Examiners’ commentaries 20171Examiners’ commentaries 2017MN3032 Management science methodsImportant noteThis commentary reflects the examination and assessment arrangementsfor this course in the academic year 2016–17. The format and structureof the examination may change in future years, and any such changeswill be publicised on the virtual learning environment (VLE).Information about the subject guide and the Essential readingreferencesUnless otherwise stated, all cross-references will be to the latest versionof the subject guide (2014). You should always attempt to use the mostrecent edition of any Essential reading textbook, even if the commentaryand/or online reading list and/or subject guide refers to an earlieredition. If different editions of Essential reading are listed, please checkthe VLE for reading supplements – if none are available, please use thecontents list and index of the new edition to find the relevant section.General remarksLearning outcomesAt the end of the course, and having completed the Essential reading andactivities, you should be able to:• discuss the main techniques and problem-structuring methods usedwithin management science• critically appraise the strengths and limitations of these techniques andproblem-structuring methods• carry out simple exercises using such techniques and problemstructuring methods themselves (or explain how they should be done)• commission more advanced exercises.Planning your time in the examinationYou are strongly advised to plan and think through your approach tothe order (sequence) in which you answer your selected questions.Candidates, especially less confident ones, can often gain ‘easier’ marksby attempting (albeit incorrectly) a numeric calculation rather thananswering an essay-type question. In general, therefore, you are advised toanswer predominantly quantitative questions before questions that arepredominantly qualitative in nature. In addition, it is logical for you toattempt ‘stronger’ questions first (ones where you are more confident thatyou can do well).The advice given to you in the Programme handbook applies here:‘Examiners who mark these papers are instructed to mark the first answersthat they get to in the answer booklet which meet the required number ofquestions. If you do extra, they will not mark all questions attempted andMN3032 Management science methods2then select the best marks.’ Candidates who may have attempted morethan the required number of questions need to clearly cross out questionsthat they do not wish to have marked by use of a line through the entiretext. Simply crossing out the question number at the head of a page oftext, for example, or omitting the question number from the questionslisted on the front of the examination answer booklet, is not sufficient.What are the examiners looking for?You need to be clear that, aside from being able to display knowledge ofthe subject guide (both in qualitative and in quantitative questions), youare expected to be able to apply that knowledge, perhaps to examinationquestions not encountered in precisely the same format before.Some questions may contain features that have not been directly examinedpreviously. It is disappointing to see how many candidates are completelyunable to cope with these features. You are reminded that success in anexamination is not simply a matter of practising all previous examinationquestions in the hope that similar questions will reoccur. Rather, theexaminers expect you to have a clear understanding of the principlesunderlying both quantitative and qualitative approaches and to be able toapply them appropriately. This may be in a question that is similar in formto a previous examination question or it may be in a question that containsfeatures that have not been encountered before.Some candidates appear to adopt the strategy for qualitative questionsof simply memorising significant quantities of text from the subject guidein the hope that reproducing that text in the examination will lead tosuccess. You are reminded that the examiners are looking for you to showunderstanding of the material presented in the subject guide. This mayentail expressing concepts and ideas in your own words.With regard to qualitative questions in particular, learning an answer toa previously asked question and then repeating it in the hope that it willcontain sufficient points to address an unfamiliar question will not be asuccessful strategy. You need to focus on the question asked, not hope thatan unfocused answer will be rewarded by the examiners.You should be clear that, for all questions, but especially for quantitativequestions, answers need to be supported by appropriate calculation/written reasoning. For example, a statement may be given by theexaminers in the question and you have to agree or disagree withthe statement. Candidates who give a correct answer, but without anappropriate reason, will not score highly.You may be aware that while previously, the examination was split intoa qualitative section and a quantitative section, this format was droppedin the 2013 examination. You need to be clear that this lack of sectionsmeans that a question can be drawn from any topic in the subject guide. Inparticular, a question may mix both qualitative and quantitative topics.Examination structureFor the examination in 2018, the structure of the examination paper willbe the same as it was in the 2017 examination. The 2018 examinationpaper will contain eight questions and candidates will have to answer anyfour of them. One of these eight questions will be a short notes questionthat is predominantly qualitative in nature. This will contain three topicsand will ask candidates to ‘Briefly explain each of the following topicsfrom the viewpoint of Operational Research/Management Science.’Examiners’ commentaries 20173Examination revision strategyMany candidates are disappointed to find that their examinationperformance poorer than they expected. This may be due to anumber of reasons. The Examiners’ commentaries suggest ways ofaddressing common problems and improving your performance.One particular failing is ‘question spotting’, that is, confining yourexamination preparation to a few questions and/or topics whichhave come up in past papers for the course. This can have seriousconsequences.We recognise that candidates may not cover all topics in the syllabusin the same depth; but you need to be aware that examiners are freeto set questions on any aspect of the syllabus. This means that youneed to study enough of the syllabus to enable you to answer therequired number of examination questions.The syllabus can be found in the Course information sheet in thesection of the