Subject Code : MATH1075
Country : Australia
Assignment Task:

 

Task:

QUANTITATIVE METHODS FOR BUSINESS (MATH 1075) 

RATIONALE: 

Problem solving exercises part 2 is structured to consolidate your understanding of Topics 3, 4 and 5 in the course, namely linear equations, break-even analysis and linear programming models in business 

Completion of this assignment will allow you to consolidate and demonstrate an understanding of mathematical and statistical knowledge in relation to business and management, in accordance with Course Objective CO1 and GQ1. 

You will analyse real life problems within a mathematical and statistical context, identify appropriate techniques, generate and evaluation solutions, thereby developing your problem-solving skills and knowledge of key concepts in mathematics and statistics (Course Objectives CO1 and CO2, GQ3). 

This assignment will also further develop your skills in communicating mathematical and statistical concepts and results, including expressing and interpreting equations, Excel output and problems (Course Objective CO2, CO3 and GQ6). 

COURSE OBJECTIVES: 

CO1. understand the basis and role of quantitative information for decision making in business and management. 

CO2. perform basic mathematical calculations and manipulations, apply descriptive statistical methods and interpret the results. 

CO3. use Excel to carry out calculations and produce diagrams for decision making in business. 

GRADUATE QUALITIES: 

GQ1. operates effectively with and upon a body of knowledge of sufficient depth to begin professional practice. 

GQ3. is an effective problem solver, capable of applying logical, critical, and creative thinking to a range of problems. 

GQ6. communicates effectively in professional practice and as a member of the community.

TASK DESCRIPTIONS: 

In this task you will be presented with a selection of problems, in business language, which you are required to model, identify the correct method of solution and then solve the problem. 

For each question, you are required to show the working out supporting your final answer to demonstrate your understanding and reasoning. 

SUGGESTED PROCEDURE: 

Work through each of the 3 problems below. Read each question carefully as some answers require computations only, others should be written in short paragraphs and some require the use of EXCEL. Attach the copy your EXCEL spreadsheet into your assignment submission. Use the provided EXCEL file for this assessment (This Link). 

ASSIGNMENT ADVICE: 

For guidance on how to layout your solutions and demonstrate your reasoning, see the solutions for the weekly review questions. Be sure to follow all EXCEL instructions and to perform your calculations in EXCEL and submit the spreadsheets. For additional support working with EXCEL, see the EXCEL supplement on the course webpage. 

PROBLEM SOLVING EXERCISES: 

3 PARTS (consist of several subsections)- DUE END OF WEEK 5 

Part 1. Motion Music is doing a financial feasibility analysis for a new album. Recordings and production are estimated at $70,000. The printing of CDs is fixed at $4,500 for set up plus $2.50 per CD. The artist’s royalty is 5% per CD of the publisher’s price to music stores. Advertising and promotion costs are budgeted at $8,000. 

(a) If the price to music stores is set at $22, how many CDs must be sold 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 CDs and in that column enter values from 0 to 5,000 in increments of 500. 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 CDs 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). Refer to Topic 3 in the EXCEL Supplement for further instructions on entering formulae and graphing in EXCEL. 

(b) The marketing department is forecasting sales of 6,000 CDs at the price of $22. Based on your graph from part (a), will there be a net profit or net loss from the project at this volume of sales? How do you know? Calculate this net profit or loss amount. (Assume no tax costs.) 

(c) If the artist requires an increase in royalty to 8%, how does this impact Motion Music? 

Assuming the fixed costs and the selling price remains as in (a), explain in a short paragraph (3-4 sentences) whether the number of CDs required to break-even will increase or decrease. Do not re-calculate the break-even quantity, x, for this question however you may quote the break-even formula to aid your explanation. 

(d) The marketing department is also forecasting that if the price is reduced by 6% then unit sales will be 8% higher. Should Motion Music continue with the original price in (a) or this reduced price? Show calculations that support your recommendation. Assume initial fixed and variable cost estimates. 

Part 2. There are 4 tasks in this part. 

Dean runs The Creamy Bar which specialises in artisan ice cream sold at a local farmer’s market. Prevailing prices in the local market are $10 for a take-home tub of Classic Vanilla and $18 for a tub of Chocolate Almond Fudge. 

