Lesson 7.6, page 767Linear Programming

Objective: Write an objective function describing a quantity that must be maximized or minimized. Use inequalities to describe limitations in a situation. Use linear programming to solve problems.

Linear Programming In many applications, we want to find a

maximum or minimum value. Linear programming can tell us how to do this.

Constraints are expressed as inequalities. The solution set of the system of inequalities made up of the constraints contains all the feasible solutions of a linear programming problem.

The function that we want to maximize or minimize is called the objective function.

See Example 1, page 767 Check point 1: A company manufactures bookshelves and desks

for computers. Let x represent the number of bookshelves manufactured daily and y the number of desks manufactured daily. The company’s profits are $25 per bookshelf and $55 per desk. Write the objective function that describes the company’s total daily profit, z, from x bookshelves and y desks.

See Example 2, page 767 Check point 2: To maintain high quality, the company in Check

Point 1 should not manufacture more than a total of 80 bookshelves and desks per day. Write an inequality that describes this constraint.

See Example 3, page 768 Check point 3: To meet customer demand, the company in Check

Point 1 must manufacture between 30 and 80 bookshelves per day, inclusive. Furthermore, the company must manufacture at least 10 and no more than 30 desks per day. Write an inequality that describes each of these sentences. Then summarize what you have described about this company by writing the objective function for its profits and the three constraints.

Linear Programming Procedure To find the maximum or minimum value of

a linear objective function subject to a set of constraints:

1. Graph the region of feasible solutions. (the system of equations)

2. Determine the coordinates of the vertices of the region.

3. Evaluate the objective function at each vertex. The largest and smallest of those values are the maximum and minimum values of the function, respectively.

Example – Watch. Graph the following system of inequalities

and find the coordinates of any vertices formed:

2 0

2

0

y

x y

x y

Example continued

We graph the related equations using solid lines. We shade the region common to all three solution sets.

2 0

2

0

y

x y

x y

Example continued

The system of equations from inequalities (1) and (2): y + 2 = 0x + y = 2

The vertex is (4, 2).

The system of equations from inequalities (1) and (3):y + 2 = 0x + y = 0The vertex is (2, 2).

The system of equations from inequalities (2) and (3):x + y = 2 x + y = 0The vertex is (1, 1).

To find the vertices, we solve three systems of equations.

Example A tray of corn muffins requires 4 cups of milk and 3 cups

of wheat flour. A tray of pumpkin muffins requires 2 cups of milk and 3 cups of wheat flour. There are 16 cups of milk and 15 cups of wheat flour available, and the baker makes $3 per tray profit on corn muffins and $2 per tray profit on pumpkin muffins. How many trays of each should the baker make in order to maximize profits?

Solution:

We let x = the number of corn muffins and y = the number of pumpkin muffins. Then the profit P is given by the function P = 3x + 2y.

Example continued We know that x muffins require 4 cups of milk and y

muffins require 2 cups of milk. Since there are no more than 16 cups of milk, we have one constraint. 4x + 2y 16

Similarly, the muffins require 3 and 3 cups of wheat flour. There are no more than 15 cups of flour available, so we have a second constraint.

3x + 3y 15

We also know x 0 and y 0 because the baker cannot make a negative number of either muffin.

Example continued

Thus we want to maximize the objective function P = 3x + 2y subject to the constraints:

4x + 2y 16,

3x + 3y 15,

x 0, and y 0.

We graph the system of inequalities and determine the vertices.

4x + 2y 16,3x + 3y 15,x 0, y 0.

Next, we evaluate the objective function P at each vertex.

P = 3(3) + 2(2) = 13(3, 2)

P = 3(0) + 2(5) = 10(0, 5)

P = 3(4) + 2(0) = 12(4, 0)

P = 3(0) + 2(0) = 0(0, 0)

Profit P = 3x+ 2yVertices

The baker will make a maximum profit when 3 trays of corn muffins and 2 trays of pumpkin muffins are produced.

Check Point 4, page 770 For the company in Check Points 1-3, how many

bookshelves and desks should be manufactured per day to obtain maximum profit? What is the maximum daily profit? x + y < 80, x > 0, y > 0, 30 < x < 80, 10 < y < 30

y

10

10

-10

-10

Check Point 4, page 770 For the company in Check

Points 1-3, how many bookshelves and desks should be manufactured per day to obtain maximum profit? What is the maximum daily profit? x + y < 80, x > 0, y > 0, 30 < x < 80, 10 < y < 30

Corner

(x,y)

Objective

Function

Check Point 5 Find the value of the

objective function z=3x + 5y subject to the constraints x>0, y>0, x+y>1, x+y < 6.

y

10

10

-10

-10