How to formula the given linear programming model?
$begingroup$
Chem Labs uses raw materials I and II to produce two domestic cleaning solutions, A and B.
The daily availabilities of raw materials I and II are 150 and 145 units, respectively. One unit of
solution A consumes 0.5 unit of raw material I and 0.6 unit of raw material II, and one unit of
solution B uses 0.5 unit of raw material I and 0.4 unit of raw material II. The profits per unit of
solutions A and B are 8 and $10, respectively. The daily demand for solution A lies between
30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production
amounts of A and B.
My attempt
Let A and B be the no. of units of A and B produced and X and Y be no. of raw materials I and II to be processed respectively.
The objective function is to maximize the profit, Z.
Z=8A+10B
The objective function is subject to the following constraints
30<=0.5X+0.6Y<=150
40<=0.5X+0.4Y<=200
X<=150
Y<=145
Is this formulation correct? If it is, how can one proceed from this point to find the maximum profit?
linear-programming
asked Dec 29 '18 at 2:56
HamadaHamada
82
$endgroup$
add a comment |
$begingroup$
Chem Labs uses raw materials I and II to produce two domestic cleaning solutions, A and B.
The daily availabilities of raw materials I and II are 150 and 145 units, respectively. One unit of
solution A consumes 0.5 unit of raw material I and 0.6 unit of raw material II, and one unit of
solution B uses 0.5 unit of raw material I and 0.4 unit of raw material II. The profits per unit of
solutions A and B are 8 and $10, respectively. The daily demand for solution A lies between
30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production
amounts of A and B.
My attempt
Let A and B be the no. of units of A and B produced and X and Y be no. of raw materials I and II to be processed respectively.
The objective function is to maximize the profit, Z.
Z=8A+10B
The objective function is subject to the following constraints
30<=0.5X+0.6Y<=150
40<=0.5X+0.4Y<=200
X<=150
Y<=145
Is this formulation correct? If it is, how can one proceed from this point to find the maximum profit?
linear-programming
asked Dec 29 '18 at 2:56
HamadaHamada
82
$endgroup$
add a comment |
$begingroup$
Chem Labs uses raw materials I and II to produce two domestic cleaning solutions, A and B.
The daily availabilities of raw materials I and II are 150 and 145 units, respectively. One unit of
solution A consumes 0.5 unit of raw material I and 0.6 unit of raw material II, and one unit of
solution B uses 0.5 unit of raw material I and 0.4 unit of raw material II. The profits per unit of
solutions A and B are 8 and $10, respectively. The daily demand for solution A lies between
30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production
amounts of A and B.
My attempt
Let A and B be the no. of units of A and B produced and X and Y be no. of raw materials I and II to be processed respectively.
The objective function is to maximize the profit, Z.
Z=8A+10B
The objective function is subject to the following constraints
30<=0.5X+0.6Y<=150
40<=0.5X+0.4Y<=200
X<=150
Y<=145
Is this formulation correct? If it is, how can one proceed from this point to find the maximum profit?
linear-programming
asked Dec 29 '18 at 2:56
HamadaHamada
82
$endgroup$
Chem Labs uses raw materials I and II to produce two domestic cleaning solutions, A and B.
The daily availabilities of raw materials I and II are 150 and 145 units, respectively. One unit of
solution A consumes 0.5 unit of raw material I and 0.6 unit of raw material II, and one unit of
solution B uses 0.5 unit of raw material I and 0.4 unit of raw material II. The profits per unit of
solutions A and B are 8 and $10, respectively. The daily demand for solution A lies between
30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production
amounts of A and B.
My attempt
Let A and B be the no. of units of A and B produced and X and Y be no. of raw materials I and II to be processed respectively.
The objective function is to maximize the profit, Z.
Z=8A+10B
The objective function is subject to the following constraints
30<=0.5X+0.6Y<=150
40<=0.5X+0.4Y<=200
X<=150
Y<=145
Is this formulation correct? If it is, how can one proceed from this point to find the maximum profit?
linear-programming
linear-programming
asked Dec 29 '18 at 2:56
HamadaHamada
82
asked Dec 29 '18 at 2:56
HamadaHamada
82
asked Dec 29 '18 at 2:56
HamadaHamada
82
asked Dec 29 '18 at 2:56
HamadaHamada
82
asked Dec 29 '18 at 2:56
HamadaHamada
82
82
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The use of raw materials $I$ and $II$ depends on the production of $A$ and $B$, so you don't need the variables $X$ and $Y$.
You can create a table of given data:
$$begin{array}{c|c|c|c}
Products&I&II&Profit&Demand\
hline
A&0.5&0.6&8&30le Ale 150\
B&0.5&0.4&10&40le Ble 200\
hline
Available&le 150&le 145&maximize&end{array}$$
Now we can formulate the LPP: let $A$ and $B$ be the numbers of units of $A$ and $B$, respectively. Then:
$$pi(A,B)=8A+10Bto text{max} text{subject to}\
0.5A+0.5Ble 150 text{(material I constraint)}\
0.6A+0.4Ble 145 text{(material II constraint)}\
30le Ale 150 text{(demand for A)}\
40le Ble 200 text{(demand for B)}\
$$
You can use graphical or Simplex methods to solve LPP.
