Fuzzy Linear Systems and Computational Methods
Summary
Fuzzy linear systems extend classical linear algebra to the treatment of imprecise data by allowing coefficients, variables or constants to be represented as fuzzy sets rather than exact values. Such systems capture uncertainty inherent in engineering design, economic modelling, control systems and decision-making under ambiguity. Computational methods for fuzzy linear systems combine principles from interval arithmetic, optimisation and parametric representation to transform fuzzy equations into tractable problems. Common strategies include α-cut decomposition, which reduces a fuzzy equation to a family of interval systems; direct methods, such as embedding techniques or matrix factorizations adapted to fuzzy data; and iterative schemes that exploit convergence properties of fuzzy arithmetic. Defuzzification procedures then convert fuzzy solutions into crisp estimates, balancing fidelity to the original uncertainty with the need for actionable outputs. Recent advances have emphasised higher-order convergence, efficient handling of nonlinear boundary conditions and methods that preserve the full multidimensional structure of fuzzy solutions. Practical applications span robotics trajectory planning under sensor noise, traffic flow prediction with uncertain demands and risk assessment in financial portfolios. By bridging theoretical rigour with computational scalability, the field offers robust tools for real-world problems where uncertainty cannot be ignored.
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Fuzzy Linear Systems and Computational Methods publication trend
The graph below shows the total number of articles in fuzzy linear systems and computational methods across all publications each year (not limited to Nature Index journals).
Technical terms
Fuzzy linear system: A set of linear equations in which coefficients, variables or constants are characterised by fuzzy sets rather than precise values.
Fuzzy number: A special type of fuzzy set on the real line with a normalised, convex membership function representing uncertain numerical values.
Membership function: A mapping from each element in the domain to a degree of membership between zero and one, quantifying fuzziness.
α-cut: A crisp set containing all elements of a fuzzy set whose membership degrees are at least α, used to reduce fuzzy problems to interval computations.
Defuzzification: The process of converting a fuzzy quantity into a single crisp value, often through optimisation or ranking methods.
Triangular fuzzy number: A fuzzy number defined by a triplet of real values forming a triangular membership function for simplified arithmetic.
Horizontal fuzzy number: A parametric form of fuzzy number with nonlinear left and right boundary functions facilitating multidimensional computations.
References
- Combined Defuzzification Under Shared Constraint. IEEE Transactions on Fuzzy Systems (2024).
- Highly efficient numerical scheme for solving fuzzy system of linear and non-linear equations with application in differential equations. Applied Mathematics in Science and Engineering (2022).
- Method with horizontal fuzzy numbers for solving real fuzzy linear systems. Soft Computing (2018).
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