Examiners’ commentaries 2016
Examiners’ commentaries 2016
MN3032 Management science methods
Important note
This commentary reflects the examination and assessment arrangements
for this course in the academic year 2015–16. 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 (2014). 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 refers 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.
General remarks
Learning outcomes
At the end of the course, and having completed the Essential reading and
activities, you should be able 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 problemstructuring methods themselves (or explain how they should be done)
• commission more advanced exercises.
Planning your time in the examination
You are strongly advised to plan and think through your approach to
the order (sequence) in which you answer your selected questions.
Candidates, especially less confident ones, can often gain ‘easier’ marks
by attempting (albeit incorrectly) a numeric calculation rather than
answering an essay-type question. In general, therefore, you are advised
to answer predominantly quantitative questions before questions that
are predominantly qualitative in nature. In addition, it is logical for you
to attempt ‘stronger’ questions first (ones where you are more confident
that you can do well).
The advice given to you in the Handbook applies here: ‘Examiners who
mark these papers are instructed to mark the first answers that they get
to in the answer booklet which meet the required number of questions.
If you do extra, they will not mark all questions attempted and then
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MN3032 Management science methods
select the best marks.’ Candidates who may have attempted more than
the required number of questions need to clearly cross out questions that
they do not wish to have marked by use of a line through the entire text.
Simply crossing out the question number at the head of a page of text, for
example, or omitting the question number from the questions listed 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 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.
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 calculations/
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.
You may be aware that while previously the examination was split into
a qualitative section and a quantitative section, this format was dropped
in the 2013 examination. You need to be clear that this lack of sections
means that a question can be drawn from any topic in the subject guide.
In particular, a question may mix both qualitative and quantitative topics.
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Examiners’ commentaries 2016
Examination structure
For the examination in 2017 the structure of the examination paper will
be the same as it was in the 2016 examination. The 2017 examination
paper will contain eight questions and candidates will have to answer any
four of them. One of these eight questions will be a short notes question
that is predominantly qualitative in nature. This will contain three topics
and will ask candidates to ‘Briefly explain each of the following topics
from the viewpoint of Operational Research/Management Science.’
Examination revision strategy
Many candidates are disappointed to find that their examination
performance is poorer than they expected. This may be due to a
number of reasons. The Examiners’ commentaries suggest ways of
addressing common problems and improving your performance.
One particular failing is ‘question spotting’, that is, confining your
examination preparation to a few questions and/or topics which
have come up in past papers for the course. This can have 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 each course. You should read the
syllabus 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. Examination papers may legitimately include
questions on any topic in the syllabus. So, although past papers can
be helpful during your 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. We strongly advise you not to adopt this
strategy.
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MN3032 Management science methods
Examiners’ commentaries 2016
MN3032 Management science methods – Zone A
Important note
This commentary reflects the examination and assessment arrangements
for this course in the academic year 2015–16. 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 (2014). 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 refers 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.
Comments on specific questions
Question 1
Describe what you understand by the following in relation to methodological
issues that arise in Operational Research/Management Science:
a. The benefits of an OR approach to decision problems. (12 marks)
b. Cost versus decision quality. (13 marks)
Reading for this question
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
have followed (either exactly or in an appropriate other form) the model
answer (for all parts of the question) given below.
For part (a)
Benefits of an OR approach are discussed on pp.25–26 of the subject
guide. The examiners would expect to see a discussion about the points
mentioned below.
• OR is particularly well suited for routine tactical decision making
where data is typically well-defined and decisions for the same
problem (e.g. how much stock to order from a supplier) must be made
repeatedly over time.
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Examiners’ commentaries 2016
• We might reasonably expect an ‘analytical’ (structured, logical)
approach to decision making to be better (on average) than simply
relying on a person’s innate decision-making ability.
• Sensitivity analysis can be performed in a systematic fashion.
• constructing an OR model helps to structure your thinking about what
is/is not important in a problem.
For part (b)
Cost versus decision quality relates to pp.22–23 of the subject guide. The
examiners would expect to see a discussion about the points mentioned
below.
• If we wish to reach good quality decisions then we have to take our
time and incur costs.
• OR projects use resources – the consultant’s, the client’s, other staff’s
– and there is always a tension between minimising the cost and time
taken to reach a decision and making the ‘perfect’ decision.
• Models are representations of the system under investigation, but are
simplified representations. The more time and cost put into building a
model, the more accurately it should represent the system.
• The issue of the time available to reach a decision.
Question 2
a. Clearly explain the Strategic Choice methodology, in particular with regard to
the role played by UE, UV and UR.
(13 marks)
b. Apply the methodology to two problems with which you are familiar.
(12 marks)
Reading for this question
This question relates to pp.40–44 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.
For part (a)
The examiners would expect you to mention the three types of uncertainty.
• Uncertainty about the working environment (UE), reduced by a
technical response (e.g. collecting data, surveys, numeric analysis).
• Uncertainty about guiding values (UV), reduced by a political response
(e.g. clarifying objectives, consulting interest groups, asking higher
authorities for their opinions).
• Uncertainty about related decision fields (UR), also known as
uncertainty about choices on related agendas, reduced by an
exploration of structural relationships (e.g. adopting a broader
perspective, negotiating/collaborating with other decision-makers,
looking at the links between a decision that might be made by
ourselves and decisions that might be made by others).
For part (b)
The examiners would expect you to discuss the four modes of working.
• Shaping – including a decision graph.
• Designing – identify options and get a list of feasible schemes.
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MN3032 Management science methods
• Comparing feasible schemes.
• Choosing – decide on a commitment package and address
uncertainties.