VLE dedicated to each course. You should read thesyllabus carefully and ensure that you cover sufficient material inpreparation for the examination. Examiners will vary the topics andquestions from year to year and may well set questions that have notappeared in past papers. Examination papers may legitimately includequestions on any topic in the syllabus. So, although past papers can behelpful during your revision, you cannot assume that topics or specificquestions that have come up in past examinations will occur again.If you rely on a question-spotting strategy, it is likelyyou will find yourself in difficulties when you sit theexamination. We strongly advise you not to adopt thisstrategy.MN3032 Management science methods4Examiners’ commentaries 2017MN3032 Management science methods – Zone AImportant noteThis commentary reflects the examination and assessment arrangementsfor this course in the academic year 2016–17. The format and structureof the examination may change in future years, and any such changeswill be publicised on the virtual learning environment (VLE).Information about the subject guide and the Essential readingreferencesUnless otherwise stated, all cross-references will be to the latest versionof the subject guide (2014). You should always attempt to use the mostrecent edition of any Essential reading textbook, even if the commentaryand/or online reading list and/or subject guide refer to an earlieredition. If different editions of Essential reading are listed, please checkthe VLE for reading supplements–if none are available, please use thecontents list and index of the new edition to find the relevant section.Comments on specific questionsQuestion 1(a) Clearly describe Soft Systems Methodology (SSM), making clear what the assumptions that underlie SSM are.(15 marks)(b) Illustrate Soft Systems Methodology by applying it to any TWO exampleproblems of which you are aware.Reading for this question(10 marks) This question relates to Chapter 2 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.The examiners would expect you to mention the SSM assumptions.• Different individuals and different groups make different evaluationsof events and this leads to their taking different actions.• Concepts and ideas from systems engineering are useful.• It is necessary when describing any human activity system to takeaccount of the particular image of the world underlying the descriptionof the system and it is further necessary to be explicit about theassumptions underlying this image.• It is possible to learn about a system by comparing pure models ofthat system with perceptions of what is happening in the real worldproblem situation.Examiners’ commentaries 20175The examiners would also expect you to mention:• CATWOE – Customers, Actors, Transformation (or Transformationprocess), Worldview (or Weltanschauung), Owner, Environmentalconstraints• root definition, giving a clear indication that it is a statement of theideal• the link between CATWOE and root definition (deduce from the rootdefinition answer to CATWOE).In addition the examiners would expect you to discuss the SSM stages.For each example problem, the examiners would expect:• a clear statement (in words) of the problem considered, not just asingle phrase• appropriate application of SSM to the problem, includingroot definition (statement of the ideal)CATWOE for the root definitionexplicit and clear check of root definition by CATWOEapplication of the stages.Question 2Briefly explain each of the following topics from the viewpoint of OperationalResearch/Management Science: (a) M/D/3/35/250(b) Benefits of simulating passenger check-in at an airport(c) OPTReading for this questionThis chapter relates to Chapters 5 and 11 of the subject guide.Approaching the question(9 marks)(9 marks)(7 marks) In a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.For part (a)This part relates to Kendall’s notational system for classifying queueingsystems, referenced at page 197 of the subject guide. Points that theexaminers would expect to see here are:• notation (not model) for a queuing system• devised by Kendal• M = Poisson arrival distribution or exponential (negative exponential)interarrival time• D = deterministic (constant) service time• 3 = 3 servers (channels)• 35 = maximum number of customers allowed in the queueing system(either being served or waiting for service)• 250 = maximum number of customers in total (population).For part (b)This part touches on simulation and relates to Chapter 11 of the subjectguide. The examiners would expect a good answer to contain thefollowing points:MN3032 Management science methods6• reference to using simulation in terms of MSM where a computermodel of a real world situation is being presented, involvingprobabilities/statistics and entities queuing for service• description of generic benefits of simulation, namely, its potential forexploring options/alternatives without the hassle/expense of changingthe real world• presentation of statistics showing data such as numbers served perminutes, variation (e.g. standard of deviation) in numbers served perminutes, queue lengths–how many people will be queuing, etc.)• consideration of how these factors might change where, for example,we change the number of or variety of check-in options at airports(such as by adopting self-service check-ins, special queues for singlepassengers/family groups, different check-in layouts, single and/ormultiple queues).For part (c)This part deals with OPT as referenced at page 108 of the subject guide.The examiners would expect a good answer to contain the followingingredients:• reference to fact that OPT = Optimised Production Technology• idea that constraints (bottlenecks) must be identified and eliminated• recognition that flow should be able to accommodate bottlenecks thatdo exist• simple example (e.g. 3 machines in a production line) to illustrate theconcept.Question 3(a) The payoff table below shows for a company the final year end profitsexpected (£m) from four possible choices (A, B, C and D) with respect to threepossible scenarios (S1 to S3 respectively) for the forthcoming year. S1S2S3A211-5B41314C5-513D1218-10 Here, for example, if the company makes choice D (and it can only choose oneof A, B, C or D) and the scenario for the forthcoming year turns out to be S3then it makes a loss of £10m.Copy the following table into your answer book and fill in the choice thatshould be made under each criterion and the associated value. CriteriaChoiceAssociated valueOptimisticConservativeRegretEqually likely Based on your analysis which choice would you recommend the companymakes? (10 marks)(b) What is the minimum value for the payoff associated