Practice Questions: Linear Programming
1.
A steel manufacturer produces metal sheet in standard size of: width 11ft and
length 11ft. After the production, the plant can cut the sheet into the sizes a
customer may want. For example, it can cut a sheet into 2 pieces of 11ft*4ft
and one piece of 11ft*3ft. A customer comes and need 100 11ft*3ft, 200
11ft*4ft, and 300 11ft*5ft sheets. What is the smallest number of metal sheets
the company must produce to satisfy all the demand?
In order to produce the 100 11ft*3ft, 200 11ft*4ft, and 300 11ft*5ft sheets
from standard size 11ft*11ft sheet, the following cut pattern can be used.
a.
having 3 3ft one from one standard sheet
b.
having 2 3ft one and 1 4ft one from one standard sheet
c.
having 1 3ft one and 2 4ft one from one standard sheet
d.
having 2 3ft one and 1 5ft one from one standard sheet
e.
having 1 4ft one and 1 5ft one from one standard sheet
f.
having 2 5ft one from one standard sheet
Assume we use x_1, x_2, x_3, x_4, x_5, x_6 sheets for all the six cutting
pattern:
Objective function: min x_1+x_2+x_3+x_4+x_5+x_6
Constraints:
3x_1+2x_2+x_3+2x_4>=100
x_2+2x_3+x_5>=200
x_4+x_5+2x_6>=300
The optimal solution is x_3=100, x_6=150, and all other xs equal to zero.
2.
The Sue All Law Firm handles two types of lawsuits: medical malpractice
suits against unscrupulous heart surgeons for performing unnecessary surgery,
and suits against hard-working math professors for failing students who do not
deserve to pass. Math professor lawsuits each require 6 person-months of
preparation and the hiring of 5 expert witnesses, whereas medical lawsuits
each require 10 person-months of preparation and the hiring of 3 expert
witnesses. The firm has a total of 30 person-months to work with and feels
that it cannot afford to hire more than 15 expert witnesses. It makes an
average profit of $1 million per math professor sued and $5 million per heart
surgeon sued. How many of each type of lawsuit should it initiate in order to
maximize its expected profits?
X
:number of lawsuits against teachers
Y
:number of lawsuits against a medic

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