Source-linked AI summary
Intuitionistic fuzzy parametrized soft set theory and its decision making
İrfan Deli, Naim Çağman
TL;DR
Many uncertain-data problems may not be successfully modeled by classical mathematics and other tools. The paper defines intuitionistic FP-soft sets and an adjustable decision-making method; in its example, candidate u2 has the highest membership degree and is selected for the position.
Problem
Many fields deal with uncertain data that may not be successfully modeled by classical mathematics and other mathematical tools.
Source: §1 · p. 1Method
The paper defines intuitionistic FP-soft sets and their operations, then constructs an adjustable decision-making method that produces a decision fuzzy set from crisp alternatives.
Source: §3 · p. 3, §4 · pp. 8–9Results
0.2720 was the maximum membership degree, so the example selected candidate u2 for the position.
Source: §4 · p. 11Takeaways & Limitations
The authors conclude that the method can be applied to uncertainty problems in fields such as computer science and game theory.
Source: §5 · p. 11
Abstract
from arXiv · showhide
In this work, we first define intuitionistic fuzzy parametrized soft sets (intuitionistic FP-soft sets) and study some of their properties. We then introduce an adjustable approaches to intuitionistic FP-soft sets based decision making. We also give an example which shows that they can be successfully applied to problems that contain uncertainties.
1. Introduction
Soft set theory was proposed as an approach to modeling vagueness and uncertainty that classical mathematics and other tools may not successfully model. Building on prior soft-set work and embeddings of fuzzy, rough, and intuitionistic fuzzy set ideas, this paper defines intuitionistic FP-soft sets and presents a decision-making method.
1. Introduction
- Soft set theory was proposed as a more universal approach to modeling vagueness and uncertainty that classical mathematics and other tools may not successfully model.The introduction names probability theory, fuzzy sets [33], and rough sets [30] among those tools.Source: §1 · p. 1
- Soft-set operations, properties, applications, and algebraic structures have been studied, and applications have expanded by embedding fuzzy, rough, and intuitionistic fuzzy set ideas.The cited examples include fuzzy sets [4, 6, 10, 14, 20, 22, 25, 27], rough sets [4, 15, 16], and intuitionistic fuzzy sets [23, 24, 29].Source: §1 · pp. 1–2
- Building on FP-soft sets and their decision-making method, the paper defines intuitionistic FP-soft sets, studies their properties, and introduces an adjustable decision-making approach.The authors state that the method can be applied to problems containing uncertainties.Source: §1 · p. 2
2. Preliminary
This section reviews soft, fuzzy, intuitionistic fuzzy, and FP-soft set definitions and operations used in later discussion. The definitions distinguish parameter-indexed object sets from fuzzy parameter grades and intuitionistic membership and non-membership grades.
2. Preliminary
- A soft set FA over universe U uses a parameter subset A, and its approximate-function values may be arbitrary sets.Each value fA(x) is called the x-element of the soft set.Source: §2 · p. 2
- A fuzzy set X over U has a membership function whose value gives each element's degree of membership.The value μX(u) is called the grade of membership of u.Source: §2 · p. 2
- An intuitionistic fuzzy set assigns each element membership and non-membership degrees whose sum is between 0 and 1.The membership and non-membership functions take values in [0, 1].Source: §2 · pp. 2–3
- Intuitionistic fuzzy set subset and equality compare both degrees, while union, intersection, addition, and product use paired operations on them.Subset and equality are defined by inequalities or equalities for both grades; the four operations combine the membership and non-membership grades using max, min, sums, or products.Source: §2 · p. 3
- An FP-soft set pairs fuzzy parameter membership grades with approximate-function values, and a zero grade requires an empty value.The approximate function maps parameters to subsets of U, and the membership function maps parameters to grades.Source: §2 · p. 3
3. Intuitionistic FP-soft sets
The section defines intuitionistic FP-soft sets by combining intuitionistic fuzzy parameter grades with approximate object sets, then specifies their basic operations and properties.
3. Intuitionistic FP-soft sets
- An intuitionistic FP-soft set associates parameters with membership and non-membership grades and an approximate set of objects.The grades encode each parameter’s degree of importance and unimportance; when they are 0 and 1, respectively, its approximate set is empty.Source: §3 · p. 3
- A set is empty when every parameter has grades 0 and 1, and universal when every parameter has grades 1 and 0 and approximate set U.Source: §3 · p. 4
- Intuitionistic FP-soft subsethood requires ordered membership and non-membership grades and inclusion of the corresponding approximate sets.The paper cautions that subsethood does not imply that every element of one intuitionistic FP-soft set is an element of the other.Source: §3 · pp. 4–5
- Two intuitionistic FP-soft sets are equal exactly when their membership grades, non-membership grades, and approximate sets agree for every parameter.Source: §3 · p. 5
- The paper defines union, intersection, OR/AND-sums, and OR/AND-products for intuitionistic FP-soft sets.It also states De Morgan-type results for these operations.Source: §3 · pp. 6–8
4. Intuitionistic FP-soft decision making method
The method makes decisions by reducing an intuitionistic FP-soft set first to an intuitionistic fuzzy set and then to a fuzzy set. In the hiring example, it selects the candidate with the highest membership degree.
Decision method
- The method reduces an intuitionistic FP-soft set to an intuitionistic fuzzy set, then reduces that set to a fuzzy set.These two reductions are presented as an adjustable approach to decision making based on intuitionistic FP-soft sets.Source: §4 · pp. 8–9
- The decision maker constructs a feasible intuitionistic fuzzy subset over parameters and an intuitionistic FP-soft set over the alternatives.The algorithm then computes the two reduced sets and chooses the alternative with maximum membership degree.Source: §4 · pp. 9–10
Example 2
- In the hiring example, candidate u2 is selected with the maximum degree, 0.2720.The example considers five applicants against four parameters: experience, computer knowledge, training, and young age.Source: §4 · pp. 10–11
Decision method
- The paper states that the method can also be applied to group decision making using Definitions 13–16.Source: §4 · p. 11
5. Conclusion
The conclusion recaps the definition of intuitionistic FP-soft sets and the decision method, then reports an example demonstrating that the method works.
Conclusion
- The paper defines intuitionistic FP-soft sets and their operations.Source: §5 · p. 11
- It presents a decision-making method based on intuitionistic FP-soft set theory.Source: §5 · p. 11
- An example demonstrates that the decision-making method can work successfully for uncertainty-related problems in fields such as computer science and game theory.Source: §5 · p. 11