with B and scenarioS1 such that under the Regret criterion you choose B? Clearly explain thesteps you take in arriving at your answer (but without enumerating differentvalues for this payoff). (11 marks)Examiners’ commentaries 20177(c) For the situation as described in (a) above discuss the use you might makeof sensitivity analysis to generate further insight into the decision as to the choice the company should make.Reading for this question(4 marks) This question relates to Chapter 4 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.The situation is:Scenario PayoffChoiceS1S2S3Equally likelyA211-52.67B4131410.33C5-5134.33D1218-106.67 Regret ChoiceS1S2S3A10719B850C7231DThe required table is0024 CriteriaChoiceAssociated valueOptimisticD18ConservativeB4RegretB8Equally likelyB10.33 So, here B is the recommended choice.Letting the value for B-S1 be x, then the payoff matrix is: S1S2S3A211-5BX1314C5-513D1218-10 To calculate regret, we need the maximum in each column; so, here we donot know the maximum in the S1 column, as it may be x or it may be12.For the payoff matrix in the first part of the question (which includes apayoff of 4 associated with B-S1), we, under regret, choose B. Hence, theminimum value ‘such that under the Regret criterion you chooseB’ must be less than (or equal to) 4; so, the payoff B-S1 may decrease (orremain the same). In other words, we now know that x≤4.If x≤4, then the maximum in the S1 column is 12Given that we now know that the maximum in the S1 column is 12, weare justified in writing the regret matrix as:MN3032 Management science methods8Regret ChoiceS1S2S3A10719B12-x50C7231D0024The maximum regrets are:A19Bmax (12-x,5,0) = max (12-x,5)C23D24 We choose B if it minimises the maximum regret, i.e. if max (12-x,5) ≤ 19The minimum value of x which satisfies this inequality is x=-7Hence, the minimum value for the payoff such that we choose B underregret is -7Technically, when x=-7 the maximum regrets are:A 19B max(12-x,5,0)=max(12-x,5)=19C 23D 24So, here we would be indifferent as between A and B. If x is slightly lessthan -7, say 6.9999, then B would be the unique choice under regret.With respect to sensitivity to give insight into the recommended choice,points that the examiners would expect you to mention are:• investigating changes in the financial calculations that led to theprofit/payoff values, thereby producing a wider range of profit/payoffvalues• investigating changes in the calculations that led to the threescenarios, thereby producing a greater number of scenarios• using an algebraic approach (i.e. replacing a number by a symbol).Question 4A company is considering using Markov theory to analyse customers switchingbetween four internet sites that are used for downloading videos. These fourinternet sites trade under the names ‘Vdown’, ‘V2down’, ‘Watch’ and ‘Relax’. Ananalysis of historic data has produced the transition matrix shown below for theprobability of a customer switching at each download between these sites. Onaverage customers download a video once every week. ToVdownV2downWatchRelaxFromVdown0.150.180.210.46V2down0.130.150.180.54Watch0.200.120.020.66Relax0.250.230.290.23 In this transition matrix, for example, there is a probability of 0.66 of a customerswitching from using Watch to using Relax when they come to download a newvideo. The current market shares for these four sites are 10%, 20%, 30% and40% for Vdown, V2down, Watch and Relax, respectively.Examiners’ commentaries 20179(a) What do you think the market share for Relax will be in two weeks’ time?(3 marks)If after two weeks severe financial problems mean that Relax stops trading,the customers who are then currently using Relax will be split equallybetween the three remaining download sites.The new transition matrix after Relax stops trading is: ToVdownV2downWatchFromVdown0.020.450.53V2down0.690.220.09Watch0.360.380.26 (b) What will be the market share for each site one week after Relax has stopped trading?(3 marks)(c) What is the long-run prediction for the market shares for each of the threeremaining download sites?Reading for this question:(19 marks) This question relates to Chapter 6 of the subject guide.Approaching the question:In a good answer, you would demonstrate to the examiners that youhave followed (either exactly or in an appropriate alternative form) themodel answer (for all parts of the question) given below.Applying the standard Markov approach, we have the state of the systemas below. Initial0.10.20.30.410.20100.17600.17900.444020.19980.18620.20620.4078 Market share for Relax in two weeks time is 0.4078If Relax then stops trading, the 0.4078 is split equally between the otherthree sites; so, the new system state (for the future) is 0.1998+0.4078/3for the first state and similarly for states 2 and 3. This is:0.3358 0.3221 0.3422With the new transition matrix, and this as the new initial state, the stateof the system after one week is:0.3521 0.3520 0.2959For the long-run prediction, we need to set up [x1,x2,x3] where[x1,x2,x3] = [x1,x2,x3](transition matrix) and x1 + x2 + x3 = 1.Expanding we get:x1 = 0.02×1 + 0.69×2 + 0.36×3×2 = 0.45×1 + 0.22×2 + 0.38×3×3 = 0.53×1 + 0.09×2 + 0.26×3×1 + x2 + x3 = 1and solving these equations simultaneouslyx1 = 0.3546×2 = 0.3490×3 = 0.2964MN3032 Management science methods10Question 5A company is planning a small project and the following table gives the variousactivities in that project, as well as their associated completion times. ActivityCompletion time (days)A4B4C3D5E1F3G3 Here, for example, activity G takes 3 days to be completed.The immediate precedence relationships are: ActivityActivityAmust be finished beforeBcan startAmust be finished beforeGcan startBmust be finished beforeGcan startCmust be finished beforeDcan startCmust be finished beforeEcan startFmust be finished beforeAcan startFmust be finished beforeCcan start In addition two other conditions (X1 and X2) must apply• Condition X1: 4 days must elapse between the end of activity C and the startof activity B• Condition X2: 2 days must elapse between the end of activity A and the startof activity E(a) Draw the network diagram and calculate the overall project completion time.State the critical path(s). (9 marks)(b) Copy the following table into your answer book and fill in the latest starttimes and float (slack) times. ActivityLatest start time (days)Float (slack) time (days)ABCDEFG (6 marks)(c) A colleague has asked you to clearly explain to them what is meant by thefollowing terms in relation to project planning:• earliest start time• latest start time• float (slack) time• beta distribution• resource smoothingWhat would you say to them? (10 marks)Examiners’ commentaries 201711Reading 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 havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.The network diagram can be seen below.Details of the calculation are: ActivityEarliestStartLatestStart FloatA 3 6 3B 10 10 0C 3 3 0D6126E9167F000G14140X1660X2 7 14 7 • completion time 17 days• critical path is F→C→X1→B→GThe required table is: ActivityLatest start time (days)Float (slack) time (days)A63B100C30D126E167F00G140 For definitions of the three different time concepts, the examiners wouldexpect you to mention the points below.