Exercise 1
(20 marks)
.
(a) Let
x
i
be 1 if block
i
is mined and 0 otherwise (
i
= 1
,
2
,
3
,
4
,
5
for A, B, C, D, E respectively).
The formulation without matrices would be
max
25
x
1
+ 13
x
2
+ 15
x
3
+ 20
x
4
+ 12
x
5
subject to
7
x
1
+ 12
x
2
+ 4
x
3
+ 12
x
4
+ 9
x
5
≥
20
2
x
1
+
x
2
+
x
3
+ 2
x
4
+
x
5
≤
4
0
≤
x
i
≤
1
(
i
= 1
,...,
5
)
x
i
integer
(
i
= 1
,...,
5
)
It was asked to formulate this in terms of matrices and vectors, which becomes
max
[25 13 15 20 12]
x
subject to
Ax
≥
≤
"
20
4
#
0
≤
x
≤
1
x
i
integer
(
i
= 1
,...,
5
)
where
x
= [
x
1
,x
2
,x
3
,x
4
,x
5
]
T
,
A
=
"
7
12
4
12
9
2
1
1
2
1
#
,
0
= [0 0 0 0 0]
T
,
1
= [1 1 1 1 1]
T
.
OR As an alternative, multiply the second inequality by

1
and replace the constraint involving
A
by
Ax
≥
[20

4]
T
where
A
is now
"
7
12
4
12
9

2

1

1

2

1
#
.
(b) Introduce a variable
x
6
such that
x
6
= 1
represents that we exercise the option of increasing the amount
of Impurity X to 5 tonnes, and
x
6
= 0
stands for not exercising the option. We just need to subtract