Page 1 of 9 UL20/0468 MN2032 Summer 2020 online assessment guidance MN2032 Management Science Methods The assessment will be an open-book take-home online assessment within a 24- hour window. The requirements for this assessment remain the same as the originally planned closed-book exam, with an expected time/effort of 3 hours. Candidates should answer FOUR of the following EIGHT questions. All questions carry equal marks. In the event that you answer more than four questions ONLY the first four questions answered will be marked. Some questions have a word/page limit. This limit is to discourage spending beyond the 3 hour expected time/effort by producing excessively lengthy responses. You should complete this paper in one of two ways: EITHER: 1. Using word processing software (eg, Microsoft Word). This should be saved as a .doc or .docx file and then uploaded to the VLE as ONE individual file including the coversheet. Each page should have your candidate number in the header. Please do not write your name anywhere on any part of your submission. OR: 2. Using pen and paper. Please use BLACK ink only. Handwritten work then needs to be scanned, converted to PDF and then uploaded to the VLE as ONE individual file including the coversheet. Each scanned sheet should have your candidate number written clearly at the top. Please do not write your name anywhere on any part of your submission The paper will be available at 12.00 pm (BST) on Tuesday 7 July 2020. You have until 12.00 pm (BST) on Wednesday 8 July 2020 to upload your file into the VLE submission portal. However, you are advised not to leave your submission to the last minute. We will deduct 5 marks for each hour (or part thereof) that your submission is overdue (e.g. we will deduct 5 marks if your submission is up to one hour late, 10 marks if your submission is more than one hour late but less than two hours late etc.). If you think there is any information missing or any error in any question, then you should indicate this but proceed to answer the question stating any assumptions you have made. Page 2 of 9 UL20/0468 The assessment has been designed with a duration of 24 hours to provide a more flexible window in which to complete the assessment and to appropriately test the course learning outcomes. As an open-book assessment, the expected amount of effort required to complete all questions and upload your answers during this window is no more than 3 hours. Organise your time well and avoid working all night. You may use any calculator for any appropriate calculations but full working must be shown to gain all marks for each question. You are assured that there will be no benefit in you going beyond the expected 3 hours of effort. Your assessment has been carefully designed to help you show what you have learned in the hours allocated. This is an open-book assessment and as such you may have access to additional materials including but not limited to subject guides and any recommended reading. But the work you submit is expected to be 100% your own. Therefore, unless instructed otherwise, you must not collaborate or confer with anyone during the assessment. The University of London will carry out checks to ensure the academic integrity of your work. Many students that break the University of London’s assessment regulations did not intend to cheat but did not properly understand the University of London’s regulations on referencing and plagiarism. The University of London considers all forms of plagiarism, whether deliberate or otherwise, a very serious matter and can apply severe penalties that might impact on your award. The University of London 2019-20 Procedure for the Consideration of Allegations of Assessment offences is available online at: https://london.ac.uk/sites/default/files/governance/assessment-offence-procedure- year-2019-2020.pdf The University of London’s Rules for Taking Online Timed Assessments have been included in an update to the University of London General Regulations and are available at: https://london.ac.uk/sites/default/files/regulations/progregs-general-2019-2020.pdf © University of London 2020 Page 3 of 9 UL20/0468 1. (a) Apply both Journey Making and Strategic Choice to one problem with which you are familiar (16 marks) (b) Apply Soft Systems Methodology to one problem with which you are familiar (where this problem must be distinctly different from the problem considered in part (a) of the question). (9 marks) Note here that for this question no credit will be obtained for using the crime problem as given in the subject guide. Your answer to part (a) should be limited to 6 pages, your answer to part (b) limited to 3 pages. 2. Briefly discuss each of the following topics from the viewpoint of Operational Research/Management Science: (a) Using simulation to estimate the effect on public bus transport of enforcing a one metre separation between individuals (9 marks) (b) Two specific and practical problem situations with which you are familiar where the assumptions underlying optimisation are false (7 marks) (c) Using Markov processes to model the progress of a new, and previously unknown, disease in a population (9 marks) Note here that for this question no credit will be obtained for repeating material which is given in the subject guide. Your answer to this question should be limited to 2 pages for each of the three parts (so at most 6 pages in total). 