• Earliest start time for any activity is the earliest time we can start theactivity such that all its (immediately) preceding activities have beenfinished.• Latest start time for any activity is the latest time we can startthe activity and still complete the project in the minimum overallcompletion timeMN3032 Management science methods12• Float (slack) time for an activity is the amount by which we canincrease the time taken to complete the activity without changing(increasing) the overall project completion time.For the beta distribution, the examiners would expect you to mention thepoints below.• It is the probability distribution for activity completion time.• Its distribution is defined by three time estimates (optimistic, mostlikely and pessimistic).• The shape of the distribution (asymmetric) is as below.For resource smoothing the examiners would expect you to mention:• resource profile not fixed• resource profile can be changed (without affecting the overall projectcompletion time) by moving the start time of non-critical activities• ideal resource profile is a constant usage of resource over time.Question 6A company makes three products (X, Y and Z) and has a number of optionsavailable as to how these products can be produced on the three machines (A, Band C) it has available. These options are tabulated below. ProductOptionTime required (minutes)ABCXA→B→C1067C→B415A→C125YC→B→A987B→C142ZA→B→C1079C→A215 This table shows for each option the machines used (and their order) as wellas the processing time in minutes required on each machine. For example theoption B→C for producing Y requires 14 minutes of processing on machineB followed by 2 minutes of processing on machine C to produce one unit ofproduct Y. The company can adopt more than one option for production of anyproduct.Examiners’ commentaries 201713The company is planning its production for the next week. Available time inminutes on machines A, B and C is 2400, 2350 and 2000 respectively. By movingworkers between machines it is possible to increase working time at anymachine, but at the expense of reducing working time at another machine. Eachminute gained on one machine reduces the time available on the machine fromwhich it is taken by 2 minutes.Products X, Y and Z sell to customers for £95, £80 and £75 per unit (respectively).(a) Formulate the problem of deciding the number of units of each product to make as an integer program with linear constraints.(20 marks)(b) What might be the principal difficulty in implementing the numeric solutionfrom this integer program? Illustrate this difficulty by example using the dataprovided in the question.(5 marks) Reading for this question:This question relates to Chapter 7 of the subject guide.Approaching the question:In a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.Let x(i) (≥0, integer) be the number of units of product X produced usingoption i (i=1,2,3) where the options are numbered as in their order inthe question, for example, option 1 for X is A→B→C and option 2 for X isC→B.Similarly, let y(i) (≥0 integer) (i=1,2) and z(i) (≥0 integer) (i=1,2) bethe number of units of products Y and Z produced.As we can move time between machines, we need t(A,B), for example, forthe time in minutes moved from machine A to machine B (≥0, integer).Other time variables (e.g. t(A,C), t(B,A)) also exist.For machine A, we have the calculations below.• 10x(1)+12x(3)+9y(1)+10z(1)+21z(2) is the time used.• t(B,A)+t(C,A) is the time gained by shifting time from other machinesto A.• 2t(A,B)+2t(A,C) is the time lost by shifting time to other machinesfrom A.So, the machine A time capacity constraint is:[10x(1)+12x(3)+9y(1)+10z(1)+21z(2)] + [t(B,A)+t(C,A)] –[2t(A,B)+2t(A,C)] ≤ 2400In a similar fashion, the machine B time capacity constraint is:[6x(1)+4x(2)+8y(1)+14y(2)+7z(1)] + [t(A,B)+t(C,B)] –[2t(B,A)+2t(B,C)] ≤ 2350In a similar fashion, the machine C time capacity constraint is:[7x(1)+15x(2)+5x(3)+7y(1)+2y(2)+9z(1)+5z(2)] + [t(A,C)+t(B,C)] –[2t(C,A)+2t(C,B)] ≤ 2000The objective is:Maximise 95[x(1)+x(2)+x(3)] + 80[y(1)+y(2)] + 75[z(1)+z(2)]The principal difficulty relates to the scheduling of the products onmachines and potential queueing/wasted time.MN3032 Management science methods14Suppose we decide to produce one unit of X using option 1 and one unit ofZ using option 1.Then both X and Z have the same route A→B→C but with differentprocessing times. If we schedule these in the order, first X followed byZ, then X will take 10 minutes on A, then start on B, finishing after 6minutes. So, B is free after 6 minutes, but the unit of Z which startedprocessing on A when X finished still has 4 minutes left before it can starton B; so, we have free time on B that cannot yet be utilised. This wastedtime on B is not accounted for in our formulation.Question 7(a) Describe the phases associated with an Operational Research project.(15 marks)(b) Discuss how you might ensure that an Operational Research project is successfully implemented.Reading for this question:(10 marks) This question relates to Chapter 1 of the subject guide.Approaching the question:In a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.For part (a)The examiners would expect you to mention the phases in an OR project,as outlined below:Phase 1: Problem identification• where not obvious, diagnosis of the problem from its symptoms• delineation of the subproblem to be studied• establishment of objectives, limitations and requirements.Phase 2: Formulation as a mathematical model• choice as to the appropriate model as a factor crucial to success• ‘data barrier’ and data-rich/data-poor environments.Phase 3: Model validation (or algorithm validation)• model validation, running the algorithm for the model on thecomputer in order to ensure:the input data/computer program is free from errorsthe computer program correctly represents the modelthe results from the algorithm seem reasonable.Phase 4: Solution of the model• use of a standard computer package, or specially developed algorithm• design of different numeric scenarios to establish sensitivity• exploitation of processing speed (turn-around time) and interactive/user-friendly nature of many PC software packages.Phase 5: Implementation• implementation of the results of the study, or• implementation of the algorithm for solving the model as anoperational tool.Examiners’ commentaries 201715For part (b)The examiners would expect you to mention:• ensuring, when formulating the problem, that all the relevanttechnological and cultural constraints are known and, whereappropriate, included in the model• undertaking continuous contact with the organisation and, specifically,the client in the organisation (the person/people who has/have theproblem). Although the quality of the work should never surpriseclients, such regular and positive communication may do so in that itmay not fit in with clients’ previously held ideas of what the solutionswould be.