Optimization Math 1010 Project

Background Information:
Linear Programming is a technique used for optimization of a real-world situation. Examples of optimization include maximizing the number of items that can be manufactured or minimizing the cost of production. The equation that represents the quantity to be optimized is called the objective function, since the objective of the process is to optimize the value. In this project the objective is to maximize the audience of a small business.
The objective is subject to limitations or constraints that are represented by inequalities. Limitations on the number of items that can be produced, the number of hours that workers are available, and the amount of land a farmer has for crops are examples of constraints that can be represented using inequalities. Manufacturing an infinite number of items is not a realistic goal. In this project some of the constraints will be based on budget.
Graphing the system of inequalities given by the constraints provides a visual representation of the possible solutions to the problem. If the graph is a closed region, it can be shown that the values that optimize the objective function will occur at one of the "corners" of the region.
The Problem:
In this project your group will solve the following situation:
Elizabeth Bailey is the owner and general manager of Princess Brides, which provides a wedding planning service in Southwest Louisiana. She uses radio advertising to market her business. Two types of ads are available -- those during prime time hours and those at other times.
Each prime time ad costs $390 and reaches 8,250 people, while the offpeak ads each cost $240 and reach 5,100 people. Bailey has budgeted $1,800 per week for advertising.
Based on comments from her customers, Bailey wants to have at least 2 prime time ads and no more than 6 off peak ads.
Your goal is to figure out how many of each ad should be aired and what is the total reach of your audience.
Modeling the Problem:
Let x be the number of prime ads that are made and y be the number of non-prime that are made.
1. Write down a linear inequality that models how the cost of ads will be kept within budget.
2. Recall that she wants least 2 prime time ads and no more than 6 off peak ads. Write down two linear inequalities to model these constraints.
3. There are two more constraints that must be met. These relate to the fact that it is impossible to buy a negative number of ads. Write the two inequalities that model these constraints:
4. Next, write down a linear equation that models the total reach/audience of the ads. This is the Objective Function for the problem.
𝑅𝑅 ��(�𝑥𝑥, �𝑦𝑦 �) = 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅ℎ 𝑜𝑜𝑜𝑜 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴
You now have five linear inequalities and an objective function. These together describe the situation. This combined set of inequalities and objective function make up what is known mathematically as a linear programming problem. This is typically written as a list of constraints, with the objective function last.
5. To solve this problem, you will need to graph the intersection of all five inequalities on one common x,y-plane. Do this on the grid below or you may use your own graph paper or graphing software if you prefer. Let the bottom left be the origin, with the horizontal axis representing x and the vertical axis representing y. Be sure to:
a. Label the axes with appropriate numbers and verbal descriptions
b. Label your lines as you graph them.
The shaded region in the graph is called the feasible region. Any (x, y) point in the region corresponds to a possible number of peak and off-peak ads that will meet all the requirements of the problem. Your region should have three corners or vertices.
6. Generally, to find which number of each type of ad that will maximize the audience you would evaluate the objective function R for each of the vertices you found. But notice that two of the vertices would have decimals and we can’t purchase a decimal number of ads. So instead, let’s list out all possible integer ordered pairs INSIDE the shaded area. (Hint: you should have 10 ordered pairs). Evaluate all 10 of the possible integer pairs that fall within the shaded region for the objective function R. Determine which ordered pair gives you the maximum audience/reach? Ordered Pair Evaluate for #4: R(x,y) = Audience Reached at Ordered Pair
Your goal is was figure out how many of each ad should be aired. Write one to two sentences describing how many of each item should be purchased to produce the greatest reach. Include the audience number that will be reached in your description.
Reflective Writing – Please type your answer/reflection on a separate sheet of paper and submit it with the pages above.
Did this project change the way you think about how math can be applied to the real world? Write at least one paragraph stating what ideas changed and why. If this project did not change the way you think, write how this project gave further evidence to support your existing opinion about applying math. Be specific.