft sets as a response to uncertainty and parametrization limitations in existing mathematical tools. The paper combines bipolar fuzzy sets with soft sets and applies the resulting concept to decision making.
- Existing tools such as fuzzy sets, rough sets, interval mathematics, and probability theory address uncertainty but have inherent limitations, including inadequate parametrization.
- Soft set theory was introduced as a mathematical tool for describing uncertainties and has supported extensive theoretical development and applications.
- Bipolar-valued fuzzy sets extend membership degrees from [0, 1] to [−1, 1], allowing bipolar information representation.
- The paper combines bipolar fuzzy sets with soft sets, studies their properties and operations, and develops a decision-making application and general algorithm.
2. Preliminaries
The preliminaries review soft sets, fuzzy soft sets, and bipolar fuzzy sets as the conceptual foundations for the paper’s later construction. They specify parameterized mappings and standard set operations used in this framework.
- The preliminaries introduce bipolar fuzzy sets, soft sets, and fuzzy soft sets before developing bipolar fuzzy soft sets.
- A soft set maps a parameter subset A of E to the power set P(U) of a universe U.
- Soft-set union uses the combined parameter set A ∪ B, while restricted intersection uses A ∩ B and intersects corresponding values.
- A fuzzy soft set maps parameters to the collection of all fuzzy subsets of the universe.
- Fuzzy soft-set union is defined for two fuzzy soft sets over a common universe using a combined parameter set.
3. Bipolar Fuzzy Soft Sets.
This section defines bipolar fuzzy soft sets as parameterized families of bipolar fuzzy subsets and introduces their basic special cases and complement operation. Examples instantiate the definitions with cars and bikes.
- A bipolar fuzzy soft set over U assigns each parameter in A ⊂ E a bipolar fuzzy subset of U.
- Examples apply the definitions to parameterized collections of cars and bikes.
- The bipolar fuzzy soft class contains all bipolar fuzzy soft sets on U with attributes drawn from E.
- A null bipolar fuzzy soft set assigns the empty set to every parameter, whereas an absolute one assigns the full collection BF U.
- The complement of a bipolar fuzzy soft set is defined by transforming its positive membership component using 1 − µ+.
4. Bipolar Fuzzy Soft Subsets
The paper defines bipolar fuzzy soft subsets and equality through parameter inclusion and bipolar fuzzy subset relations. These notions formalize comparison between parameterized bipolar fuzzy information structures.
- A bipolar fuzzy soft subset requires A ⊆ B and F(e) to be a bipolar fuzzy subset of G(e) for every e ∈ A.
- The subset relation means every element of (F, A) is presented in (G, B) independently of membership or non-membership.
- The section illustrates subset comparison using parameterized sets of men and conditions F(e) ≤ G(e).
- A bipolar fuzzy soft equality holds when each set is a bipolar fuzzy soft subset of the other.
5. Operations on Bipolar Fuzzy Soft Sets
The paper defines restricted and extended operations for bipolar fuzzy soft sets and establishes their principal algebraic properties. These include identity, absorption, commutative, associative, and distributive laws, along with operations on families of sets.
- Basic operations: Intersection uses the common parameter set C = A ∩ B and assigns H(e) = F(e) ∩ G(e) for each e ∈ C.
- Basic operations: Union uses C = A ∪ B, while extended union and restricted union provide related constructions for bipolar fuzzy soft sets.
- Fundamental properties: The operations satisfy identity and null-set laws, including (F, A) ¯∪ ∅ = (F, A) and (F, A) ¯∩ ∅ = ∅.
- Fundamental properties: Absorption holds in both forms: (F, A) ¯∪ ((F, A) ¯∩ (G, B)) = (F, A) and (F, A) ¯∩ ((F, A) ¯∪ (G, B)) = (F, A).
- Algebraic laws: Intersection and union are commutative and associative for bipolar fuzzy soft sets.
- Algebraic laws: The operations also obey both distributive laws, including intersection over union and union over intersection.
6. De Morgan’s Law of Bipolar Fuzzy Soft Sets
This section states De Morgan’s laws for bipolar fuzzy soft sets, relating complements of unions and intersections to corresponding operations on complements.
- Theorem 4 states De Morgan’s laws for bipolar fuzzy soft sets.
- The complement of a union equals the restricted intersection of the complements.
- The complement of an extended intersection equals the extended union of the complements.
- The proof evaluates parameter cases such as e ∈ A\B and e ∈ A ∩B before applying complements.
7. OR and AND Operations on Bipolar Fuzzy Soft Sets
This section defines OR and AND operations for bipolar fuzzy soft sets, including Cartesian-parameter operations, family-level operations, and idempotence properties.
- AND operation: The AND operation forms a bipolar fuzzy soft set on A × B using H(a,b) = F(a) ∩G(b).
- OR operation: The OR operation forms a bipolar fuzzy soft set on A × B using H(a,b) = F(a) ∪G(b).
- Family operations: For a family of bipolar fuzzy soft sets, intersection and union use Cartesian parameter sets with corresponding ˜∧ and ˜∨ operations.
- Idempotence: Both AND and OR are idempotent: applying either operation to the same bipolar fuzzy soft set returns that set.
- Examples: Examples instantiate these operations using parameterized universes of cars, men, and houses.
8. An Application of Bipolar Fuzzy Soft Sets in Decision Making
The paper applies bipolar fuzzy soft sets to a car-selection problem and provides a scoring algorithm that combines positive and negative information to identify the optimal choice.
- Problem formulation: The application models Mr. X’s car-selection problem using a bipolar fuzzy soft set over chosen parameters.The considered universe contains four cars, while the parameter set includes cost, beauty, fuel efficiency, modern technology, and luxury.
- Decision algorithm: The decision procedure represents the bipolar fuzzy soft set in tabular form and computes comparison tables for positive and negative information.The comparison table records, for each object pair, the number of parameters where one object’s value is at least the other’s.
- Decision algorithm: The algorithm calculates positive and negative information scores, then obtains a final score by subtracting the positive score from the negative score.The object with the maximum final score is selected as the recommended choice.
- Decision outcome: The car c3 receives the maximum score of 4 and is selected for purchase.If c3 is unavailable, c2 or c4 are identified as tied second choices.
9. Conclusion
The conclusion presents bipolar fuzzy soft sets as a framework combining bipolar fuzzy sets with soft sets, extending their operations and applying them to decision making.
- The paper combines bipolar fuzzy sets and soft sets to introduce bipolar fuzzy soft sets.
- It studies fundamental properties and basic operations, including extended union and intersection.
- It applies bipolar fuzzy soft sets to decision-making problems and gives a general solution algorithm.