The local dairy farmer delivers 50 litres of milk every Friday in preparation for market day. Classic Vanilla will need 0.5 litres per tub and Chocolate Almond Fudge requires 3 times as much. Both flavours require 500g of sugar to enhance the taste. There is a total of 22kg of sugar available per market day. For the signature velvety mouthfeel, Dean adds 0.5 litres of heavy cream to Classic Vanilla and double the amount for Chocolate Almond Fudge. He ordered 55 litres of heavy cream from the supplier. 

Part 2-Task 1: 3 sections 

Construct a mathematical model for this problem. In doing so, consider the following: 

(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 maximised or minimised? 

(c) What constraints are relevant to this problem? Using the decision variables 

from part (a), formulate those constraints. 

Part 2-Task 2: 1 section 

Use Excel Solver to obtain a solution to the mathematical problem from Task 1. Your submission should include: 

• your Excel spreadsheet 

• the Sensitivity Report 

• the Answer Report 

Part 2-Task 3: 4 sections 

Use your Excel output to answer the following questions: 

(a) Describe the linear programming solution to the Dean of The Creamy Bar in terms of: 

• The optimum number of take-home tubs of Classic Vanilla and Chocolate Almond Fudge to prepare each market day. 

• The maximum revenue per market day. 

• Whether all the milk purchased will be fully utilised. 

• Whether all the sugar allocated will be fully utilised. 

• Whether all the heavy cream ordered will be fully used. 

Which of the Solver reports helps you answer these questions? 

(b) What is the maximum profit per market day if Dean paid $1.50 per litre for milk and cream and $50 for sugar? Note that Dean also draws a $150 salary per market day. Which Solver report allows you to answer this question? (assume unused materials will be wasted) 

(c) Due to the popularity of the Chocolate Almond Fudge flavour, Dean is hoping to increase the price to $19 per take-home tub. Would the solution obtained in Task 2 still be optimal? Which of the EXCEL reports helps you answer this question? Justify your answer carefully. How would the solution and The Creamy Bars’ revenue change, if at all? 

(d) In preparation for the scorching heat in summer, Dean would like to purchase an extra 15 litres of milk to increase ice cream production. Would the solution obtained in Task 2 still be optimal? Which of the EXCEL reports helps you answer this question? Justify your answer carefully. How would the solution and The Creamy Bars' revenue change, if at all? 

Attach the new Answer Report ONLY, for the scenario in which Dean purchases 65 litres of milk, verifying your calculated maximum revenue per market day. 

Part 2.Task 4: 

Write a summary outlining the solution and discussing your findings from Task 3 (use EXCEL Text box). 

1. The first step is always to work out the mathematical set up for the problem. This means identifying decision variables, formulating the objective function and then formulating constraints. At this stage, we are not trying to solve the problem or work out interactions among constraints. We simply list all conditions that must be satisfied. 

When you complete Task 1, you should have two decision variables, the objective function written in terms of those decision variables, and five constraints, also written in terms of decision variables (some using both decision variables, others just one of them). 

2. The second step is to find a solution. Task 2 tells you specifically to use Excel Solver to find this solution. The key here is to translate all mathematical expressions from Task 1 into Excel format. Instructions for doing so can be found under Topic 5 in the Excel booklet, as well as in the Linear Programming supplement. In addition, the Lecture notes page in this website gives you access to Excel spreadsheets used to generate Excel output shown in lecture slides for Week 5. It may be worthwhile examining them before attempting Task 2. 

3. The final step is interpreting the solution that has been found, which is Task 3. 

4. The summary in Task 4 is a summary of the results from linear programming and sensitivity analysis in Tasks 2 and 3. 

Part 3. A simple random sample of 227 university students were asked what pasta they usually order and with which sauce. The preferences of these respondents are summarised below: 

(a) Use Excel to obtain a 100% stacked column chart for the data from the table. What does your chart suggest between the relationship between Pasta and Sauce? 

(b) Use the contingency table to calculate the following probabilities; include an appropriate probability statement for each case: 

i. What is the probability that a respondent prefers Gnocchi? 

ii. What is the probability that a respondent prefers Spaghetti and Bolognese sauce? 

iii. What is the probability that a respondent prefers Ravioli or Carbonara sauce? 

iv. Suppose the preferred sauce is Pesto. What then is the probability the pasta ordered is Ravioli? 

v. Suppose that the preferred Pasta is Spaghetti. What then is the probability the sauce ordered is Bolognese? 

(c) Are the events Gnocchi and Sauce statistically independent or dependent? How do you know? Show all calculations that support your answer. 

 

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