We study the paper of Hadley to compare the order quantities for inventory models using the average annual cost and the discounted cost. The purpose of this note is threefold. First, we show that the solution for the discounted cost inventory model is exist and unique. Second, we provide a formulated approximation for the discounted cost inventory model. Third, we use our approximation to explain why two solutions by the average annual cost and the discounted cost are close each other such that the numerical analysis in Hadley’s examples to demonstrate a negligible difference between them is explained. Our findings will undoubtedly improve the theoretical base for inventory models and develop new practical application for operational research.

Most inventory models dealt with deteriorated items. On the contrary, just a few papers considered inventory systems under amelioration environment. We study an amelioration inventory model with Weibull distribution. However, there are some questionable results in the amelioration paper. We will first point out those questionable results in the previous paper that did not derive the optimal solution and then provide some improvements. We will provide a rigorous analytical work for different cases dependent on the size of the shape parameter. We present a detailed numerical example for different ranges of the sharp parameter to illustrate that our solution method attains the optimal solution. We developed a new amelioration model and then provided a detailed analyzed procedure to find the optimal solution. Our findings will help researchers develop their new inventory models.