maths assignments
Quantitative Analysis BA 452 Homework 3 Questions
19. Better Products, Inc., manufactures three products on two machines. In a typical week, 40 hours are available on each machine. The profit contribution and production time in hours per unit are as follows:
Category 
Product 1 
Product 2 
Product 3 
Profit/unit 
$30 
$50 
$20 
Machine 1 time/unit 
0.5 
2.0 
0.75 
Machine 2 time/unit 
1.0 
1.0 
0.5 
Two operators are required for machine 1; thus, 2 hours of labor must ne scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of 100 laborhours is available for assignment to the machines during the coming week.
Other production requirements are that product 1 cannot account for more than 50% of the units produced and that product 3 must account for at least 20% of the units produced.
a. How many units of each product should be produced to maximize the total profit contribution? What is the projected weekly profit associated with your solution?
b. How many hours of production time will be scheduled on each machine?
c. What is the value of an additional hour of labor?
d. Assume that labor capacity can be increased to 120 hours. Would you be interested in using the additional 20 hours available for this resource? Develop the optimal product mix assuming the extra hours are made available.
20. Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 7% for home loans, 12% for personal loans, and 9% for automobile loans. The bank’s planning committee has decided that at least 40% of new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans.
a. Formulate a linear programming model that can be used to determine the amount of finds ASB should allocate to each type of loan in order to maximize the total annual return for the new funds.
b. How much should be allocated to each type of loan? What is the total annual return? What is the annual percentage return?
c. If the interest rate on home loans increased to 9%, would the amount allocated to each type of loan change? Explain.
d. Suppose the total amount of new funds available was increased by $10,000. What effect would this have on the total annual return? Explain.
e. Assume that ASB has the original $1 million in new funds available and that the planning committee has agreed to relax the requirement that at least 40% of the new funds must be allocated to home loans by 1%. How much would the annual return change? How much would the annual percentage return change?
Quantitative Analysis BA 452 Homework 3 Questions
21. Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:
Room 
Super Saver 
Deluxe 
Business 
Type I 
$30 
$35 
 
Type II 
$20 
$30 
$40 
Type I rooms do not have Internet access and are not available for the Business rental class.
Round Tree’s management makes a forecast of the demand by rental class for each m=night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 60 rental sin the Deluxe class, and 50 rentals in the Business class. Round Tree has 100 Type I rooms and 120 Type II rooms.
a. Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. Is the demand by any rental class not satisfied? Explain.
b. How many reservations can be accommodated in each rental class?
c. Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5. Should this incentive be offered?
d. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room? Why?
e. Could the linear programming model be modified to plan for the allocation of rental demand for the next night? What information would be needed and how would the model change?
Quantitative Analysis BA 452 Homework 3 Questions
22. Industrial Design has been awarded a contract design label for a new wine produced by Lake View Winery. The company estimates that 150 hours will be required to complete the project. The firm’s three graphics designers available for assignment to this project are Lisa, a senior designer and team leader; David, a senior designer; and Sarah, a junior designer. Because Lisa has worked on several projects for Lake View Winery, management specified that Lisa must be assigned at least 40% of the total number of hours assigned to the two senior designers. To provide labeldesigning experience for Sarah, Sarah must be assigned at least 15% of the total project time. However, the total number of hours assigned to Sarah must not exceed 25% of the total number of hours assigned to the two senior designers. Due to other project commitments, Lisa has a maximum of 50 hours available to work on this project. Hourly wage rates are $30 for Lisa, $25 for David, and $18 for Sarah.
a. Formulate a linear program that can be used to determine the number of hours each graphic designer should be assigned to the project in order to minimize total cost.
b. How many hours should each graphic designer be assigned to the project? What is the total cost?
c. Suppose Lisa could be assigned more than 50 hours. What effect would this have on the optimal solution? Explain.
d. If Sarah were not required to work a minimum number of hours on this project, would the optimal solution change? Explain.
23. Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a sharper and a grinder. The times (in minutes) required on each machine are as follows:
Machine 

Component 
Sharper 
Grinder 
1 
6 
4 
2 
4 
5 
3 
4 
2 
The sharper is available for 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. In fact, the company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1,2, and 3 are $8, $6, and $9, respectively.
a. Formulate and solve for the recommended production quantities.
b. What are the objective coefficient ranges for the three components? Interpret these ranges for company management.
c. What are the righthandside ranges? Interpret these ranges for company management.
d. If more time could be made available on the grinder, how much would it be worth?
e. If more units of component 3 can be sold by reducing the sales price by $4, should the company reduce the price?
Quantitative Analysis BA 452 Homework 3 Questions
24. National Insurance Associates carries an investment portfolio of stocks, bonds, and other investment alternatives. Currently $200,000 of funds are available and must be considered fro new investment opportunities. The four stock options National is considering and the relevant financial data are as follows:
Stock 

