Optimal policies in deterministic and stochastic inventory models: theory and applications

A well-designed inventory system is critical to the success of any business organisation. One of the major challenges of inventory managers is to determine an inventory optimisation strategy that ensures the right balance between keeping enough inventory on hand to meet customer demand and optimising costs related to holding inventory. This thesis focuses on providing general inventory optimisation strategies to support business organisations, military supply chain management (defence inventory) and humanitarian operations. We examine both general deterministic and stochastic models of inventory in which arbitrary functions can be used to describe the demand and deterioration rates. The demand is determined by price and the rate of deterioration of items can change over the cycle time. For this general model, we examine the profit function arising when the costs are linear with respect to the number of items purchased in the inventory cycle and the total item-time of holding. This framework encompasses a wide range of deterministic and stochastic models that have appeared in the literature and are useful in practice. Within this framework we derive optimisation results for the cycle time, price and cycle service level. We show the application of these results to particular deterioration and demand functions. This allows us to extend the existing inventory literature by deriving the solution to a more generalised problem. Our results are analytically and numerically compared with existing specific results in the inventory literature. The scope of the research is to deal with generalised inventory-control models which are linked to the well-known Economic Order Quantity (EOQ) model (for the deterministic problem), the reorder-point model and random supply model (for the stochastic problem). We develop and analyse models assuming only second-order differentiable forms of the deterioration and demand rates in an inventory cycle. These conditions allow the inclusion of various intermediate functional forms to describe, in detail, the inventory level during the cycle. In cases where no closed-form solutions exist we introduce numerical algorithms which iteratively search for the optimal solution. We show that these algorithms converge to the global solution. Understanding how the solutions of the various models developed in this thesis respond to changes in the input parameters is of vital importance. In this study, we have also undertaken rigorous mathematical analyses to study how the variation of the cost parameters impact the maximised profit-rate function. These sensitivity analyses provide deep understanding of how the profit is affected by the various cost parameters in the various models