Mathematical Programming Approach to the Optimality of the Solution for Deterministic Inventory Models with Partial Backordering

We give an alternative proof of the optimality of the solution for the deterministic EPQ with partial backordering (EPQ-PBO) [Omega, vol. 37, no. 3, pp. 624–636, 2009]. Our proof is based on the mathematical programming theory. We also demonstrate the determination of the optimal decision policy through solving the corresponding mathematical programming problem. We indicate that the same approach can be used within other inventory models with partial backordering, and we consider additional models. 1. Introduction A basic inventory control model is the economic order quantity (EOQ) model known by its simplicity and its restrictive modeling assumptions. Managing the inventory of a single item the purpose of the EOQ model is to determine how much to purchase (order quantity) and when to place the order (the reorder point). By relaxing the assumption that shortages are not allowed and allowing stockouts with partial backordering, the EOQ will extend to EOQ with partial backordering (EOQ-PBO). But when the assumption of instantaneous replenishment is replaced with the assumption that the replenishment order is received at a constant finite rate over time, EOQ is extended to the economic production quantity (EPQ) model, and by allowing stockouts with partial backordering it will extend to EPQ with partial backordering (EPQ-PBO). There are many papers that propose an inventory model with partial backordering; a very few of them are [1–7]. For a survey of the deterministic models for the EOQ and EPQ with partial backordering, see Penticio’s and Drake’s paper [8]. Pentico et al. [5] proposed EPQ-PBO model, whose main characteristics are that the derived equations are more like those for the classical EPQ model and it gives the optimal solution in a closed form that helps in understanding the behavior of the inventory system. Let us briefly give the notations from [5] that are used in the total cost function and in the solution of the Pentico et al.'s EPQ-PBO as well. They are？ : demand per year,？ : production rate per year if constantly producing,？ : the fixed cost of placing and receiving an order,？ : the cost to hold a unit in inventory for a year,？ : the cost to keep a unit backordered for a year,？ : the cost for a lost sale,？ : the backordering rate,？ : the length of an order cycle (a decision variable),？ : the fill rate (a decision variable). The derived total averaged cost per year for the Pentico et al.’s EPQ-PBO model, according to [5], that has to be minimized is where , , and . Setting the first partial derivatives of equal to zero, Pentico et al. [5]