Geoscience Reference
In-Depth Information
For weighted average:
A i ¼
S j w j A ij
and
A i ¼
S j w j A ij ;
for w j denoting the weight assigned to the jth criterion.
8.4 The Algorithm of Classification
cation procedure is based on the comparison of the differences A i +
A i .
The classi
cation rule is simple: alternative A belongs to the class i with the
smallest absolute value for A i +
The classi
A i .
Since there is the possibility of ties, the alternative may be classi
ed in two
adjacent classes. With the distributions and pro
les constructed as above described,
no more than two classes can present the same absolute value for the difference
A i +
A i .
If the criteria obey the principle of assigning values increasing with the pref-
erence for the alternative, these differences constitute a non-increasing sequence.
Then, an ef
cient algorithm for applying this rule can be developed in two steps. In
A i is
negative or null. If this difference is always positive, the alternative must belong to
the highest class. If it is never positive, this value is 1 and the alternative is placed in
the lowest class. If the difference is zero for the class with the smallest absolute
value, then the alternative is placed in that class.
If none of these cases happen, a second step consists in comparing the absolute
values of the differences A i +
es the smallest value of i for which the difference A i +
the
first step, it identi
A i for the class with the
first negative difference and
for that one preceding it. If these absolute values are the same, the alternative is
classi
ed in these two classes. If they are different, it is placed in that one of these
two classes with the smallest absolute value for the difference.
Theoretically, due to continuity, possibility of ties can be neglected. But, if the
number of criteria and the number of values for each criterion is not too small, ties
may be bring to happen in practice if approximations make small the numerical
differences, specially when the minimum is used to combine evaluations, in the
joint probabilities composition under the assumption of dependence. The possibility
of allocation in a pair of classes or even in a larger interval of acceptable values of i
is a natural consequence of the imprecision of the subjective process of determining
preferences.
A procedure to automatically determine the extremes of such an interval may be
formalized by following a descending procedure that stops at the highest possible
point, to determine the highest extreme, and an ascending procedure that stops at
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