• conveying the solution–and justification for the solution–in plainlanguage, using terms the client will understand• ensuring that the action recommended in the solution is within thepower of the client.Question 8You are working for one company and have been approached by a rival companywho wish you to come and work for them. You have three objectives in choosingwhether to stay with your current company or not:• Objective 1 – salary – the larger the salary the better• Objective 2 – earnings potential – the more you can earn over time the better• Objective 3 – vacation time – the more vacation time you are allowed thebetterYour pairwise comparison matrix for these objectives is: Objective123Objective11452–133––1 Here, for example, your judgement as to objective 1 as compared with objective3 is rated 5.(a) Are the judgements made with respect to these objectives reasonablyconsistent or not (assume the value of RI for n=3 is 0.58)? (13 marks)Currently your judgement (pairwise comparison matrices) of the relativeworth of your current employer (company A) as compared to your potentialnew employer (company B) with respect to the above three objectives is: ABObjective 1A17B–1 ABObjective 2A1–B51 ABObjective 3A13B–1 (b) Assuming that the judgements made with respect to the three objectives arereasonably consistent which company should you choose to work for?(5 marks)MN3032 Management science methods16(c) Discuss how you might apply MAUA/MAVA to the situation considered above relating to choosing which company to work for.Reading for this question:(7 marks) This question relates to Chapter 10 of the subject guide.Approaching the question:In a good answer, you would demonstrate to the examiners that youhave followed (either exactly or in an appropriate alternative form) themodel answer (for all parts of the question) given below.The details of the calculation are:Comparison matrixObjective123Objective1145 2 0.2500 1 33 0.2000 0.3333 1Column sum 1.4500 5.3333 9.0000Normalisedmatrix 1 2 3 Row average1 0.6897 0.7500 0.5556 0.66512 0.1724 0.1875 0.3333 0.23113 0.1379 0.0625 0.1111 0.1038Weights 0.6651 0.2311 0.1038Columnvector 2.10850.70880.3138lmax 3.0868 CI0.0434RI0.5800CI/RI0.0748As the ratio CI/RI is less than 0.1, the judgements are consistent.For objective 1, we have:ComparisonmatrixJob offerABA17Job offerBColumn sum0.14291.142918.0000 Normalised matrixABRow average0.8750A0.87500.8750B0.12500.12500.1250 Examiners’ commentaries 201717For objectives 2 and 3, the normalised matrices are:Normalised matrixA0.16670.8333B Row averageAB0.16670.83330.16670.8333 Normalised matrixA0.75000.2500B Row averageAB0.75000.25000.75000.2500which means that the calculation for the job offers is:ObjectiveAB10.66510.87500.1250 2 0.2311 0.1667 0.83333 0.1038 0.7500 0.2500 ScoresSo, the choice is company A.0.69830.3017 MAUA/MAVA relates to the subject guide, at pages 190-191. Theexaminers would expect a good answer to contain the following points:1. evaluation of each alternative (job offer) with respect to eachdimension (objective)2. mapping of each dimension to numeric values (typically scaled to liebetween zero and 100) using a utility/value function3. weights for each dimension (objective)4. bringing together of the factors to score the job offers.MN3032 Management science methods18Examiners’ commentaries 2017MN3032 Management science methods – Zone BImportant noteThis commentary reflects the examination and assessment arrangementsfor this course in the academic year 2016–17. The format and structureof the examination may change in future years, and any such changeswill be publicised on the virtual learning environment (VLE).Information about the subject guide and the Essential readingreferencesUnless otherwise stated, all cross-references will be to the latest versionof the subject guide (2014). You should always attempt to use the mostrecent edition of any Essential reading textbook, even if the commentaryand/or online reading list and/or subject guide refer to an earlieredition. If different editions of Essential reading are listed, please checkthe VLE for reading supplements–if none are available, please use thecontents list and index of the new edition to find the relevant section.Comments on specific questionsQuestion 1(a) Discuss the similarities between Journey Making, Soft Systems Methodology and Strategic Choice.(b) Clearly explain the Journey Making methodology.(10 marks)(9 marks) (c) Apply Journey Making to any problem with which you are familiar. (6 marks)Reading for this question:This question relates to Chapter 2 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an alternative other form) the model answer(for all parts of the question) given below.In terms of similarities, the examiners would expect you to mention thefollowing points:• primarily focus on the people involved with the problem• secondarily focus on the problem• help in structuring of (complex) problems• are mainly used with a small group of decision makers in organisations• typically use a consultant (external person) tosee that the group contains individuals with knowledge of thesituation and/or individuals who will affect the success of anyaction proposedact as a facilitator/organiser of the processExaminers’ commentaries 201719• use such methods to try to capture, verbally and in pictures/diagrams,the group’s perception of the problem and also to help the members ofthe group to: gain an understanding of the problem they facegain an understanding of the views of the problem formed by other members of the group in order to negotiate on the action to takeagree on a consensus course of action to which they arecommitted.In terms of Journey Making, the examiners would expect you to addressthe points below:• clear explanation of the elements of a cognitive map:A…BA…B → C…DA…B → C…→ with a minus at its head.In terms of the Journey Making process, the examiners would expect youto