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## Preference combination

There are several ways to combine different preference criteria--or, in other words, building new posets from existing posets. The Noos operations dealing with preference combination are methods that create new posets from (a combination of) posets--posets created by preference methods. We will show here the hierarchical union of posets, that is tantamount to consider some preferences more important (more preferred) than others. Let us imagine we prefer PCs with PowerPC chip and we prefer cheaper PCs. If we take both preference criteria as equal, we can construct the poset that is the union of both posets. Given two preference defined over the same set A we define their transitive union preference as

where is the transitive closure of .

The result in our example is that we prefer both PC-white   PC-red (the cheaper) and PC-red    PC-white (the PowerPC is preferred). This is equivalent to build a poset as the poset union of the results of cheaper-pref and ppc-pref. The problem now having a cycle in preferences is that we are incapable of deciding (preferring) PC-white or PC-red. The hierarchical union h-union of preferences can be considered as a preference over preferences such that, after some order among pairs of elements have been is added by a preference, the next preference method is allowed to add new order only among pairs that are not already ordered. Formally, given two posets defined over the same set A the hierarchical union poset is defined as follows:

Forbidding the addition of when already has been set by a prior (more preferred) preference simply we avoid cycles. Indeed, we can consider hierarchical union of preferences as a process of refining a partial order by means of new preference criteria. This is shown in the next example where the h-union method embodies the hierarchical union of higher-poset and lower-poset.

The subsumption-preference takes the unordered set of PCs and adds the preferences for PowePC computers, and returns the appropriate poset where PC-blue and PC-white are preferred to PC-red. Now this poset is the value of higher-poset feature of h-union that adds the cheaper preferences of lower-poset only when a higher preference is not already established. In this situation, only a preference among PC-blue and PC-white can be added, and since PC-blue is cheaper it is preferred to PC-white. The final result of ppc-cheaper-pref is then poset {PC-blue PC-white PC-red} that contains no cyclic preferences.

Next: Follow Up Up: Preferences Previous: Preference methods

Enric Plaza
Thu Jan 23 11:36:28 MET 1997