PAPER CODE
MATH 268
EXAMINER: Dr. A.Piunovskiy
DEPARTMENT: Mathematical Sciences
May 2020 Final Assessments
Operational research: probabilistic models
Time allowed: 24 hours
INSTRUCTIONS TO CANDIDATES: Answer all of Section A questions.
Marks for all questions in Section A and the THREE best marks for questions
in Section B will be taken into account.
The marks shown against questions, or parts of questions, indicate their rela-
tive weight. Section A carries 55% of the available marks, section B carries 45%
of the available marks.
By submitting solutions to this assessment you affirm that you have read and
understood the Academic Integrity Policy detailed in Appendix L of the Code of
Practice on Assessment.
Paper Code MATH 268 Page 1 of 5 CONTINUED
SECTION A
1. Suppose U is a standard uniform random variable on [0, 1]. Having in hand
the simulated values u1, u2, . . ., describe the method for calculating the simulated
values v1, v2, . . . of another random variable V :
(a) in case V is binomial with parameters n = 5, p = 0.3; [5 marks]
(b) in case V has the Pareto cumulative distribution function
F (x) =
 1−
(
1
x
)2
, if x ≥ 1;
0 otherwise.
[5 marks]
2. The observed values for a financial index are given in the table
x1 x2 x3
5092 5111 5132
Accept the linear trend forecasting model with the initial estimates for the
level A and slope B being x0 = x1 = 5092; T1 = 0. Apply the Holt’s method,
i.e. compute F1, L2, T2, F2, L3, T3, F3, L4, T4, F4 using the level-related and
slope-related smoothing constants α = 0.8 and β = 0.5. Give the forecast for x4
and for x5.
[10 marks]
3. Jimmy can order 10, 15, 20, 25, or 30 copies of a popular DVD. Assume
for simplicity that the demand is also restricted to one of the 5 levels specified
above. Jimmy pays £5 per copy and sells them for £10 each. Any copies that
are not sold by the end of the day are disposed off at £3 each, i.e., the cost is
£5-3=2.
(a) Write down the payoff table. [6 marks]
(b) Find the optimal decision using the Laplace’s criterion. [4 marks]
Paper Code MATH 268 Page 2 of 5 CONTINUED
4. A Markov Decision Process is given by the following elements.
S = {1, 2}, A = {1, 2} are the state and action spaces.
a 1 2
s
1 5 2
2 3 4
is the one-step cost matrix c(s, a).
From
s To y 1 2
1 1/3 2/3
2 2/3 1/3
is the transition matrix p(y|s, a) when a = 1.
From
s To y 1 2
1 1/2 1/2
2 0 1
is the transition matrix p(y|s, a) when a = 2.
(i) Write down the optimality equation for Vt(s), the minimal total cost starting
from the time t, if the state is X(t) = s. [5 marks]
(ii) Suppose V4(1) = 100; V4(2) = 150 and compute V3(s) along with the optimal
actions a∗4 at the corresponding time 4. [8 marks]
5. A community is served by two cab companies. Each company owns two
cabs, and the two companies are known to share the market almost equally. This
is evident by the fact that calls arrive at each company’s dispatching office at the
rate of eight per hour. The time intervals from the start of a call to the end of
the ride are exponentially distributed and have the average of 12 minutes. The
time intervals between the calls are again exponentially distributed. If both cabs
are busy, a company rejects the new arriving call. For each company, investigate
the corresponding queuing system.
(i) Describe the states and identify all the parameters of the system.
[3 marks]
(ii) Draw the transition diagram. [3 marks]
(iii) Write down and solve the balance equations with the normalization con-
dition. [6 marks]
Paper Code MATH 268 Page 3 of 5 CONTINUED
SECTION B
6. Consider the independent random variables X1 ∼ Exp(2), X2 ∼ Exp(3),
Y1 ∼ Uniform(1, 2), Y2 ∼ Uniform(2, 3). Let
Z = α(2X1 + Y1) + (1− α)(X2 + 3Y2),
where α ∈ [0, 1] is the decision variable.
(i) Derive E[Z] as function of α. [4 marks]
(ii) Find the range of values of α such that E[Z] ≥ 5. [2 marks]
(iii) Derive V ar[Z]. [4 marks]
(iv) Find the value of α that minimizes the expected value-variance criterion
J(α) = E[Z] + 3V ar[Z].
[5 marks]
7. Colaco currently has assets of £150,000 and wants to decide whether to
market a new chocolate-flavoured soda, Chocola. Colaco has three alternatives:
Alternative 1. Test market Chocola locally, then utilize the results of the market
study to determine whether or not to market Chocola nationally.
Alternative 2. Immediately (without test marketing) market Chocola nationally.
Alternative 3. Immediately (without test marketing) decide not to market Chocola
nationally.
In the absence of a market study, Colaco believes that Chocola has a 55%
chance of being a national success. In this case, Colaco’s asset position will
increase by £300,000, and if Chocola is a national failure, Colaco’s asset position
will decrease by £100,000.
If Colaco performs a market study (at a cost of £30,000), there is a 60%
chance that the study will yield favourable results (referred to as a local success)
and a 40% chance that the study will yield unfavourable results (referred to as a
local failure). If a local success is observed, there is an 85% chance that Chocola
will be a national success. If a local failure is observed, there is only a 10% chance
that Chocola will be a national success. Colaco wants to maximize its expected
final asset position.
(i) Draw the full decision tree for Colaco. [5 marks]
(ii) Elaborate the optimal policy for Colaco and calculate the corresponding
expected final asset position. [10 marks]
Paper Code MATH 268 Page 4 of 5 CONTINUED
8. Suppose the sales in millions of pounds for a department store were as
follows.
Year 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019
Sales S(t) 21.0 23.2 23.2 24.0 24.9 25.6 26.6 27.4 28.5 29.6
(i) Construct the linear regression model
Yt = A+Bt, t = 1, 2, . . . , 10
using the method of least squares, that is, provide formulas for A and B and do
all the calculations. (You can encode year 2010 as t = 1, 2011 as t = 2, and so
on.) [11 marks]
(ii) Hence, evaluate the forecast for sales in 2020 and 2021. [4 marks]
9. Customers arrive at a local library according to a Poisson process at the
rate of a = 2 per hour. Each person stays in the library for an average of b = 24
minutes. Assume that the time spent by a person in the library is exponentially
distributed and independent of the other customers. Suppose the library is big
enough to hold customers always. We are interested in the number L of people
in the library.
(a) Suggest an appropriate queuing model with detailed values of parameters
for this scenario. [3 marks]
(b) Is this queuing system stable? Sketch the transition diagram and write
down the balance (steady state) equations along with the normalisation condition.
[4 marks]
(c) Solve the balance equations and calculate the average number of people
in the library. [5 marks]
(d) Do there exist such values of parameters a ≥ 0 and b ≥ 0 that the system
is not stable? [3 marks]
Paper Code MATH 268 Page 5 of 5 END















































































































































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