Note on “replenishment run time problem with machine breakdown and failure in rework”

A recursive search algorithm along with mathematical analysis was used in a recent paper (Chiu, Y.-S.P., Chen, K.-K., Ting, C.-K., 2012. Replenishment run time problem with machine breakdown and failure in rework. Expert Systems with Applications, 39, 1291–1297) to determine the optimal operating policy for a specific replenishment run time problem. In its optimization procedure a conditional convexity theorem on the cost function was adopted. This paper presents a direct proof of convexity of the function to enhance quality of their solution process and eliminate unnecessary computational efforts in verifying the conditional convexity.