You are a member of an architectural firm that has been hired by a school distri

Question

You are a member of an architectural firm that has been hired by a school district to design a new classroom building. The cost to build the new school depends on many factors. Any new school needs to have a faculty consisting of teachers and aides. You can have at most 50 faculty, and you need to have at least 12 teachers. The minimum number of faculty you can have is 20 and the number of teachers must be at least half of the number of aides. If the average teacher salary is $15,000 and the average aides salary is $10,000, how many teachers and aides must you have so that the cost to the district is as small as possible? 1.

Explanation / Answer

Let x = number of teachers in faculty ; y = number of aids in faculty

Total faculty = x+y <=50

x>=12

Also 20 <= x+y <= 50

x >= y/2 => x-y/2 >=0

We have following constraints with objective function :

Min.z = min {15000x + 10000y}

subject to :

x+y >=20

x+y <=50

x-y/2>=0

x>=12

Using graphical method :

We apply corner point method :

The corner points are (12,38); (16.667,33.33) and (12,24)

At (12,38);z = 15000(12)+10000(38) = 560000
At (16.667,33,33); z=15000(16.667)+10000(33.33) = 583305

At (12,24) ; z = 15000 (12)+(10000)(24) = 420000

Hence 12 teachers and 24 aides must be there in order to have cost as small as possible

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