Abstract: The pentagonal fuzzy number (PentFN), like the triangular fuzzy number (TriFN), has various algebraic alternatives proposed by different studies. Generally, there is a difference in the algebra for addition/subtraction and multiplication by a scalar. However, for multiplication/division, numerous algebraic alternatives exist, and few provide methods for determining the inverse of a PentFN. Some studies only address multiplication for positive PentFN, with varying definitions of positivity. This variation in defining multiplication algebra prevents the existence of a consistent inverse for any PentFN $$ \widetilde{a}_{Pent}$$ = $$(α_{1}, α_{2}, α_{3}, α_{4}, α_{5}) $$ such that $$ \widetilde{a}_{Pent}\bigotimes \frac{1}{ \widetilde{a}_{Pent}}= \widetilde{l}_{Pent}$$. Based on these conditions, all applications of matrices whose elements consist of PentFN do not produce compatible results. In this paper, The author will construct an algebraic alternative for multiplication/division and inversion applicable to any PentFN, ensuring the existence of $$ \widetilde{a}_{Pent}$$ there is $$ \frac{1}{ \widetilde{a}_{Pent}}$$ such that $$ \widetilde{a}_{Pent}\bigotimes \frac{1}{ \widetilde{a}_{Pent}}= \widetilde{l}_{Pent}$$. <br>The construction process involves transforming the pentagonal fuzzy numbers into a parametric form <img src="https://wseas.com/journals/sac/2025/b025106-2109-abstarct.png" width="300" alt="" /> which can be returned back in standard form. The parametric form that we provide here also has a formula that is different from the existing forms created by various authors. To validate the proposed algebra, examples of calculating the inverse and general inverse of a trapezoidal fuzzy matrix (TraFM) are provided, using modified elementary row operation steps.

Keywords: Alternative algebra, pentagonal fuzzy number, inverse fuzzy matrix, generalized inverse, modification elementary row operations, two mid point

Pages: 501-519