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Partial ordering of OMPs

Based on the combinations of OMPs, a partial order $ \preccurlyeq$ 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 ($ =$), smaller preference ($ \prec$), greater preference ($ \succ$) or incomparable preference ($ ?$). This calculus is developed assuming the following:

From the above, it can be derived that given two OMPs $ P_1$ and $ P_2$ and an order of magnitude $ O$, $ P_1$ is less or equally preferred to $ P_2$ with respect to the order of magnitude $ O$ (denoted $ P_1\preccurlyeq_O P_2$) provided that

$\displaystyle \forall b_i\in O, \big(f_{P_1}(b_i)+\sum_{b_j\in O, b_i<b_j}f_{P_1}(b_j)\big)\leq\big(f_{P_2}(b_i)+\sum_{b_j\in O, b_i<b_j}f_{P_2}(b_j)\big)$    

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

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 $ P_1=b_{11}\oplus b_{13}$ and the preference of a scenario model with a Roger's host-parasitoid model and a Lotka-Volterra predation model is $ P_2=b_{11}\oplus
b_{15}$. The latter model is less than or equally preferred to the former within the ``host-parasitoid'' order of magnitude ($ O_1$), i.e. $ P_2\preccurlyeq_{O_1} P_1$, because

$\displaystyle f_{P_2}(b_{11})=1$ $\displaystyle \leq 1=f_{P_1}(b_{11}),$    
$\displaystyle f_{P_2}(b_{11})\oplus f_{P_2}(b_{12})=1$ $\displaystyle \leq 1=f_{P_1}(b_{11})\oplus f_{P_1}(b_{12}),$    
$\displaystyle f_{P_2}(b_{11})\oplus f_{P_2}(b_{13})=1$ $\displaystyle \leq 2=f_{P_1}(b_{11})\oplus f_{P_1}(b_{13}),$    
$\displaystyle f_{P_2}(b_{11})\oplus f_{P_2}(b_{12})\oplus f_{P_2}(b_{14})=1$ $\displaystyle \leq 1=f_{P_1}(b_{11})\oplus f_{P_1}(b_{12})\oplus f_{P_1}(b_{14}),$    
$\displaystyle f_{P_2}(b_{11})\oplus f_{P_2}(b_{12})\oplus f_{P_2}(b_{13})\oplus f_{P_2}(b_{14})=2$ $\displaystyle \leq 2=f_{P_1}(b_{11})\oplus f_{P_1}(b_{12})\oplus f_{P_1}(b_{13})\oplus f_{P_1}(b_{14}).$    

Similarly, it can be established that the reverse, i.e. $ P_1\preccurlyeq_{O_1} P_2$, is not true. Therefore, the latter scenario model is less preferred than the former within $ O_1$, i.e. $ P_2\prec_{O_1} P_1$.

The above result can be further generalised such that given two OMPs $ P_1$ and $ P_2$, $ P_1$ is less or equally preferred to $ P_2$ (denoted $ P_1\preccurlyeq P_2$) if

$\displaystyle \forall O_i\in\mathbf{O},(P_1\preccurlyeq_{O_i} P_2)\vee(\exists O_j\in\mathbf{O}, O_i\ll O_j\wedge P_1\prec_{O_j} P_2)$    

More generally, the relations $ \prec$, $ \succ$, $ =$ and $ ?$ can be derived in the same manner as with the relation $ \preccurlyeq$ where $ \prec_O$, $ \succ_O$, $ =_O$ and $ ?_O$ with $ \preccurlyeq_O$.

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:

As demonstrated earlier, it can be shown that $ P_1=_{O_1}P_2$, $ P_1\succ_{O_2}P_2$, and $ P_1\prec_{O_3}P_2$. From these relations it follows that $ P_1\preccurlyeq P_2$ because As the reverse is not true, it can be concluded that scenario model 2 is preferred over scenario model 1.


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