620-362 Assignment 2
Due:
5pm Wednesday 12th October 2005
Submit:
By hand, to A/Prof Boland in Friday’s lecture, or by leaving at her office,
Room 144 in the Richard Berry Building. It is recommended that you
keep a copy of your completed assignment.
Note: This assignment is to be entirely individual work: no collaboration, consultation or
discussion of assignment questions with others is permitted. You must give adequate and
clear explanation to receive full marks.
ASSIGNMENT QUESTIONS
1) A manufacturer of rubber products manufactures car tyres using four types of rubber:
white (W), brown (Br), black (Bl) and heat-resistant (HR). Because of the need to
clean and reset machinery, the time required to produce a batch of rubber depends on
the type of rubber last produced. For example, to switch to white rubber after black
takes longer than to switch to brown rubber. The time (in minutes) required to switch
from one type of rubber to another are shown in Table 1. Note that in this case, the
time to switch between two rubber types is independent of the order, in other words,
to switch from white to black takes the same time as to switch from black to white.
The rubber manufacturer runs a 24-hour a day operation, and wants to determine a
schedule for the production of the four different rubber types that will be repeated ad
infinitum.
a) Explain why the problem of finding the rubber production schedule that
minimized the switching time can be modeled as a Traveling Salesman Problem
(TSP).
b) Apply the Nearest Neighbour Heuristic, starting with black rubber, to obtain a
feasible schedule. What is the schedule and how much switching time is
incurred?
c) Consider the 2-opt neighbourhood of the solution you found in part (b). What are
the neighbours of this solution? Give the schedule and switching time incurred by
each neighbour.
d) For each neighbour of the Nearest Neighbour solution, found in part (c), what is
the probability that the Simulated Annealing algorithm would accept the
neighbour as the new current solution, if the current temperature is T=30?
e) Apply the Nearest Insertion Heuristic, starting with the initial schedule HR-W-Br,
to obtain a feasible schedule. What is the schedule and how much switching time
is incurred?
f) Consider the 2-opt neighbourhood of the solution you found in part (e). What are
the neighbours of this solution? Give the schedule and switching time incurred by
each neighbour.
g) Is either of the solutions you found in parts (b) or (e) a local optima with respect
to their 2-opt neighbourhood? Explain your answer in both cases.
Table 1. Switching times in minutes between production of different rubber types
Rubber to be produced next
Last rubber produced
HR
W
Bl
Br
HR
50
90
70
W
50
140
110
Bl
90
140
80
Br
70
110
80
2) The following data represents the returns recorded by each of three assets over the
last 12 months, as percentages.
Asset
1
2
3
1
12.5
7.5
-1.0
2
8.0
2.0
3.5
3
3.5
5.0
6.5
4
9.0
5.5
5.5
5
-1.5
6.0
2.5
Month
6
7
0.5
6.0
7.5
7.0
4.5
5.5
8
10.5
7.0
6.0
9
4.5
8.0
6.0
10
11.0
2.5
5.0
11
2.0
-0.5
6.0
12
8.5
5.5
5.5
An investor has $60,000 to spend and wishes to invest it in some combination of
these three assets. Throughout this question you may make use of/modify as you
require the generic Xpress-IVe portfolio optimization model available via the 620362 website.
a) Write down the expected return for each asset, correct to 4 decimal places. Write
down the covariance matrix for the returns of the 3 assets, correct to 4 decimal
places. What is the highest return the investor can expect to get from a portfolio
chosen from these three assets? What is the lowest return of any portfolio chosen
from these assets? What is the least risk, measured by the portfolio covariance,
that the investor can expect to be exposed to, if she invests entirely in only one
asset? What would her return be in this case?
b) Determine the minimum portfolio risk for five different values of the expected
return between the highest and lowest possible values (from (a)), including these
highest and lowest values. To do this, you may use the given Xpress-IVe file.
You will need to insert appropriate values in the data structures. Write down the
values you inserted, or include a printout of the file, in your answer. Plot the five
points you obtain on a graph of minimal portfolio risk versus desired return, and
sketch the graph by “joining the dots”. The resulting curve is known as the
efficient frontier.
c) The investor has decided to seek a portfolio with an expected return of 5.5
percent. What is the minimum risk portfolio achieving the expected return of 5.5
percent? How much should be invested in each of the three assets? What is the
risk?
d) In fact, for each asset the investor includes in the portfolio, a fixed overhead cost
of $2000 must be paid to the broker, which must be deducted from the budget.