For each problem the examiners would expect:
• a clear statement (in words) of the problem considered
• the four modes of working for the example
shaping
designing comparing choosing.
Question 3
Briefly explain each of the following topics from the viewpoint of Operational
Research / Management Science:
a. CATWOE (9 marks)
b. JIT (9 marks)
c. Project time/project cost tradeoff in network analysis. (7 marks)
Reading for this question
CATWOE relates to pp.36–37 of the subject guide. JIT relates to pp.106–
107 of the subject guide. Project time/project cost tradeoff in network
analysis relates to pp.60–61 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.
For part (a)
Points that the examiners would expect to see here are below.
• Mention of soft systems methodology (SSM) – covered just before
CATWOE on p.35 of the subject guide.
• CATWOE – Customers, Actors, Transformation (or Transformation
process), Worldview (or Weltanschauung), Owner, Environmental
constraints.
• Elements of CATWOE more fully explained.
• Root definition, clear indication that it is a statement of the ideal.
• Link between CATWOE and root definition (deduce the answer to
CATWOE from the root definition).
For part (b)
Points that the examiners would expect to see here are below.
• JIT = just-in-time.
• Also known as a lean production system.
• Aims to control idle inventories.
• Aims to control defective parts.
• Cycle of continual improvement.
• Requirements for a successful JIT system, e.g. uniform final assembly
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Examiners’ commentaries 2016
• Boat and rocks example.
For part (c)
Points that the examiners would expect to see here are below.
• ‘Crashing’ (time/cost tradeoff).
• Normal point and crash point.
• Linear relationship between normal point and crash point (or a diagram
showing this, such as Figure 3.4, subject guide p.60).
• Use of linear programming (LP).
• LP objective is minimise cost subject to achieving desired project
completion time.
Question 4
A company is using Markov theory to analyse consumers switching between four
different brands of cell phone. Although consumers often change their particular
cell phone model (e.g. upgrading to a later model) the issue here is the brand/
company they use (so for example those using a particular brand often stay with
the same brand when they upgrade their cell phone).
Consumers only switch brand once every two years. An analysis of data has
estimated that the transition matrix shown below applies for the probability of
switching between brands.
To brand
From brand
A
B
C
D
A
1
0
0
0
B
0
1
0
0
C
0.04
0.04
0.76
0.16
D
0.08
0.16
0.24
0.52
Here, for example, there is a probability of 0.08 that a consumer switches from
brand D to brand A. Currently all brands have equal market share.
a. What is the longrun prediction for the market shares for the brands?
(18 marks)
b. A colleague has remarked that it seems very unlikely that the transition
probability from A to A is precisely 1, since that would imply that all consumers
who switch to brand A never switch to another brand. Indeed he has an
aunt who switched to brand A, but two years later switched to brand B,
contradicting the transition matrix you have used. Hence he believes that your
analysis is invalid. How would you respond? (7 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.
For part (a)
This is a Markov process question with absorbing states.
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MN3032 Management science methods
The details of the long-run calculation are:
R
0.0400
0.0400
0.0800
0.1600
0.7600
0.1600
0.2400
0.5200
0.2400
–0.1600
–0.2400
0.4800
d
0.0768
(1/d=13.0208)
inverse
6.2500
2.0833
3.1250
3.1250
0.4167
0.5833
0.3750
0.6250
Q
I-Q
NR
The initial system state is 0.25 for each state. So the long run is:
for A = 0.25+ [0.25,0.25] × [0.4167] = 0.4479
[0.3750]
for B = 0.25+ [0.25,0.25] × [0.5833] = 0.5521
[0.6250]
Long run for brands C and D is zero.
For part (b)
Points that the examiners would expect to see here (drawing on p.123 of
the subject guide) include the following.
• We are modelling the market as a whole, not the behaviour of any one
individual, such as the aunt mentioned.
• Any transition probabilities used must be averages over a whole group
of consumers, to the two decimal place accuracy used a number in the
transition probability matrix could indeed be 1.
• Markov theory is only a simplified model of a complex decisionmaking process.
• The analysis we have presented provides insight: clearly, transition
probabilities will change over time as different brands evolve their cell
phone models.
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Examiners’ commentaries 2016
Question 5
A company has a number of different branches. The data they have collected for
these branches over the last full financial year is shown below.
Branch
Number of
customers (‘000)
Non-employee cost
(£m)
Number of
employees
A
21
19
12.6
B
36
8
14.5
C
53
13
9.9
D
39
17
6.5
E
30
10
12.0
F
15
15
35.0
Here, for example, in the last full financial year branch D had 39,000 customers
and employed 6.5 people (on average over the year). The non-employee cost for
branch D, i.e. the cost of operating the branch excluding the cost of employees,
was £17m.
a. Apply data envelopment analysis to compare the relative performance of
these branches. Copy the table shown below into your answer book and fill
in the efficiencies and reference sets (where appropriate). (14 marks)
Branch
Efficiency
Reference set
A
B
C
D
E
F
b. You are now given new information that each employee was paid £50,000
per year (on average, in the last full financial year). A colleague has argued
that this will make no difference to the analysis you have performed since
this information is simply a scaling between number of employees and
employee cost. Would you agree or not? If not what would be the new values
for the efficiencies of the branches? (8 marks)
c. Another colleague has come to you confused as to the meaning of the
terms ‘efficient’, ‘inefficient’ and ‘efficiency’ in data envelopment analysis.
You are aware that this colleague has an educational background that has
not included scientific and mathematical topics. How, in language that your
colleague could understand, would you explain the meaning of these terms
to them? (3 marks)
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.
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MN3032 Management science methods
For part (a)
Appropriate ratios are:
• customers per £m
• customers per employee.
