Linear Programming
Michelle_Michy
LinearProgramming.docx
1. a)
Introducing the slack variables:
SIMPLEX TABLE
|
BASIC |
X |
y |
Z |
|
|
|
Q |
SOLUTION |
|
|
1 |
0 |
2 |
1 |
0 |
0 |
0 |
24 |
|
|
1 |
2 |
4 |
0 |
1 |
0 |
0 |
28 |
|
|
1 |
-2 |
6 |
0 |
0 |
1 |
0 |
44 |
|
Q |
-2 |
-1 |
-6 |
0 |
0 |
0 |
1 |
0 |
b)
The highest negative number in the last row is -6. The pivot is determined by division
|
BASIC |
X |
y |
Z |
|
|
|
Q |
SOLUTION |
|
|
1 |
0 |
2 |
1 |
0 |
0 |
0 |
24 |
|
z |
|
|
1 |
0 |
|
0 |
0 |
7 |
|
|
1 |
-2 |
6 |
0 |
0 |
1 |
0 |
44 |
|
Q |
-2 |
-1 |
-6 |
0 |
0 |
0 |
1 |
0 |
|
BASIC |
X |
y |
Z |
|
|
|
Q |
SOLUTION |
|
|
|
-1 |
0 |
|
0 |
0 |
0 |
10 |
|
z |
|
|
1 |
0 |
|
0 |
0 |
7 |
|
x |
|
-5 |
0 |
0 |
|
|
0 |
2 |
|
Q |
-
|
2 |
0 |
0 |
|
0 |
1 |
42 |
|
BASIC |
X |
y |
Z |
|
|
|
Q |
SOLUTION |
|
|
|
-1 |
0 |
|
0 |
0 |
0 |
10 |
|
z |
-1 |
2 |
0 |
-2 |
1 |
0 |
0 |
8 |
|
x |
1 |
10 |
0 |
0 |
3 |
-2 |
0 |
-4 |
|
Q |
-
|
2 |
0 |
0 |
|
0 |
1 |
42 |
|
BASIC |
X |
y |
Z |
|
|
|
Q |
SOLUTION |
|
|
0 |
-6 |
0 |
1 |
-2 |
1 |
0 |
12 |
|
z |
0 |
-2 |
1 |
0 |
|
0 |
0 |
8 |
|
x |
1 |
10 |
0 |
0 |
3 |
-2 |
0 |
-4 |
|
Q |
0 |
7 |
0 |
0 |
3 |
-1 |
1 |
40 |
c)
Determining y using Q
From the table we get the variable values as:
2. a)
The problem is balanced, because the total supply is equivalent to the demand
Demand = 20+56+40=116
Supply=34+57+25=116
b)
2
S
3
S
1
S
2
S
3
S
12
2
24
1
=
=
S
7
4
28
2
=
=
S
3
.
7
6
44
3
=
=
S
2
S
1
S
4
1
2
1
4
1
3
S
z
Q
Q
z
S
S
z
S
S
6
6
2
3
3
1
1
+
=
-
=
-
=
1
S
2
1
2
1
-
4
1
2
1
4
1
2
1
-
2
3
-
1
2
1
2
3
3
S
0
0
0
0
6
2
0
,
,
,
,
,
44
6
2
28
4
2
24
2
3
2
1
3
2
1
3
2
1
=
+
+
+
-
-
-
³
=
+
+
-
=
+
+
+
=
+
+
S
S
S
z
y
x
Q
S
S
S
z
y
x
S
z
y
x
S
z
y
x
S
z
x
4
5
.
0
2
28
25
.
0
7
20
5
.
0
10
3
1
-
=
-
=
=
=
=
=
S
z
S
1
S
2
3
x
Q
Q
z
x
z
x
S
S
5
.
0
5
.
0
1
1
-
=
+
=
-
=
1
S
1
S
12
,
8
,
4
,
40
1
=
=
-
=
=
S
z
x
Q
0
44
)
8
(
6
)
0
(
2
4
44
6
2
0
28
)
8
(
4
)
0
(
2
4
28
4
2
0
40
)
8
(
6
)
4
(
2
6
2
3
3
3
2
2
2
=
=
+
+
-
-
=
+
+
-
=
=
+
+
+
-
=
+
+
+
=
=
+
+
-
=
+
+
S
S
S
z
y
x
S
S
S
Z
y
x
y
y
Q
z
y
x
40
6
2
=
+
+
z
y
x
