Solve the following linear programming problem using the two-stage simplex method.
Minimise
subject to
Minimising is the same as maximising , where
will need rewriting in the correct form for the initial tableau
Use slack, surplus and artificial variables to rewrite the constraints as equations
Introduce, find and write in the correct format for the initial tableau
Let
Construct the initial tableau
b.v. |
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|
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|
|
|
|
Value |
|
1 |
8 |
-1 |
0 |
0 |
1 |
0 |
9 |
|
1 |
1 |
0 |
1 |
0 |
0 |
0 |
9 |
|
6 |
-1 |
0 |
0 |
-1 |
0 |
1 |
5 |
|
4 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
|
-7 |
-7 |
1 |
0 |
1 |
0 |
0 |
-14 |
Apply the simplex algorithm to maximise
There is a negative entry in the () objective row
-7 is the most negative (either can be chosen, we're choosing the first)
C1 is the pivot column, the -values are
is the least positive so R3 is the pivot row
The pivot element is in cell R3C1
Pivot is 6
Apply the appropriate row operations, starting with the pivot row, R3
b.v. |
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Value |
Row Op. |
|
0 |
49/6 |
-1 |
0 |
1/6 |
1 |
-1/6 |
49/6 |
|
|
0 |
7/6 |
0 |
1 |
1/6 |
0 |
-1/6 |
49/6 |
|
|
1 |
-1/6 |
0 |
0 |
-1/6 |
0 |
1/6 |
5/6 |
|
|
0 |
8/3 |
0 |
0 |
2/3 |
0 |
-2/3 |
-10/3 |
|
|
0 |
-49/6 |
1 |
0 |
-1/6 |
0 |
-7/6 |
-49/6 |
|
There is a negative entry in the () objective row
-49/6 is the most negative
C2 is the pivot column, the -values are
is the least positive so R1 is the pivot row
The pivot element is in cell R1C2
Pivot is 49/6
Apply the appropriate row operations, starting with the pivot row, R1
b.v. |
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Value |
Row Op. |
|
0 |
1 |
-6/49 |
0 |
1/49 |
6/49 |
-1/49 |
1 |
|
|
0 |
0 |
1/7 |
1 |
1/7 |
-1/7 |
-1/7 |
7 |
|
|
1 |
0 |
-1/49 |
0 |
-8/49 |
1/49 |
8/49 |
1 |
|
|
0 |
0 |
16/49 |
0 |
30/49 |
-16/49 |
-30/49 |
-6 |
|
|
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
|
There are no negative entries in the () objective row so the first-stage is complete,
Remove from the tableau to complete the first-stage
b.v. |
|
|
|
|
|
Value |
|
0 |
1 |
-6/49 |
0 |
1/49 |
1 |
|
0 |
0 |
1/7 |
1 |
1/7 |
7 |
|
1 |
0 |
-1/49 |
0 |
-8/49 |
1 |
|
0 |
0 |
16/49 |
0 |
30/49 |
-6 |
There are no negative values in the objective row so the method is complete (no second-stage required) and the optimal solution can be read from the final tableau
are basic variables, are non-basic
Change from maximising to minimising
The solution to the linear programming problem is
with minimised at
()