A 
B 
C 
D 

Price per share 
$100 
$50 
$80 
$40 
Annual rate of return 
0.12 
0.08 
0.06 
0.10 
Risk measure per dollar 
0.10 
0.07 
0.05 
0.08 
invested 

The risk measure indicates the relative uncertainty associated with the stock in terms of it realizing the projected annual return; higher values indicate greater risk. The risk measures are provided by the firm’s top financial advisor.
National’s top management has stipulated the following investment guidelines: The annual rate of return for the portfolio must be at least 9% and o one stock can account of more than 50% of the total dollar investment.
a. Use linear programming to develop an investment portfolio that minimizes risk.
b. If the firm ignores risk and uses a maximum returnoninvestment strategy, what is the investment portfolio?
c. What is the dollar difference between the portfolios in parts (a) and (b)? Why might the company prefer the solution developed in part (a)?
Quantitative Analysis BA 452 Homework 3 Questions
25. Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. Because of a large backlog of orders for oak and cherry cabinets, the company decided to contract with three smaller cabinetmakers to do the final finishing operation. For the three cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of hours available for the final finishing operation, and the cost per hour to perform the work are shown here.
Cabinetmaker 1 
Cabinetmaker 2 
Cabinetmaker 3 

Hours required to 
50 
42 
30 
complete all the 

oak cabinets 

Hours required to 
60 
48 
35 
complete all the 

cherry cabinets 

Hours available 
40 
30 
35 
Cost per hour 
$36 
$42 
$55 
For example, Cabinetmaker 1 estimates it will take 50 hours to complete all the oak cabinets and 60 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 40 hours available for the final finishing operations. Thus, Cabinetmaker 1 can only complete 40/50 = 0.80, or 80%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/60 = 0.67, or 67%, of the cherry cabinets if it worked only on cherry cabinets.
a. Formulate a linear programming model that can be used to determine the percentage of the oak cabinets and the percentage of the cherry cabinets that should be given to each of the three cabinetmakers in order to minimize the total cost of completing both projects.
b. Solve the model formulated in part (a). What percentage of the oak cabinets and what percentage of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects?
c. If Cabinetmaker 1 has additional hours available, would the optimal solution change? Explain.
d. If Cabinetmaker 2 has additional hours available, would the optimal solution change? Explain.
e. Suppose Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change have on the optimal solution? Explain.
Quantitative Analysis BA 452 Homework 3 Questions
26. Benson Electronics manufactures three components used to produce cell telephones and other communication devices. In a given production period, demand for the three components may exceed Benson’s manufacturing capacity. Tin this case, the company meets demand by purchasing the components from another manufacturer at an increased cost per unit. Benson’s manufacturing cost per unit and purchasing cost per unit for the three components are as follows:
Source 
Component 1 
Component 2 
Component 3 
Manufacture 
$4.50 
$5.00 
$2.75 
Purchase 
$6.50 
$8.80 
$7.00 
Manufacturing times in minutes per unit for Benson’s three departments are as follows:
Department 
Component 1 
Component 2 
Component 3 
Production 
2 
3 
4 
Assembly 
1 
1.5 
3 
Testing & Packaging 
1.5 
2 
5 
For instance, each unit for component 1 that Benson manufactures requires 2 minutes of production time, 1 minute of assembly time, and 1.5 minutes of testing and packaging time. For the next production period, Benson has capacities of 360 hours in the production department, 250 hours in the assembly department, and 300 hours in the testing and packaging department.
a. Formulate a linear programming model that can be used to determine how many units of each component to manufacture and how many units of each component to purchase. Assume that component demands that must be satisfied are 6000 units for component 1, 4000 units for component 2, and 3500 units for component 3. The objective is to minimize the total manufacturing and purchasing costs.
b. What is the optimal solution? How many units of each component should be manufactured and how many units of each component should be purchased?
c. Which departments are limiting Benson’s manufacturing quantities? Use the dual value to determine the value of an extra hour in each of these departments.
d. Suppose that Benson had to obtain one additional unit of component 2. Discuss what the dual value for the component 2 constraints tells us about the cost to obtain the additional unit.

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