Resolution of a system of fuzzy polynomial equations using the Gröbner basis

The occurrence of imprecision in the real world is inevitable due to some unexpected situations. The imprecision is often involved in any engineering design process. The imprecision and uncertainty are often interpreted as fuzziness. Fuzzy systems have an essential role in the uncertainty modelling, which can formulate the uncertainty in the actual environment. In this paper, a new approach is proposed to solve a system of fuzzy polynomial equations based on the Grobner basis. In this approach, first, the h-cut of a system of fuzzy polynomial equations is computed, and a parametric form for the fuzzy system with respect to the parameter of h is obtained. Then, a Grobner basis is computed for the ideal generated by the h-cuts of the system with respect to the lexicographical order using Faugere's algorithm, i.e., F"4 algorithm. The Grobner basis of the system has an upper triangular structure. Therefore, the system can be solved using the forward substitution. Hence, all the solutions of the system of fuzzy polynomial equations can easily be obtained. Finally, the proposed approach is compared with the current numerical methods. Some theorems together with some numerical examples and applications are presented to show the efficiency of our method with respect to the other methods.

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