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I have the following exercise:

Stockco is considering four investments. Investment 1 will yield a net present value (NPV) of \$16,000; investment 2, an NPV of \$22,000; investment 3, an NPV of \$12,000; and investment 4, an NPV of \$8,000. Each investment requires a certain cash overflow at the present time: investment 1, \$5,000; investment 2, \$7,000; investment 3, \$4,000; and investment 4, \$3,000. Currently, \$14,000 is available for investment. Formulate an IP whose solution will tell Stockco how to maximize the NPV obtained from investments 1-4.

 

As in LP formulations, we begin by defining a variable for each decision that Stockco must make. This leads us to define a 0-1 variable: $$x_j(j=1,2,3,4)=\begin{cases}1\quad\text{if investment}\,j\,\text{is made}\\0\quad\text{otherwise}.\end{cases}$$

Suppose we add the following restriction to Example 1 (Stockco):

If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?

The solution is as given below:

Condition 2: If investments 2 and 3 are chosen, then investment 4 must be chosen. This condition is obtained by the constraint $x_2+x_3-2x_4\le0$. If both $x_2,x_3=1$, then $x_4$ must be $1$. Otherwise $x_4$ may be zero. Hence the condition is satisfied.

Hence the following IP is formulated. \begin{alignat}2\max&\quad16x_1+22x_2+12x_3+8x_4\\\text{s.t.}&\quad5x_1+7x_2+4x_3+3x_4&\le14\\&\quad x_2+x_3-2x_4&\le0\\&\quad x_1,x_2,x_3,x_4&\ge0\end{alignat} My question: Is the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. i.e it satisfies for one of them to be chosen for the inequality to be true.

I have the following exercise:

Stockco is considering four investments. Investment 1 will yield a net present value (NPV) of \$16,000; investment 2, an NPV of \$22,000; investment 3, an NPV of \$12,000; and investment 4, an NPV of \$8,000. Each investment requires a certain cash overflow at the present time: investment 1, \$5,000; investment 2, \$7,000; investment 3, \$4,000; and investment 4, \$3,000. Currently, \$14,000 is available for investment. Formulate an IP whose solution will tell Stockco how to maximize the NPV obtained from investments 1-4.

 

As in LP formulations, we begin by defining a variable for each decision that Stockco must make. This leads us to define a 0-1 variable: $$x_j(j=1,2,3,4)=\begin{cases}1\quad\text{if investment}\,j\,\text{is made}\\0\quad\text{otherwise}.\end{cases}$$

Suppose we add the following restriction to Example 1 (Stockco):

If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?

The solution is as given below:

Condition 2: If investments 2 and 3 are chosen, then investment 4 must be chosen. This condition is obtained by the constraint $x_2+x_3-2x_4\le0$. If both $x_2,x_3=1$, then $x_4$ must be $1$. Otherwise $x_4$ may be zero. Hence the condition is satisfied.

Hence the following IP is formulated. \begin{alignat}2\max&\quad16x_1+22x_2+12x_3+8x_4\\\text{s.t.}&\quad5x_1+7x_2+4x_3+3x_4&\le14\\&\quad x_2+x_3-2x_4&\le0\\&\quad x_1,x_2,x_3,x_4&\ge0\end{alignat} My question: Is the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. i.e it satisfies for one of them to be chosen for the inequality to be true.

I have the following exercise:

Stockco is considering four investments. Investment 1 will yield a net present value (NPV) of \$16,000; investment 2, an NPV of \$22,000; investment 3, an NPV of \$12,000; and investment 4, an NPV of \$8,000. Each investment requires a certain cash overflow at the present time: investment 1, \$5,000; investment 2, \$7,000; investment 3, \$4,000; and investment 4, \$3,000. Currently, \$14,000 is available for investment. Formulate an IP whose solution will tell Stockco how to maximize the NPV obtained from investments 1-4.

As in LP formulations, we begin by defining a variable for each decision that Stockco must make. This leads us to define a 0-1 variable: $$x_j(j=1,2,3,4)=\begin{cases}1\quad\text{if investment}\,j\,\text{is made}\\0\quad\text{otherwise}.\end{cases}$$

Suppose we add the following restriction to Example 1 (Stockco):

If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?

The solution is as given below:

Condition 2: If investments 2 and 3 are chosen, then investment 4 must be chosen. This condition is obtained by the constraint $x_2+x_3-2x_4\le0$. If both $x_2,x_3=1$, then $x_4$ must be $1$. Otherwise $x_4$ may be zero. Hence the condition is satisfied.

Hence the following IP is formulated. \begin{alignat}2\max&\quad16x_1+22x_2+12x_3+8x_4\\\text{s.t.}&\quad5x_1+7x_2+4x_3+3x_4&\le14\\&\quad x_2+x_3-2x_4&\le0\\&\quad x_1,x_2,x_3,x_4&\ge0\end{alignat} My question: Is the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. i.e it satisfies for one of them to be chosen for the inequality to be true.

