Write the dual for the following linear program
Write the dual for the following linear program:
max($3x_1 + 8x_2$)
subject to
$x_1 + 4x_2$ ≤ 20
$x_1 + x_2$ ≥ 7
$x_1$ ≥ -1
$x_1$ ≤ 5
The posted solution is as follows, but does not show the steps.
Where did I go wrong?
linear-programming
asked Dec 10 '18 at 3:56
Leonardo Lopez
1033
add a comment |
Write the dual for the following linear program:
max($3x_1 + 8x_2$)
subject to
$x_1 + 4x_2$ ≤ 20
$x_1 + x_2$ ≥ 7
$x_1$ ≥ -1
$x_1$ ≤ 5
The posted solution is as follows, but does not show the steps.
Where did I go wrong?
linear-programming
asked Dec 10 '18 at 3:56
Leonardo Lopez
1033
Welcome to MathStackExchange. I please urge you to rewrite the pictures using MathJax, here is a guide, as it reallu helps reading the questions and also helps anyone struggling with a similar problem to find your question in the future.
– Mefitico
Dec 10 '18 at 4:02
add a comment |
Write the dual for the following linear program:
max($3x_1 + 8x_2$)
subject to
$x_1 + 4x_2$ ≤ 20
$x_1 + x_2$ ≥ 7
$x_1$ ≥ -1
$x_1$ ≤ 5
The posted solution is as follows, but does not show the steps.
Where did I go wrong?
linear-programming
asked Dec 10 '18 at 3:56
Leonardo Lopez
1033
Write the dual for the following linear program:
max($3x_1 + 8x_2$)
subject to
$x_1 + 4x_2$ ≤ 20
$x_1 + x_2$ ≥ 7
$x_1$ ≥ -1
$x_1$ ≤ 5
The posted solution is as follows, but does not show the steps.
Where did I go wrong?
linear-programming
linear-programming
asked Dec 10 '18 at 3:56
Leonardo Lopez
1033
asked Dec 10 '18 at 3:56
Leonardo Lopez
1033
asked Dec 10 '18 at 3:56
Leonardo Lopez
1033
asked Dec 10 '18 at 3:56
Leonardo Lopez
1033
asked Dec 10 '18 at 3:56
Leonardo Lopez
1033
1033
Welcome to MathStackExchange. I please urge you to rewrite the pictures using MathJax, here is a guide, as it reallu helps reading the questions and also helps anyone struggling with a similar problem to find your question in the future.
– Mefitico
Dec 10 '18 at 4:02
add a comment |
Welcome to MathStackExchange. I please urge you to rewrite the pictures using MathJax, here is a guide, as it reallu helps reading the questions and also helps anyone struggling with a similar problem to find your question in the future.
– Mefitico
Dec 10 '18 at 4:02
Welcome to MathStackExchange. I please urge you to rewrite the pictures using MathJax, here is a guide, as it reallu helps reading the questions and also helps anyone struggling with a similar problem to find your question in the future.
– Mefitico
Dec 10 '18 at 4:02
Welcome to MathStackExchange. I please urge you to rewrite the pictures using MathJax, here is a guide, as it reallu helps reading the questions and also helps anyone struggling with a similar problem to find your question in the future.
– Mefitico
Dec 10 '18 at 4:02
add a comment |
2 Answers
2
active
oldest
votes
Check your definition of $z_1$ and $z_2$.
$$z_1 = x_1 + 1 iff x_1 = z_1-1$$
$$z_2 = -x_2+5 iff x_2 = -z_2+5$$
Hence $$x_1+4x_2 le 20$$ becomes
$$(z_1-1)+4(-z_2+5) le 20$$ which is equivalent to
$$z_1 -4z_2 le 1.$$
Do the same thing for the second inequality.
answered Dec 10 '18 at 4:05
Siong Thye Goh
99.4k1464117
add a comment |
If inequality is greater than you don’t multiply by $-1$, but add the constant. Note:
$$x_1ge -1 Rightarrow underbrace{x_1+1}_{z_1}ge 0;\
x_2le 5Rightarrow -x_2ge -5 Rightarrow underbrace{-x_2+5}_{z_2}ge 0.$$
Note that your conversion to dual is correct.
answered Dec 10 '18 at 4:31
farruhota
19.3k2736
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Check your definition of $z_1$ and $z_2$.
