3.4 linear programming

3-4 Linear Programming
Today’s objectives:
1. I will find the maximum and
minimum values of the objective
function for the given feasible
region.
2. I will solve linear programming
problems.

Linear Programming
Businesses use linear
programming to find out how to
maximize profit or minimize
costs. Most have constraints on
what they can use or buy.

Find the minimum and maximum
value of the function f(x, y) = 3x - 2y.

We are given the constraints:
 y≥2
 1 ≤ x ≤5
 y≤x+3

Linear Programming
 Find the minimum and maximum
values by graphing the inequalities
and finding the vertices of the
polygon formed.
 Substitute the vertices into the

function and find the largest and
smallest values.

8 1 ≤ x ≤5
7

6

5

4

3
y≥2
2
y≤x+3
1

1 2 3 4 5

Linear Programming
 The vertices of the
quadrilateral formed are:
(1, 2) (1, 4) (5, 2) (5, 8)
 Plug these points into the

function f(x, y) = 3x - 2y

Linear Programming
f(x, y) = 3x - 2y
 f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1

 f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5

 f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11

 f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1

Linear Programming
 f(1, 4) = -5 minimum
 f(5, 2) = 11 maximum

Find the minimum and maximum value
of the function f(x, y) = 4x + 3y

We are given the constraints:
 y ≥ -x + 2
1
 y ≤ x+2
4
 y ≥ 2x -5

6
y ≥ 2x -5
5 1
y≤ x+2
4
4

3
y ≥ -x + 2
2

1

1 2 3 4 5

Vertices
f(x, y) = 4x + 3y
 f(0, 2) = 4(0) + 3(2) = 6

 f(4, 3) = 4(4) + 3(3) = 25

7 1 7 1 28 25
 f( , - ) = 4( ) + 3(- ) =
3 3 3 3 3 -1 = 3

Linear Programming

 f(0, 2) = 6 minimum
 f(4, 3) = 25 maximum

3.4 linear programming

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