MGNT 3150 Management Science

Homework Assignment 1

Topic Questions Value

Chapter 1 Question 1 5 points

Question 2 5 points

Chapter 2 Question 3 6 points

Question 4 9 points

Question 5 12.5 points

Chapter 3 Question 6 12.5 points

Chapter 4*

Question 7 or Question 8 (Choose one)* 12.5 points

Chapter 7 Question 9 12.5 points

Total Questions 75 points

* You need to answer only Question 7 or Question 8 for Chapter 4.

You MUST submit your file as a ONE .doc or .pdf file. Other formats (.jpeg, .png, etc.) will not be accepted.

Chapter 1: An Introduction to Management Science

Question 1 (5 points): Micromedia offers computer training seminars on a variety of topics. In the seminars each student works at a personal computer, practicing the particular activity that the instructor is presenting. Micromedia is currently planning a two-day seminar on the use of Micro- soft Excel in statistical analysis. The projected fee for the seminar is $600 per student. The cost for the conference room, instructor compensation, lab assistants, and promotion is $9600. Micromedia rents computers for its seminars at a cost of $120 per computer per day.

Develop a model for the total cost to put on the seminar. Let x represent the number of students who enroll in the seminar. (1 point)

9600 + (2*120x) = 9600+ 240x

Develop a model for the total profit if x students enroll in the seminar. (1 point)

600x – (9,600 + 240x)= 360x – 9600

Micromedia has forecasted an enrollment of 30 students for the seminar. How much profit will be earned if their forecast is accurate? (1.5 points)

360 (30) – 9,600 = $1,200

d.Compute the breakeven point. (1.5 points)

total revenue = total cost

9,600 + 240x= 600x

360x= 9,600

X= 26.66…(27)

Question 2 (5 points): Preliminary plans are under way for the construction of a new stadium for a major league baseball team. City officials have questioned the number and profitability of the luxury corporate boxes planned for the upper deck of the stadium. Corporations and selected individuals may buy the boxes for $300,000 each. The fixed construction cost for the upper- deck area is estimated to be $4,500,000, with a variable cost of $150,000 for each box constructed.

What is the breakeven point for the number of luxury boxes in the new stadium? (2.5 points)

Profit on each box = (300,000 – 150,000) = $150,0000

Breakeven point = $4,500,000 = 30 boxes

$150,000

Preliminary drawings for the stadium show that space is available for the construction of up to 50 luxury boxes. Promoters indicate that buyers are available and that all 50 could be sold if constructed. What is your recommendation concerning the construction of luxury boxes? (1.25 points) What profit is anticipated? (1.25 points)

Total revenue – Total expenses= Profit

= 50 boxes – 30 boxes (break- even point) = 20 boxes

= 20 X $150,000 = $3,000,000 in sales

The marginal profit is positive, so the construction should be continued.

Chapter 2: An Introduction to Linear Programming

Question 3 (6 points):: Reiser Sports Products wants to determine the number of All-Pro (A) and College (C) footballs to produce in order to maximize profit over the next four-week planning ho- rizon. Constraints affecting the production quantities are the production capacities in three departments: cutting and dyeing; sewing; and inspection and packaging. For the four-week planning period, 340 hours of cutting and dyeing time, 420 hours of sewing time, and 200 hours of inspection and packaging time are available. All-Pro football’s provide a profit of $5 per unit, and College footballs provide a profit of $4 per unit. The linear programming model with production times expressed in minutes is as follows:

A portion of the graphical solution to the Reiser problem is shown in Figure.

a.Shade the feasible region for this problem. (1 point)

b.Determine the coordinates of each extreme point and the corresponding profit. Which extreme point generates the highest profit? (1 point)

c.Draw the profit line corresponding to a profit of $4000. Move the profit line as far from the origin as you can in order to determine which extreme point will provide the optimal solution. Compare your answer with the approach you used in part (b). (2 points)

d.Which constraints are binding? Explain (2 points)

Question 4 (9 points): Solve the following linear programming problem using the graphical solution procedure:

Max 5A + 5B

s.t.

