1. A company manufactures two products: large fans and medium fans. Each large fan requires 6 hours of wiring and 4 hours of drilling. Each medium fan requires 4 hours of wiring and 2 hour of drilling. There are 1920 hours of wiring time available and 11200 hours of drilling time available. Each large fan yields a profit of $30. Each medium fan yields a profit of $20. The company wants to manufacture at least 40 large fans. The objective is to maximize total profit.

a. Formulate a linear optimization model for this problem by defining the decision variables, objective function and all the constraints. What do they represent?
b. Find the optimal solution of this model by hand using the Corner Points graphical method.

2. Determine whether the following linear programming problem is infeasible, unbounded, or has multiple optimal solutions. Draw a graph and explain your conclusion.

Maximize 300x + 250y
Subject to
2x + 1y > 160
-3x + 4y < 240
x, y > 0

3. An air conditioning company manufactures three home air conditioners: a regular model, a super model, and a deluxe model. The profits per unit are $80, $105, and $160, respectively. The production requirements per unit and the availability of the three resources are given below:

Regular Model Super Model Deluxe Model Available
Number of fans 1 1 1 960
Number of cooling coils 1 2 4 1600
Manufacturing Time 12 18 24 12000 hours

How many regular models, super models, and deluxe models should the company manufacture in order to maximize profit?

a. Formulate a linear optimization model for this problem.
b. Solve this model by using the Excel Solver. Include Excel output with you answer.
c. Determine the optimal solution. Interpret and make recommendations based on the optimal solution.

4. A real estate company is considering five possible projects: a small condominium complex, a small shopping center, a warehouse, a small business office building and a sports arena. Each of these projects requires different funding over the next three years, and the net present values of the projects also varies. The following table provides the required investment amounts (in $10,000s) and the net present value, NPV (in $10,000s), of each project:

PROJECT NPV YEAR 1 YEAR 2 YEAR 3
Condominium 300 64 60 58
Shopping center 290 50 45 58
Warehouse 250 50 32 60
Business Office Building 240 55 45 38
Sports Arena 275 50 44 40

The company has $2,500,000 to invest in year 1, $1,850,000 to invest in year 2 and $2,300,000 in year 3. The company wants to select at least 3 projects. In addition, the company also wants to select at least one project from the shopping center and sports arena projects.

a. Formulate an integer optimization model for this problem to maximize the total NPV in this situation by defining the decision variables, the objective function and all the constraints. What type of integer optimization model is this? Briefly describe what the objective function and each constraint represent.

b. The optimal solution for the above problem is given below.

Variable values are X1 = 1, X2 = 1, X3 = 1, X4 = 0, X5 = 1
Objective function value is 1115

Interpret the optimal solution to make a recommendation to the company.

Instructions:

– Please include all details and steps performed to find your answers. Just writing the final answers will not get you full credit.

– Some of the questions above require drawing graphs. Here is a list of options for you to include a graph in your answer.

Draw the graph using MS Paint or CorelDraw or some other software. Copy the graph in a MS Word file and post the file on the course website in appropriate category.
OR
Draw the graph by hand on a paper, scan it or take a photo and post it with your answer.
OR
Describe the graph in words in your answer.