  We consider a production-inventory control model with finite capacity and two
different production rates, assuming that the cumulative process of customer
demand is given by a compound Poisson process. It is possible at any time to
switch over from the different production rates but it is mandatory to
switch-off when the inventory process reaches the storage maximum capacity. We
consider holding, production, shortage penalty and switching costs. This model
was introduced by Doshi, Van Der Duyn Schouten and Talman in 1978. Our aim is
to minimize the expected discounted cumulative costs up to infinity over all
admissible switching strategies. We show that the optimal cost functions for
the different production rates satisfy the corresponding
Hamilton-Jacobi-Bellman system of equations in a viscosity sense and prove a
verification theorem. The way in which the optimal cost functions solve the
different variational inequalities gives the switching regions of the optimal
strategy, hence it is stationary in the sense that depends only on the current
production rate and inventory level. We define the notion of finite band
strategies and derive, using scale functions, the formulas for the different
costs of the band strategies with one or two bands. We also show that there are
examples where the switching strategy presented by Doshi et al. is not the
optimal strategy.
