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1. Suppose that p, q ER are both constants and consider the following linear program: maximize pr+q₂ subject to a₁ + 3x₂ ≤ 6, 3x1 + x₂ ≤ 6, x₁ + x₂ ≥ 1, #1, #2 20 (a) Sketch the feasible region for these constraints. Solution:

1. Suppose that p, q ER are both constants and consider the following linear program: maximize pr+q₂ subject to a₁ + 3x₂ ≤ 6, 3x1 + x₂ ≤ 6, x₁ + x₂ ≥ 1, #1, #2 20 (a) Sketch the feasible region for these constraints. Solution:

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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What is the reason for the d) answer? I cannot understand it. Please, if possible visualize why it is correct.

1. Suppose that p, q ER are both constants and consider the following linear program: maximize pr₁+qx₂ subject to ₁ + 3x2 ≤ 6, 3x1 + x2 ≤ 6, x₁ + x₂ ≥ 1, X1, X2 20 (a) Sketch the feasible region for these constraints. Solution:
Transcribed Image Text:1. Suppose that p, q ER are both constants and consider the following linear program: maximize pr₁+qx₂ subject to ₁ + 3x2 ≤ 6, 3x1 + x2 ≤ 6, x₁ + x₂ ≥ 1, X1, X2 20 (a) Sketch the feasible region for these constraints. Solution:
X2 21 (b) Find a value for the constants p and q so that x₁ = 3/2, x₂ = 3/2 is the unique optimal solution of the resulting program. Solution: By looking at the picture, one option is to set p = 1, q = 1, which gives us a direction (1,1) that makes this the only optimal solution. (c) Find a value for the constants p and q so that x₁ = 0, x₂ = 1 is the unique optimal solution of the resulting program. Solution: Again, by looking at the picture, we see that one option is to set p= -1,q = -0.5. (d) Find a value for the constants p and q for which both x₁ = 3/2, x₂ = 3/2 and x₁ = 2,7₂ = 0 are optimal solutions of the resulting program. Solution: This can only happen if our objective points in the same direction as the normal vector of the constraint whose boundary includes these points. One such setting is to just take the normal vector itself, which gives p = 3, q=1. (e) Find a value for the constants p and q for which both x₁ = 3/2, x₂ = 3/2 and *₁ = 0,7₂ = 2 are optimal solutions of the resulting program. Solution: This can only happen if our objective points in the same direction as the normal vector of the constraint whose boundary includes these points. One such setting is to just take the normal vector itself, which gives p = 1, q=3.
Transcribed Image Text:X2 21 (b) Find a value for the constants p and q so that x₁ = 3/2, x₂ = 3/2 is the unique optimal solution of the resulting program. Solution: By looking at the picture, one option is to set p = 1, q = 1, which gives us a direction (1,1) that makes this the only optimal solution. (c) Find a value for the constants p and q so that x₁ = 0, x₂ = 1 is the unique optimal solution of the resulting program. Solution: Again, by looking at the picture, we see that one option is to set p= -1,q = -0.5. (d) Find a value for the constants p and q for which both x₁ = 3/2, x₂ = 3/2 and x₁ = 2,7₂ = 0 are optimal solutions of the resulting program. Solution: This can only happen if our objective points in the same direction as the normal vector of the constraint whose boundary includes these points. One such setting is to just take the normal vector itself, which gives p = 3, q=1. (e) Find a value for the constants p and q for which both x₁ = 3/2, x₂ = 3/2 and *₁ = 0,7₂ = 2 are optimal solutions of the resulting program. Solution: This can only happen if our objective points in the same direction as the normal vector of the constraint whose boundary includes these points. One such setting is to just take the normal vector itself, which gives p = 1, q=3.
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