3. An organisation is reviewing the performance of its branches using data envelopment analysis. The data they have collected for these branches is shown below. Branch Number of customers (‘000) Customer satisfaction score Number of employees A 52.9 14.2 22 B 48.9 14.8 18 C 40.4 20.7 23 D 11.4 12.0 13 E 26.3 18.6 17 F 35.2 21.2 23 For example branch E last year employed 17 people and had 26,300 customers. The customer satisfaction score is a measure of how satisfied a Page 4 of 9 UL20/0468 customer is with the branch (arrived at by surveying a random sample of customers, approximately 1% of the number of customers a branch has). The higher the satisfaction score, the more satisfied customers are. Branch E, for example, had a satisfaction score of 18.6. (a) Apply data envelopment analysis to compare the relative performance of these branches using the data shown above. Copy the table shown below into your answer and fill in the efficiencies and reference sets (where appropriate). (12 marks) Branch Efficiency Reference set A B C D E F (b) In a discussion with a colleague as to your analysis they have noted that you have not included the fact that customer satisfaction is scored out of 25. They feel that this information should have been included in your analysis, and would have altered the results. What would you say to them? (2 marks) (c) Another colleague has argued that customer satisfaction score is a wrong measure to use and you should use total satisfaction, defined as the current customer satisfaction score multiplied by the number of customers. What would you say to them? (5 marks) (d) The company CEO feels that only sampling approximately 1% of the customers in a branch to get a customer satisfaction score is insufficient and could result in some branches being shown to be efficient when in fact they are inefficient; or branches being classified as inefficient when they are actually efficient. In support of their argument they have produced the new customer satisfaction scores shown below based on a larger sample. What would you say to them? (6 marks) Branch Customer satisfaction score A 13.1 B 16.3 C 19.2 D 11.0 E 20.5 F 19.3 Page 5 of 9 UL20/0468 4. (a) A company faces a decision as to whether to continue or not, with a particular product it has developed. It can cease development of the product, in which case all expenditure to date will be written off in the company accounts. Alternatively it can decide to test market the product in a small geographic area. Over the next financial year test marketing is estimated to cost 5 (£’million). If the market for the product when test marketed is favourable then the company will receive 7 (£’million) in revenue. However if the market is unfavourable it will only receive 2.5 (£’million). Alternatively the company could introduce the product to a wider market. Over the next financial year this is estimated to cost 15 (£’million). If the market for the product is favourable then the company will receive 23 (£’million) in revenue. However if the market is unfavourable it will only receive 11 (£’million). Assuming that the probability of the market being favourable is 0.40 represent this problem using a decision tree. What course of action for the company would you recommend based upon this decision tree? (12 marks) (b) Copy the table shown below into your answer and fill it in with respect to perfect information for the probability of the market being favourable. Value (£’million) Expected value with perfect information Expected value of perfect information (7 marks) (c) What would the probability of the market being favourable have to be if the expected value of perfect information was one half of the expected value with perfect information? Note that if no probability exists which can achieve this you should clearly indicate why no such probability exists. (6 marks) Page 6 of 9 UL20/0468 5. (a) Solve the following linear program graphically using an iso-cost/iso-profit line, clearly indicating the feasible region. Maximise z = 8x + 6y Subject to 2x + 5y ≤ 27 Constraint 1 3x – 28y ≤ 19 Constraint 2 x,y ≥ 0 Copy the tables shown below into your answer and fill in the values required. (19 marks) Optimal value (4dp) Opportunity cost (4dp) x y z Shadow price (4dp) Constraint 1 Constraint 2 (b) In this linear program the objective, z = 8x + 6y, represents the total profit made by a company from producing two liquid chemical products (chemical product X with a profit per litre of £8, chemical product Y with a profit per litre of £6). The company can either: • Increase the profit per litre on product X to £9; or • Increase the amount of resource available associated with the right-hand side of constraint 1 from its current value of 27 to a new value of 29. Based on the linear program given above which course of action would you recommend and why? Clearly explain the reasons for your recommendation. (6 marks) 6. A company is using Markov theory to analyse switching between four different foods marketed as slimming aids. These slimming aids claim that with regular use they result in significant weight loss. Individual customers who buy these sliming aids only switch (on average) once every four months. Market research has produced the data shown below for the probability of customers switching between slimming aids. Page 7 of 9 UL20/0468 From slimming aid 1 2 3 4 To slimming aid 1 1 0.04 0.02 0 2 0 0.55 0.89 0 3 0 0.30 0.04 0 4 0 0.11 0.05 1 Here, for example, there is a probability of 0.11 that a customer switches to slimming aid 4 from slimming aid 2. The current market shares are 9%, 78%, 3% and 10% for aids 1, 2, 3 and 4 respectively. (a) Copy the following table into your answer and fill in the long-run prediction for the market shares for these four slimming aids. (20 marks) Aid 1 Aid 2 Aid 3 Aid 4 Long-run (%, 2dp) (b) The appropriate regulatory authority in the country where these four slimming aids are marketed has been concerned about the claims made for weight loss. As a result of an investigation which they have made they intend to ban sliming aid 2 from the market in six months’ time for making false claims. Giving a clear explanation of your working, copy the following table into your answer and fill in the percentage of the market that will be affected by this ban when it is imposed in six months’ time. Value Percentage affected (2dp) (5 marks) 