Abstract
Combining the study of queuing with inventory is very common and such systems are referred to as queuing-inventory systems in the literature. These systems occur naturally in practice and have been studied extensively in the literature. The inventory systems considered in the literature generally include (s,S)-type. However, in this paper we look at opportunistic-type inventory replenishment in which there is an independent point process that is used to model events that are called opportunistic for replenishing inventory. When an opportunity (to replenish) occurs, a probabilistic rule that depends on the inventory level is used to determine whether to avail it or not. Assuming that the customers arrive according to a Markovian arrival process, the demands for inventory occur in batches of varying size, the demands require random service times that are modeled using a continuous-time phase-type distribution, and the point process for the opportunistic replenishment is a Poisson process, we apply matrix-analytic methods to study two of such models. In one of the models, the customers are lost when at arrivals there is no inventory and in the other model, the customers can enter into the system even if the inventory is zero but the server has to be busy at that moment. However, the customers are lost at arrivals when the server is idle with zero inventory or at service completion epochs that leave the inventory to be zero. Illustrative numerical examples are presented, and some possible future work is highlighted.
Highlights
Models for inventory management under uncertainty have been studied extensively.The two key questions of when and how many to order have been addressed under a variety of factors such as the nature of inventory review, order quantity, lead time for an order to be fulfilled, nature of demand, and other factors to optimize a function of various costs such as ordering, carrying inventory, lost sales, etc.Most models assume a single supplier and fixed cost of replenishment
For qualitative evaluation of the models presented in this paper, we look at the system performance measures in the following table
The first two components in Z1 and Z2 represent the cost of the inventory subsystem, and the third item represents the cost of the customer subsystem
Summary
Models for inventory management under uncertainty have been studied extensively. The two key questions of when and how many to order have been addressed under a variety of factors such as the nature of inventory review (continuous or periodic), order quantity (fixed or variable), lead time for an order to be fulfilled (negligible, constant or random), nature of demand (deterministic or random), and other factors to optimize a function of various costs such as ordering, carrying inventory, lost sales, etc. Considered systems with constant demand rate, zero lead time, and special discounted opportunities occurring at exponentially distributed intervals, and obtained optimal order quantity at regular price when inventory reaches zero, and the threshold and order quantity at discounted price. In all the above models, it is assumed that the special opportunities for replenishment are always considered as a supplement to the normal replenishment process In many situations such as drugstores, groceries, small supermarkets, etc., the suppliers visit the retailers at random (but frequent) intervals to offer special sales. Our models extend this to consider demand that can occur in batches of random size, with arrivals following a very general process which allows for a broad class of inter-arrival times and autocorrelation between inter-arrival times For both types of inventory systems, it is a significant departure to manage inventory exclusively based on random replenish-. The objective of this study was to understand the behavior of such systems and compare them with the traditional (s, S)-type inventory management systems
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