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QMB Assignment 2

Question 1

Your local Chamber of Commerce periodically sponsors public service seminars. In your role as the marketing manager, you are putting together an advertising plan for this year’s program of events. The promotional budget is $18,200. Advertising alternatives include television, radio and newspaper. You have obtained the following estimates for the size of an audience and cost for each type of advertisement:
Television Radio Newspaper
Audience per ad 100,000 18,000 40,000
Cost per ad $2,000 $300 $600

Past experience suggests that at most ten TV advertisements should be aired. Similarly, no more than 10 newspaper ads will be placed. For radio, maximum usage is set at twenty advertisements. You must find the mix of advertisements that will reach a maximum audience. You have decided to use linear programming to obtain a solution to the problem before you.

a.What are the decision variables for this problem?

b.Using decision variables identified in part (a), formulate the objective function for this problem. Is the quantity of interest to be maximized or minimized? Explain briefly.

c.What is the limited resource in this problem? Using decision variables from part (a), formulate the constraint for this resource.

d.What other constraints are relevant to this problem? Using the decision variables from part (a), formulate those constraints.

e.Suppose that to ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. Using decision variables from part (a), formulate a constraint based on this requirement. Use algebra to rewrite this constraint in the standard linear format.

f. Suppose that, in addition, television should account for at least 10% of the total number of advertisements authorized. Using decision variables from part (a), formulate a constraint based on this requirement in the standard linear format.

g. Give the full mathematical model for this problem.

h. Use EXCEL to obtain a solution to the linear programming problem from part (g), 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 Solver submission.

EXCEL Instructions: All EXCEL output should have your e-mail ID on it. To ensure this, you will need to save your EXCEL file as ‘E-mail ID Advertising Plan.xls’. In addition, your constraint names should begin with your initials, e.g. JB_Budget.

Please refer to the Linear Programming Using Solver (supplement to the EXCEL Booklet) or the linear programming supplement to the textbook (pdf file posted on the course website) for detailed instructions on how to set up your spreadsheet and use Solver.

i. Management agrees to increase the budget to $28,200. Can we estimate what the change in audience will be? What report tells us this? By how much will the maximum audience change?

j. Suppose that you are not convinced that the estimated audience per television ad is accurate. You think it could be 10% higher or 10% lower than the initially obtained figure of 100,000 viewers per TV ad. Would the solution obtained in part (h) still be optimal? Which of the EXCEL reports helps you answer this question? Justify your answer carefully. What would be the total audience reached if the TV audience per ad were 10% lower or 10% higher?

k. Prepare a report to your superiors with your recommendation. Your report should discuss the following issues:

•The optimal number of advertisements of each type and the maximum total audience.
•The actual cost of this advertising campaign and the amount left in the budget, if any.
•The proportion of TV and radio advertisements.
•Your concerns from part (i).

Report Instructions: Your report should be at most one page, in 12pt Times New Roman or a similar font, with 1.5 line spacing and at least 2cm margins.

Question 2
In financial year 2003-2004, bankruptcies related to business in Australian numbered 4449 according to Insolvency and Trustee Service Australia. The economic conditions accounted for 1332 of these business bankruptcies. New South Wales accounted for 1185 of the business bankruptcies. In addition, 650 of all business bankruptcies were in New South Wales and related to economic conditions. A business-related bankruptcy from 2003-2004 is sampled randomly.
Information in the question can be interpreted as a contingency table shown below:

Due to Economic Location
Condition New South Wales Not New South Wales Total
Yes 650 1332
No
Total 1185 4449

a. Fill in the missing values in the contingency table.

b. What is the probability that the business-related bankruptcy is located in New South Wales?

c.Suppose that the chosen bankruptcy is a business located in New South Wales. What is the probability that this bankruptcy is due to economic conditions?

d. What is the probability that the business-related bankruptcy is due to economic conditions or located in New South Wales (or both)?

e. Are the two events, the business-related bankruptcy due to economic conditions and its location, independent of each other?

Question 3
A real estate company surveyed 50 of its sold properties. The following table gives the selling prices (in thousand dollars) of these properties.
61.4 27.3 26.4 37.4 30.4 47.5
63.9 46.8 67.9 19.1 81.6 47.9
73.4 54.6 65.1 53.3 71.6 58.6
57.3 87.8 71.1 74.1 48.9 60.2
54.8 60.5 32.5 61.7 55.1 48.2
56.8 60.1 52.9 60.5 55.6 38.1
76.4 46.8 19.9 27.3 77.4 58.1
32.1 54.9 32.7 40.1 52.7 32.5
35.3 39.1

a. Use Excel to obtain a histogram of this data. It should be formatted as described in the Excel Booklet which you received in your practical in Week 1. The bins start with 10 as the lowest class beginning point and use a class width of 10.

b. Briefly describe the shape (symmetry, modality and outliers) of the data based on the histogram obtained from part (a).

c. Use Excel to obtain the Descriptive Statistics, Quartile1 and Quartile 3 for the data.

d.Which measures of location and dispersion should you use for this data? What are their values? Give a brief explanation for your decision.

Question 4
A T-shirt company is interested in knowing the average retail price charged for one product sold in stores across the country. The company cannot justify a national census to generate this information. Based on the company information system’s list of all retailers who carry the product, a researcher for the company contacts 36 of these retailers and ascertains the retail prices for the product. A population standard deviation is known to be $1.13. The price data (in dollar) is listed in the following table.

22.3 21.6 21.2 20.1 19.9 22.3
21.1 23.1 20.7 18.7 21.0 21.2
19.8 21.7 21.8 20.9 20.8 22.0
23.0 21.8 22.2 20.5 21.7 21.4
22.9 23.2 21.5 21.0 18.2 21.9
20.2 21.9 22.6 22.4 21.7 21.6

a.Use Excel to calculate the sample average retail price.

b.Set up a 90% confidence interval of the population average retail price charged for this T-shirt item.

c.Does the population of retail price have to be normally distributed here? Explain briefly.

d.What is the probability that the mean retail price, in a sample of size 36, is greater than $21? Assume the population average retail price is $21.4.

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