Treating number of customers as the output and cost and the number of
employees as the input, the ratios are:
Branch
Customers per £m
Customers per employee
A
1.11
1.67
B
4.50
2.48
C
4.08
5.35
D
2.29
6.00
E
3.00
2.50
F
1.00
0.43
Then with respect to these ratios the relative efficiencies are:
Branch
Customers per £m
Customers per employee
A
0.25
0.28
B
1.00
0.41
C
0.91
0.89
D
0.51
1.00
E
0.67
0.42
F
0.22
0.07
So here B is the best performer with respect to the first ratio, and D is the
best performer with respect to the second ratio.
The data envelopment analysis (DEA) diagram is:
So note here that C is efficient even though it is not the best performer in
either of the two ratios.
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Examiners’ commentaries 2016
The required table is:
Branch
Efficiency
Acceptable range
Reference set
lower
upper
0.28
0.32
C, D
A
0.30
B
1
C
1
D
1
E
0.69
0.67
0.71
B, C
F
0.22
0.20
0.24
B
The acceptable range above is the range within which we count an answer
as correct.
For part (b)
With the information that each employee costs £50,000 the problem
reduces to one with just one output (number of customers) and one input
(total cost, non-employee cost plus employee cost). So we have just a
single ratio: customers per unit of cost. In this case the efficiencies may
potentially change, hence our colleague is incorrect.
The analysis becomes:
Customers
(‘000)
Cost
(£m)
Ratio
(customers per £m of cost)
Efficiency
A
21
19.63
1.07
0.26
B
36
8.725
4.13
1
C
53
13.495
3.93
0.95
D
39
17.325
2.25
0.55
E
30
10.6
2.83
0.69
F
15
16.75
0.90
0.22
Here the efficiencies are derived by comparison with the branch (B) with
the largest ratio of customers/cost.
For part (c)
With respect to the three words:
• ‘efficient’ for a branch means that they are one of the best performers
(overall) in terms of the set of branches considered. This is not the
same as being best in the entire world.
• ‘inefficient’ for a branch means one that is not efficient as defined in
the point above.
• ‘efficiency’ for a branch is a number between zero and 1 (equivalently
0% to 100%) and gives an intuitive understanding of how good a
branch is at converting its inputs into outputs. An ‘efficient’ branch has
efficiency of 1 (equivalently 100%).
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MN3032 Management science methods
Question 6
A company makes two products (X and Y) and has available three machines (A, B
and C) with which it can produce these products. The cost of producing one unit
of each product on each machine is shown below.
Cost per unit (£)
Product
A
B
C
X
0.8
3.2
2.3
Y
0.5
5.4
1.2
Here, for example, producing one unit of product X on machine B costs £3.2 per
unit.
The company is planning its production for the next week. The machines have a
limited weekly production capacity. For machine A its capacity is 1,000 units (X
and Y combined) per week. Machine B can produce at most 500 units of X and
250 units of Y per week. Machine C can produce at most 600 units of X and 150
units of Y per week.
If it is not possible to produce enough to meet forecast demand the company
has the option of buying additional units of X and Y from a supplier, but this
costs £10 per unit of X and £12 per unit of Y.
Products X and Y sell to customers for £9 and £7 per unit (respectively). There
are no units of either product currently in stock.
The company has a number of goals:
• they would like to buy 20 units (X and Y combined) from the supplier
• they would like to meet forecast demand for the next week, which is 1,500
units of X and 750 units of Y
• they would like their profit to equal £14,500
[For the full version of this question, please refer to the original]
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.
For part (a)
Let x(A,B,C) (≥0) be the number of units of product X produced on
machines A, B and C respectively; y(A,B,C) (≥0) be the number of units of
product Y produced on machines A, B and C respectively.
Let S(X) and S(Y) (≥0) be the number of units of X and Y respectively
bought from the supplier.
Then the capacity constraints are:
• x(A) + y(A) ≤ 1000
• x(B) ≤ 500
• y(B) ≤ 250
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Examiners’ commentaries 2016
• x(C) ≤ 600
• y(C) ≤ 150
For the supplier goal we have the constraints:
• S(X) + S(Y) = 20 + a+ – b–
where a+,b– ≥0 are the upward and downward deviation from this goal.
For the demand goal for X we have the constraints:
• x(A) + x(B) + x(C) + S(X) = 1500 + c+ – d–
where c+,d– ≥0 are the upward and downward deviation from this goal.
For the demand goal for Y we have the constraints:
• y(A) + y(B) + y(C) + S(Y) = 750 + e+ – f–
where e+,f– ≥0 are the upward and downward deviation from this goal.
For the total profit goal we have the constraints:
• 9[x(A) + x(B) +x(C) + S(X)] + 7[y(A) + y(B) + y(C) + S(Y)]
–0.8x(A)– 3.2x(B)–2.3x(C) – 0.5y(A) – 5.4y(B) – 1.2y(C) –10S(X) –
12 S(Y) = 14500 + g+ – h–
where g+,h– ≥0 are the upward and downward deviation from this goal.
The objective here is to minimise a weighted sum of deviation variables.
Take 1% of the goal values:
• Supplier goal 20/100 = 0.2
• Demand goal for X 1500/100 = 15
• Demand goal for Y 750/100 = 7.5
• Profit goal 14500/100 = 145
minimise
w1(a+/0.2) + w2(b–/0.2) + w3(c+/15) + w4(d–/15) + w5(e+/7.5) +
w6(f–/7.5) + w7(g+/145) + w8(h–/145)
where the w’s are the weights (numeric values) for the deviations.
