linear pogramming – your local chamber of commerce periodically

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 $25,000. 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:

TelevisionRadioNewspaper

Audience per ad 50000 18000 20,000

Cost per ad $2,500 $600 $500

Past experience suggests that at most ten TV advertisements should be aired. Similarly, no more than 15 newspaper ads will be placed. For radio, maximum usage is set at 15 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.

Give the full mathematical model for this problem.

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 60% 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 5 % 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.

Use SOLVER to produce the answer and sensitivity reports required to answer the following questions.

1. What is the optimal number of adverts for each of the media?

2.What is the maximum audience?

3. How much of the budget has been spent?

4. If the radio audience per advertisement doubled to 36,000 would the optimum number of advertisements change? Explain how you know.

5. If the provisional budget is increased to $30,000 what impact, if any, will this increase have on the maximum audience?

Question 2 (Total 26 marks)

In January 2002, John, having recently graduated from UNISA established a business as a financial advisor. His uncle who was planning to retire in one years time asked John to recommend a management investment fund that he can investment in by buying units on a weekly basis. The uncle also wanted the option to sell his units at any time at the current price. John’s uncle recently complained to John that he had received very poor advice and that he was not in a position financially to retire as he had planned. Thinking back to his QMB course John decided to summarize the weekly prices data for about the past two and a half years. The data file is available on the portal (Assignment 2 Future Leaders.xls)

John asks you to summarize the data

(a) Make a histogram and obtain Descriptive Statistics, including Quartile 1 and Quartile 3 for both data sets, . Attach printouts of your data, histogram and Descriptive Statistics to your assignment submission.

EXCEL Instructions:

Add your network ID to the column name ggsj001).

Create another column called PRICES with numbers from 1.50 to 5.0 in steps of 0.5 i.e. 1.5, 2.0, 2.50 and so on up to 5). Please make sure that the title of your histogram ends with your network ID

(b) Describe the shape (symmetry, modality and outliers) for the distribution.

Instructions for identifying outliers:

Whether an observation is an outlier is a matter of judgement. One rule commonly used for identifying outliers is the so-called 1.5 × IQR rule. An observation is suspected to be an outlier if it is more than 1.5 × IQR below the first quartile Q1 or above the third quartile Q3.

Apply this rule to Entry Price and identify suspected outliers (if any) by using the calculated amounts.

(C)For the distribution nominate appropriate summary measures for Entry Price ($). Give reason(s) for your choice. For the distribution, give and interpret the values of the summary measures you have chosen.

Question 3 (Total of 25 marks)

John decides that in addition to summarizing the data he should analyze the change in prices over time. In order to determine the growth or otherwise of prices over time and that he should ask you to produce 1regression plot and equation for the set of data

a) Produce a regression plot with trend line and suitable labels

b) Include the regression equation and coefficient of determination

i) What is the value of the slope? Interpret what the value means.

ii) What is value of the intercept? Interpret what the value means

iii) Calculate the coefficient of determination and explain what the value means.

iv) Using the equation predict the price of a unit on 3 January 2003.

v) Are you confident about your prediction? Give reasons for your answer.

vi) Is John’s uncle entitled to be concerned about his investment?

Question 4 (Total of 15 marks)

In June 2009, the average Australian credit card account balance was $3,127, according to data extracted from the Reserve Bank of Australia www.rba.gov.au . Assuming that these balances are approximately normally distributed with a mean of $3,127 and a standard deviation of $1,500:

a) Find the probability that an account balance is less than $2,500. Use a diagram! Clearly indicate any properties of the Normal distribution that you are using to facilitate your calculation.

b) One quarter of credit card accounts carry at least what balance? Use a diagram!

c)Suppose that 100 credit card account balances are randomly selected. What is the probability that the average account balance in this sample is greater than $3,200? Use a diagram!

d) Construct and interpret a 99 % confidence interval with a sample mean of $3180.

e) What sample size is required for the margin of error to be $120 for this confidence interval?