Quantitative Methods for Business
Question 1
A manufacturer sells a product for $10 inclusive of a government tax. Due to cost pressures, the company increases the price by 15 percent. Some time later the government reduces the tax on this product, bringing the price down by 15 percent. Is the price of the product once again $10? Show calculations that support your answer.
Question 2
A company operates a savings scheme for its employees, with interest at 5.6% per year compounded monthly.
(a) Calculate the amount saved at the end of 5 years when $200 is deposited at the end of each month.
(b) How much should be saved from a monthly salary in order to have $15,000 at the end of 5 years?
Question 3
Classic Furniture advertises a "One-Year No-Interest Sale" on any purchase over $499. They require a deposit of 10%, with the balance of the purchase price to be paid in 12 equal monthly payments with no interest charged.
(a) What is the cash price equivalent to Classic's no-interest plan on an item ticketed at $1,000? Assume an interest rate of 11.5% per year compounded monthly. Is this a good deal for a customer? Give a reason.
(b) Flemmings Furniture offers the same no interest plan but requires an establishment fee of $25 payable immediately. Flemmings also charges an account keeping fee of $3.95 per month due with each payment. What is the cash price equivalent to Flemmings' no-interest plan on an item ticketed at $1,000? Assume the same interest rate as in part (a). Is this a good deal for a customer? Give a reason.
Question 4
Air Atlantic is considering a new service between two cities. Its existing planes, each of which has a capacity of carrying 240 passengers, could be used for one flight a week with fixed costs of $30,000 and variable costs amounting to 50% of the ticket price.
(a) Air Atlantic is thinking of charging $200 per ticket. How many passengers will the airline need on each flight to break even? Find the break-even point algebraically and by using an EXCEL graph. Attach the printout or copy your EXCEL graph into your assignment submission.
EXCEL Instructions: Create a column called Number of Passengers and in that column enter values from 0 to 400 in increments of 20. Then create two more columns, one for Total Cost and another for Total Revenue. Enter appropriate formulae in EXCEL to obtain the total cost and total revenue corresponding to each value in the Number of Passengers column. Highlight the resulting three sets of numbers and go to the Insert tab (or Chart menu) to obtain an appropriate diagram. Make sure that your graph has been labelled appropriately (i.e. title, axis labels, legend). Please refer to Topic 3 in the EXCEL Booklet for further instructions on entering formulae and graphing in EXCEL.
(b) Based on your analysis from part (a), and without any additional calculations, will Air Atlantic realize a profit or a loss if 240 seats are sold for a flight? Does the break- even number of passengers from part (a) seem reasonable? Explain briefly.
(c) What price per seat should Air Atlantic charge to break-even on a full flight? Find the break-even ticket price algebraically.
Question 5
David, Diana and Lydia are the sole partners and workers in a company that produces fine clocks. David and Diana are each available to work a maximum of 40 hours per week at the company, while Lydia is available to work a maximum of 20 hours per week.
The company makes two different types of clocks: a grandfather clock and a wall clock. To make a clock, David (a mechanical engineer) assembles the inside mechanical parts of the clock while Diana (a woodworker) produces the hand carved wood casings. Lydia is responsible for taking orders and shipping the clocks. The amount of time required for each of these tasks is shown below.
|
Task
|
Time required
|
|
Grandfather clock
|
Wall clock
|
|
Assemble clock mechanism
|
6 hours
|
4 hours
|
|
Carve wood casing
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8 hours
|
4 hours
|
|
Ship finished clock
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3 hours
|
3 hours
|
Each grandfather clock built and shipped yields a profit of $300, while each wall clock yields a profit of $200.
The three partners now want to determine how many clocks of each type should be produced per week to maximize the total profit using linear programming. They have asked for your help.
(a) Formulate a linear programming model for this problem. In doing so, consider the following:
- What are the decision variables for this problem?
- What is the objective for this problem? Using your decision variables, formulate the objective function.
- What are the constraints in this problem? Using your decision variables, formulate these constraints.
(b) Use EXCEL Solver to obtain a solution to the linear programming problem from part (a), together with an Answer Report and a Sensitivity Report. Provide a printout of your EXCEL spreadsheet and of the two Excel reports with your assignment submission.
EXCEL Instructions: All EXCEL output should bear your e-mail ID. To ensure this, you will need to save your EXCEL file as ‘E-mail ID Fine Clocks.xls'. In addition, your constraint names should begin with your initials, e.g. JB_Assembly.
Please refer to Topic 5 in the Excel Booklet or the Linear Programming Supplement to the textbook (pdf file posted on the course resources website) for detailed instructions on how to set up your spreadsheet and use Solver.
(c) Grandfather clocks have become very popular and the partners feel that a unit profit per grandfather clock of $375 may be possible. Would the solution obtained in part
(b) still be optimal? Which of the EXCEL reports helps you answer this question? Justify your answer carefully. How would the total profit change, if at all?
(d) Lydia agrees to work an additional ten hours each week. What effect, if any, will this increase have on the profit received and the number of clocks produced each week?
(e) Prepare a report that summarizes your linear programming results. Your report should discuss the following:
- The optimal number of grandfather and wall clocks, as well as the maximum total profit.
- Resulting work hours for each of the partners. How do these compare to the maximum hours each partner is able to allocate to this business?
- Profit per unit considerations from part (c).