A note on "Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns"

In a recently published paper by Chiu et al. [Chiu, S.W., Wang, S.-L., Chiu, Y.-S.P., 2007. Determining the optimal run time for EPQ model with scrap, rework and stochastic breakdowns. European Journal of Operational Research 180, 664-676], a theorem on conditional convexity of the integrated total cost function was employed in their solution procedure. We reexamine this theorem and present a direct proof to the convexity of the total cost function. This proof can be used in place of Theorem 1 of Chiu et al.'s paper to enhance quality of their optimization process.