Using the convexities of total annual relevant costs to determine the optimal cycle times of inventory models for deteriorating items with permissible delay in payments

The inventory problem consists of two parts: (1) the modeling and (2) the solution procedure. Ouyang et al.’s inventory models [Ouyang, L. Y., Wu, K. S., & Yang, C. T. (2006). A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments. Computers & Industrial Engineering, 51, 637–651] are correct and interesting. However, the processes of proofs of Theorems 1 and 2 about solution procedures in Ouyang et al. (2006) are not complete such that accuracies of proofs of their theorems are questionable from the viewpoint of logic. So, this paper will use calculus to explore and adopt the convexities of the total annual relevant costs to overcome those drawbacks and present accurate and complete proofs for Theorems 1 and 2 in Ouyang et al. (2006). Consequently, this paper not only mathematically completes but also improves Ouyang et al. (2006).