Let
be the set of all BPQs with respect to a particular
decision problem. The BPQs in
are ordered with respect
to one another at two levels of granularity, by two relations
and . First,
is partitioned into orders of magnitude,
which are ordered by . Then, the BPQs within each
order of magnitude are ordered by . Formally, an
*order-of-magnitude ordering* over BPQs
is a tuple
, where
is a partition of
and is an irreflexive and
transitive binary relation over
. Any subset of BPQs
is said to be an *order of magnitude* in
. Similarly, a *within-magnitude ordering* over a
set of BPQs is a tuple
, where is an order of
magnitude in
and is an irreflexive and transitive
binary relation over .

To illustrate these ideas, consider the problem of constructing an ecological model describing a scenario containing a number of populations. Let some of the populations be parasites and others be hosts for these parasites. Also, assume that certain populations compete with others for scarce resources. In order to construct a scenario model, the compositional modeller must make a number of model design decisions: which population growth, host-parasitoid and competition phenomena are relevant, and which types of model best describe these phenomena.

Figure 1 shows a sample space of BPQs that
correspond to the selection of types of model. For the sake of
illustration, the presumption is made that the quality of a scenario
model depends on the inclusion of types of model, rather than on the
inclusion or exclusion of phenomena. Apart from and
, all BPQs correspond to standard textbook ecological
models^{1}. BPQ stands for the use of a
population growth model that is implicit in another population growth
model (the Lotka-Volterra model, for instance, implicitly includes its
own concept of growth). Finally, BPQ is the preference
associated with a competition model (say, the only one included in the
knowledge base).

The 9 BPQs in this sample space are partitioned over 3 orders of magnitude. The relation orders the orders of magnitude: and . The binary relation orders individual BPQs within an order of magnitude. In the BPQ ordering within , for instance, Rogers' host-parasitoid model () is preferred over that by Nicholson and Bailey () and the Holling predation model (). The latter two models can not be compared with one another, but they both are preferred over the Lotka-Volterra model. Furthermore, Thompson's host-parasitoid model is less preferred than that of Nicholson and Bailey, but it can not be compared with the Lotka-Volterra and Holling models.