Modelling in Management Science Questions

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EXAMINATION PAPER
Academic Year:
2019/2020
Examination Period
SPRING
Examination Paper Number:
Examination Paper Title:
MODELLING IN MANAGEMENT SCIENCE
Duration:
2 HOURS
Do not turn this page over until instructed to do so by the Senior Invigilator
Structure of Examination Paper:
There are 6 pages
This examination paper is divided into 2 Sections
There are 4 questions in total.
The maximum mark for the examination paper is 100% and the mark obtainable for a
question or part of a question is shown in brackets alongside the question.
Students to be provided with:
Formula Sheet
Statistical Table
The following items of stationery are to be provided at the start of the examination:
Answer book
Instructions to Students:
Answer THREE questions. Answer Compulsory question 1 from Section A. Answer Two
questions only from Section B.
Calculators may be used though all workings should be shown to obtain full credit.
The use of non-electronic translation dictionaries between English or Welsh and a foreign
language bearing an appropriate departmental stamp is permitted in this examination.
Please turn over
-1-
BS3619
SECTION A
Answer all of this section
1
(a)
Explain how the management science of waiting lines can be considered in a
marketing context (such as with supermarkets).
(4 marks)
(b)
Briefly describe, going beyond listing bullet points as in lecture material, what
you think the important considerations in simulation modelling are in regard to
a global virus outbreak (give your thoughts on the possible ‘reasons for’ and
possible ‘shortcomings of’ modelling specifically in this context). (6 marks)
(c)
Briefly explain the Midsquare random number generator technique, and
demonstrate its workings by performing four iterations of the technique, with
the initial seed 1536 (finding random value in each iteration). Are there any
concerns associated with this technique for random number generation?
(5 marks)
(d)
The snapshot below shows a consecutive group of six customers arriving at a
bank help desk (where customers can ask for help). For the purposes of the
simulation, the help desk is considered to open at 10.30am and closes around
2.30pm. The time periods considered here are units of 1 hour, with associated
estimated arrival and service rates of six and nine customers per time period,
respectively.
Customer
Arrival
Number
9
10
11
12
13
14
Inter
Arrival
Time
0.3682
0.0420
0.0453
0.1366
0.2030
0.0601
Arrival
Time
Service
Start
Service
Time
Service
End
Waiting
Time
System
Time
0.9928
1.0348
1.0801
1.2168
1.4198
1.4798
0.9928
1.0650
1.0903
1.2168
1.4198
??
0.0722
0.0253
0.0214
0.1747
0.0666
0.0499
1.0650
1.0903
1.1117
1.3915
1.4864
??
0.0000
0.0302
0.0102
0.0000
0.0000
??
0.0722
0.0555
0.0316
0.1747
0.0666
0.0564
Briefly describe the important points in the production of the above simulation
model (as described by the snapshot), including how the model has utilised the
negative exponential distribution (in respect to both arrivals and services).
From the above simulation snapshot:
(i)
(ii)
(iii)
(iv)
What time did the 10th customer arrive at the bank help desk?
How long after the 10th customer did the 11th customer arrive?
How long did the 11th customer have to wait before being served? Why
did they have to wait?
The details for the 14th customer are incomplete – can you fill in the
gaps denoted by ?? for this customer?
(12 marks)
Question 1 continues overleaf
-2-
BS3619
(e)
A popular bakery in a village in North Cardiff sells many different types of
bread (and cakes). The service rate for the single queue, and single serving
kiosk, in the bakery, is believed to follow a negative exponential distribution
and is estimated to be 18 customers every hour. What is the probability of;
(i)
(ii)
A customer serviced in less than or equal to 5 minutes.
Four customers served in 30 minutes.
(5 marks)
(f)
A local chemist has become increasingly popular over the last few years, and
it is part of a large chain of chemists, hence has a local and regional manager
overseeing its operation. Since it was recently renovated, exiting the chemist
means touching a big button attached to a railing to help customers balance
themselves when opening the heavy door.
The chemist currently uses a single checkout, and works with a single queue
on a first come first served basis (they do have space for a second checkout
but have not previously employed it). To the local manager of the chemist,
sometimes the checkout is busy and sometimes it is not (throughout its
opening times of between 10.00am and 6.00pm each day).
The local manager of the chemist estimates the arrival rate of customers (to
the chemist checkout or queue) is 12 customers per hour (following a Poisson
distribution). An estimate for the average service time at the checkout is four
minutes (following a negative exponential distribution). Give an analysis of
the associated waiting line system surrounding the checkout.
As mentioned previously, the door to the chemist means it is awkward to
operate. So much so that when there is a queue to the checkout – there is little
room to get past the queue to leave the chemist (the handrail the exit button is
attached to means movement is restricted around the door). Watching the
entering and leaving of customers to the chemist over a period of time has
suggested that more than two customers in the queue to the checkout can
cause a problem to someone wanting to leave the chemist.
Further, the local manager of the chemist is concerned by the amount of time a
customer spends in the chemist, hence they are keen to keep customers, on
average, spending less than four minutes in the chemist (system).
The regional manager of the chemist is also interested in developments in this
chemist, but is concentrating on costs. Based on some calculations, they have
estimated £4 and £15 (both per hour) assigned to cost of waiting and cost of
servicing, respectively, in relation to the chemist checkout.
Comment on the appropriateness of introducing the further checkout in the
chemist, based on what has been written above.
(16 marks)
Please turn over
-3-
BS3619
SECTION B
Answer two questions only from this section
2.
(a)
Give a summary of the basic EOQ Model assumptions, as given in your
lectures.
(3 marks)
(b)
Explain the construction (with the aid of a diagram) of the formula for the
