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Question 1108805: Solve using the graphical method. Choose your variables, identify the objective function and the constraints, graph the constraints, shade the feasibility region, identify all corner points, and determine the solution that optimizes the objective function. Use this information to answer the following 8-part question:
A company produces two types of shoes - casual, and athletic - at its two factories, Factory I and Factory II. Daily production at each factory for number of pairs of each type of shoe is listed below:
Factory I Factory II
Casual 100 200
Athletic 300 100
The company must produce at least 8000 pairs of casual shoes, and 9000 pairs of athletic shoes. The cost of operating Factory I is $1500 per day and the cost of operating Factory II is $2000. Management needs to determine the number of days each factory needs to operate to meet production goals at minimum cost.
(i) Which variable should x define? (Enter letter for correct answer in the blank)
a. number of pairs of athletic shoes produced
b. number of days Factory I operates
c. number of pairs of casual shoes produced
d. cost to produce a pair of athletic shoes
(ii) Which variable should y define? (Enter letter for correct answer in the blank)
a. number of pairs of athletic shoes produced
b. cost to produce a pair of casual shoes
c. number of pairs of casual shoes produced
d. number of days Factory II operates
(iii) Which of the following correctly defines the constraint on total daily production of pairs of casual shoes? (Enter letter for correct answer in the blank)
a. 200x+100y≤8000
b. 100x+200y≤8000
c. 200x+100y≥8000
d. 100x+200y≥8000
(iv) Which of the following correctly defines the constraint on total daily production of pairs of athletic shoes? (Enter letter for correct answer in the blank)
a. 300x+100y≥9000
b. 100x+300y≥9000
c. 300x+100y≤9000
d. 100x+300y≤9000
(v) Which of the following correctly defines the objective function? (Enter letter for correct answer in the blank)
a. C=8000x+9000y
b. C=9000x+8000y
c. C=1500x+2000y
d. C=2000x+1500y
(vi) Which of the following sets of corner points correctly defines the feasible region for this system of linear inequalities? (Enter letter for correct answer in the blank)
a. (0, 0) ; (40, 0) ; (20, 30) ; (30, 0)
b. (0, 40) ; (0, 90) ; (20, 30)
c. (0, 90) ; (20, 30) ; (80, 0)
d. (20, 30) ; (30, 0) ; (80, 0)
(vii) Based on your determination of the feasible region for this business process, how many days should each factory operate to meet production goals at minimum cost? (Enter letter for correct answer in the blank)
a. 30 days for Factory I and 0 days for Factory II
b. 20 days for Factory I and 30 days for Factory II
c. 0 days for Factory I and 40 days for Factory II
d. None of these answers are correct
(viii) Which of the following is the correct value for the minimum cost to meet production goals for both types of shoes?
a. C = $45000
b. C = $80000
c. C = $90000
d. C = $120000
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52847)   (Show Source):
You can put this solution on YOUR website! .
When I see such big peace of paper (or the spot in my computer screen) filled with numbers, I have complicated feelings.
It is a standard problem on the Linear Programming method.
It is not so difficult for me to solve it for you by filling all the positions.
But the question is: what you will learn from it ???
I have no any wishes to fill somebody's website.
I have no any wishes to make your work instead of you.
I think it is NOT THE METHOD OF TEACHING.
I am here to teach you.
Therefore, my instructions are THESE:
In this site, just there is a lesson on the Linear Programming method:
- Solving minimax problems by the Linear Programming method
It teaches you by examples on how to solve minimax problems using the LP method.
So, read this lesson first.
Then return to your problem and do as much as you can on your own.
Ideally, it would be good, if after reading the lesson you will be able to complete the solution to the end.
But if not, THEN send/post your work to the forum, and I (or other tutors) will help you to do the rest.
It will be much more effective way for you to learn this subject.
H a p p y l e a r n i n g ! !
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! it looks like you are comparing the cost from factory I and factory II.
that's because management wants to determine the number of days they should operate from factory I and factory II.
so let x equal the number of days to operate factory I and y equal the number of days to operate factory II.
then draw up a table as shown below:
factory I factory II total
casual 100 / day 200 / day >= 8000
athletic 300 / day 100 / day >= 9000
cost 1500 / day 2000 / day minimize
variable x y
from this table, you can draw up constraint equations.
you get:
100 * x + 200 * y >= 8000.
