1. Linear Programming Problems
Section 3.2
Linear Programming Problems

2. 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.

3. 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?

4. Organize the Information:
Table A
Table B
Time
Carpentry 3 2
16 hours
Finishing 1
8 hours
Profit/table
$50
$54
Let x = # type A and y = # type B.
The Profit to Maximize (in dollars) is given by:
P = 50x + 54y

5. The constraints are given by:
Carpentry
Finishing
Also so that the number of units is not less than 0:
So we have:

6. Ex. A particular company manufactures specialty chairs in two plants
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?

7. Organize the Information: Plant A B C I x1 x2 x3 150 II x4 x5 x6 120
Number of Chairs
Plant A B C
Max. Prod. I x1 x2 x3
150 II x4 x5 x6
120
Min. 70 80
Cost to Ship
Plant A B C I 30 32 38 II 28 26

8. 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

9. So the problem is:
Minimize:
Subject to:
C = 30x1 + 32x2 + 38x3 + 32x4 + 28x5 + 26x6