Stanislav Molchanov, UNC Charlotte

Abstract: In her PhD thesis [1], Jennifer Hill, a graduate of UNCC, analyzed the following model proposed by I. Sonin (see also [2] and [3]).

There is a firm, which uses a certain commodity for production and consumes it with a unit intensity. The price of the commodity follows a continuous time Markov chain with a finite number N of states and known transition rates. The firm can keep some of the commodity in storage. At any time point, the firm can either purchase the commodity at the current price or use some of its stored reserves. Further, it can buy the commodity either with some intensity or instantly some amount for storage. The storage cost is proportional to the amount of the commodity stored. The goal is to minimize the average (or discounted) performance cost, which equals the storage cost plus the purchase cost.

For N = 2 and for some cases with N = 3, Hill and Sonin found the minimal values of thresholds in the class of threshold strategies. We consider the general case and prove that the optimal strategy is indeed the threshold one. Further, we give an algorithm for sequential construction of optimal thresholds beginning from the smallest one.

References

[1] J. Hill (2004). A Markov-Modulated Acquisition Strategy, PhD thesis.

[2] J. Hill, I. Sonin (2006). An Inventory Optimization Model with Markov Modulated Commodity Prices, abstract, Intern. Conf. on Management Sciences, Univ.of Texas at Dallas.

[3] M. Katehakis, I. Sonin (2013). A Markov Chain Modulated Inventory Model, abstract, INFORMS, 2013.