Decomposition in multi-item inventory control

This paper considers a Lagrangian decomposition approach to a stochastic demand multi-item inventory control problem with a single resource constraint. The work is a generalization of existing decomposition methods.Three decomposition methods are proposed, and bounds on the loss of optimality for each are given in terms of the Lagrange multiplier used. One method allows the calculation of the complete decision rule in advance of the realization of the states, but is expected to perform worse than the other two methods. The second and third method allow the determination of decisions as an optimization problem as the states are realized. Since, in any problem with many states, only a small proportion will actually be realized even in a large time-horizon problem, there may be some advantage in taking this approach.