For part (b)
Sequential goal program
Priority level 1: minimise the upward deviation from the supplier goal.
minimise a+
subject to the five inequality constraints and four goal-related equality
constraints (and non-negativity constraints) seen above.
Priority level 2: minimise the downward deviation from profit goal
minimise h–
subject to the five inequality constraints and four goal related equality
constraints (and non-negativity constraints) seen above.
and a+ = A*
where A* is the value that a+ has in the solution at priority level 1.
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MN3032 Management science methods
Question 7
a. Company A wishes to develop a materials requirements plan for producing
a particular product (X) over an eight week period. The lead time between
releasing an order to the shop floor and producing a unit of product X is
two weeks. The company currently has 235 units of the product in stock. The
forecast customer demand is shown below.
Week
1
2
3
4
5
6
7
8
Demand
75
75
70
70
125
125
95
65
Here, for example, the demand for product X is 125 units in week 5. By
applying materials requirements planning decide the orders that need to
be placed for the product in each of the eight weeks. Your answer should
clearly explain the logic behind deciding the orders in materials requirements
planning. (12 marks)
b. Company B purchases a particular component from an external supplier.
The demand for the component is estimated to be 112 units per month.
The cost of placing an order with the supplier is £55. The component can be
purchased from the supplier at a price per unit of £85 if so desired. However
the supplier also offers a quantity discount such that if the company orders
125 units or more the price per unit is only £75. The cost of holding this
component (per year) is estimated to be 12% of the price. What advice can
you offer the company as to the cheapest way to purchase this component?
Clearly explain any procedure you follow in arriving at your advice, in
particular the reason as to why you consider each and every order quantity
costed in your answer. (13 marks)
Reading for this question
This question relates to Chapter 5 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.
For part (a)
This part of the question relates to materials requirements planning (MRP)
(pp.102–106 of the subject guide).
Week
1
2
3
4
5
6
7
8
Demand
75
75
70
70
125
125
95
65
On hand at end of week
160
85
15
0
0
0
0
0
Order quantity
0
55
125
125
95
65
n/k
n/k
In MRP (subject guide, p.104) the logic is that we always order as late as
possible and as little as possible to avoid a stockout.
For part (b)
R = 112 × 12 = 1,344
co = 55
P = 85
ch =0.12(85) = 10.2 per year
EOQ = √(2Rco/ch) = √(2×1344×55/10.2)
= 120.39 = 120 (or 121)
Total cost per year is RP + (chQ/2) + (coR/Q)
1,344×85 + (10.2×120/2) + (55×1,344/120) = 115,468
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Examiners’ commentaries 2016
Here we cost the EOQ as this (by definition) is the most economical
(least cost) order quantity (see Figure 5.6 on p.97 of the subject guide).
With the discount we have
P=75
ch=0.12(75) = 9 per year
So the new EOQ is
√(2Rco/ch) = √(2×1,344×55/9) = 128.17 = 128 (or 129)
Total cost per year is RP + (chQ/2) + (coR/Q)
1,344×75 + (9×128/2) + (55×1,344/128) = 101,953.5
This is the cheapest option evaluated so advise the company to order 128.
Here, the reason why we cost the EOQ rather than the minimum order
quantity (MOQ) of 125 units is that the procedure to follow when you
have a quantity discount is that if the EOQ ≥ MOQ you cost the EOQ as it
is guaranteed to be cheaper than ordering the MOQ (see Figure 5.6 on
p.97 of the subject guide).
Question 8
A company has formulated a linear program relating to the quantities of two
products (X and Y) to produce. Each unit of product X gives a contribution to
total profit of £2, each unit of product Y gives a contribution to total profit of
£5. Production of the two products is constrained by available working time on
two machines (A and B). In the forthcoming week there are 15 hours available
on machine A, 22 hours available on machine B. In addition there are two
constraints that arise from other considerations (constraints T1 and T2, as seen
below). The linear program the company have formulated is:
Maximise 2x + 5y
Subject to
2x + y ≤ 15
Machine A
3x + 4y ≤ 22
Machine B
3x – 2y ≥ -7
T1
-2x + 3y ≥ -9
T2
x,y ≥ 0
a. Solve this linear program graphically using an iso-cost/iso-profit line, clearly
indicating the feasible region. (13 marks)
b. For each of the four constraints in this linear program calculate the shadow
price. (8 marks)
c. Discuss robustness and planning with reference to the specific linear program
considered in this question. (4 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.
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MN3032 Management science methods
For part (a)
The LP diagram is
The optimal solution is at the intersection of the T1 and machine B lines,
i.e. at x=0.8889, y=4.8333.
The associated objective function solution value is 25.9444.
For part (b)
The shadow price is given by changing the right-hand side of a constraint
by 1 and resolving.
The shadow price for both the machine A constraint and the T2 constraint
is zero (equivalently not defined/unspecified), since they are not binding.
For machine B if we increase the right-hand side by 1 the new solution is
x=1, y=5 of value 27; giving the shadow price as 27–25.9444 = 1.0556.
For T1 if we increase the right-hand side by 1 the new solution is
x=1.1111, y=4.6667 of value 25.5556; giving the shadow price as
25.5556–25.9444 = = –0.3888, so 0.3888 if we ignore the minus sign, as
is customary for shadow prices.
For part (c)
Robustness and planning relate to pp.144–145 of the subject guide.
Robustness:
• Change in solution for small data changes, e.g. an objective function
coefficient (in this case profit per unit) or a constraint coefficient
(here, number of required hours per product on machines A, B).
Planning:
• Change in solution as we change the price of the product, hence
changing the amount produced and so the profit gained.
• Change in solution as we change the machine capacity (number of
working hours), hence changing the amount produced and so the
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Examiners’ commentaries 2016
profit gained.