7. A company manufactures three products (A, B and C) using three machines (X, Y and Z). The time required (in minutes) to produce one unit of product on these machines is as shown below: Product A B C Machine X 1.1 1.8 2.3 Y 0.5 0.9 1.6 Z 3.3 2.7 4.3 For example producing one unit of product A on machine X requires 1.1 minutes. Units of A can also be produced on machines Y and Z: each unit of A produced on machine Y requires 0.5 minutes, each unit of A produced on machine Z requires 3.3 minutes. In the forthcoming production period it is estimated that there will be (at most) 1500 minutes available on machine X, 2700 minutes available on machine Y Page 8 of 9 UL20/0468 and 2400 minutes available on machine Z. These machines are very reliable, with only 2% of available time being lost due to machine breakdown. Technological constraints mean that for every 5 units of A produced at most 8 units of B can be produced. The company would like (if possible) to meet forecast demand, which for the forthcoming production period is 3950 units of A, 4400 units of B and 970 units of C. Should it not be possible to meet forecast demand then the most important product is A, with the second most important product being B and the third most important product being C. (a) Formulate this problem as a weighted goal program with linear constraints. Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give. (20 marks) (b) How does your formulation change if you use a sequential goal programming approach? Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give. (5 marks) 8. A company is planning a small project and the following table gives the various activities in that project, as well as their associated completion times. Activity Completion time (days) A 5 B 6 C 7 D 4 E 2 F 1 G 5 Here, for example, activity D takes 4 days to be completed. The immediate precedence relationships are: Activity Activity A must be finished before B can start B must be finished before C,E can start D must be finished before E can start E must be finished before F can start G must be finished before D,B can start In addition: • 4 days must elapse between the end of activity D and the start of activity C • 2 days must elapse between the end of activity F and the start of activity C Page 9 of 9 UL20/0468 (a) Draw the network diagram and calculate the overall project completion time. State the critical path(s). (12 marks) (b) Copy the following table into your answer and fill in the latest start times. Activity Latest start time (days) A B C D E F G (4 marks) (c) A colleague who has considered the same network as that given above believes that the project completion time is 27 days. They have also produced the following table of latest start times. Activity Latest start time (days) A 0 B 5 C 20 D 11 E 15 F 17 G 0 Your immediate superior believes that your colleague has made a mistake in their consideration of the network, namely either that they have used a wrong value for one of the activity completion times, or that they have not included one of the precedence relationships. Clearly explaining your reasoning what mistake (if any) has your colleague made? (9 marks) END OF PAPER Examiners’ commentaries 2020 1 Examiners’ commentaries 2020 MN2032 Management science methods Important note General remarks Learning outcomes At the end of this course and having completed the Essential reading and Activities, you should be able to: • to discuss the main techniques and problem-structuring methods used within management science • critically appraise the strengths and limitations of these techniques and problem-structuring methods • carry out simple exercises using such techniques and problem-structuring methods themselves (or explain how they should be done) • commission more advanced exercises. What the examiners are looking for You need to be clear that, aside from being able to display knowledge of the subject guide (both in qualitative and in quantitative questions), you are expected to apply that knowledge, perhaps to examination questions not encountered in precisely the same format before. This commentary reflects the examination and assessment arrangements for this course in the academic year 2019–2020. The format and structure of the examination may change in future years, and any such changes will be publicised on the virtual learning environment (VLE). Information about the subject guide and the Essential reading references Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2018). You should always attempt to use the most recent edition of any Essential reading textbook, even if the commentary and/or online reading list and/or subject guide refer to an earlier edition. If different editions of Essential reading are listed, please check the VLE for reading supplements – if none are available, please use the contents list and index of the new edition to find the relevant section. MN2032 Management science methods 2 Some questions may contain features that have not been directly examined previously. It is disappointing to see how many candidates are completely unable to cope with these features. You are reminded that success in an examination is not simply a matter of practising all previous examination questions in the hope that similar questions will recur. Rather, the examiners expect you to have a clear understanding of the principles underlying both quantitative and qualitative approaches and to be able to apply them appropriately. This may be in a question that is similar in form to a previous examination question, or it may be in a question that contains features that have not been encountered before. Some candidates appear to adopt the strategy for qualitative questions of simply memorising significant quantities of text from the subject guide in the hope that reproducing that text in the examination will lead to success. You are reminded that the examiners