We present a model for globally optimizing arborescent multilevel inventory-production problems. State-space constraints as well as Poisson distributed demands arriving at any node are considered. For the numerical solution a rate of convergence of order h.Inh is obtained.We introduce a new algorithm for the fast computation of the numerical solution which allows the resolution of complex problems. Some numerical results are shown. We also study the advantages of cooperative behaviour of nodes versus decentralized optimization.