Question

The manager of a department store in Seattle is attempting to decide on the types and amounts of advertising the store should use. He has invited representatives from the local radio station, television station, and newspaper to make presentations in which they describe their audiences.
The television station representative indicates that a TV commercial, which costs $15,000, would reach 25,000 potential customers. The breakdown of the audience is as follows:

   Male   Female
Senior   5,000   5,000
Young   5,000   10,000

The newspaper representative claims to be able to provide an audience of 10,000 potential customers at a cost of $4,000 per ad. The breakdown of the audience is as follows:

   Male   Female
Senior   4,000   3,000
Young   2,000   1,000

The radio station representative says that the audience for one of the station’s commercials, which costs $6,000, is 15,000 customers. The breakdown of the audience is as follows:

   Male   Female
Senior   1,500   1,500
Young   4,500   7,500
The store has the following advertising policy:
Use at least twice as many radio commercials as newspaper ads.
Reach at least 100,000 customers.
Reach at least twice as many young people as senior citizens.
Make sure that at least 30% of the audience is female.

Available space limits the number of newspaper ads to seven. The store wants to know the optimal number of each type of advertising to purchase to minimize total cost.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
c. Suppose a second radio station approaches the department store and indicates that its commercials, which cost $7,500, reach 18,000 customers with the following demographic breakdown:

   Male   Female
Senior   2,400   3,600
Young   4,000   8,000

If the store considered this station along with the other media alternatives, how would this affect the solution?
Hint: we have 4 variables only

quick +1

Answer 1

A departmental store wants to advertise in the City S. the manager of the store has invited representative of various media, such as radio station, television station, and newspaper in order to make presentation.

(a)

Formulate the linear programming model and consider the following decision variables:

Decision variable:

Objective function is to minimize the total cost associated with advertisement of the department store:

…… (1)

Constraints are subjected to:

Constraint 1: The Store could use at least twice of radio commercial as newspaper ads

Or

Constraint 2: Total target audience of the store to reach through advertisements:

Constraint 3: The store should reach twice of young audience as senior audience

Simplify the above equation, as shown below:

Constraint 4: The store should target at least 30% of female audience

Or

Constraint 5: The store could use only 7 newspaper ads

Non-negativity constraints are:

(b)

Formulate an excel spreadsheet for the department store, as shown below:

Figure 1: Screenshot of formulated spreadsheet

The following spreadsheet is obtained:

Figure 2

Select “Solver” from the “Data” menu bar, as shown below:

Figure 3: Screenshot to access solver option

A pop-up window would appear after clicking on “Solver”. Fill in the objective function and constraints as shown below:

Figure 4: Screenshot of solver

Click on “Solve” option. Then the following pop-up would appear:

Figure 5: Screenshot of solver results

The following results are obtained:

Figure 6: Screenshot of final results

From the above results, the department store should use 2.5 times newspaper ads and 5 times radio commercials to minimize the total cost of advertising ($40,000)

(c)

Formulate the linear programming model and consider the following decision variable:

Decision variable:

Objective function is to minimize the total cost associated with advertisement of the department store along with second radio station:

…… (2)

Constraints are subjected to:

Constraint 6: The Store could use at least twice of radio commercial as newspaper ads

Or

Constraint 7: Total target audience of the store to reach through advertisements along with second radio station:

Constraint 8: The store should reach twice of young audience as senior audience

Simplify the above equation, as shown below:

Constraint 9: The store should target at least 30% of female audience

Or

Constraint 10: The store could use only 7 newspaper ads

Formulate an excel spreadsheet for the department store, along with second radio station constraint, as shown below:

Figure 7: Screenshot of formulated spreadsheet

The following spreadsheet is obtained:

Figure 8

Access “solver option” as shown in figure 3, and enter the constraints in the solver, as shown below:

Figure 9: Screenshot of solver

Click on “Solve” option. Then the following pop-up would appear:

Figure 10: Screenshot of solver results

The following results are obtained:

Figure 11: Screenshot of final results

From the above figure, the value of objective function does not change by including the second radio station. Hence, additional radio station will have no effect on the situation.