For example, if the investor wishes to split her portfolio 50-50 between the first
two assets, she must first pay the broker $4000, which leaves her $28000 to invest
in each of assets 1 and 2. Explain how to include such overhead costs in the
general portfolio optimization model, in such a way that all constraints in the
model remain linear in all variables. (Hint: you will need to introduce new extra
binary variables.) Do not attempt to implement or solve your resulting model.
Use xi to denote the amount invested in asset i, ri to denote the expected return of
asset i, and cij to denote the covariance of the returns of assets i and j, for each i, j
= 1,…,n, where there are n assets. Let v denote the return required of the
portfolio, and use a budget scaled to 1.
3) The assignment problem can be stated as follows. Given two sets of items, A and B,
of equal cardinality, i.e. with |A| = |B|, for each item i  A, we seek to assign it to a
unique item j  B. The cost of assigning i to j is denoted by cij. Each item in B must
be assigned exactly one item in A, so as to minimize the total cost.
For example, a freight railway company likes to ensure that each train driver
experiences different parts of the track network with regularity, so keeping up their
knowledge and familiarity with the track. Furthermore, drivers have been found to be
less fatigued if they avoid too-frequent repetition of trips. The company has
developed a numerical system for rating the incompatibility of a driver with a
particular trip, based on the trips he or she has done recently. The degree of
incompatibility of a trip with a driver is measured by the number of track segments in
the trip that also appeared in trips he or she has driven recently. The company has
assessed the incompatibility of each of its five drivers with each of a set of five
upcoming trips, shown in the table below. The company must assign each driver to
one trip, and every trip must be assigned a driver.
Incompatibility
Drivers
Trips
1
2
3
4
5
Taylor
5
15
20
25
10
Landman
10
12
5
15
19
Xia
5
17
18
9
11
Borovkov
8
9
10
5
12
Abad
9
10
5
11
7
One possible solution to this problem would be to assign driver Taylor to trip 5,
driver Landman to trip 3, driver Xia to trip 1, driver Borovkov to trip 2, and driver
Abad to trip 4. Every driver has been assigned a trip to drive, and every trip has been
assigned a driver. The “cost” of this solution is its total incompatibility rating: 10 + 5
+ 5 + 9 + 11 = 40.
There are several very efficient ways to solve assignment problems, but here we will
consider heuristics, and branch-and-bound.
a) Design a greedy heuristic to solve the assignment problem. Describe your
heuristic as clearly and concisely as possible, illustrated with the train driver
example. What is the solution to the train driver problem you obtain with your
heuristic, and what is its total incompatibility rating?
b) The branch-and-bound method can be used to solve assignment problems. For
example, using the train driver problem to illustrate, we may begin by branching
on the driver assigned to trip 1. This creates five branches: driver Taylor assigned
to trip 1, driver Landman assigned to trip 1, driver Xia assigned to trip 1, driver
Borovkov to assigned trip 1, and driver Abad assigned to trip 1. How can we
obtain a lower bound on the total cost associated with a branch? Consider the
branch with driver Landman assigned to trip 1. On this branch, Landman cannot
be assigned to any trip other than 1, and no other driver can be assigned to trip 1.
Thus no other assignments can come from row two or column one of the table.
Now we cannot do better for the other trips than to assign them their least
incompatible driver, so assigning drivers to the other trips will cost at least the
sum of the minimum costs in each column, excluding those in row two. Thus the
cost is at least 9 (trip 2) + 5 (trip 3) + 5 (trip 4) + 7 (trip 5) = 26. Since assigning
Landman to trip 1 costs 10, the total cost for this assignment must be at least 36.
Similarly, we cannot do better for the other drivers than to assign them to their
least incompatible trip, so assigning the other drivers will cost at least the sum of
the minimum costs in each row, excluding those in column one. Thus the cost is
at least 10 (for Taylor) + 9 (for Xia) + 5 (for Borovkov) + 5 (for Abad) = 29.
Since assigning Landman to trip 1 costs 10, the total cost for this assignment must
be at least 39. Since the cost of assigning Landman to trip 1 is at least 36 and is at
least 39, we can take the greater of these as the (better) lower bound, i.e. 39 is a
lower bound at this node.
Continue this idea to solve the train driver assignment problem by branch-andbound. Clearly show the branch-and-bound tree, indicating the lower bound at
each node. Clearly label each branch, to show how it is defined. Number the
nodes in the order explored. Note that you may use the feasible solution you
found with your heuristic in part (a) to prune the tree. If you pruned the tree
below a particular node, explain clearly your reasons for doing so: did you use
your heuristic solution to prune the node, or did you use a feasible solution found
at another node to prune it, and if so, what node was that?