address the points below:• individual interviews and maps• merged maps• discussion at workshop of the merged map and individual mapsincluding:analysis of content and structureidentification of any ‘emerging themes’ and ‘core concepts’discussion of key goals, inter-related problems, key options andassumptions.For the example problem, the examiners would expect:• clear definition of problem, not just a few words• two individual maps and a merged map• discussion of the process and words linking the maps together.Question 2Briefly explain each of the following topics from the viewpoint of OperationalResearch/Management Science: (a) MRP(b) Kendall’s notational system for classifying queueing systems(c) Value judgements in DEAReading for this question:This question relates to Chapters 5, 9 and 11 of the subject guide.Approaching the question(9 marks)(9 marks)(7 marks) In a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.For part (a)MRP relates to pages 102-106 of the subject guide and points that theexaminers would expect to see here are:• MRP=Materials Requirements Planning• Bill of Materials (BOM) mentionedMN3032 Management science methods20• Bill of Materials (BOM) clearly explained (e.g. example)• Master Production schedule (MPS)• timing and quantity with respect to inventory orders• structural and tactical information.For part (b)This part relates to Kendall’s notational system for classifying queueingsystems, at page 197 of the subject guide. Points that the examiners wouldexpect to see here are:• standard notation system as P1/P2/P3/P4/P5• P1 probability distribution for the arrival process• P2 probability distribution for the service process• P3 number of channels (servers)• P4 maximum number of customers allowed in the queueing system(either being served or waiting for service)• P5 maximum number of customers in total.Common options for P1 and P2 are:• M for a Poisson arrival distribution (exponential interarrivaldistribution) or an exponential service time distribution• D for a deterministic or constant value• G for a general distribution (but with a known mean and variance).For part (c)This part relates to value judgements in DEA, at pages 168-169 of thesubject guide. Points that the examiners would expect to see here are:• adding additional constraints (typically to the weights) so as to betterrepresent the situation being modelled• introducing issues relating to solving nonlinear problems.Question 3(a) The payoff table below shows for a company the final year end profitsexpected (£m) from four possible choices (A, B, C and D) with respect to threepossible scenarios (S1 to S3 respectively) for the forthcoming year. S1S2S3A211-5B41314C5-513D1218-10 Here, for example, if the company makes choice D (and it can only choose oneof A, B, C or D) and the scenario for the forthcoming year turns out to be S3then it makes a loss of £10m.Copy the following table into your answer book and fill in the choice thatshould be made under each criterion and the associated value. CriteriaChoiceAssociated valueOptimisticConservativeRegretEqually likely Examiners’ commentaries 201721Based on your analysis which choice would you recommend the companymakes? (10 marks)(b) What is the minimum value for the payoff associated with B and scenarioS1 such that under the Regret criterion you choose B? Clearly explain thesteps you take in arriving at your answer (but without enumerating different values for this payoff).(11 marks)(c) For the situation as described in (a) above discuss the use you might makeof sensitivity analysis to generate further insight into the decision as to thechoice the company should make.(4 marks) Reading for this questionThis question relates to Chapter 4 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.The situation is:Scenario PayoffChoiceS1S2S3Equally likelyA211-52.67B4131410.33C5-5134.33D1218-106.67 Regret ChoiceS1S2S3A10719B850C7231D0024 The required table is CriteriaChoiceAssociated valueOptimisticD18ConservativeB4RegretB8Equally likelyB10.33 So, here B is the recommended choice.Letting the value for B-S1 be x, then the payoff matrix is: S1S2S3A211-5Bx1314C5-513D1218-10 MN3032 Management science methods22To calculate regret, we need the maximum in each column; so, here we donot know the maximum in the S1 column as it may be x or it may be 12.For the payoff matrix in the first part of the question (which includes apayoff of 4 associated with B-S1), we, under regret, choose B. Hence, theminimum value ‘such that under the Regret criterion you chooseB’ must be less than (or equal to) 4; so, the payoff B-S1 may decrease (orremain the same). In other words, we now know that x≤4.If x≤4, then the maximum in the S1 column is 12Given that we now know that the maximum in the S1 column is 12, weare justified in writing the regret matrix as:Regret ChoiceS1S2S3A10719B12-x50C7231D0024The maximum regrets are:A19Bmax(12-x,5,0)=max(12-x,5)C23D24 We choose B if it minimises the maximum regret, i.e. if max(12-x,5) ≤ 19The minimum value of x which satisfies this inequality is x=-7Hence, the minimum value for the payoff such that we choose B underregret is -7Technically, when x=-7 the maximum regrets are:A 19B max(12-x,5,0)=max(12-x,5)=19C 23D 24So, here we would be indifferent as between A and B. If x is slightly lessthan -7, say -6.9999, then B would be the unique choice under regret.With respect to sensitivity to give insight into the recommended choice,points that the examiners would expect you to mention are:• investigating changes in the financial calculations that led to theprofit/payoff values, thereby producing a wider range of profit/payoffvalues• investigating changes in the calculations that led to the threescenarios, thereby producing a greater number of scenarios• using an algebraic approach (i.e. replacing a number by a symbol).Question 4(a) A company is using Markov theory to analyse consumers switching betweenfour different brands of gaming console. An analysis of data has estimatedthat the transition matrix shown below applies for the probability ofswitching between brands.Examiners’ commentaries 201723 To brandABCDFrom brandA1000B0100C0.120.270.160.45D0.190.140.070.60 Here, for example, there is a probability of 0.27 that a consumer switchesfrom brand C to brand B. The current market shares for brands A,B,C,D are10%, 20%, 25% and 45% respectively.What is the longrun prediction for the market shares for each brand?