are looking for you to show understanding of the material presented in the subject guide. This may entail expressing concepts and ideas in your own words. With regard to qualitative questions in particular, learning an answer to a previously asked question and then repeating it in the hope that it will contain sufficient points to address an unfamiliar question, will not be a successful strategy. You need to focus on the question asked, not hope that an unfocused answer will be rewarded by the examiners. You should be clear that for all questions, but especially for quantitative questions, answers need to be supported by appropriate calculation/written reasoning. For example, a statement may be given by the examiners in the question and you have to agree or disagree with the statement. Candidates who give a correct answer, but without an appropriate reason, will not score highly. A question can be drawn from any topic in the subject guide. In particular, a question may mix both qualitative and quantitative topics. Examiners’ commentaries 2020 3 Examination revision strategy Many candidates are disappointed to find that their examination performance is poorer than they expected. This can be due to a number of different reasons and the Examiners’ commentaries suggest ways of addressing common problems and improving your performance. We want to draw your attention to one particular failing – ‘question spotting’, that is, confining your examination preparation to a few question topics which have come up in past papers for the course. This can have very serious consequences. We recognise that candidates may not cover all topics in the syllabus in the same depth, but you need to be aware that examiners are free to set questions on any aspect of the syllabus. This means that you need to study enough of the syllabus to enable you to answer the required number of examination questions. The syllabus can be found in the Course information sheet in the section of the VLE dedicated to this course. You should read the syllabus very carefully and ensure that you cover sufficient material in preparation for the examination. Examiners will vary the topics and questions from year to year and may well set questions that have not appeared in past papers – every topic on the syllabus is a legitimate examination target. So although past papers can be helpful in revision, you cannot assume that topics or specific questions that have come up in past examinations will occur again. If you rely on a question spotting strategy, it is likely you will find yourself in difficulties when you sit the examination paper. We strongly advise you not to adopt this strategy. MN2032 Management science methods 4 Examiners’ commentaries 2020 MN2032 Management science methods Important note Comments on specific questions Candidates should answer FOUR of the following EIGHT questions. All questions carry equal marks. Question 1 a) Apply both Journey Making and Strategic Choice to one problem with which you are familiar. (16 marks) b) Apply Soft Systems Methodology to one problem with which you are familiar (where this problem must be distinctly different from the problem considered in part (a) of the question). (9 marks) Reading for this question This question relates to Chapter 2 of the subject guide. This commentary reflects the examination and assessment arrangements for this course in the academic year 2019–2020. The format and structure of the examination may change in future years, and any such changes will be publicised on the virtual learning environment (VLE). Information about the subject guide and the Essential reading references Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2018). You should always attempt to use the most recent edition of any Essential reading textbook, even if the commentary and/or online reading list and/or subject guide refer to an earlier edition. If different editions of Essential reading are listed, please check the VLE for reading supplements – if none are available, please use the contents list and index of the new edition to find the relevant section. Examiners’ commentaries 2020 5 Approaching the question a) In a good answer, you would demonstrate to the examiners that you have followed (either exactly or in an appropriate other form), the model answer (for all parts of the question) given below. The examiners would expect you to give a clear definition of the problem considered. For Journey Making the examiners would expect you to give: • two individual maps and a merged map • discussion of the process. For Strategic Choice, the examiners would expect you to give the four modes of working for the example: • Shaping. • Designing. • Comparing. • Choosing. b) For Soft Systems Methodology (SSM) the examiners would expect you to give: • a clear statement (in words) of the problem considered • appropriate application of SSM to the problem: o root definition (statement of the ideal) o CATWOE for the root definition. • explicit and clear check of root definition by CATWOE • application of the stages. Note: for this question, no credit will be obtained for using the crime problem as given in the subject guide. Your answer to part (a) should be limited to six pages. Your answer to part (b) should limited to three pages. Question 2 Briefly discuss each of the following topics from the viewpoint of Operational Research/Management Science: a) Using simulation to estimate the effect on public bus transport of enforcing a one metre separation between individuals. (9 marks) b) Two specific and practical problem situations with which you are familiar where the assumptions underlying optimisation are false. (7 marks) c) Using Markov processes to model the progress of a new, and previously unknown, disease in a population. (9 marks) MN2032 Management science methods 6 Reading for this question Part (a) relates to Chapter 11 of the subject