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n
2
Þþ
n
i
1
Þþ
n
3
Þ
¼
P
ð
X
i
¼
P
ð
X
j
¼
P
ð
X
k
¼
1
n
j
2
n
2
!
n
i
1
n
i
1
!
n
k
3
n
3
!
)
k
e
k
þ
k
e
k
þ
k
e
k
¼
1
n
j
2
n
i
1
n
k
3
n
3
!
Þ
¼
e
k
ð
k
n
i
1
!
þ
k
n
2
!
þ
k
)
1
n
2
n
3
n
3
!
Þ
|
{z
}
P
ðkÞ
n
i
1
)ð
k
n
i
1
!
þ
k
n
2
!
þ
k
e
k
¼
)
P
ðkÞ
¼
e
k
:
Thus to have a feasible system, it is necessary to
find the value of
k
, i.e., that is to
e
k
. In this case, we restraint
solve the equation: P
ðkÞ
¼
to the mathematical
problem said problem of the
fixed point (f(x) = x), or the problem of the contracting
function f: E
fixes of f.
The purpose of this work is to establish this theorem, said about
E which contracts on a point, the unique point
→
fixed point.
Here, we have:
e
k
P
ðkÞ
¼
e
k
Þ
¼
k
)
Log
ð
P
ðkÞÞ
¼
Log
ð
)
f
ðkÞ
¼
k;
with f
¼
Log
ð
P
ðkÞÞ:
Diagrammatically the unique value of
is the intersection of both curves y = P
λ
¼
e
k
; it is the focused solution.
(
λ
) and z
Running Example
n
2
¼
n
i
1
¼
To more
illustrate
this
point, we
assume
that
1
;
2
and
2
2
þ
k
3
6
.
n
3
¼
ðkÞ
¼
k þ
k
3
)
P
So,
2
2
þ
k
3
6
¼
e
k
) k þ
k
e
k
:
ðkÞ
¼
P
= 0 is the unique solution and even by graphic
resolution we obtain the intersection of both curves y
In this case and mathematically,
λ
2
2
þ
k
3
6
ðkÞ
¼
k þ
k
¼
P
and
e
k
, is the unique point
z
0. Thus to have a feasible system with one task of
type i, two tasks of type j and three tasks of type k, i
¼
k
¼
fix the value of lambda to 0 in
my program. In this case the program would return the optimal solution.
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