Linear Programming Problems

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Section 3.2

• Linear Programming Problems

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Linear Programming Problem
A linear programming problem consists of a linear
objective function to be maximized or minimized,
subject to certain constraints in the form of
linear equalities or inequalities.
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Ex. A small company consisting of two carpenters
and a finisher produce and sell two types of
tables type A and type B. The type-A table
will result in a profit of 50, and each type-B
table will result in a profit of 54. A type-A
table requires 3 hours of carpentry and 1 hour of
finishing. A type-B table requires 2 hours of
carpentry and 2 hours of finishing. Each day
there are 16 hours available for carpentry and 8
hours available for finishing. How many tables
of each type should be made each day to maximize
profit?
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Organize the Information
Let x type A and y type B. The Profit to
Maximize (in dollars) is given by P 50x
54y
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The constraints are given by Carpentry Finishin
g
Also so that the number of units is not less than
0
So we have
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Ex. A particular company manufactures specialty
chairs in two plants. Plant I has an output of
at most 150 chairs/month. Plant II has an output
has an output of at most 120 chairs/month. The
chairs are shipped to 3 possible warehouses - A,
B, and C. The minimum monthly requirements for
warehouses A, B, and C are 70, 70, and 80
respectively. Shipping charges from plant I (to
A, B, and C) are 30, 32, and 38/chair and from
plant II (to A, B, and C) are 32, 28, 26. How
many chairs should be shipped to each warehouse
to minimize the monthly shipping cost?
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Organize the Information
Number of Chairs
Cost to Ship
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We want to minimize the cost function C 30x1
32x2 38x3 32x4 28x5 26x6
Production constraints Plant I Plant II
Warehouse constraints A B C
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So the problem is Minimize Subject to
C 30x1 32x2 38x3 32x4 28x5 26x6