Online Resource Allocation with Limited Flexibility

We consider a class of online resource allocation problems in which there are n types of resources with limited initial inventory and n demand classes. The resources are flexible in that each type of resources can serve more than one demand class. In this paper, we focus on a special class of structures with limited flexibility, the long chain design, which was proposed by Jordan and Graves (1995) and has been an important concept in the design of sparse flexible processes. We study the long chain design in an online stochastic environment where the requests are drawn repeatedly and independently from a known probability distribution over the different demand classes. Also, the decision on how to address each request must be made immediately upon its arrival. We show the effectiveness of the long chain design in mitigating supply-demand mismatch under a simple myopic online allocation policy. In particular, we provide an upper bound on the expected total number of lost sales that is irrespective of how large the market size is.