Examiners’ commentaries 2016
MN3032 Management science methods – Zone B
Important note
This commentary reflects the examination and assessment arrangements
for this course in the academic year 2015–16. 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 (2014). 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 refers 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.
Comments on specific questions
Question 1
a. Clearly describe Soft Systems Methodology (SSM) with particular emphasis
on the underlying assumptions. (15 marks)
b. Illustrate SSM by applying it to any two examples with which you are
familiar. (10 marks)
Reading for this question
This question relates to pp.35–40 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.
For part (a)
The examiners would expect you to mention the SSM assumptions, that:
• different individuals and groups make different evaluations of events
and this leads to them taking different actions
• concepts and ideas from systems engineering are useful
• it is necessary when describing any human activity system to take
account of the particular image of the world underlying the description
of the system and it is necessary to be explicit about the assumptions
underlying this image
• it is possible to learn about a system by comparing pure models of
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MN3032 Management science methods
that system with perceptions of what is happening in the real world
problem situation.
The examiners would expect you to address the points below.
• CATWOE – Customers, Actors, Transformation (or Transformation
process), Worldview (or Weltanschauung), Owner, Environmental
constraints.
• Elements of CATWOE should be more fully explained.
• Root definition, and a clear indication that it is a statement of the
ideal.
• Link between CATWOE and root definition (deduce the answer to
CATWOE from the root definition).
The examiners would also expect you to discuss the stages in SSM.
For part (b)
For each problem the examiners would expect:
• a clear statement (in words) of the problem considered, not just a
single phrase • appropriate application of SSM to the problem, including
root definition (statement of the ideal)
CATWOE for the root definition
explicit and clear check of root definition by CATWOE
application of the stages.
Question 2
Briefly explain each of the following topics from the viewpoint of Operational
Research/Management Science:
a. SCM (9 marks)
b. Benefits of simulating customer checkout operations at a supermarket.
(9 marks)
c. Utilities, with reference to a risk-seeking decision maker. (7 marks)
Reading for this question
SCM relates to supply chain management (pp.108–109 of the subject
guide). Simulation relates to Chapter 11 of the subject guide. Utilities are
dealt with on pp.83–85 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.
For part (a)
Points that the examiners would expect to see here are:
• SCM = supply chain management
• strategic decisions and tactical decisions
strategic – facility location and overall distribution pattern
tactical – levels of stock (inventory) and patterns of local delivery
• an example of a supply chain.
For part (b)
Points that the examiners would expect to see here are below.
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Examiners’ commentaries 2016
• Using simulation in terms of management science methods where it
is a computer model of a real-world situation involving probabilities/
statistics and entities queuing for service.
• The usual generic benefits of simulation, in that you can explore
options/alternatives without the hassle and expense of changing things
in the real world.
• Statistics such as number served per minute, variation (e.g. standard
deviation) in number served per minute, queue lengths (e.g. how
many people will be queuing).
• How these factors might change, for example as we change the
number of servers in the context of a supermarket (e.g. self-service
tills, only five items or fewer tills, supermarket layout).
For part (c)
Points that the examiners would expect to see here are below.
Utilities:
• Involved with decision trees as discussed in further detail in the
subject guide, pp.83–84.
• Replace monetary values in decision tree calculations by
points=utilities.
Risk seeker:
• use of formulae (x–a)R/(b–a)R with R>1 (as on pp.83–84 of the
subject guide)
• appropriate graph (as Figure 4.5 on p.84 of the subject guide).
Question 3
a. The payoff table below shows for a company the final year end profits
expected (£m) from four possible choices (A, B, C and D) with respect to three
possible scenarios (S1 to S3 respectively) for the forthcoming year.
S1
S2
S3
A
8
3
7
B
1
18
1
C
5
-6
18
D
10
14
-4
Here, for example, if the company makes choice B (and it can only choose one
of A, B, C or D) and the scenario for the forthcoming year turns out to be S2
then it makes a profit of £18m. Scenario S1 is believed to have probability
0.3 and scenario S3 is believed to have probability 0.2.
Copy the following table into your answer book and fill in the choice that
should be made under each criteria and the associated value.
Criteria
Choice
Associated value
Optimistic
Conservative
Regret
Equally likely
Expected monetary value
Based on your analysis, which choice would you recommend the company
makes? (15 marks)
b. In a single server queueing system with an exponential service time
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MN3032 Management science methods
distribution and a Poisson arrival distribution customers arrive at an
(average) rate of 100 per hour. The server takes (on average) 25 seconds to
serve a customer.
i. What is the traffic intensity for this system?
Assuming that the system is in a steady state calculate:
I
The average number of customers in the queue.
II The probability that a customer has to wait for service.
III The average time that a customer spends in the system.
IV The probability that there are four customers in the system.(8 marks)
ii. Explain how knowing the average number of customers in the queue
could help you manage the system. (2 marks)
Reading for this question
This question relates to Chapters 4 and 11 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.
For part (a)
The situation is:
Probabilities
0.3
0.5
0.2
Scenario
Payoff
Choice
S1
S2
S3
Equally likely
EMV
A
8
3
7
6.00
5.30
B
1
18
1
6.67
9.50
C
5
-6
18
5.67
2.10
D
10
14
-4
6.67
9.20
Regret
Choice
S1
S2
S3
A
2
15
11
B
9
0
17
C
5
24
0
D
0
4
22
The required table is
Criteria
Choice
Associated value
Optimistic
B or C
18
Conservative
A
3
Regret
A
15
Equally likely
B or D
6.67
Expected monetary value
B
9.50
So here B is the majority choice since there are three criteria in which it is
a winner (or equal winner).