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TheSimpliFire ♦
TheSimpliFire ♦
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enter image description here

I have anthe following exercise:

Stockco is considering four investments. Investment 1 will yield a net present value (NPV) of \$16,000; investment 2, an NPV of \$22,000; investment 3, an NPV of \$12,000; and investment 4, an NPV of \$8,000. Each investment requires a certain cash overflow at the present time: investment 1, \$5,000; investment 2, \$7,000; investment 3, \$4,000; and investment 4, \$3,000. Currently, \$14,000 is available for investment. Formulate an IP whose solution will tell Stockco how to maximize the NPV obtained from investments 1-4.

As in LP formulations, we begin by defining a variable for each decision that Stockco must make. This leads us to define a 0-1 variable: $$x_j(j=1,2,3,4)=\begin{cases}1\quad\text{if investment}\,j\,\text{is made}\\0\quad\text{otherwise}.\end{cases}$$

Suppose we add the following restriction to Example 1 (Stockco):

If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?

The solution is as given below:

enter image description here enter image description hereCondition 2: If investments 2 and 3 are chosen, then investment 4 must be chosen. This condition is obtained by the constraint $x_2+x_3-2x_4\le0$. If both $x_2,x_3=1$, then $x_4$ must be $1$. Otherwise $x_4$ may be zero. Hence the condition is satisfied.

Hence the following IP is formulated. \begin{alignat}2\max&\quad16x_1+22x_2+12x_3+8x_4\\\text{s.t.}&\quad5x_1+7x_2+4x_3+3x_4&\le14\\&\quad x_2+x_3-2x_4&\le0\\&\quad x_1,x_2,x_3,x_4&\ge0\end{alignat} My question: Is the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. i.e it satisfies for one of them to be chosen for the inequality to be true.

enter image description here

I have an exercise:

Suppose we add the following restriction to Example 1 (Stockco):

If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?

The solution is as given below:

enter image description here enter image description here

My question: Is the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. i.e it satisfies for one of them to be chosen for the inequality to be true

I have the following exercise:

Stockco is considering four investments. Investment 1 will yield a net present value (NPV) of \$16,000; investment 2, an NPV of \$22,000; investment 3, an NPV of \$12,000; and investment 4, an NPV of \$8,000. Each investment requires a certain cash overflow at the present time: investment 1, \$5,000; investment 2, \$7,000; investment 3, \$4,000; and investment 4, \$3,000. Currently, \$14,000 is available for investment. Formulate an IP whose solution will tell Stockco how to maximize the NPV obtained from investments 1-4.

As in LP formulations, we begin by defining a variable for each decision that Stockco must make. This leads us to define a 0-1 variable: $$x_j(j=1,2,3,4)=\begin{cases}1\quad\text{if investment}\,j\,\text{is made}\\0\quad\text{otherwise}.\end{cases}$$

Suppose we add the following restriction to Example 1 (Stockco):

If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?

The solution is as given below:

Condition 2: If investments 2 and 3 are chosen, then investment 4 must be chosen. This condition is obtained by the constraint $x_2+x_3-2x_4\le0$. If both $x_2,x_3=1$, then $x_4$ must be $1$. Otherwise $x_4$ may be zero. Hence the condition is satisfied.

Hence the following IP is formulated. \begin{alignat}2\max&\quad16x_1+22x_2+12x_3+8x_4\\\text{s.t.}&\quad5x_1+7x_2+4x_3+3x_4&\le14\\&\quad x_2+x_3-2x_4&\le0\\&\quad x_1,x_2,x_3,x_4&\ge0\end{alignat} My question: Is the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. i.e it satisfies for one of them to be chosen for the inequality to be true.

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EhsanK ♦
EhsanK ♦
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enter image description here

I have an exercise:

Suppose we add the following restriction to Example 1    (Stockco): 

If investments 2 and 3 are chosen, then investment 4 4 must be chosen. What constraints would be added to the formulation formulation given in the text?

The solution is as given below:

enter image description here enter image description here

My question isMy question: isIs the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. Ii.e it satisfies for one of them to be chosen for the inequality to be true

enter image description here

I have an exercise:

Suppose we add the following restriction to Example 1  (Stockco): If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?

The solution is as given below:

enter image description here enter image description here

My question is: is the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. I.e it satisfies for one of them to be chosen for the inequality to be true

enter image description here

I have an exercise:

Suppose we add the following restriction to Example 1  (Stockco): 

If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?

The solution is as given below:

enter image description here enter image description here

My question: Is the solution correct? It seems incorrect since it's a "$x_2$ or $x_3$ are chosen" statement, not a "$x_2$ and $x_3$ are chosen" statement. i.e it satisfies for one of them to be chosen for the inequality to be true

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Slim Shady
Slim Shady
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