# linear programming

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A manufacturer produces two items, bookcases and library tables. Each item requires processing in each of two departments. Department I has 40 hours available and department II has 36 hours available each week for production. To manufacture a bookcase requires 2 hours in department I and 4 hours in department II, while a library table requires 3 hours in department I and 2 hours in department II. Profits on the items are \$6.00 for a bookcase and \$7.00 for a library table.
Identify all constraints.
Identify all applicable corner points of the feasibility region.
If all units produced can be sold, how many of each should be made in order to maximize profits?

• linear programming -

We are not going to do all your linear programming problems for you. I will rough out one of them. Then you try and post your work if you want feedback.

b = number of bookcases
t = number of tables

constraints:

Department 1:
2 b + 3 t </= 40
Department 2:
4 b + 2 t </= 36

Profit to maximize:
p = 6 b + 7 t

corner points: (b,t)
(0,0)
(0,13.33) because 13.33<36/2
(9,0) because 9<20
then the intersection
t = 11 and b = 3.5 or (3.5,11)

check corners for p =6 b + 7 t
(0,0) gives 0 profit
(0,13.33) gives 6*0+7*13.33=93.31
(9,0) gives 6*9 = 54
(3.5,11) gives 6*3.5+7*11 = 98 winner
I am assuming fractional bookcases and tables can be built because you presumably can spread this out over many weeks.

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