On Kemeny Optimization Scheme for Fuzzy Set of Relations
Abstract
:1. Introduction
- By utility values;
- By ordering the alternatives from the best to the worst;
- By using a preference relation.
- Fuzzy preference relations (FPRs) [4,5]. These are characterized by the fact that the elements of the preference matrix directly express the preference degree or the preference intensity of the alternative over . The equality indicates an indifference between the alternatives and . The equality means that the alternative is absolutely preferred to . The inequality indicates that the alternative is preferred to . In addition, the strict consistency of an FPR requires the equality for all i and k, which guarantees reflexivity and anti-symmetry. The property of additive transitivity is modeled (see, for instance [6]) by the equalities for all , and k.
- Multiplicative preference relations [7]. Each element of the preference matrix represents the ratio of the preference intensities of the alternatives and . This is interpreted as times better than ; the equality indicates an indifference between and ; the equality indicates that is unanimously preferred to , and the case indicates intermediate evaluations. Usually, the property of multiplicative reciprocity for all i and k is also assumed. The property of multiplicative transitivity means that for all , and k.
- The resulting FPR may not match the opinion of the majority of DMs; this happens when a significantly high or significantly low rating from one expert can greatly bias the final rating;
- It is impossible to construct a group FPR with specified properties, which may differ from the properties of individual FPRs;
- The dependence of a group FPR on which a subset of DMs (generally fuzzy) is involved in decision-making may remain outside the scope of attention.
2. Materials and Methods
2.1. Type-2 Fuzzy Relations
2.2. T2FRs with Constant Secondary Grades
3. Formulation of the Problem and Main Idea
3.1. The Group FPR
3.2. Fuzzy Set of Decision-Makers
4. The Group FPR for a Fuzzy Set of Decision-Makers
4.1. The Group T2FR
4.2. Decomposition of the Group T2FR
4.3. Calculation of the Group T2FR
4.4. Properties of the Group T2FR
- 1
- For any , , and inequality , where is the minimum Kemeny distance.
- 2
- If is the membership degree of the alternative over for the -cut , of the FS of DMs, then the pair of alternatives has the primary membership degree to the group T2FR with the secondary grade (the degree of truth) no smaller than , that is , , .
5. Numerical Examples
- The T1FR with the degree of truth being equal to 1;
- The T1FR with the degree of truth being equal to 0.4.
6. Results and Discussion
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
DM | decision-maker |
MF | membership function |
FS | fuzzy set |
T2FR | Type-2 fuzzy relation |
T1FR | Type-1 fuzzy relation |
FPR | fuzzy preference relation |
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Mashchenko, S.O.; Kapustian, O.A.; Rubino, B. On Kemeny Optimization Scheme for Fuzzy Set of Relations. Axioms 2023, 12, 1067. https://doi.org/10.3390/axioms12121067
Mashchenko SO, Kapustian OA, Rubino B. On Kemeny Optimization Scheme for Fuzzy Set of Relations. Axioms. 2023; 12(12):1067. https://doi.org/10.3390/axioms12121067
Chicago/Turabian StyleMashchenko, Serhii O., Olena A. Kapustian, and Bruno Rubino. 2023. "On Kemeny Optimization Scheme for Fuzzy Set of Relations" Axioms 12, no. 12: 1067. https://doi.org/10.3390/axioms12121067
APA StyleMashchenko, S. O., Kapustian, O. A., & Rubino, B. (2023). On Kemeny Optimization Scheme for Fuzzy Set of Relations. Axioms, 12(12), 1067. https://doi.org/10.3390/axioms12121067