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ETF5650: Business Optimisation Skills

Assignment 1, 2025

(Forecasting and quality control)

Instructions

· This is an individual assignment.

· Answer all questions showing all workings. All calculations must be done in Excel.

· VERY IMPORTANT: Submit answers in ONE EXCEL file.

· Provide worded answers in a text box within the spreadsheet.

· Label the sheets with the question number and part or parts of the question. For example, Question 1 Part (a), Question 2 Parts (a)&(c).

· Submit the .xlsx file via “Assignment 1 submission” link on the Moodle site. Submission is not allowed until one week before the deadline.

· Submit the .xlsx file before 11.55pm, 9 April 2025. Late submission incurs penalties.

· VERY IMPORTANT: Copy and paste the declaration given in page 2 in a text box on a separate sheet in the .xlsx file and sign or print your name with date.

Format of the file name for submission in Moodle:

Your surname_initials_(ID number).xlsx

E.g. Tran_A_(12345678).xlsx

Upload your .xlsx file to Moodle as follows;

Click on the “Assignment 1 Submission” link given under ASSESSMENTS and upload your file.

You have to accept the Submission Statement before clicking the Submit button. Once you upload the file, the following message will appear “File uploaded successfully.” To confirm that your upload was successful, click on the “Assignment Submission” uploading link. You can then see the uploaded file name.

The aim of this assignment is to give you more insights on some analytical concepts and for you to practice application of the techniques that fall within the areas examined.

NOTE: You are allowed to submit only once. So, submit only the final version.

Total mark is 50

Signature: Date:

You must not use generative artificial intelligence (AI) to generate any material or content in relation to the assessment task.

Question 1. 3+4+6+(1+1+1)+4 = 20 marks

Table 1 provide the number of mergers and acquisitions (M&A) realised from 2012 to 2023.

Year	M&A	Year	M&A
2012	325	2018	469
2013	393	2019	406
2014	322	2020	334
2015	391	2021	345
2016	423	2022	460
2017	427	2023	283

(a) Fit a three-year moving average to the data and plot the results together with the observed time series. (3 marks)

NOTE: Moving average is placed at the middle of the time span. Moving average forecast is placed at the time period immediately after the time span.

(b) Using smoothing constant , exponentially smooth the series. (4 marks)

(d) What is your forecast for M&A for the years 2024 and 2025 based on,

(i) the three-year moving average method (1 mark)

(ii) the exponential smoothing method with (1 mark)

(iii) the linear trend line (1 mark)

(e) We can not really expect a perfectly fitting model for any time series data. However, we can consider the first differences, second differences and percentage differences for a given time series as guides in choosing an appropriate model. Based on the differences, what would you consider as the most appropriate model (linear, quadratic and exponential) for the given data? (4 marks)

Question 2. 5+2+1+2+5 = 15 marks

The data in Table 2 give the results of inspecting a sample of size 100 each day for the past 20 days.

Table 2

Sample number	Number of nonconforming units	Sample number	Number of nonconforming units
1	7	11	6
2	4	12	15
3	1	13	0
4	3	14	9
5	6	15	5
6	8	16	1
7	10	17	4
8	5	18	5
9	2	19	7
10	7	20	12

(a) Construct a fraction non-conforming control chart with 99.7% control limits for the given data. (5 marks)

(b) Does the process appear to be in control? (2 marks)

NOTE: When a process appears to be out of control, the points must be investigated to see whether an assignable cause can be determined. When assignable causes can be found, we revise the control limits by excluding the corresponding samples from control limit calculations. However, we do not drop those samples from the plot (chart). In other words, all observed sample data will be included in the plot.

(c) Suppose assignable causes cannot be found for all points outside the control limits. How would you proceed with this quality control process? (1 mark)

(d) Suppose assignable causes can be found for all points outside the control limits. How would you proceed with this quality control process? (2 marks)

Now, assume that, for all instances (samples) where the fraction non-conforming lies outside the control limits, assignable causes can be found.

(e) Plot the 99.7% control chart that may be used in future quality control checks. (5 marks)

Question 3. 2+(2+2+2+2)+3+2 = 15 marks

(a) When setting up and R control charts, it is best to begin with the R chart. Why?

(b) A quality characteristic of a part is normally distributed with mean =10 and standard deviation .

(i) Specification limits for the quality characteristic of the part are and . This means, a part is defective if its quality characteristic lies outside these limits. What is the probability of producing a defective part? Use EXCEL to do the calculations. Give the final answer to four decimal places.

(ii) If the specification limits for the quality characteristic of the part are and , how many defective parts may be expected out of 1 million parts?

(iii) Seventy-five independent parts are used in manufacturing a product. All the seventy-five independent parts must be non-defective for the product to function satisfactorily. If the specification limits for the quality characteristic of the part are and , what percentage of the products will be defective? Give the final answer to two decimal places. Use EXCEL to do the calculations. Do not approximate at the intermediate steps.

(iv) Suppose the new specification limits for the quality characteristic of the part are and This is the six-sigma standard. Compute the probability of a defective part and determine how many parts will be defective out of 1 billion parts.? Use EXCEL to do the calculations. Do not approximate at the intermediate steps. Give the final answer to six decimal places.

(c) Explain, with reference to the underlying statistical concepts, what is meant by the statement “A process has the potential to operate at the six-sigma standard”.

APPENDIX

You can determine areas under the normal distribution using the following commands.