Competitive analysis of the online inventory problem

We consider a real-time version of the inventory problem with deterministic demand in which decisions as to when to replenish and how much to buy must be made in an online fashion without knowledge of future prices. We suggest online algorithms for each of four models for the problem and use competitive analysis to obtain algorithmic upper and lower bounds on the worst-case performance of the algorithms compared to an optimal offline algorithm. These bounds are closely related to the tight M/m-bound obtained for the simplest of the models, where M and m are the upper and lower bounds on the price fluctuation.