SOLUTION: ) A brewery manufactures three types of beer - lite, regular, and dark. Each vat of lite beer requires 6 bags of barley, 1 bag of sugar and 1 bag of hops. Each vat of regular be

A brewery manufactures three types of beer - lite, regular, and dark. Each vat
of lite beer requires 6 bags of barley, 1 bag of sugar and 1 bag of hops. Each
vat of regular beer requires 4 bags of barley, 3 bag of sugar and 1 bag of
hops. Each vat of dark beer requires 2 bags of barley, 2 bag of sugar and 4
bags of hops. Each day the brewery has 800 bags of barley, 600 bag of sugar and
300 bag of hops. The brewery realizes a profit of $10 per vat of lite beer, $20
per vat of regular beer, and $30 per vat of dark beer. For this linear
programming problem: 
(a) What are the decision variables?

x = the number of vats of lite beer to make each day
y = the number of vats of regular beer to make each day
z = the number of vats of dark beer beer to make each day

(b) What is the objective function?

Profit = $10x + $20y + $30z or

P = 10x + 20y + 30z 

(c) What are the constraints?

There are six constraints (inequalities)

1.
>>...Each vat of lite beer requires 6 bags of barley...<<
>>...Each vat of regular beer requires 4 bags of barley...<<
>>...Each vat of dark beer requires 2 bags of barley...<<
>>...Each day the brewery has 800 bags of barley...<<

So this inequality is the barley constraint:  6x + 4y + 2z <= 800
It ensures that the brewery does not run out of barley.

2.
>>...Each vat of lite beer requires...1 bag of sugar...<<
>>...Each vat of regular beer requires...3 bags of sugar...<<
>>...Each vat of dark beer requires...2 bags of sugar...<<
>>...Each day the brewery has...600 bags of sugar...<<

So this inequality is the sugar constraint:  1x + 3y + 2z <= 600
It ensures that the brewery does not run out of sugar.

3.
>>...Each vat of lite beer requires...1 bag of hops...<<
>>...Each vat of regular beer requires...1 bag of hops...<<
>>...Each vat of dark beer requires...4 bags of hops...<<
>>...Each day the brewery has...300 bags of hops...<<

So this inequality is the hops constraint:  1x + 1y + 4z <= 300
It ensures that the brewery does not run out of hops.

4.  The trivial constraint that says the amount of barley cannot be a
negative number.

      x >= 0

5.  The trivial constraint that says the amount of sugar cannot be a
negative number.

      y >= 0

6.  The trivial constraint that says the amount of hops cannot be a
negative number.

      z >= 0

Edwin
AnlytcPhil@aol.com