Problem 1:
You work at an advertising firm and you're managing the account of a client. You're helping them plan their advertising for the next month. They have a budget of $30,000 and want you to design an ad campaign that:
- Includes at least 10 TV commercials
- Reaches at least 50,000 people
- Allocates no more than $18,000 to TV ads
- Maximizes the total Exposure Quality Units
The following table shows the number of people reached per ad, the cost per ad, the number of available ads, and the Exposure Quality Units (which measures the ad's effectiveness) per ad for the five available types of media.
|
|
Reach
|
Cost
|
Available
|
Exposure Quality Units
|
|
Daytime TV
|
1000
|
$1500
|
15
|
65
|
|
Evening TV
|
2000
|
$3000
|
10
|
90
|
|
Daily Paper
|
1500
|
$400
|
25
|
40
|
|
Sunday Paper
|
2500
|
$1000
|
4
|
60
|
|
Radio
|
300
|
$100
|
30
|
20
|
Write out the formulation of a linear optimization model to help you determine the number of each type of ad to purchase in order to maximize the total Exposure Quality Units of the campaign. (You do not have to find the optimal solution.)
Problem 2:
Consider the linear optimization model:
Minimize x - 3y
Subject to 6x - y ≥ 18
3x + 2y ≤ 24
x, y ≥ 0
(a) Graph the constraints and identify the feasible region.
(b) Choose a value and draw the isoquant representing all combinations of x and y that make the objective function equal to that value.
(c) Determine the optimal solution.
(d) Label the optimal solution on your graph.
(e) Calculate the optimal value of the objective function.
Problem 3:
Consider the linear optimization model:
Maximize 6x + 4y
Subject to x + 2y ≤ 12
3x + 2y ≤ 24
x, y ≥ 0
(a) Graph the constraints and identify the feasible region.
(b) Choose a value and draw the isoquant representing all combinations of x and y that make the objective function equal to that value.
(c) Determine the optimal solutions.
(d) Label the optimal solutions on your graph.
(e) Calculate the optimal value of the objective function.
Problem 4:
You would like to make a nutritious meal of steak and potatoes. The meal should provide at least 50 g of carbohydrates, at least 40 g of protein, and no more than 60 g of fat. A piece of steak contains 5 g of carbohydrates, 20 g of protein, and 15 g of fat. A potato contains 15 g of carbohydrates, 5 g of protein, and 2 g of fat. A steak costs $4 and a potato costs $2. You have formulated the following linear optimization model to determine the number of pieces of steak and the number of potatoes that should be in the meal in order to meet the nutrition requirements at minimal cost.
Let S be the number of pieces of steak to eat P be the number of potatoes to eat
|
Minimize
|
4S + 2P
|
|
|
Subject to
|
5S + 15P ≥ 50
|
(Carbohydrates)
|
|
|
20S + 5P ≥ 40
|
(Protein)
|
|
|
15S + 2P ≤ 60
|
(Fat)
|
S, P ≥ 0
(a) Enter the model in Excel and use the Solver to find the optimal solution. Turn in your Excel file.
(b) Report the optimal solution.
(c) Report the optimal value of the objective function.
(d) Report the slack/surplus for each constraint.
Problem 5:
Recall the Sonoma Apple problem from Homework 3. (You can find the formulation in the Home- work 3 Solutions.)
(a) Enter the model in Excel and use the Solver to find the optimal solution. Turn in your Excel file.
(b) How many jars of applesauce and bottles of apple juice should they produce?
(c) How much should they spend on advertising for applesauce and apple juice?
(d) What will their profit be?