We study new online versions of a broad class of supply chain management and logistics models, where in addition to production decisions, one also has to decide on which customers to serve. Specifically, customers arrive sequentially during a selection phase, and one has to decide whether to accept or reject each customer upon arrival. If a customer is rejected, then a lost-sales cost is incurred. Once the selection decisions are all made, one has to satisfy all of the accepted customers with minimum possible production cost. The goal is to minimize the total cost of lost-sales and production. A key feature of the model is that customers arrive in an online manner, and the decision maker does not have any information about future arrivals.' We provide two novel algorithms for online customer selection problems which are based on repeatedly solving offline sub-problems that 'ignore' previously made decisions. For many important settings, our algorithms achieve constant competitive ratio guarantees. That is, for any sequence of arriving customers, the cost incurred by the online algorithm is within a constant factor of the cost incurred by the respective optimal offline solution that has full knowledge upfront on the sequence of arriving customers. Finally, we provide a computational study on the performance of these algorithms when applied to the economic lot sizing problem with online customer selection.'