(17 marks)(b) In a single server queueing system with an exponential service timedistribution and a Poisson arrival distribution customers arrive at an(average) rate of 150 per hour. The server takes (on average) 20 seconds toserve a customer.Copy the following table into your answer book and fill in the correspondingnumeric values. (5 marks) FactorValue (four decimal places)Average number of customers in thequeueAverage time that a customer spendsin the system (in minutes)Probability that there are threecustomers in the system (5 marks)A colleague has argued that values for the factors you have tabulated wouldbe better calculated by using discrete-event simulation. Would you agree or not and why?Reading for this question(3 marks) This question relates to Chapters 6 and 11 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.For part (a)This is a Markov process question with absorbing states.The details of the long-run calculation are: R0.12000.19000.27000.1400 Q0.16000.07000.8400-0.07000.30450.45000.6000-0.45000.4000(1/d=3.2841)I-QD inverse1.31360.22991.47782.7586 NR0.43840.55170.56160.4483 MN3032 Management science methods24Given the initial system state, the long run is: for A =0.10+ [0.25,0.45] ×[0.4384] = 0.4579[0.5517][0.5616] = 0.5421[0.4483]for B =0.20+ [0.25,0.45] × Long run for brands C and D is zeroFor part (b)Here we will have λ=150 per hour, µ=3600/20=180 per hour, where theformulae used in the required table are as below FactorFormulaValue (four decimal places)Average number of customers inthe queueλ2/[µ(µ- λ)]4.1667Average time that a customerspends in the system (inminutes)1/(µ-λ)2Probability that there are threecustomers in the system(λ/µ)3(1-λ/µ)0.0965 We would not use discrete event simulation as these are steady-statevalues and so calculating these via simulation would not be appropriate(as it would require running the simulation over a long time period).Question 5A company is planning a small project and the following table gives the variousactivities in that project, as well as their associated completion times. ActivityCompletion time (days)A4B4C3D5E1F3G3 Here, for example, activity G takes 3 days to be completed.The immediate precedence relationships are: ActivityActivityAmust be finished beforeBcan startAmust be finished beforeGcan startBmust be finished beforeGcan startCmust be finished beforeDcan startCmust be finished beforeEcan startFmust be finished beforeAcan startFmust be finished beforeCcan start In addition two other conditions (X1 and X2) must apply• Condition X1: 4 days must elapse between the end of activity C and the startof activity B• Condition X2: 2 days must elapse between the end of activity A and the startof activity EExaminers’ commentaries 201725(a) Draw the network diagram and calculate the overall project completion time.State the critical path(s). (9 marks)(b) Copy the following table into your answer book and fill in the latest starttimes and float (slack) times. ActivityLatest start time (days)Float (slack) time (days)ABCDEFG (6 marks)(c) A colleague has asked you to clearly explain to them what is meant by thefollowing terms in relation to project planning:• earliest start time• latest start time• float (slack) time• beta distribution• resource smoothing What would you say to them?Reading for this question(10 marks) This question relates to Chapter 3 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.The diagram can be seen below.Details of the calculation are: Activity Earliest Start Latest Start FloatA 3 6 3B 10 10 0C 3 3 0D6126E9167F000G14140X1660X2 7 14 7 MN3032 Management science methods26• completion time 17 days• critical path is F→C→X1→B→GThe required table is: ActivityLatest start time (days)Float (slack) time (days)A63B100C30D126E167F00G140 For definitions of the three different time concepts, the examiners wouldexpect you to mention the points below.• Earliest start time for any activity is the earliest time we can start theactivity such that all its (immediately) preceding activities have beenfinished.• Latest start time for any activity is the latest time we can startthe activity and still complete the project in the minimum overallcompletion time.• Float (slack) time for an activity is the amount by which we canincrease the time taken to complete the activity without changing(increasing) the overall project completion time.For the beta distribution, the examiners would expect you to mentionthe points below.• It is the probability distribution for activity completion time.• Its distribution is defined by three time estimates (optimistic, mostlikely and pessimistic).• The shape of the distribution (asymmetric) is as below.For resource smoothing the examiners would expect you to mention:• resource profile not fixed• resource profile can be changed (without affecting the overall projectcompletion time) by moving the start time of non-critical activities• ideal resource profile is a constant usage of resource over time.Question 6A company makes three products (X, Y and Z) and has a number of optionsavailable as to how these products can be produced on the three machines (A, Band C) it has available. These options are tabulated below.Examiners’ commentaries 201727 ProductOptionTime required (minutes)ABCXA→B→C1067C→B415A→C125YC→B→A987B→C142ZA→B→C1079C→A215 This table shows for each option the machines used (and their order), as wellas the processing time in minutes required on each machine. For example theoption B→C for producing Y requires 14 minutes of processing on machineB followed by 2 minutes of processing on machine C to produce one unit ofproduct Y. The company can adopt more than one option for production of anyproduct.The company is planning its production for the next week. Available time inminutes on machines A, B and C is 2400, 2350 and 2000 respectively. By movingworkers between machines it is possible to increase working time at anymachine, but at the expense of reducing working time at another machine. Eachminute gained on one machine reduces the time available on the machine fromwhich it is taken by 2 minutes.Products X, Y and Z sell to customers for £95, £80 and £75 per unit (respectively).(a) Formulate the problem of deciding the number of units of each product to make as an integer program with linear constraints.(20 marks)(b) What might be the principal difficulty in implementing the numeric solutionfrom this integer program? Illustrate this difficulty by example using the dataprovided in the question.(5 marks) Reading for this questionThis question relates to Chapter 7 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.Let x(i) (≥0, integer) be the number of units of product X produced usingoption i (i=1,2,3) where the options are numbered as in their order in thequestion, for example, option 1 for X is A→B→Cand option 2 for X is C→B.Similarly, let y(i) (≥0 integer) (i=1,2) and z(i) (≥0 integer) (i=1,2) bethe number of units of products Y and Z produced.As we can move time between machines, we need t(A,B), for example, forthe time in minutes moved from machine A to machine B (≥0, integer).Other time variables (e.g. t(A,C), t(B,A)) also exist.For machine A, we have the calculations below.