Replenishment run time problem with machine breakdown and failure in rework

Recently, Chung published a paper in Expert Systems with Applications providing a correct solution procedure to the inventory system for the replenishment run time problem with machine breakdown and failure in rework in Lin and Chiu 〔2〕. In this short note, we provide a detailed discussion for the convex property in Chiu et al. 〔1〕, Lin and Chiu〔2〕and Chung〔3〕. Moreover, we will put forth an exact expression for the concave up and concave down sub-intervals to clearly solve the non-convex phenomenon. Our results will help researchers realize the importance of the findings of Chiu et al.〔1〕, Lin and Chiu 〔2〕and Chung〔3〕.