1. Korey is a business student at state U. She has just completed a course in decision models, which had a midterm exam, a final exam, individual assignments, and class participation. She earned an 86% on the midterm, 94% on the final, 93% on the individual assignments, and 85% on participation. The benevolent instructor is allowing his students to determine their own weights for each of the four grade components; of course, with some restrictions:
The participation grade can be no more than 15% of the total grade.
The midterm grade must count at least twice as much as the individual assignment score.
The final exam grade must count at least three times as much as the individual assignment score.
Each of the four components must count for at least 10% of the course grade.
The weights must sum to 1.0 and be nonnegative.
a. Develop a model that will yield a valid set of weights to maximize Korey's score for the course.
b. Implement your model on a spreadsheet and find a good solution using only your intuition.
c. Find an optimal solution using Solver.
2. Rosenberg Land Development (RLD) is a developer of condominium properties in the Southwest United States. RLD has recently acquired a 40.625 acre site outside of Phoenix, Arizona. Zoning restrictions allow at most 8 units per acre. Three types of condominiums are planned: one, two, three bedroom units.
The average contruction costs for each type of unit are $450,000, $600,000, and $750,000. These units will generate a net profit of 10%. The company has equity and loans totaling $180 million dollars for this project. From prior development projects, senior managers have determine that there must be a minimum of 15% one-bedroom units, 25% two-bedroom units, and 25% three-bedroom units.
a. Developa linear optimization model to determine how many each type of unit the developer should build.
b. Implement your model on a spreadsheet and find an optimal solution.
c. Explain the value of increasing the budget fhe project.
3. The Children's Theater Company is a nonprofit corporation managed by Shannon Board. The theater performs in two venues: Kristin Marie Hall and the Lauren Elizabeth Theater. For the upcoming season, seven shows have been chosen. The question Shannon faces is how many performances of each of the seven shows should be scheduled.
A financial analysis has estimated revenues for each performance of the seven shows, and Shannon has set the minimum number of performances of each show based upon union agreements with Actor's Equity Association and the popularity of the shows in other markets:
| Show |
Revenue |
Cost |
Minimum Number of Performances |
| 1 |
$2,217 |
$968 |
32 |
| 2 |
$2,330 |
$1,568 |
13 |
| 3 |
$1,993 |
$755 |
23 |
| 4 |
$3,364 |
$1,148 |
34 |
| 5 |
$2,868 |
$1,180 |
35 |
| 6 |
$3,851 |
$1,541 |
16 |
| 7 |
$1,836 |
$1,359 |
21 |
Kristin Marie Hall is available for 60 performances during the season, while Lauren Elizabeth Theater is available for 150 performances. Shows 3 and 7 must be performed in Kristi Marie Hall, while the other shows are performed in the Lauren Elizabeth Theater. The company wants to achieve revenues of at least $550,000 while minimizing its production costs. Develop and solve linear optimization model to determine the best way to schedule the shows. Is it possible to achieve revenues of $600,000? What is the highest amount of revenue that can be achieved?