Computational Algorithm of Fuzzy Arithmetic Based on the Principle of Maximum Entropy
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DOI:
https://doi.org/10.55121/mmds.v1i1.1245Abstract
Fuzzy numbers are a powerful tool for working with numerical uncertainty, but they remain little used in solving practical problems in finance and economics. The main problem lies in the significant extension of the support of a fuzzy number when a large number of arithmetic transformations are performed. For example, when calculating profit, the minimum and maximum support values can differ by several times. In such cases, making an investment decision is extremely difficult. Therefore, we developed a new algorithm that generates the result of an arithmetic operation as a projection of a fuzzy binary relation, selected based on the principle of maximum entropy. The proposed algorithm generates a matrix of a fuzzy binary relation in accordance with Zadeh's extension principle. The principle of maximum entropy implies choosing the relation projection (row or column) that contains the cell with the highest degree of membership in the matrix and that has the highest entropy. Compared to standard fuzzy arithmetic, the proposed algorithm provides a much smaller extension of the domain of the resulting fuzzy number than existing methods. In addition, the proposed algorithm works with operands whose support crosses zero. The algorithm is implemented as a Microsoft Excel add-in, which readers can download free of charge and use for applied calculations. We also described an example of calculating the profit and currency risk of options using the proposed algorithm.
Keywords
Algorithm, Fuzzy Set, Fuzzy Number, Arithmetic OperationReferences
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