Consider the case of Super Builders, Inc., a residential builder in Louisiana and Texas. The company's advertising department is developing a media plan and has decided to use three different types of advertising media, television, radio and direct mail. The characteristics of the foregoing three media are as follows:
1. The unit costs of advertising are $2,600, $1,000 and $1,500 for television, radio and direct mail, respectively.
2. No more than 12 units of TV, 40 units of radio and 25 units of direct mail are available. A minimum of 9 units of radio and 5 units of direct mail must be purchased.
3. The total number of effective exposures per unit of advertising medium depends on the region reached. In Louisiana, the number of effective exposures is 70,000, 100,000 and 75,000 for television, radio and direct mail, respectively. In Texas, the number of effective exposures is 650,000, 300,000 and 450,000 for television,radio and direct mail, respectively.
4. The number of exposures to potential single (unmarried) customers per unit of advertising is 100,000, 80,000 and 95,000 for the three media.
5. The number of exposures to potential married customers per unit of advertising is 400,000, 50,000 and 250,000 for the three media.
As a result of the discussions between the media planners and the advertiser, it was agreed that the following conditions should be satisfied:
a) No more than $60,000 worth of advertising can be spent on this campaign.
b) At least 2.5 million and 10 million exposures are to be achieved in Louisiana and Texas, respectively.
c) No more than 3.5 million exposures are to take place among single customers.
d) At least 2.8 million exposures are to take place among married customers.
The objective of Super Builders, Inc. is to find the optimal number of advertisement units to be purchased in each medium to maximize the total number of effective exposures.
Define the control variables and formulate the mathematical model for the problem?