MA 3231
Linear Programming
Assignment 2
1. Solve the following linear program using the simplex algorithm:
max z = 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?)
min z = −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:
max z = 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?)
min z = −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?
