EPM945
CITY UNIVERSITY
London
Optimization and Decision Making
2016
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. 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) Show that the expected revenue from the portfolio is
∑n
i=1wiµi, and
find an expression for the variance of the portfolio revenue. [8]
(b) Formulate the general n stock problem using Lagrange multipliers,
and hence write down a set of equations for the optimal value of the
wis. Hence use this set of equations to show that the optimal solution
for the 3 stock problem is given by the following equations:
[µ2(σ
2
1 − σ13) + µ1(σ13 − σ12) + µ3(σ12 − σ
2
1)]w1+
[µ3(σ
2
2 − σ12) + µ2(σ12 − σ23) + µ1(σ23 − σ
2
2)]w2+
[µ1(σ
2
3 − σ23) + µ3(σ23 − σ13) + µ2(σ13 − σ
2
3)]w3 = 0,
w2 =
(µP − µ3)− w1(µ1 − µ3)
µ2 − µ3
,
w3 =
(µ2 − µP ) + w1(µ1 − µ2)
µ2 − µ3
.
[12]
(c) Now consider a problem with three stocks where the means, variances
and covariances are as follows:
µ1 = 0.14, µ2 = 0.10, µ3 = 0.08, σ
2
1 = 0.6, σ
2
2 = 0.3, σ
2
3 = 0.4, σ12 =
0.2, σ13 = −0.2 and σ23 = 0.1.
Find the optimal portfolio (i.e. the one with the minimum variance)
if the targeted expected rate of return is 0.10. [10]
2 Turn over . . .
2. A construction company must decide whether to invest money in buying a
derelict commercial building and potentially developing it into residential
accommodation. The company estimates that there is a 70% chance that
permission for a change of use will be obtained to develop the site within
five years.
If change of use permission has not been obtained at the end of this period,
the company would abandon the project and resell the site, with a loss of
£3 million.
If permission has been obtained, the type of development would have to be
decided upon, Houses or Flats. The returns generated would depend upon
the strength of the economy, which for simplicity have been categorized
as either Strong or Weak.
If the company opted for Flats and demand was Strong, a £6 million profit
would be obtained. If the demand was Weak in this case, however, there
would be a loss of £1 million.
If the company decided to build Houses, Strong demand would generate
a profit of £4 million and Weak demand would generate a profit of £1
million.
The company estimates that there is a 65% 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 change of use per-
mission may not be correct. Assuming that all other problem ele-
ments 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) -3 -1 0 1 4 6
Utility 0 0.3 0.5 0.7 0.9 1
3 Turn over . . .
3. Consider the primal linear program below.
Maximise 3×1 + 2×2 + 2×3,
subject to
x1 + 4×2 + 2×3 ≤ 9,
6×1 + x2 + 2×3 ≤ 12,
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 9 replaces 12 in the second constraint in the primal
problem, i.e. we have 6×1 + x2 + 2×3 ≤ 9. Find the solution to this
new problem. [6]
4. A luxury furniture manufacturer makes four different types of chair; Din-
ing Chairs, Armchairs, Reclining Chairs and Sofas. There are different
materials, but the only one in short supply is locally produced wood which
is the only wood they use. The manufacturer has broken down the require-
ments into two for each chair, Labour and Wood. The table below lists
the four types, the Labour and Wood required for each chair, the total
current (weekly) supply of Labour and Wood to the manufacturer and the
profit per chair.
Chair type Dining Armchair Reclining Sofa Supply
Labour (hours) 2 5 10 10 3000
Wood (kg) 10 15 15 25 6000
Profit (pounds) 40 100 120 150
(a) Set up the problem as a linear program. [12]
(b) Solve the problem using the simplex method. [18]
4 Turn over . . .
5. Three distribution centres send a popular brand of television to four
branches of an electrical retailer. The quantities that the distribution
centres can deliver each week is limited because the manufacturers stock
them weekly. The sales at each branch and the capacities of the distribu-
tion centres are as follows:
Centres A, B and C have stocks of 20, 30 and 50 respectively.
Branches 1, 2, 3 and 4 require a supply of 35, 25, 20 and 15 respectively.
The transportation costs (in pounds) of supplying the branches are given
in the following table. The aim is to minimise the total cost of transporting
the television sets.
1 2 3 4
A 10 12 8 14
B 6 18 14 17
C 16 10 7 9
(a) Formulate, but do not solve, the problem in the form of a linear
program. [7]
(b) Now write the problem in the form of a transportation array, explain-
ing how you deal with the fact that the total demand and the total
supply are not equal. [7]
(c) Find the optimal transportation solution. [16]
Internal Examiner: Professor M. Broom
External Examiners: Dr E.J. Collins
Dr L. Jackson
5