For part (b)
Here we will have λ=100 per hour, µ=3600/25=144 per hour.
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Examiners’ commentaries 2016
Traffic intensity = λ/µ = 100/144 = 0.6944
Average number of customers in the queue = λ2/[µ(µ–λ)] = 1.5783
• Probability a customer has to wait for service = λ/µ = 0.6944
• Average time a customer spends in the system = 1/(µ–λ) = 0.0227
hours (or 1.3636 minutes or 81.82 seconds)
• Probability that there are four customers in the system = (λ/µ)
n
(1–λ/µ), with n=4 = 0.0711.
If you know the average number of customers in the queue it helps you to
decide:
• whether you should consider changing the space allocated for queuing
customers
• whether you should consider changing the number of servers.
Question 4
a. A company X streaming films over the Internet uses Markov theory to try
and understand how its market share changes over time. The company has
two main competitors (Y and Z) and past estimates of the probability of a
customer making a transition between companies when they come to stream
a film are:
X
From company
To company
Y
Z
X
0.88
0.05
0.07
Y
0.69
0.02
0.29
Z
0.78
0.15
0.07
Here, for example, the probability that a customer switches from company Y
to company Z is 0.29. On average a customer streams a film once a week.
What will be the market share for company X after two weeks if the current
market shares are 40%, 40% and 20% for companies X, Y and Z respectively?
(2 marks)
b. What is the longrun prediction for the market shares for each of the three
companies?
(15 marks)
c. Company X is considering merging with company Y to create a new company
(XY). In this case they estimate that the matrix for the probability of a
customer making a transition between companies when they come to stream
a film will be:
XY
From company
Z
To company
XY
0.40
0.78
Z
0.60
0.22
Here, for example, the probability that a customer switches to company Z
from company XY is 0.60.
What is the longrun prediction for the market shares for each of the two
companies? (5 marks)
d. Considering the transition matrices given above do you think it is plausible
that the proposed merged company XY only retains 40% of its customers at
each switch?
(3 marks)
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MN3032 Management science methods
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.
For part (a)
Applying the standard Markov approach we have the state of the system as
below.
Period
X
Y
Z
Initial
0.4000
0.4000
0.2000
1
0.7840
0.0580
0.1580
2
0.8532
0.0641
0.0828
Market share for X in two weeks’ time (after two transitions) is 0.8532.
For the long-run prediction 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 = 0.88x1 + 0.69x2 + 0.78x3
x2 = 0.05x1 + 0.02x2 + 0.15x3
x3 = 0.07x1 + 0.29x2 + 0.07x3
x1 + x2 + x3 = 1
and solving these equations simultaneously
x1 = 0.8610
x2 = 0.0565
x3 = 0.0824
Here the transition matrix is given in the question in a [To, From] form so
it needs to be transposed into a [From, To] form, as:
XY
To company
Z
From company
XY
0.40
0.60
Z
0.78
0.22
For part (c)
For the long-run prediction we need to set up [x1,x2] where
[x1,x2 = [x1,x2]( above transition matrix) and x1 + x2 = 1. Expanding we
get
x1 = 0.40x1 + 0.78x2
x2 = 0.60x1 + 0.22x2
x1 + x2 = 1
and solving these equations simultaneously
x1 = 0.5652
x2 = 0.4348
For part (d)
In terms of whether the transition XY to XY can plausibly be 40% there
22
Examiners’ commentaries 2016
are a number of different approaches (of varying sophistication) that are
possible. The simplest of these is:
• Looking at the long run for X,Y,Z as calculated above it is very close to
the state of the system after two weeks (as calculated in the first part
of the question), so we can reasonably work with the long run as the
system state.
• Recall that the original transition probability X to X was 88% and
X has by far the largest market share (the long run of 0.8610, as
calculated above). Hence for the merged company XY the vast majority
(around 86%) of the customers will come from company X. So a very
approximate estimate of the transition probability for XY to XY can be
given by just considering the transition probability of X to X, namely
88%, so 40% does not seem plausible at all. This assumes that the
customers of X retain their brand loyalty to X after the merger with
company Y to create XY.
Question 5
A company has a number of different branches. The data they have collected for
these branches over the last full financial year is shown below.
Branch
Number of
customers (‘000)
Non-employee cost
(£m)
Number of
employees
A
21
19
12.6
B
36
8
14.5
C
53
13
9.9
D
39
17
6.5
E
30
10
12.0
F
15
15
35.0
Here, for example, in the last full financial year branch D had 39,000 customers
and employed 6.5 people (on average over the year). The non-employee cost for
branch D, i.e. the cost of operating the branch excluding the cost of employees,
was £17m.
a. Apply data envelopment analysis to compare the relative performance of
these branches. Copy the table shown below into your answer book and fill
in the efficiencies and reference sets (where appropriate). (14 marks)
Branch
Efficiency
Reference set
A
B
C
D
E
F
[For the full version of this question, please refer to the original]
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
23
MN3032 Management science methods
have followed (either exactly or in an appropriate other form) the model
answer (for all parts of the question) given below.
For part (a)
Appropriate ratios are:
• Customers per £m.
• Customers per employee.
Treating number of customers as the output and cost and the number of
employees as the input, the ratios are:
Branch
Customers per £m
Customers per employee
A
1.11
1.67
B
4.50
2.48
C
4.08
5.35
D
2.29
6.00
E
3.00
2.50
F
1.00
0.43
Then with respect to these ratios the relative efficiencies are:
Branch
Customers per £m
Customers per employee
A
0.25
0.28
B
1.00
0.41
C
0.91
0.89
D
0.51
1.00
E
0.67
0.42
F
0.22
0.07
So here B is the best performer with respect to the first ratio, and D is the
best performer with respect to the second ratio.
