Interpretation of exact solution for fuzzy fractional non-homogeneous differential equation under the Riemann–Liouville sense and its application on the inventory management control problem

Abstract

This article is presented for manifestation of the non-homogeneous linear fractional differential equation under fuzzy uncertainty. Taking the initial values and the coefficients of the fractional differential equations to be fuzzy numbers, the solutions are categorized into different problems and sub-problems. Based on the three different combinations of the fuzzy initial values and fuzzy coefficients, the theoretical foundation in this article is primarily divided into three major problems. Then, each problem again contains two different sub-problems according to the sign of the coefficients. Finally, the solutions for each of the sub-problems are given in the sense of two different cases of ${}^{\text{RL}}[\left(i\right)-\alpha ]$ and ${}^{\text{RL}}[\left(ii\right)-\alpha ]$ differentiability. The strong and the weak solution criteria have been discussed here. Also, the existence and uniqueness criterion for solution of the initial-valued fuzzy fractional differential equation has been established in this article. Finally, an EPQ model of deteriorated items is discussed for its production phase only as a proper validation of proposed theory. The greatness of the consideration of fuzzy fractional differential equation over crisp integer differential equation, fuzzy integer differential equation and crisp fractional differential equation to describe the EPQ model is established in this current article. In this context, a new defuzzification technique is developed to compare the fuzzy and crisp phenomena related to the EPQ model.