Technical Note—A Permutation-Dependent Separability Approach for Capacitated Two-Echelon Inventory Systems

We consider optimal inventory replenishment policies for capacitated 2-echelon serial inventory systems, where the capacity of upstream echelon can be the bottleneck. We show that the optimal replenishment decisions in each period can be made one echelon at a time by introducing a procedure that can sequentially decompose a multidimensional optimization problem to a series of one-dimensional problems. We also introduce the notion of permutation-dependent separability. A permutation-dependent separable function is a function that can be decomposed as a sum of single-variable component functions under each nondecreasing order of variables. We find that the value function for the capacitated 2-echelon system in each period is permutation-dependent separable, and that, for each echelon, a permutation-dependent echelon base stock policy is optimal.