The data envelopment analysis (DEA) diagram is:
So note here that C is efficient even though it is not the best performer in
either of the two ratios.
24
Examiners’ commentaries 2016
The required table is:
Branch
Efficiency
Acceptable range
Reference set
lower
upper
0.28
0.32
C, D
A
0.30
B
1
C
1
D
1
E
0.69
0.67
0.71
B, C
F
0.22
0.20
0.24
B
The acceptable range above is the range within which we count an answer
as correct.
For part (b)
With the information that each employee costs £50,000 the problem
reduces to one with just one output (number of customers) and one input
(total cost, non-employee cost plus employee cost). So we have just a
single ratio: customers per unit of cost. In this case the efficiencies may
potentially change, hence our colleague is incorrect.
The analysis becomes:
Customers
(’000)
Cost
(£m)
Ratio
(customers per £m of cost)
Efficiency
A
21
19.63
1.07
0.26
B
36
8.725
4.13
1
C
53
13.495
3.93
0.95
D
39
17.325
2.25
0.55
E
30
10.6
2.83
0.69
F
15
16.75
0.90
0.22
Here the efficiencies are derived by comparison with the branch (B) with
the largest ratio of customers/cost.
For part (c)
With respect to the three words:
• ‘Efficient’ for a branch means that they are one of the best performers
(overall) in terms of the set of branches considered. This is not the
same as being best in the entire world.
• ‘Inefficient’ for a branch means one that is not efficient as defined in
the point above.
• ‘Efficiency’ for a branch is a number between zero and 1 (equivalently
0% to 100%) and gives an intuitive understanding of how good a
branch is at converting its inputs into outputs. An ‘efficient’ branch has
efficiency of 1 (equivalently 100%).
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MN3032 Management science methods
Question 6
A company makes two products (X and Y) and has available three machines (A, B
and C) with which it can produce these products. The cost of producing one unit
of each product on each machine is shown below.
Cost per unit (£)
Product
A
B
C
X
0.8
3.2
2.3
Y
0.5
5.4
1.2
Here, for example, producing one unit of product X on machine B costs £3.2 per
unit.
The company is planning its production for the next week. The machines have a
limited weekly production capacity. For machine A its capacity is 1,000 units (X
and Y combined) per week. Machine B can produce at most 500 units of X and
250 units of Y per week. Machine C can produce at most 600 units of X and 150
units of Y per week.
If it is not possible to produce enough to meet forecast demand the company
has the option of buying additional units of X and Y from a supplier, but this
costs £10 per unit of X and £12 per unit of Y.
Products X and Y sell to customers for £9 and £7 per unit (respectively). There
are no units of either product currently in stock.
The company has a number of goals:
• they would like to buy 20 units (X and Y combined) from the supplier
• they would like to meet forecast demand for the next week, which is 1,500
units of X and 750 units of Y
• they would like their profit to equal £14,500
a. Formulate this problem as a weighted goal program with linear constraints.
Note here that you are not expected to simplify any linear equations that you
give. (19 marks)
b. If the company has two priorities:
• priority level 1: minimise the upward deviation from the supplier goal
• priority level 2: minimise the downward deviation from the profit goal
then formulate this problem as a sequential goal program with linear
constraints. Note here that you are not expected to simplify any linear
equations that you give. (6 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.
For part (a)
Let x(A,B,C) (≥0) be the number of units of product X produced on
machines A, B and C respectively; y(A,B,C) (≥0) be the number of units of
product Y produced on machines A, B and C respectively.
Let S(X) and S(Y) (≥0) be the number of units of X and Y respectively
bought from the supplier.
26
Examiners’ commentaries 2016
Then the capacity constraints are:
• x(A) + y(A) ≤ 1000
• x(B) ≤ 500
• y(B) ≤ 250
• x(C) ≤ 600
• y(C) ≤ 150
For the supplier goal we have the constraint:
• S(X) + S(Y) = 20 + a+ – b–
where a+,b- ≥0 are the upward and downward deviation from this goal.
For the demand goal for X we have the constraints:
• x(A) + x(B) + x(C) + S(X) = 1500 + c+ – d–
where c+,d– ≥0 are the upward and downward deviation from this goal.
For the demand goal for Y we have the constraints:
• y(A) + y(B) + y(C) + S(Y) = 750 + e+ – f–
where e+,f– ≥0 are the upward and downward deviation from this goal.
For the total profit goal we have the constraint:
• 9[x(A) + x(B) +x(C) + S(X)] + 7[y(A) + y(B) + y(C) + S(Y)] –
0.8x(A) – 3.2x(B) –2.3x(C) – 0.5y(A) – 5.4y(B) – 1.2y(C) – 10S(X)
– 12 S(Y) = 14500 + g+ – h–
where g+,h– ≥0 are the upward and downward deviation from this goal.
The objective here is to minimise a weighted sum of deviation variables.
Take 1% of the goal values:
• Supplier goal 20/100 = 0.2
• Demand goal for X 1500/100 = 15
• Demand goal for Y 750/100 = 7.5
• Profit goal 14500/100 = 145
minimise
w1(a+/0.2) + w2(b–/0.2) + w3(c+/15) + w4(d–/15) + w5(e+/7.5) +
w6(f–/7.5) + w7(g+/145) + w8(h–/145)
where the w’s are the weights (numeric values) for the deviations.
For part (b)
Sequential goal program
Priority level 1: minimise the upward deviation from the supplier goal
minimise a+
subject to the five inequality constraints and four goal-related equality
constraints (and non-negativity constraints) seen above.