1A ≤ 100

1B ≤ 80

2A + 4B ≤ 400

A, B ≥0

Question 5 (12.5 points): Kelson 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 cut- ting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table:

Assuming that the company is interested in maximizing the total profit contribution, answer the following: a.What is the linear programming model for this problem? (2.5 points)

b.Find the optimal solution using the graphical solution procedure. How many gloves of

each model should Kelson manufacture? (5 points)

c. What is the total profit contribution Kelson can earn with the given production quantities? (2.5 points)

d.How many hours of production time will be scheduled in each department? (1.25 points)

e.What is the slack time in each department? 1.25 points)

Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution

Question 6 (12.5 points). Consider the following linear program:

Min8X + 12Y

s.t.1X + 3Y ≥ 9

2X + 2Y ≥10

6X + 2Y ≥18

X, Y ≥ 0

a.Use the MS Excel Solver to find the optimal solution with sensitivity analysis and provide the reports by copying or taking screenshot from excel. After providing report calculate the optimal solution. (5 points)

P.S: If you will miss to provide the MS Excel Solver, you will be graded for zero for this question.

b.Based on the computer solution (sensitivity analysis report) for the linear program in part (a):

Assume that the objective function coefficient for X changes from 8 to 6. Does the optimal solution change? (1.5 points)

c.Based on the computer solution (sensitivity analysis report) for the linear program in part (a):

Assume that the objective function coefficient for X remains 8, but the objective function coefficient for Y changes from 12 to 6. Does the optimal solution change? (1.5 points)

d. Based on the computer solution (sensitivity analysis report) for the linear program in part (a):

Determine the dual value for constraint 1. (1.5 points)

e. Based on the computer solution (sensitivity analysis report) for the linear program in part (a):

Suppose that the right-hand side for constraint 1 is increased from 9 to 10. Find the new optimal solution. What does the right-hand-side range information for constraint 1 tell you about the dual value for constraint 1? (1.5 points)

f. The dual value for constraint 2 is 3. Using this dual value and the right-hand-side range information in part (a), what conclusion can be drawn about the effect of changes to the right-hand side of constraint 2? (1.5 points)

Chapter 4 – Linear Programming Applications in Marketing, Finance and Operations Management

In this part you need to answer only Question 7 or Question 8. (12.5 points)

Question 7. The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year’s program. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown.

To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized.

What is the linear programming model for this problem? (4 points)

If the promotional budget is limited to $18,200, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media, and what is the total audience reached? For solution, use Excel Solver and provide the screenshots of results from Excel. Use the MS Excel Solver to find the optimal solution with sensitivity analysis and provide the reports by copying or taking screenshot from excel. After providing report calculate the optimal solution. (6 points)

P.S: If you will miss to provide the MS Excel Solver, you will be graded for zero for this question.

By how much would audience contact increase if an extra $100 were allocated to the

promotional budget? (2.5 points)

LetT = number of television spot advertisements

R = number of radio advertisements

O = number of online advertisements

Question 8. The management of Hartman Company is trying to determine the amount of each of two products to produce over the coming planning period. The following information concerns labor availability, labor utilization, and product profitability:

Develop a linear programming model of the Hartman Company problem. What is the linear programming model for this problem? (4 points)

Solve the model to determine the optimal production quantities of products 1 and 2. For solution, use Excel Solver and provide the screenshots of results from Excel. Use the MS Excel Solver to find the optimal solution with sensitivity analysis and provide the reports by copying or taking screenshot from excel. After providing report calculate the optimal solution. (6 points)

P.S: If you will miss to provide the MS Excel Solver, you will be graded for zero for this question.

c. In computing the profit contribution per unit, management doesn’t deduct labor costs because they are considered fixed for the upcoming planning period. However, suppose that overtime can be scheduled in some of the departments. Which departments would you recommend scheduling for overtime? How much would you be willing to pay per hour of overtime in each department? (2.5 points)

Letx1 = units of product 1 produced

x2 = units of product 2 produced

Chapter 7: Integer Linear Programming

Question 9 (12.5 points):Consider the following all-integer linear program:

Max 1×1 + 1×2

s.t.4×1 + 6×2 ≤22

1×1 +5×2 ≤15

2×1 + 1×2 ≤ 9

x1, x2 ≥ 0 and integer

a.Graph the constraints for this problem. Use dots to indicate all feasible integer solutions. (4 points)

b.Solve the LP Relaxation of this problem. (4 points)

c.Find the optimal integer solution. (4.5 points)