2. A beer company makes a lager beer and a light beer and are sold in 12 ounces
ID: 468333 • Letter: 2
Question
2. A beer company makes a lager beer and a light beer and are sold in 12 ounces bottles. They mix imported beer and domestic beer to make a hybrid blend. (This is their niche!) The imported beer and domestic beer are made of hops and barley, as given in Table 2.1. Demand is at least 10,000 bottles of the lager and at least 12,000 bottles of the light beer. Each bottle of lager must consist of at most 43% of hops and each bottle of light beer must consist of at least 54% of barley. Assume that one bottle imported or domestic beer will yield one bottle of lager or light beer, and that the rest of the beer is made of insignificant ingredients. How much imported beer and domestic beer should be put into the lager and the light beer to meet demand but minimizes cost?
Table 2.1
(d) Write the linear program, in standard form, that is needed to solve the problem.
(e) Put the linear program from (a) into Excel to find the optimal solution.
Hopes (%) Barley(%) Cost/Bottle Imported 38 60 0.52 Domestic 45 50 0.40Explanation / Answer
One bottle contains 12 ounces of beer.
So if company makes lager beer,, 1 bottle bear should have atmost 43% hops which is equal to 12 ( bottle capacity) * 43% = 5.16 ounces not beyond that.
So if company makes lighter beer, 1 bottle bear should have atleast 54% barley which is equal to 12 * 54% = 6.48 ounces.
Now if lets mix 1 bottle of imported bear and 1 bottle of domestic beer
As per above table there will be 43% hops which is atmost quantity to serve beer as Lager bear.
So maximum cost of lager bear will be ( cost of 1 bottle of imported beer + cost of 1 bottle of domestic beer ) /2 = ( 0.52 + 0.4 ) /2 = 0.46
consider the below ratio in which imported and domestic bears are mixed to form Lager or lighter.
Based on percentage of hopes and Barley we can figure out whether the mixture is Lighter or Lager.
As per abouve table if the imported and domestic beer is mixed in 1:2 , it will form a Lager and if these beers are mixed in 2:1 ratio, then it will form Lighter.
minimum cost of Lager beer will be 0.46 and minimu cost of lighter beer is 0.48
So the tota cost on demand is of 10000 bottles of lager + 12000 bottles of light beer = 0.46 * 10000 + 0.48 *12000= 10360.
For more understanding
Linear Equaltion can be formed like mentioned below
Lets assume X is the QTY of Imported beer Mixed and Y is QTY if Domestic beer mixed.
Lets asume Hopes qty in Imported beer Him and that in Domestic beer is Hd.
THen Hopes QTY in mixture ( HM ) = ( X * Him + Y * Hd) / ( X + Y)
similarly Barley qty in imported beer is Bim and that is domestic beer is Bd
then Barley QTY in mixture ( BM ) = ( X * Bim + Y * Bd ) / ( X + Y)
So the total cost of mixture per bottle = ( X * 0.52 + Y * 0.4 )/(X + Y )
If the Lager beer cost is calculated by above formula is Lg
and the lighter beer cost is calculated by aboive formula is Lt.
The total cost as per demand = 10000 * Lg + 12000 * Lt.
Hopes (%) Barley(%) QTY in ounces Hopes weight Barley weight Imported 38% 60% 12 4.56 7.2 Domestic 45% 50% 12 5.4 6 If mixed 43% 53% 24 10.3 12.7Related Questions
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