On the convexity of the cost function for the (Q,R) inventory model

In this paper we investigate the convexity of the annual total cost function for the Hadley & Whitin (1963) continuous review (Q,R) inventory model with backorders and fixed lead-time, when the reorder point is non-negative and the cost per unit backordered is used to calculate the expected annual shortage cost. Efficient procedures for finding the order quantity and reorder point which ensure a global minimum are provided when the demand in the lead time follows the Normal and Log-Normal distributions. Convexity conditions are obtained for each distribution and numerical examples are given to explain how the values of the model cost parameters affect the optimal solution. The results indicate strong interaction between convexity and cost parameters as their values determine when the minimum cost obtained from solving the first order conditions is global and when the global minimum is attained setting the reorder point to zero.