Solutions to fuzzy relation inequality A ∘X ∘ B ≤ C

For traditional matrix product, there may exist infinite solutions, in a sense that some matrix equations or inequalities can be solved. Furthermore, it is very difficult to solve them directly. Similarly, for max-min composition in finite course, fuzzy relational equations or inequalities may also have the same trouble. Unfortunately, there are few papers referring to the problem. This paper devotes to deriving a new method of solving fuzzy relation inequality (FRI) in terms of A∘X ∘B ≤ C. First of all, two important formulas are proved. Then, the considered FRIs are converted into simplified ones taking use of the two transformations. While for solvability of FRIs, a necessary and sufficient condition is obtained. It illustrates that the solutions of considered FRIs can be depicted by finite ones. Via semi-tensor product (STP) of matrices, a concrete algorithm is derived. Finally, with FRIs constraints latticized linear programming is presented to demonstrate effectiveness of the proposed methods. © 2016, Editorial Department of Control Theory & Applications South China University of Technology. All right reserved.