A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures
Abstract
:1. Introduction
- In rough set theory, the common decision-making method is using decision rules. It is difficult to find the best decision rules, because different methods can obtain different rules, which will influence the decision result. Hence, a new decision-making method based on rough sets should be presented, which will be independent of decision rules.
- In decision-making theory, attribute weights are needed in almost all decision-making methods, such as the WA, OWA and TOPSIS methods. However, it is difficult to obtain the optimal weight value, and many weight values are given artificially. To solve this problem, Choquet integrals can be used to aggregate decision information without attribute weights.
2. Basic Definitions
2.1. Pawlak’s Rough Sets
2.2. Fuzzy Measures and Choquet Integrals
- (1)
- , ;
- (2)
- , , which implies .
3. Fuzzy Rough Measures and Choquet Integrals
3.1. Fuzzy Rough Measures Based on Attribute Importance Degrees
- The conditional attribute ‘’ has values: “Clear = 1”, “Cloudy = 2”, “Rain = 3”.
- The conditional attribute ‘’ has values: “Hot = 1”, “Warm = 2”, “Cool = 3”.
- The conditional attribute ‘’ has values: “Yes = 0”, “No = 1”.
- The conditional attribute ‘’ has values: “Wet = 1”, “Normal = 2”, “Dry = 3”.
- The conditional attribute ‘D’ has values: “Yes = 1”, “No = 0”.
- (1)
- and ;
- (2)
- For any , implies .
3.2. Choquet Integrals under Fuzzy Rough Measures
4. A Novel Decision-Making Method Based on Fuzzy Rough Measures and Choquet Integrals
4.1. The Problem of Decision Making
4.2. The Novel Decision-Making Method
Algorithm 1 The MCDM algorithm by fuzzy rough measures and Choquet integrals |
Input: A decision information system and a new decision object , where , . Output: The decision value of . (1) for (2) for (3) Compute ; (4) end (5) Compute ; (6) end (7) for (8) Obtain the ranking of all ; (9) end (10) Give the decision value of by the ranking of all . |
5. Comparison and Analysis
5.1. Hiring Dataset
- The conditional attribute ‘Diploma’ has values: “MBA”, “MSc”, “MCE”.
- The conditional attribute ‘Experience’ has values: “High”, “Low”, “Medium”.
- The conditional attribute ‘French’ has values: “Yes”, “No”.
- The conditional attribute ‘Reference’ has values: “Excellent”, “Good”, “Neutral”.
- The conditional attribute ‘Decision’ has values: “Accept”, “Reject”.
5.2. An Applied Example
- The conditional attribute ‘Diploma = ’ has values: “MBA = 1”, “MSc = 2”, “MCE = 3”.
- The conditional attribute ‘Experience = ’ has values: “Medium = 1”, “High = 2”, “Low = 3”.
- The conditional attribute ‘French = ’ has values: “Yes = 1”, “No = 0”.
- The conditional attribute ‘Reference = ’ has values: “Excellent = 1”, “Neutral = 2”, “Good = 3”.
- The conditional attribute ‘Decision = D’ has values: “Accept = 1”, “Reject = 0”.
5.3. Comparison with Other Methods
6. Conclusions
- The notion of the attribute measure is presented based on the importance degree in rough sets, which can illustrate the non-additive relationship of two attributes in rough sets. By the new notion, we can find that attributes are related to each other in information systems. It can also be used to construct the corresponding Choquet integral.
- Then, a type of nonlinear aggregation operator (i.e., Choquet integral) is constructed, which can aggregate all information between two objects in a decision information system. Moreover, a method based on the Choquet integral is proposed to deal with the problem of MCDM, which is inspired by case-based reasoning theory. This novel method can address the deficiency of the existing methods well. It can solve the issue of attribute association in MCDM.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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U | D | ||||
---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | |
2 | 2 | 1 | 2 | 1 | |
2 | 2 | 1 | 1 | 1 | |
1 | 2 | 0 | 3 | 1 | |
1 | 3 | 1 | 2 | 0 | |
3 | 3 | 1 | 3 | 0 | |
2 | 1 | 1 | 2 | 0 |
U | ⋯ | D | |||
---|---|---|---|---|---|
⋯ | |||||
⋯ | |||||
⋮ | ⋮ | ⋮ | ⋯ | ⋮ | ⋮ |
⋯ |
U | ⋯ | |||
---|---|---|---|---|
⋯ | ||||
⋯ | ||||
⋮ | ⋮ | ⋮ | ⋯ | ⋮ |
⋯ |
U | Diploma | Experience | French | Reference | Decision |
---|---|---|---|---|---|
MBA | Medium | Yes | Excellent | Accept | |
MSC | High | Yes | Neutral | Accept | |
MSC | High | Yes | Excellent | Accept | |
MBA | High | No | Good | Accept | |
MBA | Low | Yes | Neutral | Reject | |
MCE | Low | Yes | Good | Reject | |
MSC | Medium | Yes | Neutral | Reject | |
MCE | Low | No | Excellent | Reject |
U | Diploma () | Experience () | French () | Reference () | Decision (D) | |
---|---|---|---|---|---|---|
An information system | MBA (1) | Medium (1) | Yes (1) | Excellent (1) | Accept (1) | |
MSC (2) | High (2) | Yes (1) | Neutral (2) | Accept (1) | ||
MSC (2) | High (2) | Yes (1) | Excellent (1) | Accept (1) | ||
MBA (1) | High (2) | No (0) | Good (3) | Accept (1) | ||
MBA (1) | Low (3) | Yes (1) | Neutral (2) | Reject (0) | ||
MCE (3) | Low (3) | Yes (1) | Good (3) | Reject (0) | ||
MSC (2) | Medium (1) | Yes (1) | Neutral (2) | Reject (0) | ||
A decision object | MCE (3) | Low (3) | No (0) | Excellent (1) | “?” |
U | ||||
---|---|---|---|---|
1 | 0 | 0 | ||
1 | 0 | 0 | ||
1 | 0 | 0 | ||
1 | 0 | |||
1 | ||||
1 | 0 | |||
1 | 0 | 0 |
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Wang, J.; Zhang, X. A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures. Axioms 2022, 11, 275. https://doi.org/10.3390/axioms11060275
Wang J, Zhang X. A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures. Axioms. 2022; 11(6):275. https://doi.org/10.3390/axioms11060275
Chicago/Turabian StyleWang, Jingqian, and Xiaohong Zhang. 2022. "A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures" Axioms 11, no. 6: 275. https://doi.org/10.3390/axioms11060275
APA StyleWang, J., & Zhang, X. (2022). A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures. Axioms, 11(6), 275. https://doi.org/10.3390/axioms11060275