Section 3 2 Linear Programming Problems Linear Programming

Section 3. 2 Linear Programming Problems

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.

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?

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

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

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 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, 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?

Organize the Information: Number of Chairs Plant A B C Max. Prod. I x 1 x 2 x 3 150 II x 4 x 5 x 6 120 Min. 70 70 80 Cost to Ship Plant I II A 30 32 B 32 28 C 38 26

We want to minimize the cost function: C = 30 x 1 + 32 x 2 + 38 x 3 + 32 x 4 + 28 x 5 + 26 x 6 Production constraints: Plant II Warehouse constraints: A B C

So the problem is: Minimize: C = 30 x 1 + 32 x 2 + 38 x 3 + 32 x 4 + 28 x 5 + 26 x 6 Subject to: