Geoscience Reference
In-Depth Information
If {i: A i
A i +
0} is empty, then the alternative belongs to the lowest class,
UA
ðÞ¼
LA
ðÞ¼
C 1 :
If UP(A) = r, then the upper end of the interval is given by class C r ,
UA
ðÞ¼
C r :
For {i: A i A i +
0} not empty and UP(A) < r,
+
+
if A UP(A)
A UP(A)
<A UP(A)+1
A UP(A)+1
, then
UA
ðÞ¼
C UP ðÞ :
Otherwise,
if UP(A) + 1 = r, then
UA
ðÞ¼
C r ;
and,
if UP(A) + 1 < r,
compare A UP(A) 1
+
+
A UP(A) 1
with A UP(A) 2
A UP(A)+2
.
If A UP(A)+1 A UP(A)+1
+
+
<A UP(A)+2
A UP(A)+2
, then
UA
ðÞ¼
C UP ðÞþ 1 :
A UP(A)+ , then substitute UP(A) + 2 for
UP(A) + 1 as the provisional lower extreme and repeat the preceding step.
It is easy to see that the benevolent procedure so de
+
+
If A UP(A)+1
A UP(A)+1
=A UP(A)+2
ned will stop at a higher or
equal classi
cation than the hostile one. In fact, the descending procedure stops at
the highest i minimizing A i +
A i and the ascending procedure stops at the lowest i
minimizing that difference.
As already explained, unless rough approximations are employed, the minimum
will be unique and the benevolent and the hostile classi
cation will coincide. As
this uniqueness of the classi
cation may be unsatisfactory, because uncertainty is a
feature of preference measurements, one way to provide more information on the
subjacent uncertainty consists in enlarging the interval by making more benevolent
the benevolent procedure and more hostile the hostile procedure.
More benevolent and more hostile classi
cations will be based respectively on
more stringent rules for classifying below and above the pro
les of each class. Such
rules may be determined by applying
fixed rates of reduction respectively to the
probability of being below the representative pro
les of each class in the
descending procedure and to the probability of being above in the ascending
procedure.
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