Reliability Dependent Imperfect Production Inventory Optimal Control Fractional Order Model for Uncertain Environment Under Granular Differentiability

In this paper, we present a general formulation for the production inventory optimal control problem to a class of fractional fuzzy differential systems with reliability-dependent imperfect production. we investigate this model under the uncertain environment of fuzzy numbers with the rate of change of stock level expressed by granular Caputo fractional fuzzy derivative of order . The demands depend on the selling price, product quality, and stock level of the product displayed in the store. The unit production cost depends on development cost due to reliability, raw material cost, and wear tear cost. Here state, co-state variables and parameters are taken as uncertain variables with triangular fuzzy numbers and using granular dierentiability the existence and uniqueness of solution to the fractional dierential system with fuzzy variables and parameters. The production inventory optimal control problem for the fractional system is proposed and a necessary condition for optimality is obtained. Finally, this optimal control problem was solved by using the Pontryagin Maximum principle and for numerical results and graphical representation.