Graphical method.
1) Draw the feasible (green) region from the constraint inequalities:
2) Find the corner points: $A,B,C,D,E,F$.
3) Evaluate the objective (profit) function at the corner points and choose the maximum.
Can you do it?
Answer:
$pi(100,200)=2800.$ WolframAlpha answer.
answered Dec 29 '18 at 7:44
farruhotafarruhota
20.4k2739
$endgroup$
add a comment |
$begingroup$
We just have to decide how many units of $A$ and $B$ are to be produced.
You are right that the profit is $8A+10B$ and we want to maximize it.
Now, let's examine the constraint imposed by material I.
$$0.5A+0.5B le 150$$
Now, let's examine the constraint imposed by material II.
$$0.6A + 0.4B le 145$$
The demand informations also gives us
$$30 le A le 150$$
and
$$40 le Ble 200.$$
Now, we have a $2$-dimensional linear programming problem, you can use a graphical method to solve the problem if you wish.
answered Dec 29 '18 at 5:11
Siong Thye GohSiong Thye Goh
102k1466118
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3055494%2fhow-to-formula-the-given-linear-programming-model%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The use of raw materials $I$ and $II$ depends on the production of $A$ and $B$, so you don't need the variables $X$ and $Y$.
You can create a table of given data:
$$begin{array}{c|c|c|c}
Products&I&II&Profit&Demand\
hline
A&0.5&0.6&8&30le Ale 150\
B&0.5&0.4&10&40le Ble 200\
hline
Available&le 150&le 145&maximize&end{array}$$
Now we can formulate the LPP: let $A$ and $B$ be the numbers of units of $A$ and $B$, respectively. Then:
$$pi(A,B)=8A+10Bto text{max} text{subject to}\
0.5A+0.5Ble 150 text{(material I constraint)}\
0.6A+0.4Ble 145 text{(material II constraint)}\
30le Ale 150 text{(demand for A)}\
40le Ble 200 text{(demand for B)}\
$$
You can use graphical or Simplex methods to solve LPP.
Graphical method.
1) Draw the feasible (green) region from the constraint inequalities:
2) Find the corner points: $A,B,C,D,E,F$.
3) Evaluate the objective (profit) function at the corner points and choose the maximum.
Can you do it?
Answer:
$pi(100,200)=2800.$ WolframAlpha answer.
answered Dec 29 '18 at 7:44
farruhotafarruhota
20.4k2739
$endgroup$
add a comment |
$begingroup$
The use of raw materials $I$ and $II$ depends on the production of $A$ and $B$, so you don't need the variables $X$ and $Y$.
You can create a table of given data:
$$begin{array}{c|c|c|c}
Products&I&II&Profit&Demand\
hline
A&0.5&0.6&8&30le Ale 150\
B&0.5&0.4&10&40le Ble 200\
hline
Available&le 150&le 145&maximize&end{array}$$
Now we can formulate the LPP: let $A$ and $B$ be the numbers of units of $A$ and $B$, respectively. Then:
$$pi(A,B)=8A+10Bto text{max} text{subject to}\
0.5A+0.5Ble 150 text{(material I constraint)}\
0.6A+0.4Ble 145 text{(material II constraint)}\
30le Ale 150 text{(demand for A)}\
40le Ble 200 text{(demand for B)}\
$$
You can use graphical or Simplex methods to solve LPP.
Graphical method.
1) Draw the feasible (green) region from the constraint inequalities:
2) Find the corner points: $A,B,C,D,E,F$.
3) Evaluate the objective (profit) function at the corner points and choose the maximum.
Can you do it?
Answer:
$pi(100,200)=2800.$ WolframAlpha answer.
answered Dec 29 '18 at 7:44
farruhotafarruhota
20.4k2739
$endgroup$
add a comment |
$begingroup$
The use of raw materials $I$ and $II$ depends on the production of $A$ and $B$, so you don't need the variables $X$ and $Y$.
You can create a table of given data:
$$begin{array}{c|c|c|c}
Products&I&II&Profit&Demand\
hline
A&0.5&0.6&8&30le Ale 150\
B&0.5&0.4&10&40le Ble 200\
hline
Available&le 150&le 145&maximize&end{array}$$
Now we can formulate the LPP: let $A$ and $B$ be the numbers of units of $A$ and $B$, respectively. Then:
$$pi(A,B)=8A+10Bto text{max} text{subject to}\
0.5A+0.5Ble 150 text{(material I constraint)}\
0.6A+0.4Ble 145 text{(material II constraint)}\
30le Ale 150 text{(demand for A)}\
40le Ble 200 text{(demand for B)}\
$$
You can use graphical or Simplex methods to solve LPP.
Graphical method.
