On the convexity for the expected total cost per unit time of the EPQ model with scrap, rework and stochastic machine breakdown

This study explores the economic production quantity model with scrap, rework and stochastic machine breakdown. The main purpose of this paper is twofold:(P1) This paper will adopt the rigorous methods of mathematics to demonstrate that the expected total cost per unit time is convex on all positive numbers to improve the conditional convexity in Theorem 1 of Chiu et al. (2010) [7].(P2) This paper gives the concrete proof to provide bounds for the optimal production run time to remove the logical shortcomings of mathematics presented in proof of Theorem 2 of Chiu et al. (2010) [7].