Priority level 2: minimise the downward deviation from profit goal
minimise h–
subject to the five inequality constraints and four goal related equality
constraints (and non-negativity constraints) seen above
and a+ = A*
where A* is the value that a+ has in the solution at priority level 1.
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MN3032 Management science methods
Question 7
a. Company A wishes to develop a materials requirements plan for producing
a particular product (X) over an eight week period. The lead time between
releasing an order to the shop floor and producing a unit of product X is
two weeks. The company currently has 235 units of the product in stock. The
forecast customer demand is shown below.
Week
1
2
3
4
5
6
7
8
Demand
75
75
70
70
125
125
95
65
Here, for example, the demand for product X is 125 units in week 5. By
applying materials requirements planning, decide the orders that need to
be placed for the product in each of the eight weeks. Your answer should
clearly explain the logic behind deciding the orders in materials requirements
planning.
(12 marks)
b. Company B purchases a particular component from an external supplier.
The demand for the component is estimated to be 112 units per month.
The cost of placing an order with the supplier is £55. The component can be
purchased from the supplier at a price per unit of £85 if so desired. However,
the supplier also offers a quantity discount such that if the company orders
125 units or more the price per unit is only £75. The cost of holding this
component (per year) is estimated to be 12% of the price. What advice can
you offer the company as to the cheapest way to purchase this component?
Clearly explain any procedure you follow in arriving at your advice, in
particular the reason as to why you consider each and every order quantity
costed in your answer. (13 marks)
Reading for this question
This question relates to Chapter 5 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.
For part (a)
This part of the question relates to materials requirements planning (MRP)
(pp.102–06 of the subject guide).
Week
1
2
3
4
5
6
7
8
Demand
75
75
70
70
125
125
95
65
On hand at
end of week 160
85
15
0
0
0
0
0
Order
quantity
55
125
125
95
65
n/k
n/k
0
In MRP (subject guide p.104) the logic is that we always order as late as
possible and as little as possible to avoid a stockout.
For part (b)
R = 112 × 12 =1,344
co = 55
P = 85
ch =0.12(85) = 10.2 per year
EOQ = √(2Rco/ch) = √(2 × 1,344 × 55/10.2)
= 120.39 = 120 (or 121)
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Examiners’ commentaries 2016
Total cost per year is RP + (chQ/2) + (coR/Q)
1,344 × 85 + (10.2 × 120/2) + (55 × 1,344/120) = 115,468
Here we cost the EOQ as this (by definition) is the most economical
(least cost) order quantity (see Figure 5.3 on p.92 of the subject guide).
With the discount we have
P=75
ch=0.12(75) = 9 per year
So the new EOQ is
√(2Rco/ch) = √(2×1,344×55/9) = 128.17 = 128 (or 129)
Total cost per year is RP + (chQ/2) + (coR/Q)
1,344 × 75 + (9 × 128/2) + (55 × 1,344/128) = 101,953.5
This is the cheapest option evaluated so advise the company to order 128.
Here, the reason why we cost the EOQ rather than the minimum order
quantity (MOQ) of 125 units is that the procedure to follow when you
have a quantity discount is that if the EOQ ≥ MOQ you cost the EOQ as it
is guaranteed to be cheaper than ordering the MOQ (see Figure 5.6 on
p.97 of the subject guide).
Question 8
A company has formulated a linear program relating to the quantities of two
products (X and Y) to produce. Each unit of product X gives a contribution to
total profit of £2, each unit of product Y gives a contribution to total profit of
£5. Production of the two products is constrained by available working time on
two machines (A and B). In the forthcoming week there are 15 hours available
on machine A, 22 hours available on machine B. In addition there are two
constraints that arise from other considerations (constraints T1 and T2, as seen
below). The linear program the company have formulated is:
Maximise 2x + 5y
Subject to
2x + y ≤ 15
Machine A
3x + 4y ≤ 22
Machine B
3x – 2y ≥ –7
T1
–2x + 3y ≥ –9
T2
x,y ≥ 0
a. Solve this linear program graphically using an iso-cost/iso-profit line, clearly
indicating the feasible region.
(13 marks)
b. For each of the four constraints in this linear program calculate the shadow
price.
(8 marks)
c. Discuss robustness and planning with reference to the specific linear program
considered in this question.
(4 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.
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MN3032 Management science methods
For part (a)
The LP diagram is
The optimal solution is at the intersection of the T1 and machine B lines,
i.e. at x=0.8889, y=4.8333
The associated objective function solution value is 25.9444.
For part (b)
The shadow price is given by changing the right-hand side of a constraint
by 1 and resolving.
The shadow price for both the machine A constraint and the T2 constraint
is zero (equivalently not defined/unspecified), since they are not binding.
For machine B if we increase the right-hand side by 1 the new solution is
x=1, y=5 of value 27; giving the shadow price as 27–25.9444 = 1.0556.
For T1 if we increase the right-hand side by 1 the new solution is
x=1.1111, y=4.6667 of value 25.5556; giving the shadow price as
25.5556–25.9444 = = –0.3888, so 0.3888 if we ignore the minus sign, as
is customary for shadow prices..
For part (c)
Robustness and planning relate to pp.144–145 of the subject guide.
Robustness:
• Change in solution for small data changes, for example an objective
function coefficient (so in this case, profit per unit) or a constraint
coefficient (here, number of required hours per product on machines A
and B).
Planning:
• Change in solution as we change the price of the product, hence
changing the amount produced and so the profit gained.
• Change in solution as we change the machine capacity (number of
working hours), hence changing the amount produced and so the
profit gained.
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