The present paper is devoted to modeling and solving the inventory and production planning in a four-echelon supply chain (SC) with reverse logistics, from supplier to after-sales service center and repair center. To the best of our knowledge, this is the first scientific attempt for modeling and resolving the inventory and production planning in an SC with reverse logistics that the problem is modeled by a disturbance optimal control problem (OCP). Both recycling and reworking are assumed in the reverse logistic. There was a disturbance because we considered the demand time-dependent. From the former points the presented model is so close to the real-world problem in the field of SC. The Pontryagin minimum principle is applied to reformulate the OCP into a system of equations. In continuous, the system of equations is solved by an artificial neural network. In the final step, an example and a case study are presented to depict the performance and validity of the model and the method.

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