1. Solve the following linear program using the simplex algorithm:
maxz = 10x1 + 6x2 + 4x3
subject to
4x1 + 5x2 + 2x3 + x4 ≤ 20 3x1 + 4x2 − x3 + x4 ≤ 30 x1, x2, x3, x4 ≥ 0
2. Solve the following linear program using the simplex algorithm: (careful: is this linear program in standard form?)
minz = −7x1 − 8x2
subject to
4x1 + x2 ≤ 100 −2x1 − 2x2 ≥−160 x1 ≤ 40 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.
2
3. Solve the following linear program using the simplex algorithm and a suitable auxiliary program:
maxz = 2x1 + 6x2
subject to
−x1 − x2 ≤−3 −3x1 + 3x2 ≤ 3 x1 + 2x2 ≤ 2 x1, x2 ≥ 0 optional: Use the graphical method to find the region of feasible solutions.
4. Solve the following linear program using the simplex algorithm and a suitable auxiliary program: (careful: is this linear program in standard form?)
minz = −2x1 − 3x2 − 4x3
subject to
2x2 + 3x3 ≥ 5 x1 + x2 + 2x3 ≤ 4 x1 + 2x2 + 3x3 ≤ 7 x1, x2, x3 ≥ 0
5. Explain why 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 which it came?