guide; part (b) relates to Chapter 1 of the subject guide and part (c) relates to Chapter 6 of the subject guide. Approaching the question In a good answer, you would demonstrate to the examiners that you have followed (either exactly or in an appropriate other form), the model answer (for all parts of the question) given below. a) Points that the examiners would expect to see here are: • using simulation in terms of MSM where it is a computer model of a real- world situation involving probabilities/statistics and entities queuing for service • investigating effect on capacity (e.g. in terms of passengers dealt with per hour): o on buses o at bus stops (where passengers are also separated) • statistics such as: o queue lengths o average waiting times. b) Points that the examiners would expect to see here are: • description of two specific and practical problems • discussion describing where/why a variety of the optimisation assumptions (as listed in the subject guide) are false with regard to these problems. c) Points that the examiners would expect to see here are: • states representing patients at particular stages of illness (e.g. new infection, old infection, recovered, dead) • transition probabilities between states representing progression of the disease e.g. probability go from new infection directly to death • timescale for transition probabilities (e.g. daily? weekly?) • updating of the probability information as new information about the disease and its progression emerges. Note: for this question, no credit will be obtained for repeating material which is given in the subject guide. Your answer to this question should be limited to two pages for each of the three parts (so at most six pages in total). Question 3 An organisation is reviewing the performance of its branches using data envelopment analysis. The data they have collected for these branches is shown below. Examiners’ commentaries 2020 7 Branch Number of customers (‘000) Customer satisfaction score Number of employees A 52.9 14.2 22 B 48.9 14.8 18 C 40.4 20.7 23 D 11.4 12.0 13 E 26.3 18.6 17 F 35.2 21.2 23 For example branch E last year employed 17 people and had 26,300 customers. The customer satisfaction score is a measure of how satisfied a customer is with the branch (arrived at by surveying a random sample of customers, approximately 1% of the number of customers a branch has). The higher the satisfaction score, the more satisfied customers are. Branch E, for example, had a satisfaction score of 18.6. a) Apply data envelopment analysis to compare the relative performance of these branches using the data shown above. Copy the table shown below into your answer and fill in the efficiencies and reference sets (where appropriate). (12 marks) Branch Efficiency Reference set A B C D E F b) In a discussion with a colleague as to your analysis they have noted that you have not included the fact that customer satisfaction is scored out of 25. They feel that this information should have been included in your analysis, and would have altered the results. What would you say to them? (2 marks) c) Another colleague has argued that customer satisfaction score is a wrong measure to use and you should use total satisfaction, defined as the current customer satisfaction score multiplied by the number of customers. What would you say to them? (5 marks) MN2032 Management science methods 8 d) The company CEO feels that only sampling approximately 1% of the customers in a branch to get a customer satisfaction score is insufficient and could result in some branches being shown to be efficient when in fact they are inefficient; or branches being classified as inefficient when they are actually efficient. In support of their argument they have produced the new customer satisfaction scores shown below based on a larger sample. What would you say to them? (6 marks) Branch Customer satisfaction score A 13.1 B 16.3 C 19.2 D 11.0 E 20.5 F 19.3 Reading for this question This question relates to Chapter 9 of the subject guide. Approaching the question In a good answer, you would demonstrate to the examiners that you have followed (either exactly or in an appropriate other form), the model answer (for all parts of the question) given below. Appropriate ratios are: • Customers per employee • Customer satisfaction per employee. So, treating customers and customer satisfaction as the outputs and the number of employees as the input, hence the ratios are: Branch Customers per employee Customer satisfaction per employee A 2.40 0.65 B 2.72 0.82 C 1.76 0.90 D 0.88 0.92 E 1.55 1.09 F 1.53 0.92 Then with respect to these ratios the relative efficiencies are: Examiners’ commentaries 2020 9 Branch Customers per employee Customer satisfaction per employee A 0.89 0.59 B 1.00 0.75 C 0.65 0.82 D 0.32 0.84 E 0.57 1.00 F 0.56 0.84 So here B is the best performer with respect to the first ratio, E is the best performer with respect to the second ratio. DEA diagram is: The required table is: Branch Efficiency Acceptable range Reference set lower upper A 0.89 0.87 0.91 B B 1 C 0.90 0.88 0.92 B,E D 0.84 0.82 0.86 E E 1 F 0.88 0.86 0.90 B,E MN2032 Management science methods 10 The acceptable range above is the range within which the examiners regards an answer as correct. That customer satisfaction is scored out of 25 is simply a scaling issue and would make no difference to the efficiencies and reference sets previously found. Redoing the analysis with the two output measures: • Number of customers. • Total satisfaction = current customer satisfaction score multiplied by the number of customers. Risks introducing an element of double counting as the number of customers is then directly affecting both output measures. The CEO argument is focused on ‘could result in some branches being shown to be efficient when in fact they are inefficient; or branches being classified as inefficient when they are