EPM945
CITY UNIVERSITY
London
MSc in Project Management, Finance and Risk
Optimization and Decision Making
2014
Time allowed: 2 hours
Full marks may be obtained for correct answers to
THREE of the FIVE questions.
All necessary working must be shown.
1 Turn over . . .
1. A property company must decide whether to invest money in buying a
plot of land and potentially developing it into a housing estate. The
company estimates that there is a 50% chance that planning permision
will be obtained to develop the site within five years.
If planning permission has not been obtained at the end of this period,
the company would abandon the project with a loss of £2 million.
If planning permission has been obtained, the type of development would
have to be decided upon, Luxury or Economy. The returns generated
would depend upon the nature and level of demand, which for simplicity
have been categorized as either Strong or Weak.
If the company opted for Luxury development and demand was Strong,
a £6 million profit would be obtained. If the demand was Weak in this
case, however, the profit would be £1 million.
If the company decided to go for the Economy development, Strong de-
mand would generate a profit of £4 million and Weak demand would
generate a profit of £2 million.
The company estimates that there is a 60% chance that Strong demand
will occur.
(a) Construct a decision tree to represent this decision problem. [15]
(b) Assuming that the company’s objective is to maximize its expected
profit, determine the policy that it should adopt. [5]
(c) The company’s estimated probability of obtaining planning permis-
sion may not be correct. Assuming that all other problem elements
remain the same, determine how low this probability would have to
be before the option of not developing the site should be chosen. [5]
(d) Before the final decision is taken the company is taken over by a new
owner, who has utilities shown below for the sums of money involved
in the decision. The new owner has no interest in other attributes
which may be associated with the decision. What implications does
this have for the policy that you found from b) and why? [5]
Value (£ million) -2 0 1 2 4 6
Utility 0 0.4 0.5 0.7 0.9 1
2 Turn over . . .
2. Suppose that you have to choose an optimal portfolio from a list of n
stocks. Stock i has expected revenue rate µi with variance σ
2
i for i =
1, . . . , n, and the covariance of the revenues of stocks i and j is given by
σij for i 6= j, i, j = 1, . . . , n. The proportion of stock i in the portfolio is
denoted by wi.
(a) Showing your working carefully, show that the expected revenue from
the portfolio is
∑n
i=1wiµi, and find an expression for the variance of
the portfolio revenue, again showing your working carefully. [8]
(b) Still for a general number of n stocks, formulate this as an opti-
mization problem using Lagrange multipliers, and find a set of linear
equations for the optimal values of the wis. You do not need to solve
the problem at this stage. [7]
(c) Now consider a problem with three stocks, where the means, vari-
ances and covariances are as follows:
µ1 = 0.15, µ2 = 0.10, µ3 = 0.08, σ
2
1 = 0.6, σ
2
2 = 0.2, σ
2
3 = 0.3, σ12 =
0.3, σ13 = −0.1 and σ23 = 0.1.
Find the optimal portfolio (i.e. the one with the minimum variance)
for the expected rate of return of 0.10. [15]
3. A small specialist toy manufacturer makes four different types of soft toy; a
Bear, a Dog, a Horse and a Squirrel. The construction of each animal takes
part in two phases, Fabrication (making the fabric animal) and Completion
(stuffing and sewing up of the animal). The Fabrication times for each
animal are 10 minutes for a Bear, 15 minutes for a Dog, 20 minutes for a
Horse and 20 minutes for a Squirrel. The Completion times for each animal
are 6 minutes for a Bear, 6 minutes for a Dog, 10 minutes for a Horse
and 8 minutes for a Squirrel. There are five workers in the Fabrication
department, giving a total of 2400 minutes of labour time per day. There
are two workers in the Completion department, giving a total of 1000
minutes of labour time per day. The proft for each animal is 6 pounds for
a Bear, 6.5 pounds for a Dog, 10 pounds for a Horse and 9 pounds for a
Squirrel. The company wish to maximise their profit, subject to the time
constraints from their current workforce.
(a) Set up the problem as a linear program. [12]
(b) Solve the problem using the simplex method. [18]
3 Turn over . . .
4. Four different factories send a given product to three depots (for further
distribution) each in a different city. The quantities that the factories can
deliver each week is limited by their production. The production at each
factory and the demand at each depot are as follows:
Factories A, B, C and D have production levels of 10, 15, 20 and 15 respec-
tively. Depots 1, 2 and 3 require a supply of 15, 25 and 20 respectively.
The transportation costs (in pounds) of supplying the depots are given in
the following table. The aim is to minimise the total cost of transporting
the items.
1 2 3
A 5 10 11
B 13 9 20
C 12 17 19
D 16 12 22
(a) Formulate, but do not solve, the problem as a linear program. [7]
(b) Now write the problem in the form of a transportation array and
hence find the optimal transportation solution. [16]
(c) How would the solution to the above problem change if the cost of
transportation from factory B to depot 1 fell to 7? [7]
5. Consider the primal linear program below.
Maximise 2×1 + 4×2 + 2×3,
subject to
x1 + 6×2 + 4×3 ≤ 24,
4×1 + 6×2 + x3 ≤ 21,
x1, x2, x3 ≥ 0.
(a) Construct the dual problem and solve it graphically. [12]
(b) Determine a solution to the primal problem, stating clearly any re-
sults that you use. [12]
(c) Now suppose that 90 replaces 24 in the first constraint in the primal
problem, i.e. we have x1 + 6×2 + 4×3 ≤ 90. Find the solution to this
new problem. [6]
Internal Examiner: Professor M. Broom
External Examiner: Dr L. Jackson
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