For each of the following questions, justify your answer using complementary slack- 1 ness and if the given solution is optimal, give an optimal solution to the dual as well. (a) Determine whether or not x = (0, 4, 0, 2) is an optimal solution to the following linear program. maximize subject to 9x1 + 3x2 + 5x3 + 22x4 2x1 - x2 + 2x3 + 6x4≤ 8, 5x1 + 3x2 + x3 + 2x4≤ 16, 4x1 + x2-x3 + 3x4≤ 12, I1, I2, I3, I4 ≥0 (b) Determine whether x¹ = (0, 2, 3) is an optimal solution to the following linear program. maximize subject to 1/2x₁ + x₂ + x3 2x1 +7x2 + x3 ≤ 8, x1 + 3x2 + 3x3 ≤7, 2x1 +4x3 ≤ 6, ₁+ 3x₂ + x3 ≤ 9, I1, I2, I3 20 (c) Determine whether x¹ = (0, 1, 0) is an optimal solution to the following linear program. maximize subject to 3x1 + 8x2 + x3 2x₁ + 8x2 + 2x3 ≤ 4, 3x1 +4x2+2x3 ≤ 4, 1x₁ + 2x2 + 1x3 ≤ 1, T1, T2, T30
For each of the following questions, justify your answer using complementary slack- 1 ness and if the given solution is optimal, give an optimal solution to the dual as well. (a) Determine whether or not x = (0, 4, 0, 2) is an optimal solution to the following linear program. maximize subject to 9x1 + 3x2 + 5x3 + 22x4 2x1 - x2 + 2x3 + 6x4≤ 8, 5x1 + 3x2 + x3 + 2x4≤ 16, 4x1 + x2-x3 + 3x4≤ 12, I1, I2, I3, I4 ≥0 (b) Determine whether x¹ = (0, 2, 3) is an optimal solution to the following linear program. maximize subject to 1/2x₁ + x₂ + x3 2x1 +7x2 + x3 ≤ 8, x1 + 3x2 + 3x3 ≤7, 2x1 +4x3 ≤ 6, ₁+ 3x₂ + x3 ≤ 9, I1, I2, I3 20 (c) Determine whether x¹ = (0, 1, 0) is an optimal solution to the following linear program. maximize subject to 3x1 + 8x2 + x3 2x₁ + 8x2 + 2x3 ≤ 4, 3x1 +4x2+2x3 ≤ 4, 1x₁ + 2x2 + 1x3 ≤ 1, T1, T2, T30
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:2. For each of the following questions, justify your answer using complementary slack-
ness and if the given solution is optimal, give an optimal solution to the dual as
well.
(a) Determine whether or not x¹ = (0, 4, 0, 2) is an optimal solution to the following
linear program.
maximize
subject to
9x1 + 3x2 + 5x3 + 22x4
2x1 - x2 + 2x3 + 6x4 ≤ 8,
5x1 + 3x2 + x3 + 2x4≤ 16,
4x1 + x2 = x3 + 3x4≤ 12,
I1, I2, I3, I4 ≥ 0
(b) Determine whether x¹ = (0, 2) is an optimal solution to the following linear
X
program.
1/1/2x₁
maximize ₁ + x₂
subject to
maximize
subject to
x2 + x3
2x1 + 7x2 + x3 ≤ 8,
x₁ + 3x2 + 3x3 ≤ 7,
2x1
+ 4x3 ≤ 6,
x₁ + 3x2 + x3 ≤ 9,
I1, I2, I3 > 0
X
(c) Determine whether x¹ = (0,2,0) is an optimal solution to the following linear
program.
3x1 + 8x₂ + x3
2x₁ + 8x2 + 2x3 ≤ 4,
3x₁ +4x2+2x3 ≤ 4,
1x₁ + 2x₂ + 1x3 ≤ 1,
I1, I2, I3 > 0
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