actually efficient’. If we use these new customer satisfaction scores the customers per employee ratio is as before and the new customer satisfaction per employee ratio is as below: Branch Customer satisfaction per employee A 0.60 B 0.91 C 0.83 D 0.85 E 1.21 F 0.84 Examiners’ commentaries 2020 11 With the new DEA diagram being: Hence we can see that the efficient branches remain as before, and there are no new efficient branches (so the inefficient branches remain inefficient, although their efficiencies may have changed). Question 4 a) A company faces a decision as to whether to continue or not, with a particular product it has developed. It can cease development of the product, in which case all expenditure to date will be written off in the company accounts. Alternatively it can decide to test market the product in a small geographic area. Over the next financial year test marketing is estimated to cost 5 (£’million). If the market for the product when test marketed is favourable then the company will receive 7 (£’million) in revenue. However if the market is unfavourable it will only receive 2.5 (£’million). Alternatively the company could introduce the product to a wider market. Over the next financial year this is estimated to cost 15 (£’million). If the market for the product is favourable then the company will receive 23 (£’million) in revenue. However if the market is unfavourable it will only receive 11 (£’million). Assuming that the probability of the market being favourable is 0.40 represent this problem using a decision tree. What course of action for the company would you recommend based upon this decision tree? (12 marks) MN2032 Management science methods 12 b) Copy the table shown below into your answer and fill it in with respect to perfect information for the probability of the market being favourable. Value (£’million) Expected value with perfect information Expected value of perfect information (7 marks) c) What would the probability of the market being favourable have to be if the expected value of perfect information was one half of the expected value with perfect information? Note that if no probability exists which can achieve this you should clearly indicate why no such probability exists. (6 marks) Reading for this question This question relates to Chapter 4 of the subject guide. Approaching the question In a good answer, you would demonstrate to the examiners that you have followed (either exactly or in an appropriate other form), the model answer (for all parts of the question) given below. The decision tree is: Examiners’ commentaries 2020 13 The financial values (£’m) for A to E are: Terminal node Value (£’m) A 0 B 7-5 = 2 C 2.5-5 = -2.5 D 23-15 = 8 E 11-15 = -4 The EMV for the test market node is: 0.40(2)+0.60(-2.5) = -0.70 The EMV for the wider market node is: 0.40(8)+0.60(-4) = 0.80 Hence the recommended course of action is to introduce the product to a wide market with an EMV of £0.80m. We have the following (subject guide, p.79): Original probability EMV (initial decision node) if probability is one Market favourable 0.4 8 Market unfavourable 0.6 0 The required table therefore is: Value (£’million) Expected value with perfect information 0.4(8)+0.6(0) = 3.2 Expected value of perfect information 3.2-0.8 = 2.4 If the probability of the market being favourable is x then: The EMV for the test market node is: x(2)+(1-x)(-2.5) = 4.5x-2.5 The EMV for the wider market node is: x(8)+(1-x)(-4) = 12x-4 The EMV for the decision tree is max [0, 4.5x-2.5, 12x-4] Original probability EMV (initial decision node) if probability is one Market favourable x 8 Market unfavourable 1-x 0 MN2032 Management science methods 14 Value (£’million) Expected value with perfect information x(8)+(1-x)(0) = 8x Expected value of perfect information 8x-max[0, 4.5x-2.5, 12x-4] So, we require 8x-max [0, 4.5x-2.5, 12x-4] = 0.5(8x) Hence 4x = max [0, 4.5x-2.5, 12x-4] By inspection a solution to this is x=0, there is also another solution with x=0.5 Question 5 a) Solve the following linear program graphically using an iso-cost/iso-profit line, clearly indicating the feasible region. Maximise z = 8x + 6y Subject to 2x + 5y ≤ 27 Constraint 1 3x – 28y ≤ 19 x,y ≥ 0 Constraint 2 Copy the tables shown below into your answer and fill in the values required. (19 marks) Optimal value (4dp) Opportunity cost (4dp) x y z Shadow price (4dp) Constraint 1 Constraint 2 b) In this linear program the objective, z = 8x + 6y, represents the total profit made by a company from producing two liquid chemical products (chemical product X with a profit per litre of £8, chemical product Y with a profit per litre of £6). The company can either: • Increase the profit per litre on product X to £9; or Examiners’ commentaries 2020 15 • Increase the amount of resource available associated with the right- hand side of constraint 1 from its current value of 27 to a new value of 29. Based on the linear program given above which course of action would you recommend and why? Clearly explain the reasons for your recommendation. (6 marks) Reading for this question This question relates to Chapter 8 of the subject guide. Approaching the question In a good answer, you would demonstrate to the examiners that you have followed (either exactly or in an appropriate other form), the model answer (for all parts of the question) given below. The LP diagram is: with the optimal solution being at the intersection of the two constraints, namely: Optimal value (4dp) x 11.9859 y 0.6056 z 99.5208 MN2032 Management science methods 16 For both x and y the opportunity cost (reduced cost) is zero (allow undefined) since they are both already > 0, so we have: Opportunity cost (4dp) x 0 (allow undefined) y 0 (allow undefined) For constraint 1, the shadow price is given by changing the right-hand side by one (say to 28) and resolving. For constraint 2, the shadow price is given by