What's the definition of social welfare in a procurement auction? In single item forward-auctions, the social welfare is defined as:
$$\sum_{i=1}^{n} v_i x_i$$
where $v_i$ is buyer $i$'s valuation for the item, and $x_i$ is a binary variable indicating whether or not $i$ gets the item.
What is definition of social welfare in a reverse auction(procurement auction)?
You need to add $x_0 v_0$ to your social welfare, where $v_0$ is the value the seller assigns to keeping the good and $x_0=1$ in that case. Then, the efficient (social-welfare-maximizing) allocation is that the highest-value buyer gets the good if her value is above $v_0$ - otherwise it is efficient that the seller keeps the good.
Similarly, in a procurement auction, we have one buyer who values the good to be procured at $v_0$ and $n$ sellers who can produce the good at private cost $c_i$. Social welfare is then
$$\sum_{i=1}^n x_i (v_0 - c_i),$$
and, hence, it is efficient to let the lowest-cost firm produce the good given its cost is below $v_0$ - otherwise it is efficient that the good is not produced.
For sufficiently high values $v_0$, you can also define it as $\sum_{i=1}^n x_i (- c_i).$