This paper develops an economic production quantity (EPQ) inventory model with rework process for a single stage production system with one machine. The existence of a unique machine results in limited production capacity. The aim of this research is to determine both the optimal cycle length and the optimal production quantity for each product to minimize the expected total cost (holding, production, setup, rework costs). The convexity of the inventory model is derived. Also the objective function is proved to be convex. The proposed inventory model is validated with illustrating numerical examples and the optimal period length and the total system cost are analyzed.

In this paper, a multi products single machine economic production quantity model with
discrete delivery is developed. A unique cycle length is considered for all produced items with
an assumption that all products are manufactured on a single machine with a limited capacity.
The proposed model considers different items such as production, setup, holding, and
transportation costs. The resulted model is formulated as a mixed integer nonlinear
programming model. Harmony search algorithm, extended cutting plane and particle swarm
optimization methods are used to solve the proposed model. Two numerical examples are used
to analyze and to evaluate the performance of the proposed model.