• 10x(1)+12x(3)+9y(1)+10z(1)+21z(2) is the time used.• t(B,A)+t(C,A) is the time gained by shifting time from other machinesto A.• 2t(A,B)+2t(A,C) is the time lost by shifting time to other machinesfrom A.So, the machine A time capacity constraint is:MN3032 Management science methods28[10x(1)+12x(3)+9y(1)+10z(1)+21z(2)] + [t(B,A)+t(C,A)] –[2t(A,B)+2t(A,C)] ≤ 2400In a similar fashion, the machine B time capacity constraint is:[6x(1)+4x(2)+8y(1)+14y(2)+7z(1)] + [t(A,B)+t(C,B)] –[2t(B,A)+2t(B,C)] ≤ 2350In a similar fashion, the machine C time capacity constraint is:[7x(1)+15x(2)+5x(3)+7y(1)+2y(2)+9z(1)+5z(2)] + [t(A,C)+t(B,C)] –[2t(C,A)+2t(C,B)] ≤ 2000The objective is:Maximise 95[x(1)+x(2)+x(3)] + 80[y(1)+y(2)] + 75[z(1)+z(2)]The principal difficulty relates to the scheduling of the products onmachines and potential queueing/wasted time.Suppose we decide to produce one unit of X using option 1 and one unitof Z using option 1. Then both X and Z have the same route A→B→C butwith different processing times. If we schedule these in the order, first Xfollowed by Z, then X will take 10 minutes on A, then start on B, finishingafter 6 minutes. So, B is free after 6 minutes, but the unit of Z whichstarted processing on A when X finished still has 4 minutes left before itcan start on B; so, we have free time on B that cannot yet be utilised. Thiswasted time on B is not accounted for in our formulation.Question 7(a) One methodological issue that arises in Operational Research work relates tooptimisation. What do you understand by optimisation within the context ofOperational Research/Management Science? In particular your answer should make clear what the assumptions behind optimisation are.(17 marks)(b) Illustrate your answer by reference to TWO relevant problems of which youare aware.Reading for this question(8 marks) This question relates to Chapter 1 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.The examiners would expect your answer to contain the following:• discussion of mathematical model, decision variables, optimal solution• description of technique (e.g. LP)• consideration of sensitivity analysis• reference to robustness.The examiners would expect you to mention the assumptions behindoptimisation below.• The model accurately represents the system, and therefore the optimalsolution for the model is also the optimal solution for the system. (Thismay not be the case, however, as models rarely represent accuratelythe system under investigation).• There is one objective or a number of objectives that can be translatedinto a common unit, usually monetary values. For instance, time canbe converted into a monetary value thereby facilitating optimisation ofboth cost and time.Examiners’ commentaries 201729• There is consensus over what the objective of the system is.• The problem will not change over time (at least in the shortterm) andtherefore one optimal solution can be found (i.e. the solution given isthe best advice available now for the nearfuture).• All data can be quantified (i.e. assigned numerical values). In somecases, it is not possible to quantify some factors, and therefore onlyqualitative information can be provided.For each problem, the examiners would expect you to:• make a clear statement of the problem being considered• mention a particular specific optimisation technique that could beapplied to that problem.Question 8You are working for one company and have been approached by a rival companywho wish you to come and work for them. You have three objectives in choosingwhether to stay with your current company or not:• Objective 1 – salary – the larger the salary the better• Objective 2 – earnings potential – the more you can earn over time the better• Objective 3 – vacation time – the more vacation time you are allowed thebetterYour pairwise comparison matrix for these objectives is: Objective123Objective11452–133––1 Here, for example, your judgement as to objective 1 as compared with objective3 is rated 5.(a) Are the judgements made with respect to these objectives reasonablyconsistent or not (assume the value of RI for n=3 is 0.58)? (13 marks)Currently your judgement (pairwise comparison matrices) of the relativeworth of your current employer (company A) as compared to your potentialnew employer (company B) with respect to the above three objectives is: ABObjective 1A17B–1 ABObjective 2A1–B51 ABObjective 3A13B–1 (b) Assuming that the judgements made with respect to the three objectives arereasonably consistent which company should you choose to work for?(5 marks)(c) Discuss how you might apply MAUA/MAVA to the situation considered aboverelating to choosing which company to work for. (7 marks)MN3032 Management science methods30Reading for this questionThis question relates to Chapter 10 of the subject guide.Approaching the questionIn a good answer, you would demonstrate to the examiners that you havefollowed (either exactly or in an appropriate alternative form) the modelanswer (for all parts of the question) given below.The details of the calculation are:Comparison matrixObjective123Objective1145 2 0.2500 1 33 0.2000 0.3333 1Column sum 1.4500 5.3333 9.0000Normalisedmatrix 1 2 3 Row average1 0.6897 0.7500 0.5556 0.66512 0.1724 0.1875 0.3333 0.23113 0.1379 0.0625 0.1111 0.1038Weights 0.6651 0.2311 0.1038Columnvector 2.10850.70880.3138lmax 3.0868 CI0.0434RI0.5800CI/RI0.0748as the ratio CI/RI is less than 0.1, the judgements are consistent.For objective 1, we have:ComparisonmatrixJob offerABA17Job offerBColumn sum0.14291.142918.0000 Normalised matrixABRow average0.8750A0.87500.8750B0.12500.12500.1250 Examiners’ commentaries 201731For objectives 2 and 3, the normalised matrices are:Normalised matrixA0.16670.8333B0.16670.8333Row average0.16670.8333AB Normalised matrixA0.75000.2500B0.75000.2500Row average0.75000.2500ABwhich means that the calculation for the job offers is:ObjectiveAB10.66510.87500.1250 2 0.2311 0.1667 0.83333 0.1038 0.7500 0.2500Scores 0.6983 0.3017So, the choice is company A.MAUA/MAVA relates to the subject guide, at pages 190-191. Theexaminers would expect a good answer to contain the following points:1. evaluation of each alternative (job offer) with respect to eachdimension (objective)2. mapping of each dimension to numeric values (typically scaled to liebetween zero and 100) using a utility/value function3. weights for each dimension (objective)4. bringing together the factors together to score the job offers.

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