Description

Department of Mathematical Sciences
MA 3231
Linear Programming
Assignment 2
1. Solve the following linear program using the simplex algorithm:
maxz = 10×1 + 6×2 + 4×3
subject to
4×1 + 5×2 + 2×3 + x4 ≤ 20 3×1 + 4×2 − 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 = −7×1 − 8×2
subject to
4×1 + x2 ≤ 100 −2×1 − 2×2 ≥−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 = 2×1 + 6×2
subject to
−x1 − x2 ≤−3 −3×1 + 3×2 ≤ 3 x1 + 2×2 ≤ 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 = −2×1 − 3×2 − 4×3
subject to
2×2 + 3×3 ≥ 5 x1 + x2 + 2×3 ≤ 4 x1 + 2×2 + 3×3 ≤ 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 −2×2

x1 = 3 −2×2
w2 = 1 +w1 −2×2
Hint: If it could, what was the original problem from which it came?