Solving a class of two-stage robust inventory management problems

In this paper we address the problem of optimizing multi-period inventory in the case of uncertain demands. At each time period, the company produces a certain quantity of goods which is used to serve a client demand. The unit selling price was fixed in advance by contract for the time horizon and an expected value of the quantity to deliver at each time period is known. In case of overproduction, goods are added to the stock. In case of underproduction the missing goods are either taken from the stock or bought on the international market. In addition at each time period the manager can decide to buy more goods and add them to the stock or to sell a part of the goods in stock, on the international market. The unit price of purchase (or sale) of these goods on the international market are estimated in advance for every period, according to the previous years. But in fact, the demand and the purchasing costs on the international market are uncertain and may differ from their expected values. Following the Betsimas and Thiele approach, we assume that there is no known probabilistic distribution of these values, but each one may vary in a given interval. We also assume that the variation of the purchasing or selling costs is small, while the real demand can be far enough from its expected value. Then the prices on the international market can be approximate in the following way: the unit purchasing cost is set to its expected value plus the maximum possible gap and the unit selling cost is set to its expected value minus the maximum possible gap. Doing so, we guarantee a lower bound on the profits. The manager takes decisions in two stages: first he before discovering the actual value taken by the demand, second once uncertainty has been revealed. In this paper we address the problem of optimizing multi-period inventory in the case of uncertain demands. We consider a wholesaler who purchases goods on the international market and stocks them in a warehouse before selling them to local customers. To serve the demand he can either demand at each time period, the manager decides the quantity to buy His decisions are made in two stages: first before discovering the actual value taken by the demand, second once uncertainty has been revealed.