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Linear Optimization - The Schutzberg Brewery Problem

6. Making Beer. The Schutzberg Brewery has received an order for 1,500 gallons of 3-percent beer (that is, 3 percent alcoholic content). This is a custom order because Schutzberg does not produce a 3-percent product. They do brew the following products.

Product Percent Alcohol Cost per Gallon
Free 0.25 0.55
Light 2.50 0.65
Amber 4.50 0.80
Dark 6.00 0.75

There are 500 gallons of each of these products on hand. Rather than brewing a 3-percent beer from scratch, the brewmaster has decided to mix existing stocks, perhaps with some water (0 percent alcoholic content), to satisfy this small order in the shortest possible time, hoping that the taste will be adequate. In case the taste is bad and he has to throw the mixture out, he would like to minimize the cost of the mix.

a. What are the components of the least-cost blend that will result in a 3-percent beer?
b. What is the total cost for the 100 gallons of product?
c. Use the pattern in (b) to trace the effects of increasing the requirement by 10 percent. How will the optimal mix change? How will the optimal cost change?