Operations Management question

Question

The Biggs Department Store chain has hired an advertising firm to determine the types and amount of advertising it should invest in for its stores. The three types of advertising available are television and radio commercials and newspaper ads. The retail chain desires to know the number of each type of advertisement it should purchase in order to maximize exposure. It is estimated that each ad or commercial will reach the following potential audience and cost the following amount:

	Exposure	(people/ad or commercial)	Cost (US$)
TV commercial	40,000	16,000
Radio commercial	28,000	14,000
Newspaper ad	29,000	8,000

 

The company must consider the following resource constraints:

1. The budget limit for advertising is $200,000. 
2. The television station has time available for 6 commercials.
3. The radio station has time available for 12 commercials.
4. The newspaper has space available for 5 ads.
5. The advertising agency has time and staff available for producing no more than a total of 20 commercials and/or ads. 

 

If this problem is formulated as an integer and also a linear programming problem (separately), the combination of commercial and advertisements that would maximize the exposure would be, respectively:

a.Integer Programming: x₁ = 8; x₂ = 1; x₃ = 6;

Linear Programming: x₁ = 8; x₂ = 1.71 ; x₃ = 6;

b. Integer Programming: x₁ = 6; x₂ = 4; x₃ = 5;

Linear Programming: x₁ = 6; x₂ = 4.57 ; x₃ = 5;

c. Integer Programming: x₁ = 6; x₂ = 5; x₃ = 6;

Linear Programming: x₁ = 6; x₂ = 4.57 ; x₃ = 5;

d. Integer Programming: x₁ = 6; x₂ = 2; x₃ = 8;

Linear Programming: x₁ = 6; x₂ = 1.71 ; x₃ = 8;

e. None of the above

Answer 1

To solve this problem, we can set up the following linear programming problem:

Let \( x_1, x_2, x_3 \) be the number of TV commercials, radio commercials, and newspaper ads, respectively.

The objective is to maximize the exposure, which is given by the expression:

\[ \text{Maximize } Z = 40,000x_1 + 28,000x_2 + 29,000x_3 \]

Subject to the constraints:

1. Budget constraint: \( 16,000x_1 + 14,000x_2 + 8,000x_3 \leq 200,000 \)

2. Television station constraint: \( x_1 \leq 6 \)

3. Radio station constraint: \( x_2 \leq 12 \)

4. Newspaper constraint: \( x_3 \leq 5 \)

5. Advertising agency constraint: \( x_1 + x_2 + x_3 \leq 20 \)

Now, to solve the linear programming problem:

a. The correct solution is:

\[ \text{Linear Programming: } x_1 = 6; \, x_2 = 4.57; \, x_3 = 5; \]

b. The correct answer is:

\[ \text{Linear Programming: } x_1 = 6; \, x_2 = 4.57; \, x_3 = 5; \]