DSCI/DATA 353 University of Mary Washington Optimization Models

Problems

1.(25 points) Fred Publishing Company is considering publishing a paperback textbook on spreadsheet applications for business. The fixed cost of manuscript preparation and production setup is estimated to be 150,000. Variable production and material costs are $6 per book. The publisher plans to sell the text to college and university bookstore for $45 each.

a.

Write a math model for total cost.

b.

Write a math model for total revenue.

c.

Write a math model for total profit.

d.

Find the break-even point.

2. (25 points) Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows:

Number of fans

Number of cooling Coils

Manufacturing

Time (hours)

Economy

1

1

8

Standard

1

2

12

Deluxe

1

4

14

For the coming production period, the company has 200 fan motors, 320 cooling coils, and 2400 hours of manufacturing time available. How many economy models, standard models, and deluxe models should the company produce in order to maximize profit? Formulate the problem and answer the following questions.

a.Show LP model.

b.What is the optimal solution, and what us the value of the objective function?

c.Which constraints are binding?

d.Which constraints show extra capacity? How much?

e.If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change? Explain.

3. (25 points) Olympic Bike is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from special aluminum and steel alloys. The anticipated unit profits are $15 for the Deluxe and $17 for the Professional. The number of pounds of each alloy needed per frame is summarized in the table. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly.

Aluminum Alloy Steel Alloy

Deluxe 1 2/3

Professional 2 1

a.Develop and solve LP model to decide how many Deluxe and Professional frames should Olympic produce each week to maximize the total profit.

b.What are the dual values (shadow prices)? Interpret each.

c.Which constraints show extra (unused) alloy? How much?

4.(25 points) Virginia Sporting Equipment, Inc. makes two different types of baseball gloves: a regular model and a catcher’s model. The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department and 100 hours in its packaging department. The production time requirements and the profit contribution per glove are given in the following table:

Production Time (hours)

Model

cutting and sewing

finishing

packaging

profit per glove

Regular

1

1.5

0.5

5

Catcher’s

1.5

2

1

8

The firm is interested in maximizing the total profit. Answer the following questions.

a.What is the LP model for the problem?

b.Find the optimal solution and the total profit.

c.What is the slack time in each department?

d.What are the dual values for the resources? Interpret each.

e.If over time can be scheduled in one of the department, where would you recommend? Why?

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