  Process flexibility is widely adopted as an effective strategy for responding
to uncertain demand. Many algorithms for constructing sparse flexibility
designs with good theoretical guarantees have been developed for balanced and
symmetrical production systems. These systems assume that the number of plants
equals the number of products, that supplies have the same capacity, and that
demands are independently and identically distributed. In this paper, we relax
these assumptions and consider a general class of production systems. We
construct a simple flexibility design to fulfill $(1-\epsilon)$-fraction of
expected demand with high probability (w.h.p.) where the average degree is
$O(\ln(1/\epsilon))$. To motivate our construction, we first consider a natural
weighted probabilistic construction from Chou et al. (2011) where the degree of
each node is proportional to its expected capacity. However, this strategy is
shown to be sub-optimal. To obtain an optimal construction, we develop a simple
yet effective thresholding scheme. The analysis of our approach extends the
classical analysis of expander graphs by overcoming several technical
difficulties. Our approach may prove useful in other applications that require
expansion properties of graphs with non-uniform degree sequences.
