  We study the inventory placement problem of splitting $Q$ units of a single
item across warehouses, in advance of a downstream online matching problem that
represents the dynamic fulfillment decisions of an e-commerce retailer. This is
a challenging problem both in theory, because the downstream matching problem
itself is computationally hard, and in practice, because the fulfillment team
is constantly updating its algorithm and the placement team cannot directly
evaluate how a placement decision would perform.
  We compare the performance of three placement procedures based on optimizing
surrogate functions that have been studied and applied: Offline, Myopic, and
Fluid placement. On the theory side, we show that optimizing inventory
placement for the Offline surrogate leads to a $(1-(1-1/d)^d)/2$-approximation
for the joint placement and fulfillment problem. We assume $d$ is an upper
bound on how many warehouses can serve any demand location and that stochastic
arrivals satisfy either temporal or spatial independence. The crux of our
theoretical contribution is to use randomized rounding to derive a tight
$(1-(1-1/d)^d)$-approximation for the integer programming problem of optimizing
the Offline surrogate. We use statistical learning to show that rounding after
optimizing a sample-average Offline surrogate, which is necessary due to the
exponentially-sized support, does indeed have vanishing loss.
  On the experimental side, we extract real-world sequences of customer orders
from publicly-available JD.com data and evaluate different combinations of
placement and fulfillment procedures. Optimizing the Offline surrogate performs
best overall, even compared to simulation procedures, corroborating our theory.
