Question 1199207: A company manufactures two products, A and B, and each of these products must be processed on two different machines. Product A requires 2 min of work time per unit on machine 1 and 4 min of work time per unit on machine 2. Product B requires 3 min of work time per unit on machine 1 and 1 min of work time per unit on machine 2. Each day 200 min are available on machine 1 and 300 min are available on machine2. To satisfy certain customers, the company must produce at least 8 units per day of product A and at least 10 units per day of product B. If the profit of each unit of product A is Php50 and the profit of each unit of product B is Php60, how many units of each product should be produced daily in order to maximize the company's profits?
Provide the following:
The objective function: blank
The machine 1 constraint: blank
The machine 2 constraint: blank
The production constraints: blank
The implicit constraints: blank
The number of product A and product B to give an optimum profit: blank
The optimum cost:
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The objective function...
The profit is 50 for each product A and 60 for each product B:
P = 50A+60B
The machine 1 constraint...
2 minutes for each product A and 3 minutes for each product B; 200 total minutes available:
2A+3B <= 200
The machine 2 constraint...
4 minutes for each product A and 1 minute for each product B; 300 total minutes available:
4A+B <= 300
The production constraints...
At least 8 of product A and at least 10 of product B:
A >= 8
B >= 10
The implicit constraints...
A >= 0; B >= 0
(not needed, because the production constraints are more restrictive)
The number of product A and product B to give an optimum profit...
I'll leave it to you or another tutor to make a graph of the feasibility region and find the answer by whatever method.
Note that the standard process described in virtually all references is NOT required. The corner of the feasibility region where the objective function is maximized can be determined by comparing the slopes of the constraint boundary lines to the slope of the objective function; it is NOT necessary to evaluate the objective function at every corner of the feasibility region.
The slopes of the constraint boundary lines are -4 and -2/3; the slope of the constraint function is -5/6. Because the slope of the objective function is between the slopes of the constraint boundary lines, the objective function will be maximized where the constraint boundary lines intersect (as long as the production constraints are satisfied).
2A+3B=200
4A+B=300
4A+6B=400
5B=100
B=20
A=70
That satisfies the production constraints, so
ANSWER: The maximum profit is when 70 of product A and 20 of product B are produced.
The optimum  profit:
ANSWER: 70(50)+20(60) = 3500+1200 = 4700
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Given Facts:- A company manufactures two products, A and B, and each of these products must be processed on two different machines.
- Product A requires 2 min of work time per unit on machine 1 and 4 min of work time per unit on machine 2.
- Product B requires 3 min of work time per unit on machine 1 and 1 min of work time per unit on machine 2.
- Each day 200 min are available on machine 1 and 300 min are available on machine2.
- To satisfy certain customers, the company must produce at least 8 units per day of product A and at least 10 units per day of product B.
- The profit of each unit of product A is Php50 and the profit of each unit of product B is Php60.
Let
x = number of units of product A
y = number of units of product B
Fact 6 tells us that the objective function is
P = 50x+60y
where P is the profit in Php.
Using fact 2 through fact 4, we can form this table to determine time constraints for each machine.
| Product A | Product B | Total | Constraint | | Machine 1 | 2x | 3y | 2x+3y |  | | Machine 2 | 4x | 1y | 4x+1y |  |
Each time value is in minutes.
The key thing to pull out of that table are these two constraints:
Another two constraints are:
Refer to fact 5.
Something like "at least 8" means "8 or more".
System of inequalities
Use graphing software or do so by hand to create the feasibility region.
I used GeoGebra to make the diagram shown below. The feasibility region is in blue.
The points on the boundary are part of the feasibility region.
It turns out that the objective function reaches its highest and lowest point at the corners of the feasibility region.
To find the location of each corner, use algebra.
For instance, the location of point B is found by solving this system
since B is where these two lines cross.
Here are the corner points:
A = (8, 61.3333)
B = (70, 20)
C = (72.5, 10)
D = (8, 10)
The y coordinate of point A is approximate.
Let's check each corner.
Checking point A
P = 50x+60y
P = 50*8+60*61.3333
P = 4079.998
P = 4080
Checking point B
P = 50x+60y
P = 50*70+60*20
P = 4700
Checking point C
P = 50x+60y
P = 50*72.5+60*10
P = 4225
Checking point D
P = 50x+60y
P = 50*8+60*10
P = 1000
The largest result for P was when we got P = 4700 (for point B)
Therefore, the largest profit possible is Php4700.
This profit is reached when 70 units of product A are made, and 20 units of product B are made.
|
|
|
