Resolution of Fuzzy Relational Inequalities with Boolean Semi-Tensor Product Composition
Abstract
:1. Introduction
2. Preliminaries
2.1. Semi-Tensor Product of Matrices
- (i)
- (ii)
2.2. Boolean Semi-Tensor Product Composition
- (i)
- Let P, Q, R be three real matrices with arbitrary dimensions. Then
- (ii)
- Let , . Then
3. Problem Formulation
- Problem 1: Solve the following FRI:
- Problem 2: Solve the following SFRIs:
4. Resolution of FRI (20)
- (i)
- If holds for any , then is called the maximum solution;
- (ii)
- If for any , implies , then is called a minimal solution.
5. Illustrative Examples
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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To obtain all the solutions to FRI (11) (resp. SFRIs (12)), one can proceed |
by the following steps: |
(1) Calculate the equivalent form of FRI (11) (resp. SFRIs (12)) by Theorem 1 |
(resp. Theorem 2); |
(2) Construct the parameter set of the j-th FRI in SFRIs (16) (resp. (19)), and |
give the vector form of every element in ; |
(3) Calculate by Lemma 4; |
(4) Obtain all the minimal solutions and the unique maximum solution to the |
j-th FRI in (16) (resp. (19)) by comparing the finite number of elements in ; |
(5) Obtain the solution set of the j-th FRI in (16) (resp. (19)) by (21); |
(6) Obtain the solution set of FRI (11) (resp. SFRIs (12)). |
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Wang, S.; Li, H. Resolution of Fuzzy Relational Inequalities with Boolean Semi-Tensor Product Composition. Mathematics 2021, 9, 937. https://doi.org/10.3390/math9090937
Wang S, Li H. Resolution of Fuzzy Relational Inequalities with Boolean Semi-Tensor Product Composition. Mathematics. 2021; 9(9):937. https://doi.org/10.3390/math9090937
Chicago/Turabian StyleWang, Shuling, and Haitao Li. 2021. "Resolution of Fuzzy Relational Inequalities with Boolean Semi-Tensor Product Composition" Mathematics 9, no. 9: 937. https://doi.org/10.3390/math9090937