Advertising The Fancy Fashions, an independent, local...

Advertising The Fancy Fashions, an independent, local boutique, has $8000 available each month for advertising. News- paper ads cost $400 each, and no more than 30 can run per month. Internet banner ads cost $20 each, and no more than 60 can run per month. TV ads cost $2000 each, with a maximum of 10 available each month. Approximately 4000 women will see each newspaper ad, 3000 will see each Internet banner, and 10,000 will see each TV ad. How much of each type of advertising should be used if the store wants to maximize its ad exposure?

Advertising The Fancy Fashions, an independent, local boutique, has $8000 available each month for advertising. News- paper ads cost $400 each, and no more than 30 can run per month. Internet banner ads cost $20 each, and no more than 60 can run per month. TV ads cost $2000 each, with a maximum
of 10 available each month. Approximately 4000 women will see each newspaper ad, 3000 will see each Internet banner, and 10,000 will see each TV ad. How much of each type of advertising should be used if the store wants to maximize its ad exposure?

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Key Concepts

Feasible Region

The feasible region is the collection of all points (solutions) that satisfy the given constraints in a linear programming problem. The optimal solution for the objective function must be found within this set of possible solutions.

Constraints

Constraints are the limitations or restrictions imposed on the decision variables. They include budget limits, maximum allowed units of ads, and any other resource limitations that ensure the solution is practical and feasible.

Linear Programming

Linear programming is a mathematical technique used for optimizing a linear objective function, subject to a set of linear equality and inequality constraints. It is widely used in operations research to find the best allocation of limited resources.

Decision Variables

Decision variables are the quantities that need to be determined in a linear programming problem. They represent the levels of activity, such as the number of different advertising mediums to use, that will optimize the objective function.

Objective Function

The objective function in linear programming defines what needs to be maximized or minimized—in this case, the ad exposure. It quantifies the goal by relating the decision variables to the overall performance measure.

Transcript

00:01 Okay, in this problem, we're given some information about the advertising budget of a company, and then we're going to figure out how to write the equations and what we're trying to solve, and then we're going to set it up and get it ready to do the simplex method on it.

00:18 Okay, first of all, we have three kinds of ads, print ads, internet ads, and tv ads.

00:26 So those will be our initial variables.

00:29 X1 will be number of print ads.

00:37 X2 will be the number of internet ads.

00:45 And x3 will be the number of tv ads.

00:56 Z will be the total number of viewers.

01:07 And we're going to have four slack variables.

01:13 S2, s3, s4 slack variables.

01:17 All right, first we know that for print ads, the number of the limit is no more than 30.

01:31 So we get x1 has to be less than or equal to 30.

01:37 Internet ads no more than 60.

01:44 Oops.