Linear Programming

Many problems involve quantities that must be maximized or minimized. Businesses are interested in maximizing profit. An operation in which bottled water and medical kits are shipped to earthquake victims needs to maximize the number of victims helped by this shipment. An objective function is an algebraic expression in two or more variables describing a quantity that must be maximized or minimized.

Objective Functions in Linear Programming

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

Text Example

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 second 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.

Text Example

Solving a Linear Programming Problem• Let z=ax + by be an objective function that depends

on x and y. Furthermore, z is subject to a number of constraints on x and y. If a maximum or minimum value of z 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.

Example• Given the objective function and a system of linear

inequalities. Objective function z = 3x+2y Constraints x>0, y>0, 2x+y<8, x+y>4

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.

Example cont.

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

-10

-8

-6

-4

-2

2

4

6

8

10

• Graph x=0, y=02x+y=8x+y=4

Example cont.

• x>0

Example cont.

• y>0

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• (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.

Objective Function: z = 2x + yAt (0, 0): z = 2 • 0 + 0 = 0At (2, 0): z = 2 • 2 + 0 = 4At (3, 1): z = 2 • 3 + 1 = 7At (0, 2.5): z = 2 • 0 + 2.5 = 2.5

The maximum value of z.

Text Example

Linear Programming