1
15.093J/2.098J
Optimization
Methods
Assignment
2
Solutions
Exercise
2.1
BT,
Exercise
3.17.
The
initial
tableau
in
Phase
I
is
−
5
−
1
−
1
−
3
−
1
−
2000
x
6
=2
1
∗
3
0
4
1
1
0
0
x
7
1
2
0
−
3
1
0
1
0
x
8
=1
−
1
−
4
3
0
0
0
0
1
The
ﬁnal
tableau
in
Phase
I
is
0
0
1
0
7
0
2
0
1
x
1
1
3
0
4
1
1
0
0
x
7
=0
0
−
1
∗
0
−
7
0
−
11
0
x
3
01
/
314
/
31
/
/
301
/
3
Drive
the
artiﬁcal
variable
x
7
out
of
the
basis
0
0
0
0
0
0
1
1
1
x
1
100
−
17
1
−
2
3
0
x
2
010 7
0
1
−
1
0
x
3
0011
1
/
/
32
/
3
−
1
/
/
3
The
initial
tableau
in
Phase
II
is
−
7
000
3
−
5
x
1
−
17
1
∗
x
2
010
7
0
x
3
1
/
/
3
3
5
8
2
/
7
x
5
1
1
7
/
7
001
x
4
0
1
/
7
x
3
/
3
−
1
/
3
−
4
/
3100
The
ﬁnal
tableau
in
Phase
II
is
Exercise
2.2
BT,
Exercise
3.19.
−
10
δ
−
4
1
η
1
α
−
4010
β
γ
3
(a)
The
current
solution
is
optimal
but
the
current
basis
is
not.
Thus
the
current
solution
is
a
degenerate
optimal
solution.
So
we
have
β
=
0.
Update
the
tableau
using
one
simplex
iteration
−
10
δ
+
2
γ
0002
/
3
3
−
1
−
γη
x
3
=4
−
η
3
3
x
4
α
+
4
γ
0014
/
3
3
γ
x
2
1001
/
3
3
There
are
multiple
optimal
solutions,
thus
δ
+
2
3
γ
=
0.
In
addition,
we
need
to
have
a
feasible
direction,
which
requires
γ
≤
0.
3
The
conditions
could
be
β
,
δ
+
2
3
γ
,
and
γ
≤
0.
(b)
The
optimal
cost
is
−∞
when
we
have
a
feasible
solution
in
the
current
tableau,
a
nonbasic
variable
x
i
with
c
i
<
0and
u
i
=
B
−
1
A
i
≤
0
.
We
need
β
≥
0
for
problem
feasiblity.
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The
variable
x
2
cannot
satisfy
all
the
conditions
for
the
−∞
cost.
For
the
variable
x
1
,
the
conditions
then
can
be
expressed
as
follows:
α
≤
0,
λ
≤
0,
and
δ<
0.
(c)
The
current
solution
is
feasible
if
β
≥
0.
If
the
solution
is
not
degenerate
then
the
current
solution
is
deﬁnitely
not
optimal.
Thus
a
condition
could
be
simply
β>
0.
Exercise
2.3
BT,
Exercise
3.31.
(a)
The
set
of
all
(
b
1
,b
2
)
is
the
convex
hull
of
four
points
A
1
(2
,
1),
A
2
(3
,
2),
A
3
(1
,
1),
and
A
4
(1
,
3).
(b)
There
are
more
than
one
basis
corresponding
to
a
degenerate
basic
feasible
solution.
The
set
of
all
(
b
1
2
)
is
the
union
of
six
segments
A
1
A
2
,
A
1
A
3
,
A
1
A
4
,
A
2
A
3
,
A
2
A
4
,and
A
3
A
4
.
(c)
If
(
b
1
2
)
is
one
of
the
four
points
A
i
,
then
there
is
a
basic
feasible
solution
associated
with
it
and
this
basic
feasible
solution
has
three
bases.
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 Spring '08
 Johanning
 Operations Research, Linear Programming, Optimization, basic feasible solution, medium size yarn, De Blasi

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