Existence and uniqueness of solutions to neutrosophic fractional differential equations and their implications for inventory controls model

The fractional differential equation is one of the important tools to realize the importance of the fractional calculus. Also, neutrosophic set provides a more comprehensive framework for handling uncertainty by truth, indeterminacy, and falsity membership function. This study aims to develop fractional differential equation in neutrosophic uncertain environment. A rigorous mathematical theorem has been formulated and proven, which establishes the existence and uniqueness of the solution to the initial-valued neutrosophic fractional differential equation (NFDE). The weak and strong characteristics of the solutions to NFDEs within the context of the Caputo fractional derivative framework is presented. A non-homogeneous linear NFDE is manifested by taking two types of neutrosophic fractional derivative in Caputo’s sense. An economic lot-sizing inventory model of deteriorating items with green level and stock-level dependent demand is presented as an application of the proposed theory by taking various inventory related parameters as neutrosophic numbers. Several cases of the proposed problems are also presented. The result observed that the fractional order model in a neutrosophic environment yields significantly better results compared to integer order models in the neutrosophic environment, as well as integer or fractional order models in a crisp environment.