300 * x + 100 * y >= 9000.
your objective function is to minimize the cost.
the objective function becomes cost = 1500 * x + 2000 * y.
addition constraints are that x >= 0 and y >= 0, because the number of days can't be negative.
using the desmos.com calculator, you would graph the opposite of these constraints.
the area of the graph that is not shaded is your region of feasibility.
you calculate the objective function at the corner points of the feasible region.
your graph will look like this:
the corner points of the feasible region are:
(0,90)
(20,30)
(80,0)
evalutate the objective function at these corner points.
the points are in (x,y) format.
at (0,90), your cost is 0 * 1500 + 90 * 2000 = 180,000.
at (20,30), your cost is 20 * 1500 + 30 * 2000 = 90.000.
at (80,0), your cost is 20 * 80 + 0 * 30 = 160,000.
your minimum cost is at (20,30).
that means 20 days utilization of factory I and 30 days utilization of factory II.
your constraint must be satisfied at x = 20 and y = 30.
100 * x + 200 * y becomes 100 * 20 + 200 * 30 = 2000 + 6000 = 8000, which is less than or equal to 8000.
300 * x + 100 * y becomes 300 * 20 + 100 * 30 = 6000 + 3000 = 9000, which is less than or equal to 9000.
both x and y are greater than or equal to 0.
all the constraints are satisfied.
your minimum cost solution is 20 days from factory I and 30 days from factory II.
now to answer your questions.
they are reproduced below:
(i) Which variable should x define? (Enter letter for correct answer in the blank)
a. number of pairs of athletic shoes produced
b. number of days Factory I operates
c. number of pairs of casual shoes produced
d. cost to produce a pair of athletic shoes
selection b.
number of days factory I operates.
(ii) Which variable should y define? (Enter letter for correct answer in the blank)
a. number of pairs of athletic shoes produced
b. cost to produce a pair of casual shoes
c. number of pairs of casual shoes produced
d. number of days Factory II operates
selection d.
number of days factory II operates.
(iii) Which of the following correctly defines the constraint on total daily production of pairs of casual shoes? (Enter letter for correct answer in the blank)
a. 200x+100y≤8000
b. 100x+200y≤8000
c. 200x+100y≥8000
d. 100x+200y≥8000
selection d.
100x + 200y >= 2000
(iv) Which of the following correctly defines the constraint on total daily production of pairs of athletic shoes? (Enter letter for correct answer in the blank)
a. 300x+100y≥9000
b. 100x+300y≥9000
c. 300x+100y≤9000
d. 100x+300y≤9000
selection a.
300x + 100y >= 9000
(v) Which of the following correctly defines the objective function? (Enter letter for correct answer in the blank)
a. C=8000x+9000y
b. C=9000x+8000y
c. C=1500x+2000y
d. C=2000x+1500y
selection c.
1500x + 2000y
(vi) Which of the following sets of corner points correctly defines the feasible region for this system of linear inequalities? (Enter letter for correct answer in the blank)
a. (0, 0) ; (40, 0) ; (20, 30) ; (30, 0)
b. (0, 40) ; (0, 90) ; (20, 30)
c. (0, 90) ; (20, 30) ; (80, 0)
d. (20, 30) ; (30, 0) ; (80, 0)
selection c.
(0,90), (20,30), (80,0)
(vii) Based on your determination of the feasible region for this business process, how many days should each factory operate to meet production goals at minimum cost? (Enter letter for correct answer in the blank)
a. 30 days for Factory I and 0 days for Factory II
b. 20 days for Factory I and 30 days for Factory II
c. 0 days for Factory I and 40 days for Factory II
d. None of these answers are correct
selection b.
20 days for factory I and 30 days for factory II.
(viii) Which of the following is the correct value for the minimum cost to meet production goals for both types of shoes?
a. C = $45000
b. C = $80000
c. C = $90000
d. C = $120000
selection d.
cost = $90,000.
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