Question

4. Consider the two-dimensional linear optimization problem minimize x1 + x2 subject to x1 + x2 ? 4 x1 - x2 ? 5 x1 ? 0 x2 ? 0 (a) Draw the feasible region. (b) Is the feasible region bounded, unbounded, or empty? (c) Draw the objective vector and a few level sets of the objective function. (d) Is the objective function bounded or unbounded above over the feasible solution? (e) Is the objective function bounded or unbounded below over the feasible solution? (f) Does the linear optimization problem admit an optimal solution? If so, is this optimal solution unique or are there many optimal solutions? Mark the position of the optimal solution(s) on your drawing. (g) Mark the location of all basic solutions with crosses (%). (h) Compute the coordinates of each basic solution. (i) In your list of basic solutions, indicate which basic solutions are feasible.

          4. Consider the two-dimensional linear optimization problem

minimize x1 + x2
subject to x1 + x2 ? 4
x1 - x2 ? 5
x1 ? 0
x2 ? 0

(a) Draw the feasible region.
(b) Is the feasible region bounded, unbounded, or empty?
(c) Draw the objective vector and a few level sets of the objective function.
(d) Is the objective function bounded or unbounded above over the feasible solution?
(e) Is the objective function bounded or unbounded below over the feasible solution?
(f) Does the linear optimization problem admit an optimal solution? If so, is this optimal solution unique or are there many optimal solutions? Mark the position of the optimal solution(s) on your drawing.
(g) Mark the location of all basic solutions with crosses (%).
(h) Compute the coordinates of each basic solution.
(i) In your list of basic solutions, indicate which basic solutions are feasible.

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4. Consider the two-dimensional linear optimization problem

minimize x1 + x2
subject to x1 + x2 ? 4
x1 - x2 ? 5
x1 ? 0
x2 ? 0

(a) Draw the feasible region.
(b) Is the feasible region bounded, unbounded, or empty?
(c) Draw the objective vector and a few level sets of the objective function.
(d) Is the objective function bounded or unbounded above over the feasible solution?
(e) Is the objective function bounded or unbounded below over the feasible solution?
(f) Does the linear optimization problem admit an optimal solution? If so, is this optimal solution unique or are there many optimal solutions? Mark the position of the optimal solution(s) on your drawing.
(g) Mark the location of all basic solutions with crosses (%).
(h) Compute the coordinates of each basic solution.
(i) In your list of basic solutions, indicate which basic solutions are feasible.

Added by Jacqueline J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Frank Deng

09/28/2022

Step 1

Step 1:** Draw the feasible region by graphing the inequalities: - \(X1 + X2 < 4\) - \(X1 - X2 < 5\) - \(X1 > 0\) - \(X2 > 0\) ** Show more…

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Consider the two-dimensional linear optimization problem minimize x1 + x2 subject to x1 + x2 ≤ 4 x1 - x2 ≥ 5 x1 ≥ 0 x2 ≥ 0 (a) Draw the feasible region. (b) Is the feasible region bounded, unbounded, or empty? (c) Draw the objective vector and a few level sets of the objective function. (d) Is the objective function bounded or unbounded above over the feasible solution? (e) Is the objective function bounded or unbounded below over the feasible solution? (f) Does the linear optimization problem admit an optimal solution? If so, is this optimal solution unique or are there many optimal solutions? Mark the position of the optimal solution(s) on your drawing. (g) Mark the location of all basic solutions with crosses (%). (h) Compute the coordinates of each basic solution. (i) In your list of basic solutions, indicate which basic solutions are feasible.

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Transcript

00:05 Here we look at a two -dimensional linear optimization problem.

00:10 The objective is to minimize the sum of x1, x2, and it's subject to this constraints, right? so what are the feasible region? we can enjoy this, actually.

00:21 Let me try to do it.

00:22 So let's do this.

00:24 So suppose this is x1 and this is x2, right? and we need to look at these four constraints.

00:32 Let's read the first one x1, x2, together less than four, right? so actually if you put four here, no, i don't want to put four that higher.

00:40 So let's first say four is here and the four is also here, right? and then you draw a line and then that actually indicates the region underneath this line, right? and the signal is x1, x2 larger than five, right? so if you put x1, quos zero, that means five, x2 would be x2 would be minus five, right? so that would be, so let's look at the right.

01:05 So let's look at the right.

01:05 So that's somewhere here.

01:06 Let me draw it a bit longer.

01:09 And i'll actually find this to be actually somewhere here, right? so that would be a point.

01:15 There would be point here.

01:17 And if you say it's x2 equals 0, the x1 would be 5, right? so 5 would be somewhere here, right? supposed here.

01:24 And if you drew this line, right? and this has to be larger than 5...

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