following constraints from my information.
I will draw the graph with the x axis representing the number of small shirts,
and the y axis representing the number of large shirts.
4x + 5y < 800
This equation is due to the fact that I have a budget of £800. Therefore the
total cost of making x small t-shirts, at £4 each, and making y large shirts,
at £5 each, must be less than or equal to the budget of £800.
x > 3y
This constraint is because the rough survey has shown that we need 3 times as
many small shirts, x, than large shirts, y.
y < 65
My rough estimate from the house survey was that we did not need more than 65
large t-shirts. Some people were not present during the survey and we did not
have a long time to conduct the survey but definately not more than 65 people
need large t-shirts.
x < 50
I believe that we need more than 50 small shirts, as a large proportion of the
house seem to be on the small size, probably due to the fact that there is a
large amount of year 7s and year 8s in my house who are small.
The linear graph drawn shows the number of small t-shirts and the number of
large t-shirts we need to make to make the maximum profit, with regard to the
constraints shown above.
Results
The critical region indicated on my graph is outlined, and the verticies are
investigated to see if they give the optimum amount of profit. The critical
region is shaded in red, and the outside region is shaded in blue. The
verticies are indicated as crosses, and are labelled.
In order to find the exact location of the vertices, I have had to use
simultaneous equations:
To find vertice 1:
It is where the line x > 50 and the x axis cross.
Therefore the point is (50, 0).
Vertice 2:
This is where the lines x > 50 and x > 3y cross.
Therefore the x co-ordinate is 50.
Substituting 50 into the equation to find the y co-ordinate: 50 = 3y
y = 16 2/3
Therefore the point is (50, 16 2/3).
Vertice 3:
This is where the lines x > 3y and 4x + 5y < 800 cross.
To find the y co-ordinate: x = 3y
4x = -5y + 800
Multiply the first equation by 4 4x = 12y
4x = -5y + 800
= -17y +800
y = 800
17
y = 47.06 (to 2 d.p.)
Substituting the y co-ordinate into the first equation: x = 3 (47.06)
x = 141.18
Therefore the point is (141.18, 47.06).
Vertice 4
This is the point where the equation 4x + 5y < 800 meets the x axis.
Therefore y is 0 so to find the y co-ordinate: 4x = 800
x = 800
4
x = 200
Therefore the point is (200, 0).
In order to calculate profit, the equation:
6x + 8y+ = m where m equals the amount of money we would make from selling the
t shirts.
We must calculate first how much the t-shirts cost to make, and subtract this
from how much money we would make by selling the t-shirts.
(6x + 8y) – (4x + 5 y) = p
Simplified: 2x +3y = p
where p equals profit.
With Vertice 1 (50, 0)
p = (50 x 2) + (3 x 0) = 100
Using this point, we would have 50 small t-shirts but only £100 profit.
Vertice 2 (50, 16 2/3)
P = (2 x 50) + (3 x 16 2/3) = 150
Using this point, we would have 50 small t-shirts and 16 2/3 large t-shirts, and
have £150 profit.
Vertice 2 b
As we cannot have a 2/3 amount of a shirt, we have to use the integer value of
the number, without rounding it up, as we cannot create more shirt than we
originally have. Therefore this vertices must change to (50, 16).
P = (2 x 50) + (3 x 16) = £148
Using this point, we would have 50 small t-shirts and 16 large t-shirts, and
have a £148 profit.
Vertice 3 (141.18, 47.06)
P = (2 x 141.18) + (3 x 47.06) = £ 423.54
We would have 141.18 small t-shirts and 47.06 t-shirts, with £423.54 profit.
Vertice 3b
As with vertice 2, we would not be able to have 0.18 of a t-shirt, therefore we
have to take the integer number, so the co-ordinate would be (141, 47).
P = (2 x 141) + (3 x 47) = £423
Therefore we would have 141 small t-shirts and 47 large t-shirts, with £423
profit.
Vertice 4 (200, 0)
P = (2 x 200) + (3 x 0) = £400
We would have 200 small t-shirts and £400 profit.
My critical region on my graph showed that the line y < 65 is irrelavent as the
critical region does not even reach the line y < 65. We would not be able to
afford more than 65 large t-shirts anyway, with £800 as the critical path also
has the constraint of
x > 3y, which means that we need 3 times as many small t-shirt than large
t-shirts.
The Optimum Solution
From my results, at (141, 47), I would get my optimum solution. This would
involve the manufacture of 141 small t-shirts and 47 large t-shirts being made.
This would cost: (4 x 141) + (5 x 47) = £799 to make. We are selling them for:
(6 x 141) + (8 x 47) = £ 1222. Therefore we have a maximum profit of £422.
The Amended Problem
We have been given a sponsorship of more money by a local business as the
manager’s daughter is in my house. However, they would like their logo on the
t-shirts as well. This would make the t-shirts more expensive to make as well.
We are given £250 extra. Either t-shirt would cost £0.50 extra, therefore we
would sell our shirts for £0.50 extra.
Therefore with the modifications:
250 (new money given) + 800 (original money) = £1050
To make a small t-shirt: £4.50
large t-shirt: £ 8.50
This forms the new equation:
4.50 x + 5.50 y < 1050
This is used instead of the equation 4x + 5y < 800.
We will sell the small t-shirt: £ 6.50
large t-shirt: £ 8.50
The other constraints will remain the same, as they are not related to the cost
of the t-shirts: x > 3y
x < 50
y < 65
We would also have another profit equation:
profit = (6.5-x + 8.50y) – (4.50x + 5.50y)
Simplified: p = 2x +3y
This is the same as the original as we have
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