Expert: Steve Holleran - 12/3/2007

Question
Write the objective function and constraints for the following linear program, then carefully graph the system, clearly identify the feasible region, and the corner points, then determine the solution that optimizes the system.

An orphanage houses teens and preschoolers. The operator of the orphanage wants to house no more than 16 children, including no more than 10 preschoolers. It has been determined that it costs $15 a day to feed and clothe the preschoolers, and $45 a day to feed and clothe the teens. Also the operator knows she has a budget for food and clothing of no more than $540 each day.

The orphanage also puts the kids to work, and each day it's determined the preschoolers rake in $7 in profit and the teens, $20. How many of each type of child should the operator invite into the orphanage to maximize the cash the children bring in?

Answer
Hi Alex,

Okay, let's let x = # teens and y = # preschoolers

Then the objective function we want to maximize is

                   C = 20x + 17y

The constraints are:

x >= 0

y >= 0

y <= 10

x + y <= 16 ------------> y <= -x + 16

45x + 15y <= 540--------> y <= -3x + 36

If you graph all the constraints, you should find the following points defining the feasible region:

A(0, 10)     B(6,10)     C(10, 6)    D(12, 0)   E(0,0)

Evaluating the objective function at each point gives :

C at A = 170

C at B = 290

C at C = 202

C at D = 240

C at E = 0

So for a maximum amt of cash, x = 6 teens and y = 10 preschoolers would do the trick.

I hope you are able to graph the lines and find the intersection points without any trouble.  Let me know if anything is not clear
Steve