Posted by LIZ on Friday, January 28, 2011 at 10:04pm.
h=number of homeless patients
H=number of high-income patients
Constraints:
1. h≥770 (minimum)
2. H≤900 (maximum)
3. H≤2h (maximum ratio)
Capacity
4. H+h=2100
Z=profit = 9700h + 7900H
Obviously the objective function Z is maximized when h is at its maximum possible.
The response can be found by inspection, namely h=1400, H=700, which satisfies all 4 constraints.
If the answer is not obvious, try equal numbers (1050 each) and change for as many homeless as possible without violating any of the constraints.
I had these as the equations but how do u set them up to find how many of each to maximise profit
It depends on what your teacher expects you to know. Have you done simplex method, graphics, or trial and error?
You can plot the constraints on a graph as a line, and shade the non-feasible regions (usually on one side of the line). After all four constraints have been constructed and the appropriate sides shaded, the remaining unshaded region is the feasible region.
Evaluate the objective function at each of the corners of the feasible polygon, and find the maximum.
P.S.
I left out two other (trivial) constraints, namely h≥0 and H≥0.
noo
Sorry, there was a mistake in the formulation of the constraints.
The line H≤2h should have read h≤2H to reflect "the number of homeless patients should not exceed twice the number of high-income patients"
So a summary of the constraints now reads:
1. h≥770 (minimum)
2. H≥0 (minimum H)
3. H≤900 (maximum)
4. H≥h/2 (minimum ratio for H, maximum ratio for h)
5. H+h=2100 (capacity)
Z=profit = 9700h + 7900H
Z is to be evaluated at corners of the polygon.
Check:
http://img232.imageshack.us/img232/4710/1296270284.png
Legend:
red - max value of H
blue - capacity of 2100
green - min. ratio H/h
We see that the corners of the polygon of feasibility are at
(770,900), (1200,900), (1400,700), (770,385).
Evaluate the objective function Z at these points and make your choice.
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