We study online demand fulfillment in a class of networks with limited flexibility and arbitrary numbers of resources and request types. We show analytically that such a network is both necessary and sufficient to guarantee a performance gap independent of the market size compared with networks with full flexibility, extending the previous literature from the long chains to more general sparse networks. Inspired by the performance bound, we develop simple inventory allocation rules and guidelines for designing such network structures. Numerical experiments including one using some real data from Amazon China are conducted to confirm our findings as well as some of the flexibility principles conjectured in the literature. This paper was accepted by Chung Piaw Teo, optimization.

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