Media Selection A company has budgeted a maximum of ...

Media Selection A company has budgeted a maximum of $\$ 1,000,000$ for national advertising of an allergy medication. Each minute of television time costs $\$ 100,000$ and each one-page newspaper ad costs $\$ 20,000 .$ Each television ad is expected to be viewed by 20 million viewers, and each newspaper ad is expected to be seen by 5 million readers. The company's market research department recommends that at most 80$\%$ of the advertising budget be spent on television ads. What is the optimal amount that should be spent on each type of ad? What is the optimal total audience?

Media Selection A company has budgeted a
maximum of $\$ 1,000,000$ for national advertising of an
allergy medication. Each minute of television time costs
$\$ 100,000$ and each one-page newspaper ad costs
$\$ 20,000 .$ Each television ad is expected to be viewed by
20 million viewers, and each newspaper ad is expected
to be seen by 5 million readers. The company's market
research department recommends that at most 80$\%$ of
the advertising budget be spent on television ads. What
is the optimal amount that should be spent on each type
of ad? What is the optimal total audience?

Key Concepts

Feasible Region

The feasible region is the set of all possible values of the decision variables that satisfy all the constraints. This region represents all the potential solutions from which the optimal solution should be chosen.

Optimal Solution

The optimal solution in a linear programming problem is the specific set of decision variable values that maximizes (or minimizes) the objective function while satisfying all the constraints. In this context, it corresponds to the exact allocation of the advertising budget to maximize total audience reach.

Objective Function

The objective function is a mathematical expression that defines the goal of the optimization, such as maximizing the total audience reached by the advertisements. It is formulated as a linear combination of the decision variables, with each variable weighted according to its contribution to the objective.

Linear Programming

Linear programming is a mathematical optimization technique used to maximize or minimize a linear objective function subject to a set of linear equality or inequality constraints. In planning problems such as media selection, linear programming helps determine the best allocation of limited resources among competing activities.

Decision Variables

Decision variables represent the quantities that need to be determined in the problem – in this case, the amounts of money to be spent on different types of ads. They serve as the unknowns in the linear programming model, and establishing them correctly is crucial for formulating the model.

Constraints

Constraints are the restrictions or limitations placed on the decision variables. In the media selection problem, constraints include the overall budget limit, the per-unit cost of each advertising medium, and additional spending restrictions like ensuring that no more than a certain percentage of the budget is spent on television ads. These constraints define the feasible solutions for the problem.

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