##### Linear Programming: Bakery Problem

label Mathematics
account_circle Unassigned
schedule 1 Day
account_balance_wallet \$5

Bakery has bought 250 pounds of muffin dough. They want to make waffles or  muffins in half dozen packs out of it. Half a dozen of muffins requires 1 lb of dough and a pack of waffles uses 3/4 lb of dough. It take bakers 6 minutes to make a half-dozen of waffles and 3 minutes to make a half-dozen of muffins. Their profit will be \$1.50 on each pack of waffles and \$2.00 on each pack of muffins. How many of each should they make to maximize profit, if they have just 20 hours to do everything?

May 23rd, 2015

Let the number of packs of muffins be x

and the number of packs of waffles be y.

Amount of dough required for a pack of muffins = 1 lb

Amount of dough required for a pack of waffles = 3/4 lb

Total amount of dough available = 250 pounds = 250 lb

So, the inequality can be written as

$1*x+\frac{3}{4}y\le250\\ \\ x+\frac{3}{4}y\le250\\ \\ 4x+3y\le1000$

Time taken to make a pack of muffins = 3 min

Time taken to make a pack of waffles = 6 min

Total time available = 20 hours = 20*60 min = 1200 min

So, the inequality can be written as

$3x+6y\le1200\\ \\ x+2y\le400\\$

Profit for a pack of muffins = \$2

Profit for a pack of waffles = \$1.5

Total profit, P = 2x + 1.5y

P has to be maximised.

So, the constraints to be considered are

$4x+3y\le1000$

$x+2y\le400\\$

$x\ge0\\$

$y\ge0\\$

The solution space is the region OABC.

Its vertices can be used to find the maximum profit.

May 23rd, 2015

Considering C(0,  200)

x = 0, y = 200

then, P = 2*0 +  1.5*200 = 0 + 300 = \$300

Considering B(160, 120)

x = 160 , y = 120

then, P = 2*160 +  1.5*120 = 320 + 180 = \$500

Considering A(250, 0)

x = 250, y = 0

then, P = 2*250 +  1.5*0 = 500 + 0 = \$500

So there are two possible ways to maximize profit

i) Number of muffins = 160 and Number of waffles = 120

ii) Number of muffins = 250 and Number of waffles = 0

May 23rd, 2015

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May 23rd, 2015
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May 23rd, 2015
Oct 23rd, 2017
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