Math 1010 Optimizing An Advertising Campaign

Optimizing an Advertising Campaign
Math 1010 Intermediate Algebra Group Project

Background Information:

Linear Programming is a technique used for optimization of a real-world situation. Examples of
optimization include maximizing the number of items that can be manufactured or minimizing
the cost of production. The equation that represents the quantity to be optimized is called the
objective function, since the objective of the process is to optimize the value. In this project the
objective is to maximize the number of people who will be reached by an advertising campaign.
The objective is subject to limitations or constraints that are represented by inequalities.
Limitations on the number of items that can be produced, the number of hours that workers are
available, and the amount of land a farmer has for crops are examples of constraints that can be
represented using inequalities. Broadcasting an infinite number of advertisements is not a
realistic goal. In this project one of the constraints will be based on an advertising budget.
Graphing the system of inequalities based on the constraints provides a visual representation of
the possible solutions to the problem. If the graph is a closed region, it can be shown that the
values that optimize the objective function will occur at one of the "corners" of the region.

The Problem:

A local business plans on advertising their new product by purchasing advertisements on the
radio and on TV. The business plans to purchase at least 60 total ads and they want to have at
least twice as many TV ads as radio ads. Radio ads cost $20 each and TV ads cost $80 each.
The advertising budget is $4320. It is estimated that each radio ad will be heard by 2000
listeners and each TV ad will be seen by 1500 people. How many of each type of ad should be
purchased to maximize the number of people who will be reached by the advertisements?

Modeling the Problem:

Let X be the number of radio ads that are purchased and Y be the number of TV ads.

1. A linear inequality for the total number of desired ads. x+y>=66

2. A linear inequality for the cost of the ads. 20x+80y<=4320

3. The business wants at least twice as many TV ads as radio ads. a
linear inequality that expresses this fact. y>=2x

4. There are two more constraints that must be met. These relate to the fact that there
cannot be s negative numbers of advertisements. the two inequalities that model these
constraints: x>=0
y>=0

5. The function for the number of people that will be exposed to the
advertisements. This is the Objective Function for the problem.
P =2000y+1500x

These together describe the situation. This combined set of inequalities and objective function
make up what is known mathematically as a linear programming problem. Write all of the
inequalities and the objective function together below. This is typically written as a list of
constraints, with the objective function last.
x+y>=66
20x+80y<=4320
y>=2x
x>=0
y>=0
P =2000y+1500x

6. The graph, showing the points of intersection.

7. The shaded region in the above graph is called the feasible region. Any (x, y) point in the region
corresponds to a possible number of radio and TV ads that will meet all the requirements of the
problem. However, the values that will maximize the number of people exposed to the ads will
occur at one of the vertices or corners of the region.
.

1: (7.5, 52.5)
2: (24, 48)
3: (20, 40)

8. To find which number of radio and TV ads will maximize the number of people who are exposed to
the business advertisements, evaluate the objective function P=200y+1500x

1: 2000 (52.5) +1500 (7.5) = 62,250
2: 2000 (48) + 1500 (24) = 132,000
3: 2000 (40) +1500 (20 ) = 110,000

In order to maximize the number of people who are exposed to the advertisements the business
should purchase 24 radio ads and 48 TV ads exposing 132,000 people

Yes my view have changed, I did not realize how math is applied in advertising, TV program
ratings and so on. It is a necessary tool for business and life in general. I am now aware that I use math in
more aspects of my life than I thought.

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Optimizing an Advertising Campaign

Math 1010 Intermediate Algebra Group Project

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