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the lowest possible point, to determine the lowest one. The ascending procedure
will be named in what follows the benevolent procedure, and the descending one
the hostile procedure.
Denoting by U(A) the class in the upper extreme of the set where alternative A is
classi
ed and by L(A) the class in the lower extreme, let us start by the hostile
procedure.
The
first step consists in assigning a provisional lower classi
cation
A
i
A
i
LP A
ðÞ¼
min
f
i
0
g:
:
To determine this value, start with LP(A) = r; if A
r
+
A
r
−
≤
−
0, try r
−
1 and keep
reducing the value of i until this is no longer true.
If A
r
+
A
r
−
> 0, then{i: A
i
+
A
i
−
≤
−
−
0} is empty and the alternative belongs to
class C
r
, the highest class,
LA
ðÞ¼
C
r
:
The classi
cation is then punctual and the benevolent procedure needs no longer
be applied.
If LP(A) = 1, then the lower end of the interval is given by class C
1,
LA
ðÞ¼
C
1
:
For {i: A
i
+
A
i
−
≤
−
0} not empty and LP(A) > 1,
−
+
+
−
if A
LP(A)
−
−
A
LP(A)
<A
LP(A)
−
1
A
LP(A)
−
1
, then
LA
ðÞ¼
C
LP
ðÞ
:
Otherwise, if LP(A)
−
1 = 1, then the process
finishes with
LA
ðÞ¼
C
1
;
+
−
+
−
and, if LP(A)
−
1 > 1, compare A
LP(A)
−
1
−
A
LP(A)
−
1
with A
LP(A)
−
2
−
A
LP(A)
−
2
.
+
−
+
−
If A
LP(A)
−
1
−
A
LP(A)
−
1
<A
LP(A)
−
2
−
A
LP(A)
−
2
, then
LA
ðÞ¼
C
LP
ðÞ
1
:
+
−
+
−
If A
LP(A)
−
1
−
A
LP(A)
−
1
=A
LP(A)
−
2
−
A
LP(A)
−
2
, then substitute LP(A)
−
2 for
LP(A)
1 as the provisional lower extreme and repeat the preceding step.
The benevolent procedure follows the same route, starting by a provisional upper
bound
−
i: A
i
A
i
UP A
ðÞ¼
max
f
0
g:
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