  Budget-feasible procurement has been a major paradigm in mechanism design
since its introduction by Singer (2010). An auctioneer (buyer) with a strict
budget constraint is interested in buying goods or services from a group of
strategic agents (sellers). In many scenarios it makes sense to allow the
auctioneer to only partially buy what an agent offers, e.g., an agent might
have multiple copies of an item to sell, they might offer multiple levels of a
service, or they may be available to perform a task for any fraction of a
specified time interval. Nevertheless, the focus of the related literature has
been on settings where each agent's services are either fully acquired or not
at all. The main reason for this, is that in settings with partial allocations
like the ones mentioned, there are strong inapproximability results. Under the
mild assumption of being able to afford each agent entirely, we are able to
circumvent such results in this work. We design a polynomial-time,
deterministic, truthful, budget-feasible $(2+\sqrt{3})$-approximation mechanism
for the setting where each agent offers multiple levels of service and the
auctioneer has a discrete separable concave valuation function. We then use
this result to design a deterministic, truthful and budget-feasible mechanism
for the setting where any fraction of a service can be acquired and the
auctioneer's valuation function is separable concave (i.e., the sum of concave
functions). The approximation ratio of this mechanism depends on how `nice' the
concave functions are, and is $O(1)$ for valuation functions that are sums of
$O(1)$-regular functions (e.g., functions like $\log(1+x)$). For the special
case of a linear valuation function, we improve the best known approximation
ratio for the problem from $1+\phi$ (by Klumper & Sch\"afer (2022)) to $2$.
This establishes a separation between this setting and its indivisible
counterpart.
