1.
(a)
[2 marks]
Define the term
extreme point
.
A point
x
is an extreme point
of a convex set
C
if it does not lie on any open linesegment
in
C
. Algebraically: there do not exist distinct points
y, z
∈
C
and a number
λ
∈
(0
,
1)
satisfying
x
=
λy
+ (1

λ
)
z
.
Now consider a system of constraints
Ax
=
b
x
≥
0
.
Suppose that the vector [2
,
3
,
0
,
4
,
0]
T
is a basic feasible solution, and that the vector
[0
,
1
,
2
,
2
,
4]
T
is a feasible solution that may or may not be basic. Answer the following
questions,
justifying your answers carefully
.
(b)
[2 marks]
Exactly one of the following statements is correct. State which one, and
explain why it is correct.
(i)
A
must have no more than 3 rows.
(ii)
A
must have at least 3 rows.
Three of the components of the basic feasible solution
[2
,
3
,
0
,
4
,
0]
T
are nonzero, so
the corresponding basis must contain the three indices 1,2, and 4. Hence the basis
matrix has at least three columns. Since the basis matrix is square, it must contain
at least three rows, and hence so does the matrix
A
. So statement (ii) is correct.
(c)
[2 marks]
Find an extreme point of the feasible region.
Basic feasible solutions are the same as extreme points.
Hence
[2
,
3
,
0
,
4
,
0]
T
is an
extreme point.
(d)
[2 marks]
Find a feasible solution that is not an extreme point of the feasible
region.
Consider the point halfway between the two given points:
[1
,
2
,
1
,
3
,
2]
T
=
1
2
[2
,
3
,
0
,
4
,
0]
T
+
1
2
[0
,
1
,
2
,
2
,
4]
T
.
According to the definition, this point cannot be an extreme point.
1
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2.
[6 marks]
Use the simplex method to solve the linear program
maximize
x
1

x
2

x
3
subject to
2
x
1
+
x
2

x
3
≤
2
3
x
1
+
2
x
2
≤
4

x
1
+
x
3
≤
0
x
1
,
x
2
,
x
3
≥
0
.
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 '08
 TODD
 Optimization, basic feasible solution

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