Inclusion of firms on the efficient frontier of a Production Possibility Set I'm working through Cooper's Data Envelopment Analysis, and they are going over Production Possibility Sets.

They present 9 firms, each with two inputs and one output:

Graphically it easy to see why firms (E,D,C) make up the efficient frontier, however I am struggling to find a mathematical motivation for the inclusion of firm D? How do you show its input/output ratios warrant its inclusion algebraically as opposed to graphically.
Though I echo many of the sentiments posed by @AlecosPapadopoulos, maybe this would be helpful for your understanding. Ultimately, we can consider the efficiency frontier as the firms that make up the most efficient ways (or methods of production) of generating a particular output. For example, if our initial bundle of inputs was (listing them in $(x_1, x_2)$ pairs) $(2,5)$, the only firm that can generate a unit of output is firm E, hence its inclusion on the efficiency frontier.

But what about any bundle like $(4,3)$ or $(5,2)$? Clearly firms A and D can produce one unit of output with the first bundle, and firms D and F can generate one unit of output with the second bundle. However, in both cases, only firm D is the most efficient producer. 

(Why do we care? Perhaps if we're trying to consider the firm that satisfies a cost minimization problem. Suppose the price of each unit of $x_1$ is $1$ and the price of each unit of $x_2$ is $2$. Which firm will produce with the lowest cost? Firm D's production costs are only $\$8 $ per unit of $y$ (while both C and E's costs are $\$10$ per unit of output!), and hence if all goods are undifferentiated the cost curves are constant, firm D will dominate the market). 

Hope that clarifies a bit- let me know if I totally misunderstood your confusion.