changing the right-hand side by one (say to 20) and resolving. When this is done the resultant shadow prices are: Constraint Shadow price (4dp) 1 3.4085 2 0.3944 With the change in the objective function coefficient value for x the optimal solution remains as before (just a change in the slope of the iso-profit line), so we have: Optimal value (4dp) x 11.9859 y 0.6056 z 111.5067 For the change in the right-hand side we can make use of the shadow price for the first constraint of 3.4085, giving a profit increase of 6.8170 So, the new profit is 99.5208+6.8170 =106.3378 Hence recommend the course of action associated with increasing the profit per unit on product X as leads to the higher profit of 111.5067. Question 6 A company is using Markov theory to analyse switching between four different foods marketed as slimming aids. These slimming aids claim that with regular use they result in significant weight loss. Examiners’ commentaries 2020 17 Individual customers who buy these sliming aids only switch (on average) once every four months. Market research has produced the data shown below for the probability of customers switching between slimming aids. From slimming aid 1 2 3 4 To slimming aid 1 1 0.04 0.02 0 2 0 0.55 0.89 0 3 0 0.30 0.04 0 4 0 0.11 0.05 1 Here, for example, there is a probability of 0.11 that a customer switches to slimming aid 4 from slimming aid 2. The current market shares are 9%, 78%, 3% and 10% for aids 1, 2, 3 and 4 respectively. a) Copy the following table into your answer and fill in the long-run prediction for the market shares for these four slimming aids. (20 marks) Aid 1 Aid 2 Aid 3 Aid 4 Long-run (%, 2dp) b) The appropriate regulatory authority in the country where these four slimming aids are marketed has been concerned about the claims made for weight loss. As a result of an investigation which they have made they intend to ban sliming aid 2 from the market in six months’ time for making false claims. Giving a clear explanation of your working, copy the following table into your answer and fill in the percentage of the market that will be affected by this ban when it is imposed in six months’ time. Value Percentage affected (2dp) (5 marks) Reading for this question This question relates to Chapter 6 of the subject guide. Approaching the question In a good answer, you would demonstrate to the examiners that you have followed (either exactly or in an appropriate other form), the model answer (for all parts of the question) given below. This is a Markov process question with absorbing states. The transition matrix given in the question must be transformed into an appropriate form: MN2032 Management science methods 18 • as a [From,To] matrix • with the first two states as the absorbing states. Once this is done the transition matrix and initial market shares are: To slimming aid 1 4 3 2 From slimming aid 1 1 0 0 0 4 0 1 0 0 3 0.02 0.05 0.04 0.89 2 0.04 0.11 0.30 0.55 Initial 0.09 0.10 0.03 0.78 The details of the long-run calculation (see pages 108-110 of the subject guide) are: R 0.0200 0.0500 0.0400 0.1100 Q 0.0400 0.8900 0.3000 0.5500 I-Q 0.9600 -0.8900 -0.3000 0.4500 d 0.1650 (1/d= 6.0606 ) inverse 2.7273 5.3939 1.8182 5.8182 NR 0.2703 0.7297 0.2691 0.7309 The long-run market shares are: Examiners’ commentaries 2020 19 The ban after six months means that we will assume that only one transition will have occurred (namely after four months). So we need the state of the system (specifically for slimming aid 2) after one transition and this is given by the matrix multiplication [0.09, 0.10, 0.03, 0.78][P], so that we have: Question 7 A company manufactures three products (A, B and C) using three machines (X, Y and Z). The time required (in minutes) to produce one unit of product on these machines is as shown below: Product A B C Machine X 1.1 1.8 2.3 Y 0.5 0.9 1.6 Z 3.3 2.7 4.3 For example producing one unit of product A on machine X requires 1.1 minutes. Units of A can also be produced on machines Y and Z: each unit of A produced on machine Y requires 0.5 minutes, each unit of A produced on machine Z requires 3.3 minutes. In the forthcoming production period it is estimated that there will be (at most) 1500 minutes available on machine X, 2700 minutes available on machine Y and 2400 minutes available on machine Z. These machines are very reliable, with only 2% of available time being lost due to machine breakdown. Technological constraints mean that for every 5 units of A produced at most 8 units of B can be produced. The company would like (if possible) to meet forecast demand, which for the forthcoming production period is 3950 units of A, 4400 units of B and 970 units of C. Should it not be possible to meet forecast demand then the most important product is A, with the second most important product being B and the third most important product being C. Aid 1 Aid 2 Aid 3 Aid 4 Long-run (%, 2dp) 30.80 0 0 69.20 Value Percentage affected (2dp) 45.57 MN2032 Management science methods 20 a) Formulate this problem as a weighted goal program with linear constraints. Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give. (20 marks) b) How does your formulation change if you use a sequential goal programming approach? Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give. (5 marks) Reading for this question This question relates to Chapter 10 of the subject guide. Approaching the question In a good answer, you would demonstrate to the examiners that you have followed (either exactly or in an appropriate other form), the model answer (for all parts of the question) given below. Let x(i,j) be the number of units of product i (i=A,B,C) produced on machine j (j=X,Y,Z) The production time constraint is: 1.1x(A,X) + 1.8x(B,X) + 2.3x(C,X) ≤ 0.98(1500) = 1470 m/c X 0.5x(A,Y) + 0.9x(B,Y) + 1.6x(C,Y) ≤ 0.98(2700) = 2646 m/c Y 3.3x(A,Z) + 2.7x(B,Z) + 4.3x(C,Z) ≤ 0.98(2400) = 2352 m/c Z The technological constraint is: x(B,X) + x(B,Y) +x(B,Z) ≤ (8/5)[x(A,X) + x(A,Y) + x(A,Z)] For the demand goal we define upward deviation and downward deviation variables in this goal programming formulation. Let A+, B+ and C+ be the upward deviation variables for products A, B and C respectively (where all of these variables are ≥ 0) Let A-, B- and C- be the downward deviation variables for products A, B and C respectively (where all of these variables are ≥ 0) Note: we need the upward deviation variables here since the technological constraint between products A and B may mean we over-produce one product to satisfy the demand for the other product. The demand constraints are then: x(A,X) + x(A,Y) + x(A,Z) = 3950 + A+ - A- x(B,X) + x(B,Y) + x(B,Z) = 4400 + B+ - B- x(C,X) + x(C,Y) + x(C,Z) = 970 + C+ - C- Examiners’ commentaries 2020 21 Here we have three goals relating to the demand for each of the three products. Variable Current goal Weight for one per cent deviation from this goal One percent of goal A+ 3950 w1 39.5 A- 3950 w2 39.5 B+ 4400 w3 44 B- 4400 w4 44 C+ 970 w5 9.7 C- 970 w6 9.7 To satisfy the wording of the question ‘the most important product is A, with the second most important product being B and the third most important product being C.’ we need: weights for A >> the weights for B >> the weights for C, so w1 and w2 are much greater than w3 and w4 which is turn are much greater then then w5 and w6. The logic for these restrictions on the weights is that since (below) we are minimising a weighted sum of deviation variables then with all weights >0 we aim (if possible) to satisfy demand exactly. If we cannot meet demand exactly then we will minimise the deviation from demand, giving most weight to A, then to B, then to C. Objective here is to minimise a weighted sum of deviation variables, so minimise: w1(A+/39.5) + w2(A-/39.5) + w3(B+/44) + w4(B-/44) + w5(C+/9.7) + w6(C-/9.7) If we have a sequential goal programming (priority) approach the first program to solve is: Minimise A+ + A- subject to all the constraints seen above To minimise the total deviation from the product A goal Let the solution to this be A* (say) The second program to solve is then Minimise B+ + B- subject to all the constraints seen above and A+ + A- = A* Let the solution to this be B* (say) The third program to solve is then Minimise C+ + C- MN2032 Management science methods 22 subject to all the constraints seen above and A+ + A- = A* and B+ + B-= B* Question 8 A company is planning a small project and the following table gives the various activities in that project, as well as their associated completion times. Activity Completion time (days) A 5 B 6 C 7 D 4 E 2 F 1 G 5 Here, for example, activity D takes 4 days to be completed. The immediate precedence relationships are: In addition: • 4 days must elapse between the end of activity D and the start of activity C • 2 days must elapse between the end of activity F and the start of activity C a) Draw the network diagram and calculate the overall project completion time. State the critical path(s). (12 marks) b) Copy the following table into your answer and fill in the latest start times. (4 marks) Activity Activity A must be finished before B can start B must be finished before C,E can start D must be finished before E can start E must be finished before F can start G must be finished before D,B can start Activity Latest start time (days) A B C D E F G Examiners’ commentaries 2020 23 c) A colleague who has considered the same network as that given above believes that the project completion time is 27 days. They have also produced the following table of latest start times. Your immediate superior believes that your colleague has made a mistake in their consideration of the network, namely either that they have used a wrong value for one of the activity completion times, or that they have not included one of the precedence relationships. Clearly explaining your reasoning what mistake (if any) has your colleague made? (9 marks) Reading for this question This question relates to Chapter 3 of the subject guide. Approaching the question In a good answer, you would demonstrate to the examiners that you have followed (either exactly or in an appropriate other form), the model answer (for all parts of the question) given below. The network diagram is: Note: we do not need an End node as for this particular network as we know activity C will be the final activity. Activity Latest start time (days) A 0 B 5 C 20 D 11 E 15 F 17 G 0 MN2032 Management science methods 24 The detailed calculation is: Activity Early Start Late Start Slack A 0 0 0 B 5 5 0 C 16 16 0 D 5 7 2 E 11 11 0 F 13 13 0 G 0 0 0 H 9 12 3 I 14 14 0 • the project duration is 23 days • two distinct critical paths: o A-B-E-F-I(wait 2)-C o G-B-E-F-I(wait 2)-C The required table is: Activity Latest start time (days) A 0 B 5 C 16 D 7 E 11 F 13 G 0 There are only two possible mistakes specified in the question: • wrong value for one of the activity completion times; or • they have not included one of the precedence relationships. Since the colleague’s project completion time of 27 is > the actual project completion time of 23 we can immediately reject the second of these two errors, since not including one of the precedence relationships could only reduce (or leave unchanged) the project completion time. Comparing the correct latest start times and the colleague latest start times we have: Examiners’ commentaries 2020 25 Activity Correct latest start time Colleague latest start time A 0 0 B 5 5 C 16 20 D 7 11 E 11 15 F 13 17 G 0 0 So, the change (as highlighted above) is an increase of 4 associated with C, D, E, F. Examining the project network it is clear that such an increase can come from an increase of 4 in the completion time for B, so this is the error. The colleague has used a completion time for B of 10.
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