Question

What is the maximum profit? Thanks!

Answer 1

Breakfast tea, 75lbs, $112.50
Afternoon tea, 40lbs, $80.00

Total $192.50

Explanation:

One way to approach this is to set up a chart:

`(("","A grade"=45lb,"B grade"=70lb),("Breakfast"=$1.50,1/3lb,2/3lb),("Afternoon"=$2.00,1/2lb,1/2lb))`

Let's first do this by looking at the profits of the teas.

Let's first try Since we get more profit from the Afternoon tea, we want to make as much of that as possible. We can make 90 pounds of it (there being 45 lbs of A grade tea):

Trial 1
Afternoon tea, 90 pounds, $180 - 25 lbs of Grade B tea left over.

Can we do better than this? Since we have more Grade B than Grade A and it takes more Grade B to make the Breakfast blend, let's try making that. We have enough A grade to make `45/(1/3)=135lbs` and enough B grade to make `70/(2/3)=210/2=105lbs`, so let's make 105 lbs of Breakfast:

Trial 2
Breakfast tea, 105lbs, $157.50 - 10 lbs of Grade A leftover.

Notice that if I were to have made 30 less pounds of Breakfast, we'd have 20 pounds of A Grade and 20 pounds of B grade left over. So let's try making 30 less pounds of Breakfast and instead using all the raw ingredients in making an additional 40 pounds of Afternoon tea:

Trial 3
Breakfast tea, 75lbs, $112.50
Afternoon tea, 40lbs, $80.00

Total $192.50

Answer 2

See below.

Explanation:

Calling

`x_A =` tea `A` amount.
`x_B =` tea `B` amount.
`y_1 =` breakfast blend amount
`y_2 =` afternoon blend amount
`c_1 = 1.50` Profit for breakfast blend
`c_2 = 2.0` Profit for afternoon blend

we have

`y_1 = 1/3x_A+2/3 x_B`
`y_2 = 1/2 x_A + 1/2 x_B`

`f = c_1 y_1+c_2 y_2`

So we have the maximization problem

`max f`

subject to

`x_A le 45`
`x_B le 70`
`y_1 + y_2 le x_A+x_B`

The solution is for

`x_A = 45, x_B = 66.43` with a total profit of `200.36` pounds or
`x_A =40.24, x_B = 70` with the same profit.

As can be observed in the feasible region (light blue) there is a slanted corner due to the restriction `y_1+y_2 le x_A+x_B` so any combination

`(45,66.43)lambda + (40.24,70)(1-lambda)` for `lambda in [0,1]` is a valid solution with the same profit which is `200.36` pounds.