planned shortage economic order quantity model, including the expression for
the total cost of inventory in this case. Include also an explanation of other
factors associated within this model in the diagram constructed.
(5 marks)
(c)
A chemical factory produces medicine for use in hospitals. An important key
ingredient for one medicine in high demand (with hospitals needing the
medicine as soon as it is produced) is here called Nitrium (not its real name).
Demand for this chemical is around 50 capsules per day, with each pack of 5
capsules costing £20.00 (but can be bought individually also at same relative
price). The chemical plant operates all 364 days of the considered year. Their
current inventory management scheme for this chemical (capsules), is through
the ordering of 4550 capsules (or 910 packs) roughly every three months (an
order all arrives at the same time and no shortages). Also associated with this
model are the estimated cost of ordering at £20 per order and cost of holding is
15% the unit price.
Comment on the current inventory management position regarding these packs
of chemicals. Is there room for improvement in their inventory policy based
on the evidence given (using an inventory management model you are
acquainted with from your lectures)?
A newly appointed operations manager, who came from the car industry, has
suggested they work towards a planned shortages model (as they had in the car
industry). They have done some research and believe the cost of backordering
this chemical is £6 per backorder (per capsule). What further advice can you
give to the managers of the chemical plant, in regard to the inventory
management of the capsules of the chemical?
For only the most cost-effective model in your analysis, knowing there is a
lead time of 20 days for ordering this chemical, what is the associated reorder
point?
Draw a well labelled inventory-time diagram for the most cost-effective
inventory model.
What other thoughts do you have on your answers and subsequent conclusions
on this problem.
(13 marks)
(d)
Describe mathematically and in words the conditions that certain inventory
models converge to the basic EOQ model, which you have considered?
(5 marks)
-4-
BS3619
3.
(a)
Describe the assumptions made to move from Stochastic Process model to a
Markov Chain type model.
(3 marks)
(b)
Within a supermarket context, how would managers of one supermarket chain
utilise Markov chain models in their analysis of their market share in a city.
A supermarket’s advertisements on TV often compare their supermarket costs
of products against another specific supermarkets’ costs of products – how
would a Markov Chain model (and in particular an associated transition
matrix) be used to identify which competitor supermarket to target in this way.
(5 marks)
(c)
Draw a transition matrix T for the following system diagram of six states;
What features are included
in the presented transition
matrix (using also the X
and 0 notation to include
what the structure of the
associated T may look
like)?
(8 marks)
(d)
Many companies worry about the payment behaviour of their customers,
including whether a debt gets repaid or not. In this problem, a company has
looked at the transition of debts from customers on a monthly basis.
Month on month, a debt can be paid (P), be less than one month in existence
(M1), between 1 and 2 months in existence (M2) and an account currently not
paid after two months is classed as bad-debt (B). The table below describes
the company’s debt problem in terms of transient probabilities between states
(on occasions debt can be kept at the same month state month on month).
State
Paid (P)
Less than 1 month (M1)
Between 1 and 2 months (M2)
Bad-debt (B)
P
1
0.2
0.5
0
M1
0
0.2
0.0
0
M2
0
0.6
0.3
0
B
0
0.0
0.2
1
Draw an event diagram over two time periods from the M2 (‘Between 1 and 2
months’) transient state.
Given that there are currently £35,000.00, in debt which is less than one
month old, in the long run, what amount of this debt will be paid and what
amount will be bad-debt?
(10 marks)
Please turn over
-5-
BS3619
4.
(a)
What is Management Science, including the associated three levels as
described in your lectures?
Comment on how you think the three levels of management science could
have been considered regarding the current global virus pandemic.
(7 marks)
(b)
A Welsh museum sells medium-sized Welsh flags with current year
embroidered (sewn) onto it. They are bought by both tourists as well as sports
fans who wave them at sporting events. They currently estimate demand for
this Welsh flag is estimated to follow a normal distribution with mean 2,000
flags and standard deviation 300 flags.
The medium-sized flags usually cost the museum £23.00 per flag. Unsold
medium-sized flags, since they have the current year sewn on them, have a
salvage costs estimated to be £8.00 each. If demand for flags exceeds
available stock, quantities can be purchased at an additional cost of £3.00 each
on their price to purchase a flag.
By using the method of incremental analysis, which can be used to determine
the optimal order quantity for a single period inventory model that has
probabilistic demand, what is the recommended order quantity for the
medium-sized flags?
Draw a normal distribution graph to illustrate your evaluation of your answer.
(5 marks)
(c)
Describe advantages and disadvantages associated with the Quantity Discount
Model, as discussed in the lectures.
(4 marks)
(d)
A local hospital uses a lot of face masks to protect their workers when dealing
with infectious patients. The estimated annual demand for packs of 10 face
mask is 15,000 packs. The hospital management are investigating the
application of a quantity discount model to manage the inventory of these
packs of face masks. Annual holding costs are 35% of the price of a pack of
face masks and the ordering costs are £8.00 per order. The following discount
price schedule is offered to the hospital manager by their supplier:
Order quantity of packs
0 − 499
500 − 4,999
5,000 +
Price per pack of face masks
£4.00
£3.79
£3.69
Determine the optimal order quantity and the respective total annual inventory
cost.
Sketch a graph showing how the costs change for the different order quantity
options expressed.
(10 marks)
-6x-

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chemistry

Management Science

negative exponential distribution

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