1) Draw the feasible (green) region from the constraint inequalities:
2) Find the corner points: $A,B,C,D,E,F$.
3) Evaluate the objective (profit) function at the corner points and choose the maximum.
Can you do it?
Answer:
$pi(100,200)=2800.$ WolframAlpha answer.
answered Dec 29 '18 at 7:44
farruhotafarruhota
20.4k2739
$endgroup$
The use of raw materials $I$ and $II$ depends on the production of $A$ and $B$, so you don't need the variables $X$ and $Y$.
You can create a table of given data:
$$begin{array}{c|c|c|c}
Products&I&II&Profit&Demand\
hline
A&0.5&0.6&8&30le Ale 150\
B&0.5&0.4&10&40le Ble 200\
hline
Available&le 150&le 145&maximize&end{array}$$
Now we can formulate the LPP: let $A$ and $B$ be the numbers of units of $A$ and $B$, respectively. Then:
$$pi(A,B)=8A+10Bto text{max} text{subject to}\
0.5A+0.5Ble 150 text{(material I constraint)}\
0.6A+0.4Ble 145 text{(material II constraint)}\
30le Ale 150 text{(demand for A)}\
40le Ble 200 text{(demand for B)}\
$$
You can use graphical or Simplex methods to solve LPP.
Graphical method.
1) Draw the feasible (green) region from the constraint inequalities:
2) Find the corner points: $A,B,C,D,E,F$.
3) Evaluate the objective (profit) function at the corner points and choose the maximum.
Can you do it?
Answer:
$pi(100,200)=2800.$ WolframAlpha answer.
answered Dec 29 '18 at 7:44
farruhotafarruhota
20.4k2739
answered Dec 29 '18 at 7:44
farruhotafarruhota
20.4k2739
answered Dec 29 '18 at 7:44
farruhotafarruhota
20.4k2739
answered Dec 29 '18 at 7:44
farruhotafarruhota
20.4k2739
20.4k2739
add a comment |
add a comment |
$begingroup$
We just have to decide how many units of $A$ and $B$ are to be produced.
You are right that the profit is $8A+10B$ and we want to maximize it.
Now, let's examine the constraint imposed by material I.
$$0.5A+0.5B le 150$$
Now, let's examine the constraint imposed by material II.
$$0.6A + 0.4B le 145$$
The demand informations also gives us
$$30 le A le 150$$
and
$$40 le Ble 200.$$
Now, we have a $2$-dimensional linear programming problem, you can use a graphical method to solve the problem if you wish.
answered Dec 29 '18 at 5:11
Siong Thye GohSiong Thye Goh
102k1466118
$endgroup$
add a comment |
$begingroup$
We just have to decide how many units of $A$ and $B$ are to be produced.
You are right that the profit is $8A+10B$ and we want to maximize it.
Now, let's examine the constraint imposed by material I.
$$0.5A+0.5B le 150$$
Now, let's examine the constraint imposed by material II.
$$0.6A + 0.4B le 145$$
The demand informations also gives us
$$30 le A le 150$$
and
$$40 le Ble 200.$$
Now, we have a $2$-dimensional linear programming problem, you can use a graphical method to solve the problem if you wish.
answered Dec 29 '18 at 5:11
Siong Thye GohSiong Thye Goh
102k1466118
$endgroup$
add a comment |
$begingroup$
We just have to decide how many units of $A$ and $B$ are to be produced.
You are right that the profit is $8A+10B$ and we want to maximize it.
Now, let's examine the constraint imposed by material I.
$$0.5A+0.5B le 150$$
Now, let's examine the constraint imposed by material II.
$$0.6A + 0.4B le 145$$
The demand informations also gives us
$$30 le A le 150$$
and
$$40 le Ble 200.$$
Now, we have a $2$-dimensional linear programming problem, you can use a graphical method to solve the problem if you wish.
answered Dec 29 '18 at 5:11
Siong Thye GohSiong Thye Goh
102k1466118
$endgroup$
We just have to decide how many units of $A$ and $B$ are to be produced.
You are right that the profit is $8A+10B$ and we want to maximize it.
Now, let's examine the constraint imposed by material I.
$$0.5A+0.5B le 150$$
Now, let's examine the constraint imposed by material II.
$$0.6A + 0.4B le 145$$
The demand informations also gives us
$$30 le A le 150$$
and
$$40 le Ble 200.$$
Now, we have a $2$-dimensional linear programming problem, you can use a graphical method to solve the problem if you wish.
answered Dec 29 '18 at 5:11
Siong Thye GohSiong Thye Goh
102k1466118
answered Dec 29 '18 at 5:11
Siong Thye GohSiong Thye Goh
102k1466118
answered Dec 29 '18 at 5:11
Siong Thye GohSiong Thye Goh
102k1466118
answered Dec 29 '18 at 5:11
Siong Thye GohSiong Thye Goh
102k1466118
102k1466118
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3055494%2fhow-to-formula-the-given-linear-programming-model%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
