Linear Programming

A linear programming problem is made up of an objective function and a system of constraints.

•The objective function is an algebraic expression in two or more variables describing a quantity that must be maximized or minimized. It has the form

z = Ax + By•The system of constraints are made up of a system of linear inequalities. In applications, the variables are also limited to positive values.

The solution to the linear programming problem, if it exists, will occur at the vertices (corner points) of the solution region obtained from the system of constraints.

Linear Programming Problems

Bottled water and medical supplies are to be shipped to victims of an earth-quake by plane. Each container of bottled water will serve 10 people and each medical kit will aid 6 people. If x represents the number of bottles of water to be shipped and y represents the number of medical kits, write the objective function that describes the number of people that can be helped.

Solution Because each bottle of water serves 10 people and each medical kit aids 6 people, we have

= 10x + 6y.Using z to represent the objective function, we have

z = 10x + 6y.Unlike the functions that we have seen so far, the objective function is an equation in three variables. For a value of x and a value of y, there is one and only one value of z. Thus, z is a function of x and y.

6 times the number of medical kits.

10 times the number of bottles of water plusis

The number ofPeople helped

Example

ExampleEach plane can carry no more than 80,000 pounds. The bottled

water weighs 20 pounds per container and each medical kit weighs 10 pounds. If x represents the number of bottles of water to be shipped and y represents the number of medical kits, write an inequality that describes this constraint .

Solution Because each plane can carry no more than 80,000 pounds,

The plane's volume constraint is described by the inequality 20x + 10y < 80,000.

20x + 10y < 80,000.

80,000 pounds

The total weight ofthe medical kits

must be less than or equal toplus

The total weight ofthe water bottles

Each bottle is 20 pounds.

Each kit is 10 pounds.

Planes can carry a total volume for supplies that does not exceed 6000 cubic feet. Each water bottle is 1 cubic foot and each medical kit also has a volume of 1 cubic foot. With x still representing the number of water bottles and y the number of medical kits, write an inequality that describes this constraint.

Solution Because each plane can carry a volume of supplies that does not exceed 6000 cubic feet, we have

The plane's volume constraint is described by the inequality x + y < 6000.

lx + ly < 6000.

6000 cubic feet.

The total volume ofthe medical kits

must be less than or equal toplus

The total volume ofthe water bottles

Each bottle is 1 cubic foot.

Each kit is 1 cubic foot.

Example

Solving a Linear Programming Problem

• If a maximum or minimum value of z = Ax + By exists, it can be determined as follows:

a. Graph the system of inequalities representing the constraints.

b. Find the value of the objective function at each corner of the graphed region.

c. Use the values that you found in the prior step to determine the maximum or minimum value of the objective function and the values of x and y for which the maximum or minimum occurs.

Example

• Solve:

Steps:a. Graph the system of inequalities representing the constraints.b. Find the value of the objective function at each corner of the graphed region.c. Use the values that you found in the prior step to determine the maximum

value of the objective function and the values of x and y for which the maximum occurs.

3 2

: 2 8

4

0, 0

Max z x y

subject to x y

x y

x y

Example cont.

• Graph the constraints:

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

0, 0

2 8

4

x y

x y

x y

Example cont.

x > 0

y > 0

Example cont.

2x+y<8

Example cont.

x+y>4

Example cont.

• Corners:

(4,0), (0,4), (0,8)

Example cont.

Objective function z = 3x+2y

Corners Objective function

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

(0,4) z = 3(0) + 2(4) = 8

(0,8) z = 3(0) + 2(8) = 16

The maximum value of the objective function is 16 and it occurs when x = 0 and y = 8

-1-3 -2 1 3 4 5 6 7

5

4

2

1

-3

-4

-5

-1

-2

x – y = 2

x + 2y = 5(0, 2.5)

(0, 0)(3, 1)

(2, 0)

Find the maximum value of the objective function z = 2x + y subject to the constraints: x > 0, y > 0, x + 2y < 5, x – y < 2.

Solution We begin by graphing the region in quadrant I (x > 0, y > 0) formed by the constraints.

Thus, the maximum value of z is 7, and this occurs when x = 3 and y = 1.

Now we evaluate the objective function at the four vertices of this region.

Corners Obj. Func.: z = 2x + y(0, 0) z = 2 • 0 + 0 = 0(2, 0) z = 2 • 2 + 0 = 4(3, 1) z = 2 • 3 + 1 = 7(0, 2.5) z = 2 • 0 + 2.5 = 2.5

The maximum value of z.

Example