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- Find all the values of the parameter such that the following linear program has a finite optimal solution:
maxz = x1 + 2x2 x3
subject to
2x1 x2 + 3x3 4
x1 + x2 2x3 8 3x1 3x3 2 x1, x2, x3 0
- Solve the following linear program using the simplex algorithm and a suitable auxiliary program:
maxz = x1 + 3x2
subject to
x1 x2 3 x1 + x2 1 x1 + 2x2 2
x1, x2 0
Draw the region of feasible solution to this problem and indicate the solution you get at each step of the simplex algorithm (only for the original problem, i.e., Phase II).
- Solve the following linear program using the simplex algorithm and a suitable auxiliary program:
maxz = 2x1 + 3x2 + 4x3
subject to
2x2 3x3 5 x1 + x2 + 2x3 4 x1 + 2x2 + 3x3 7 x1, x2, x3 0
- Show that the following dictionary cannot be the optimal dictionary for any linear programming problem in which w1 and w2 are the initial slack variables:
| z | = | 4 | w1 | 2x2 |
| x1 | = | 3 | 2x2 | |
| w2 | = | 1 | +w1 | 2x2 |
Hint: If it could, what was the original problem from whence it came?
- Consider the following linear programming problem:
maxz = 2x1 3x2 + 2x3 + 12x4
subject to
4x1 + 5x2 + 2x3 10
2x1 x3 + x4 30 4x2 + 2x3 + x4 20 x1, x2, x3, x4 0
Solve it numerically using the built-in simplex algorithm of a spreadsheet program (such as MS Excel). Please just submit the .xlsx file as submission comment to your homework on Canvas.
- Reconsider the degenerate dictionary of Lecture 2.8 that led to the cycling issue:
| z | = | x1 | 2x2 | 2x4 | ||||||
| w1 | = | 0.5x1 | + | 3.5x2 | + | 2x3 | 4x4 | |||
| w2 | = | 0.5x1 | + | x2 | + | 0.5x3 | 0.5x4 | |||
| w3 | = | 1 | x1 |
Write down the corresponding linear programming problem and find the optimal solution by using
- a) the lexicographic method (comment on the choice of the entering variable). b) Blands rule
8 points per problems
Problems to be discussed in the Office Hours
- Solve the following linear program using the lexicographic method to avoid degeneracy:
maxz = 2x1 + 3x2 + 4x3
subject to
4x1 x2 0
x1 + x3 0 2x2 3x3 0 x1, x2, x3 0
How about Blands rule?
- Solve the following linear programming problem by initializing it with an auxiliary program.
maxz = 3x1 + 2x2
subject to
x1 x2 1 x1 + x2 2 x1, x2 0
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