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### Partial ordering of OMPs

Based on the combinations of OMPs, a partial order over the OMPs can be computed by exploiting the constituent BPQs of the OMPs considered. This partial order implies that a comparison of any pair of OMPs either returns equal preference (), smaller preference (), greater preference () or incomparable preference (). This calculus is developed assuming the following:
• Prioritisation: A combination of BPQs is never an order of magnitude greater than its constituent BPQs. That is, given the set of BPQs belonging to the same order of magnitude and a BPQ belonging to a higher order of magnitude, i.e. , then

With respect to the ongoing example, this means that any BPQ taken from the order of magnitude is preferred over any combination of BPQs taken from . In other words, the choice of a model to describe a host-parasitoid phenomenon is considered more important than the choice of population growth model (see Figure 1).

Prioritisation also means that distinctions at higher orders of magnitude are considered to be more significant than those at lower orders of magnitude. Consider a number of BPQs taken from one order of magnitude and a pair of BPQs taken from an order of magnitude that is higher than . If , then (irrespective of the ordering of the BPQs taken from )

• Strict monotonicity: Even though distinctions at higher orders of magnitude are more significant, distinctions at lower orders of magnitude are not negligible. That is, given an OMP and two BPQs and taken from the same order of magnitude with , then (irrespective of the orders of magnitude of the BPQs that constitute )

For instance, the preference ordering depicted in Figure 1 shows that a scenario model with a Roger's host-parasitoid model and two logistic predation models is preferred over one with a Roger's host-parasitoid model and two exponential predation models:

Note that this is a departure from conventional order-of-magnitude reasoning. If the OMPs associated with two (partial) outcomes contain equal BPQs at a higher order of magnitude, it is usually desirable to compare both solutions further in terms of the (less important) constituent BPQs at lower orders of magnitude, as the example illustrated. However, conventional order-of-magnitude reasoning techniques can not handle this.

• Partial ordering maintenance: Conventional order-of-magnitude reasoning is motivated by the need for abstract descriptions of real-world behaviour, whereas the OMP calculus is motivated by incomplete knowledge for decision making. As opposed to conventional order-of-magnitude reasoning and real numbers, OMPs are not necessarily totally ordered. This implies that, when the user states, for example, that and that , the explicit absence of ordering information between the BPQs in and those in means that the user is unable to compare them (e.g. because they are entirely different things). Consequently, would be deemed incomparable to (i.e. ), rather than roughly equivalent.

From the above, it can be derived that given two OMPs and and an order of magnitude , is less or equally preferred to with respect to the order of magnitude (denoted ) provided that

Thus, comparing two OMPs within an order of magnitude can yield four possible results:

• is less preferred than with respect to ( ) iff ,
• is more preferred than with respect to ( ) iff ,
• is equally preferred than with respect to ( ) iff , and
• is incomparable to with respect to ( ) iff .

In the ongoing example of Figure 1, for instance, the preference of a scenario model with a Roger's host-parasitoid model and a Holling predation model is and the preference of a scenario model with a Roger's host-parasitoid model and a Lotka-Volterra predation model is . The latter model is less than or equally preferred to the former within the host-parasitoid'' order of magnitude (), i.e. , because

Similarly, it can be established that the reverse, i.e. , is not true. Therefore, the latter scenario model is less preferred than the former within , i.e. .

The above result can be further generalised such that given two OMPs and , is less or equally preferred to (denoted ) if

More generally, the relations , , and can be derived in the same manner as with the relation where , , and with .

To illustrate the utility of such orderings, consider the scenario of one predator population that feeds on two prey populations while the two prey populations compete for scarce resources. The following are two plausible scenario models for this scenario:

• Model 1 contains two Holling predation models and three logistic population growth models, and has preference .
• Model 2 contains one competition model, two Holling predation models, two logistic population growth models and an exponential population growth model, and has preference .
As demonstrated earlier, it can be shown that , , and . From these relations it follows that because
• for : since ,
• for : there exists an order of magnitude where and ,
• for : since .
As the reverse is not true, it can be concluded that scenario model 2 is preferred over scenario model 1.

Next: Solving aDPCSPs Up: Order-of-magnitude preferences (OMPs) Previous: Combinations of OMPs
Jeroen Keppens 2004-03-01