6 Supply chain management We consider an inventory management problem over multiple discrete time periods. The problem consists in making ordering, stocking, and storage decisions over a given time horizon composed of T stages, with the objective of minimizing cost and satisfying the demand. Cost is due to actual purchase cost, as well as holding and shortage cost. Considering one single good, and denoting with x(k) the stock level of this good at time k, the basic evolution of the stock level in (discrete) time can be written as x(k+1) x(k) + u(k) w(k), k= 0, 1, ...,T -1, where x(0) = xo is a given initial stock level, u(k) is the stock ordered at time k, and w(k) is the demand during the period from k to k+1. Assuming that we incur a unit holding cost h for storage, a unit shortage cost p (assume that storage level could be negative), and that the unitary cost for buying the good is c, at each time k the storage cost is given by cu(k) + max(hx(k + 1), -pr(k + 1)). Further we also would like take into account an upper bound M on the size of orders in a stage: 0