Objectives:Set up a Linear Programming ProblemSolve a Linear Programming Problem

Linear Programming

Objective function: an algebraic expression (linear expression) in two variables describing a quantity that must be maximized or minimized

Constraints: a collection of linear inequalities (system of linear inequalities) involving the variables that must be satisfied simultaneously

Feasible point(s): those point(s) that maximize or minimize the objective function

Linear Programming:

a method for solving problems in which a

particular quantity that must be

maximized or minimized is limited

by other factors

Solving a Linear Programming Problem

Let be an objective function that depends on . Furthermore, is subject to a number of constraints on . If a maximum or minimum value of exists, it can be determined as follows:

1. Graph the system of inequalities representing the constraints

2. Find the value of the objective function at each corner, or vertex, of the graphed region. The maximum and minimum of the objective function occur at one or more of the corner points.

z ax by and x y z

and x y

z

Steps:

1. Write an expression for the quantity to maximized or minimized. This expression is the objective function.

2. Write all the constraints as a system of linear inequalities and graph the system.

3. List the corner points of the graph of the feasible points.

4. List the corresponding values of the objective function at each corner point. The largest or smallest of these values is the solution.

Maximize

Subject to

EX 1: Solve the Linear Programming Problem

3z x y

0

0

3

5

7

x

y

x y

x

y

Minimize

Subject to

EX 2: Solve the Linear Programming Problem

3 4z x y

0

0

2 3 6

8

x

y

x y

x y

EX 3: A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model are given in the following table:

The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours respectively. The profits per unit are $25 for model A and $15 for model B. How many of each type should be produced to maximize profit?

Model A Model B

Assembling 5 4

Painting 2 3