SOLUTION: Farmer Billy Cauer has two farms in Dallas to grow wheat and barley. There are differences in the yields and costs of growing crops due to soil conditions at two farms: McKi

Question 1164591: Farmer Billy Cauer has two farms in Dallas to grow wheat and barley. There are differences in the yields and costs of growing crops due to soil conditions at two farms:

McKinney Farm : Barley yield/acre - 400 bushels ; Cost/acre of barley - PhP 90; Wheat yield/acre- 350 bushels; Cost/acre of wheat- PhP 110

Addison Farm : Barley yield/acre - 700 bushels; Cost/acre of barley - PhP 80; Wheat yield/acre- 300 bushels; Cost/acre of wheat- PhP 100

McKinney and Addison farms have 70 and 120 acres for cultivation. At least 20,000 bushels of barley and 30,000 bushels of wheat must be grown. Formulate an LP to minimize the cost of meeting wheat and barley demand.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i did the following:

a = number of acres of barley from mckinney farm.
b = number of acres of wheat from mckinney farm.

c = number of acres of barley from addison farm.
d = number of acres of wheat from addison farm.

the total cost is e = 90a + 110b + 80 c + 100d

the total yield of barley needs to be greater than or equal to 20,000 bushels.
the total yield of wheat needs to be greater than or equal to 30,000 bushels.
the inequalities to satisfy that are:
400a + 700c >= 20000
350b + 300d >= 30000

the total acres for mckinney farm need to be less than 70.
the total acres for addison farm need to be less than 120.
the inequalities to satisfy that are:
a + b <= 70
c + d <= 120

a,b,c,d are all assumed to be greater than or equal to 0.

i used the following simplex method linear programming tool to provide the solution:

https://www.zweigmedia.com/RealWorld/simplex.html

the results from the use of this tool are shown below:

the tool says:

total cost is 248200/21 = 11819.04762 (e)
total acres from mckinney farm are 0 for barley + 70 for wheat = 70 (a + b)
total acres from addison farm are 200/7 + 55/3 = 28.57142857 for barley + 18.33333333 = 46.9047619 for wheat (c + d)
total bushels of barley are 0 * 400 from mckinney farm + 200/7 * 700 from addison for a total of 20,000.
total bushels of wheat are 70 * 350 + from mckinney farm + 55/3 * 300 from addison for a total of 30,000.

it looks like all the constraints have been satisfied.
since they only wanted you to formulate the requirements to provide the solution, your solution is more than likely as shown below:

a = number of acres of barley from mckinney farm.
b = number of acres of wheat from mckinney farm.
c = number of acres of barley from addison farm.
d = number of acres of wheat from addison farm.
e = total cost from boh farms.

objective function that you want to minimize:

e = 90a + 110b + 80c + 100d

constraints:

a + b <= 70
c + d <= 120
400a + 700c >= 20000
350b + 300d >= 30000

a,b,c,d are all assumed to be greater than or equal to 0.

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