An Exact Aggregation for Inventory Distribution in an Automatic Warehousing System

We consider a simple automatic warehousing sytem with a lot of storage spaces called slots in which only a single type of items are stored and retrieved. The purpose of this paper is to develop an efficient method for obtaining the marginal probability that each of the slots in the warehouse is full. The set of such probabilities for all the slots constitutes the spatial inventory distribution of the items in the system. This distribution, which we call the inventory distribution in short, enables us to evaluate key performance characteristics of the system such as the mean travel time of a crane for a single operation of storage or retrieval of items. Such characteristics cannot be obtained from the distribution of the total number of full slots alone. We assume that inventories are controlled by an (S, S) reordering policy, and that received items are stored from the closest open slot to an 110 point and retrieved items are chosen randomly among currently full slots. Furthermore the time between retrievals and the time between placing of order and receipt of items are exponentially distributed. Under these assumptions the system can be modeled as a Markov chain. If we use joint inventory levels of slots, the number of states of the Markov chain amounts t o 2m, where m is the total number of slots. Here we devise exact aggregation methods of the states to reduce the size of the Markov chain. By exploiting the special structure the total computational complexity for obtaining the inventory distribution is reduced to 0 ( m 4 ) .