Question

A linear programming computer package is needed.

The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year's program. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown.
Constraint Television Radio Online
Audience per advertisement 300,000 54,000 120,000
Cost per advertisement $2,000 $300 $600
Maximum media usage 10 20 10
To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized.
(a) If the promotional budget is limited to $29,700, how many commercial messages should be run on each medium to maximize total audience contact?
Television_____
Radio____
Online____
What is the allocation of the budget among the three media, and what is the total audience reached?
Television Budget_____
Radio Budget______
Online Budget______
Total Audience______
(b) By how much would audience contact increase if an extra $100 were allocated to the promotional budget? (Round your answer to the nearest whole number.)

Answer 1

To maximize the total audience contact within the given budget and constraints, we can use linear programming. Linear programming is a mathematical technique used to optimize a linear objective function subject to linear equality and inequality constraints.

(a) To determine the number of commercial messages on each medium, we need to set up the objective function and constraints.

Let:

x = number of television advertisements

y = number of radio advertisements

z = number of online advertisements

Objective function: Maximize Total Audience Contact

The total audience contact can be calculated by multiplying the audience per advertisement by the number of advertisements on each medium and summing them up:

Total Audience = 300,000x + 54,000y + 120,000z

Constraints:

1. Budget Constraint: The total cost of advertisements should not exceed the budget of $29,700:

2,000x + 300y + 600z ≤ 29,700

2. Media Usage Constraint: The maximum number of advertisements allowed on each medium:

x ≤ 10 (for television)

y ≤ 20 (for radio)

z ≤ 10 (for online)

3. Radio Usage Constraint: Radio advertisements should not exceed 50% of the total number of advertisements authorized:

y ≤ 0.5(x + y + z)

4. Television Usage Constraint: Television should account for at least 10% of the total number of advertisements authorized:

x ≥ 0.1(x + y + z)

By solving these constraints, we can determine the optimal number of commercial messages on each medium.

Television advertisements: x = 7

Radio advertisements: y = 10

Online advertisements: z = 10

(b) To determine how much the audience contact would increase if an extra $100 were allocated to the promotional budget, we can use the same objective function and constraints. We need to solve the linear programming problem again with the new budget constraint of $29,800 ($29,700 + $100).

By comparing the total audience contact between the previous and new solutions, we can find the increase in audience contact.

Total Audience Contact (with new budget): 300,000x + 54,000y + 120,000z

Note: The increase in audience contact would depend on the new optimal allocation of the budget among the three media.

Explanation: