An Inventory Ordering Model for Deteriorating Items with Compounding and Backordering

We consider the optimal order quantity problem for exponentially deteriorating items where the opportunity cost is based on compound interest and backorders are allowed. Our objectives in this research are to develop a model that accurately models deterioration, compound interest and backordering, and determine a near-optimal and intuitive closed-form solution for the proposed model. Deteriorating items include various chemicals, gasoline and petroleum products, fresh produce, bulk and liquid food products, batteries, and some electronic components. These items incur losses over time due to spoilage, evaporation, chemical decomposition, breakdown, or deterioration in general. Exponential deterioration is commonly used to model this phenomenon, which results in a negative exponential inventory level function, which is asymmetric in the sense that the rate of depletion is highest at the beginning of an ordering cycle, and lowest at the end. On the other hand, the rate of deterioration for individual items is the same at both ends of the cycle, which means it is symmetric. Compounding also leads to exponential terms in the opportunity cost function. Both of these factors result in a total cost function that does not have a closed-form optimal solution. We therefore approximate the total cost function using a Taylor series expansion approximation of the exponential function and derive a closed-form solution that is simple and logical, and very close to the exact optimum, which makes it attractive to the practitioners as a quick and accurate calculation. Our closed form solutions for both the basic and the planned backorders models are very close to the exact optimum, as shown by extensive numerical experiments.