Question

A committee in charge of promoting a Ladies Professional Golf Association tournament is trying to determine how best to advertise the event during the 2 weeks prior to the tournament. The committee obtained the following information about the three advertising media they are considering using. Category (Audience Reached per Advertisement ) (Cost per Advertisement) (Maximum Number of Advertisements) TV 200,000 $ 2500 10 Radio 50,000 $ 400 15 Newspaper 100,000 $ 500 20 The last column in this table shows the maximum number of advertisements that can be run during the next 2 weeks; these values should be treated as constraints. The committee has established the following goals for the campaign. Priority Level 1 Goal Goal 1: Reach at least 4 million people. Priority Level 2 Goal Goal 2: The number of television advertisements should be at least 30% of the total number of advertisements. Priority Level 3 Goal Goal 3: The number of radio advertisements should not exceed 20% of the total number of advertisements. Priority Level 4 Goal Goal 4: Limit the total amount spent for advertising to $20,000 Formulate a goal programming model for this problem.

Answer 1

To formulate a goal programming model for this problem, consider the committee's goals and constraints. Define decision variables and an objective function, and set up equations for each goal and constraint.

Step-by-step explanation:

To formulate a goal programming model for this problem, we need to consider the committee's goals and constraints. We can define the decision variables as the number of advertisements for each media (TV, radio, newspaper), and the objective function as minimizing the deviation from the goals. The constraints will be based on the maximum number of advertisements, the total number of people reached, the percentage of TV and radio advertisements, and the total cost. We can then set up an equation for each goal and constraint and solve the model using goal programming techniques.

Answer 2

The committee can use goal programming to optimize their advertising campaign for a Ladies Professional Golf Association tournament. They need to reach a certain number of people, allocate a specific percentage of advertisements to TV and radio, and limit the total cost. By formulating the goals as constraints and setting up an objective function, the committee can use optimization techniques to find the optimal number of advertisements for each medium.

Step-by-step explanation:

This problem can be modeled using goal programming, which is a mathematical technique used to optimize multiple goals simultaneously. The committee wants to reach at least 4 million people, have the number of TV advertisements be at least 30% of the total number of advertisements, ensure that the number of radio advertisements does not exceed 20% of the total number of advertisements, and limit the total amount spent to $20,000. To formulate the goal programming model, we need to assign variables to represent the number of advertisements for each medium and set up constraints and the objective function.

Let's assign the variables:

• xTV: Number of TV advertisements
• xRadio: Number of radio advertisements
• xNewspaper: Number of newspaper advertisements

The constraints are:

• Audience reached: 200,000*xTV + 50,000*xRadio + 100,000*xNewspaper ≥ 4,000,000
• Total number of advertisements: xTV + xRadio + xNewspaper ≤ 45
• Percentage of TV advertisements: xTV ≥ 0.3 * (xTV + xRadio + xNewspaper)
• Percentage of radio advertisements: xRadio ≤ 0.2 * (xTV + xRadio + xNewspaper)
• Total cost: 2500*xTV + 400*xRadio + 500*xNewspaper ≤ 20000

The objective function is to minimize the deviation from the goals. We can assign weights to the goals based on their priority levels and calculate the deviation as follows:

• Deviation = w1 * (200,000*xTV + 50,000*xRadio + 100,000*xNewspaper - 4,000,000)^2 + w2 * (xTV - 0.3 * (xTV + xRadio + xNewspaper))^2 + w3 * (xRadio - 0.2 * (xTV + xRadio + xNewspaper))^2 + w4 * (2500*xTV + 400*xRadio + 500*xNewspaper - 20000)^2

Where w1, w2, w3, w4 are weights assigned to each goal. The goal programming model can be solved using optimization techniques to find the values of xTV, xRadio, xNewspaper that minimize the deviation from the goals while satisfying the constraints.