Linear programming models have been used in the advertising field as a decision aid in selecting an effective media mix. Sometimes the technique is employed in allocating a fixed or limited budget across various media, which might include radio or television commercials, newspaper ads, direct mailings, social media, and so on. In other applications, the objective is the maximization of audience exposure. Restrictions on the allowable media mix might arise through contract requirements, limited media availability, or company policy. An example follows.

The Win Big Gambling Club promotes gambling junkets from a large midwestern city to casinos in the Bahamas. The club has budgeted up to $8,000 per week for local advertising. The money is to be allocated among four promotional media: TV spots, newspaper ads, and two types of radio advertisements. Win Big’s goal is to reach the largest possible high-potential audience through the various media. The following table presents the number of potential gamblers reached by making use of an advertisement in each of the four media. It also provides the cost per advertisement placed and the maximum number of ads that can be purchased per week.

Win Big’s contractual arrangements require that at least five radio spots be placed each week. To ensure a broad-scoped promotional campaign, management also insists that no more than $1,800 be spent on radio advertising every week.

In formulating this as an LP problem, the first step is to completely understand the problem. Sometimes asking what-if questions will help understand the situation. In this example, what would happen if exactly five ads of each type were used? What would the ads cost? How many people would they reach? Certainly the use of a spreadsheet for the calculations can help with this, since formulas can be written to calculate the cost and the number of people reached. Once the situation is understood, the objective and the constraints are stated:

Objective:

Constraints:

• (1) No more than 12 TV ads can be used. (2) No more than 5 newspaper ads can be used.

• (3) No more than 25 of the 30-second radio ads can be used. (4) No more than 20 of the 1-minute radio ads can be used. (5) Total amount spent cannot exceed $8,000. (6) Total number of radio ads must be at least 5. (7) Total amount spent on radio ads must not exceed $1,800.

Next, define the decision variables. The decisions being made here involve the number of ads of each type to use. Once these are known, they can be used to calculate the amount spent and the number of people reached. Let

Next, using these variables, write the mathematical expression for the objective and the constraints that were identified. The nonnegativity constraints are also explicitly stated.

Objective:

The solution to this can be found using Excel’s Solver. Program 8.1 gives the inputs to the Solver Parameter dialog box, the formula that must be written in the cell for the objective function value, and the cells where this formula should be copied. The results are shown in the spreadsheet. This solution is

This produces an audience exposure of 67,240 contacts. Because $X_1$ and $X_3$ are fractional, Win Big would probably round them to 2 and 6, respectively. Problems that require all-integer solutions (e.g., one can’t exactly buy 1.97 TVs or 6.2 radio spots) are discussed in detail in Chapter 10.

Linear programming has also been applied to marketing research problems and the area of consumer research. The next example illustrates how statistical pollsters can reach strategy decisions with LP.

Management Sciences Associates (MSA) is a marketing and computer research firm based in Washington, D.C., that handles consumer surveys. One of its clients is a national press service that periodically conducts political polls on issues of widespread interest. In a survey for the press service, MSA determines that it must fulfill several requirements in order to draw statistically valid conclusions on the sensitive issue of new U.S. immigration laws:

1. Survey at least 2,300 U.S. households in total.

2. Survey at least 1,000 households whose heads are 30 years of age or younger.

3. Survey at least 600 households whose heads are between 31 and 50 years of age.

4. Ensure that at least 15% of those surveyed live in a state that borders on Mexico.

5. Ensure that no more than 20% of those surveyed who are 51 years of age or over live in a state that borders on Mexico.

MSA decides that all surveys should be conducted in person. It estimates that the costs of reaching people in each age and region category are as follows:

MSA would like to meet the five sampling requirements at the least possible cost.

In formulating this as an LP, the objective is to minimize cost. The five requirements about the number of people to be sampled with specific characteristics result in five constraints. The decision variables come from the decisions that must be made, which involve the number of people sampled from each of the two regions in each of the three age categories. Thus, the six variables are

Objective function:

subject to

The computer solution to MSA’s problem costs $15,166 and is presented in the following table and in Program 8.2, which presents the input and output from Excel 2016. Note that the variables in the constraints are moved to the left-hand side of the inequality.