The multi-period single-sourcing problem that we address in this paper can be used as a tool for evaluating logistics network designs in a dynamic environment. We consider the assignment of retailers to facilities, taking into account the timing, location, and size of production and inventories, in the presence of various types of constraints. We formulate the problem as a nonlinear assignment problem, and develop efficient algorithms for solving the capacitated lot-sizing subproblems that form the objective function of this formulation. We propose a greedy heuristic, and prove that this heuristic is asymptotically optimal in a probabilistic sense when retailer demands share a common seasonality pattern. In addition, we develop an efficient implementation of the very-large-scale-neighborhood-search method that can be used to improve the greedy solution. We perform extensive tests on a set of randomly generated problem instances, and conclude that our approach produces very high quality solutions in limited time.