1. Field of the Invention
The present invention generally relates to computer software for business management and, more particularly, to a computer implemented method for fast and accurate evaluation of periodic review inventory policy for large scale inventory systems. The invention implements a novel computation of the renewal function using rational polynomial approximants.
2. Background Description
With the drive to increase profitability, manufacturers and retailers are increasingly focusing on minimizing their inventory costs. As a result, manufacturers and retailers have invested in computer-based inventory management systems. Such inventory management systems aim to pare inventories to the minimum level necessary for satisfying customer service requirements. In order to attain this objective, we are interested in being able to monitor the performance of the inventory system and to optimize the policy parameters.
The (s,S) policy and its variants are among the most commonly used inventory policies. Under this type of inventory policy, one places an order with the supplier once the inventory position (or the level) falls below the re-order point, s. The size of the order placed is such that the inventory level is raised to the order-up-to limit, S. We shall focus on the periodic-review (s,S) policy with complete back ordering. The continuous review version of this policy can be derived as a special case of the policy studied here.
For (s,S) inventory systems there are two questions to one who would be interested in the performance of the systems according to various metrics, and the optimal values of parameters s and S. The two questions are interlinked because the performance measures of an (s,S) inventory system are functions of s and S. On the other hand, the performance measures are also used to optimize the values of the policy parameters.
Some of the metrics used to evaluate the performance of an inventory system are:
Average stock on-hand and its variance. PA1 Average back order level and its variance. PA1 Service-level provided by the system. PA1 Average cost per period of ordering and holding inventory.
(s,S) inventory systems, like applications in the areas of queuing and reliability, have underlying stochastic processes that exhibit regenerative behavior. Regenerative stochastic processes are appealing for analysis purposes because one can apply renewal theory to model such systems. The use of renewal theory facilitates derivation of analytical expressions and/or the design simulation experiments to further characterize the regenerative stochastic process. In the (s,S) inventory system, the inventory position process is regenerative, and therefore, one can derive analytical expressions and/or design simulation experiments to study the characteristics of this process. However, more germane to the discussion here is the fact that the inventory position process is related to many other stochastic processes such as the on-hand inventory process, back order level process, average cost process, etc., in the inventory system. As a result, one can also model these related stochastic processes using renewal theory. Therefore, renewal theory lies at the heart of performance modeling of (s,S)-type inventory systems.
In the past, researchers have derived the exact analytical expressions for the performance measures for the steady-state analysis of (s,S)-type systems. These expressions are in terms of the renewal function and have not been used in most systems because of the difficulties in evaluating these expressions numerically. This is largely due to the fact that the renewal function cannot be expressed in closed-form for most common probability distributions. Hence one needs to expend a considerable amount of computation time in order to evaluate the renewal function and its derivative. Therefore, the current practice is to use approximate analytical expressions or simulation to evaluate the performance of (s,S)-type inventory systems.