$$z_1 = x_1 + 1 iff x_1 = z_1-1$$
$$z_2 = -x_2+5 iff x_2 = -z_2+5$$
Hence $$x_1+4x_2 le 20$$ becomes
$$(z_1-1)+4(-z_2+5) le 20$$ which is equivalent to
$$z_1 -4z_2 le 1.$$
Do the same thing for the second inequality.
answered Dec 10 '18 at 4:05
Siong Thye Goh
99.4k1464117
add a comment |
Check your definition of $z_1$ and $z_2$.
$$z_1 = x_1 + 1 iff x_1 = z_1-1$$
$$z_2 = -x_2+5 iff x_2 = -z_2+5$$
Hence $$x_1+4x_2 le 20$$ becomes
$$(z_1-1)+4(-z_2+5) le 20$$ which is equivalent to
$$z_1 -4z_2 le 1.$$
Do the same thing for the second inequality.
answered Dec 10 '18 at 4:05
Siong Thye Goh
99.4k1464117
add a comment |
Check your definition of $z_1$ and $z_2$.
$$z_1 = x_1 + 1 iff x_1 = z_1-1$$
$$z_2 = -x_2+5 iff x_2 = -z_2+5$$
Hence $$x_1+4x_2 le 20$$ becomes
$$(z_1-1)+4(-z_2+5) le 20$$ which is equivalent to
$$z_1 -4z_2 le 1.$$
Do the same thing for the second inequality.
answered Dec 10 '18 at 4:05
Siong Thye Goh
99.4k1464117
Check your definition of $z_1$ and $z_2$.
$$z_1 = x_1 + 1 iff x_1 = z_1-1$$
$$z_2 = -x_2+5 iff x_2 = -z_2+5$$
Hence $$x_1+4x_2 le 20$$ becomes
$$(z_1-1)+4(-z_2+5) le 20$$ which is equivalent to
$$z_1 -4z_2 le 1.$$
Do the same thing for the second inequality.
answered Dec 10 '18 at 4:05
Siong Thye Goh
99.4k1464117
answered Dec 10 '18 at 4:05
Siong Thye Goh
99.4k1464117
answered Dec 10 '18 at 4:05
Siong Thye Goh
99.4k1464117
answered Dec 10 '18 at 4:05
Siong Thye Goh
99.4k1464117
99.4k1464117
add a comment |
add a comment |
If inequality is greater than you don’t multiply by $-1$, but add the constant. Note:
$$x_1ge -1 Rightarrow underbrace{x_1+1}_{z_1}ge 0;\
x_2le 5Rightarrow -x_2ge -5 Rightarrow underbrace{-x_2+5}_{z_2}ge 0.$$
Note that your conversion to dual is correct.
answered Dec 10 '18 at 4:31
farruhota
19.3k2736
add a comment |
If inequality is greater than you don’t multiply by $-1$, but add the constant. Note:
$$x_1ge -1 Rightarrow underbrace{x_1+1}_{z_1}ge 0;\
x_2le 5Rightarrow -x_2ge -5 Rightarrow underbrace{-x_2+5}_{z_2}ge 0.$$
Note that your conversion to dual is correct.
answered Dec 10 '18 at 4:31
farruhota
19.3k2736
add a comment |
If inequality is greater than you don’t multiply by $-1$, but add the constant. Note:
$$x_1ge -1 Rightarrow underbrace{x_1+1}_{z_1}ge 0;\
x_2le 5Rightarrow -x_2ge -5 Rightarrow underbrace{-x_2+5}_{z_2}ge 0.$$
Note that your conversion to dual is correct.
answered Dec 10 '18 at 4:31
farruhota
19.3k2736
If inequality is greater than you don’t multiply by $-1$, but add the constant. Note:
$$x_1ge -1 Rightarrow underbrace{x_1+1}_{z_1}ge 0;\
x_2le 5Rightarrow -x_2ge -5 Rightarrow underbrace{-x_2+5}_{z_2}ge 0.$$
Note that your conversion to dual is correct.
answered Dec 10 '18 at 4:31
farruhota
19.3k2736
answered Dec 10 '18 at 4:31
farruhota
19.3k2736
answered Dec 10 '18 at 4:31
farruhota
19.3k2736
answered Dec 10 '18 at 4:31
farruhota
19.3k2736
19.3k2736
add a comment |
add a comment |
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Welcome to MathStackExchange. I please urge you to rewrite the pictures using MathJax, here is a guide, as it reallu helps reading the questions and also helps anyone struggling with a similar problem to find your question in the future.
– Mefitico
Dec 10 '18 at 4:02