  We consider a natural extension to the metric uncapacitated Facility Location
Problem (FLP) in which requests ask for different commodities out of a finite
set $S$ of commodities. Ravi and Sinha (SODA'04) introduced the model as the
Multi-Commodity Facility Location Problem (MFLP) and considered it an offline
optimization problem. The model itself is similar to the FLP: i.e., requests
are located at points of a finite metric space and the task of an algorithm is
to construct facilities and assign requests to facilities while minimizing the
construction cost and the sum over all assignment distances. In addition,
requests and facilities are heterogeneous; they request or offer multiple
commodities out of $S$. A request has to be connected to a set of facilities
jointly offering the commodities demanded by it. In comparison to the FLP, an
algorithm has to decide not only if and where to place facilities, but also
which commodities to offer at each.
  To the best of our knowledge we are the first to study the problem in its
online variant in which requests, their positions and their commodities are not
known beforehand but revealed over time. We present results regarding the
competitive ratio. On the one hand, we show that heterogeneity influences the
competitive ratio by developing a lower bound on the competitive ratio for any
randomized online algorithm of $\Omega(\sqrt{|S|}+\frac{\log n}{\log\log n})$
that already holds for simple line metrics. Here, $n$ is the number of
requests. On the other side, we establish a deterministic
$O(\sqrt{|S|}\cdot\log n)$-competitive algorithm and a randomized
$O(\sqrt{|S|}\cdot\frac{\log n}{\log\log n})$-competitive algorithm. Further,
we show that when considering a more special class of cost functions for the
construction cost of a facility, the competitive ratio decreases given by our
deterministic algorithm depending on the function.
