THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH3161 OPTIMIZATION Term 1, 2021
ASSIGNMENT
Your answers to this assignment must be submitted via Moodle before 5:00pm, April
23, 2021. Late assignments will not be accepted except on documented medical or compassionate
grounds.
Assignments must include a signed cover sheet available from the School of Mathematics and
Statistics web site at:
http://www.maths.unsw.edu.au/sites/default/files/assignment.pdf
School date stamp is not needed on the cover sheet .
Marking of this Assignment
Each question on this assignment is worth 25 marks. A full mark of 50 on this as-
signment is worth 5% of the total marks for MATH3161. Present your work as a self
contained, well-written report including the problem statement, solution summary, clear
interpretation of solutions, model formulation, definition of problem variables and your
computer output. State clearly the assumptions that you make. Any Matlab files that
you modify/write should be included as an appendix.
1. Minimum Cost Pizza Problem. Using only the items given in the tables below, formulate
an optimization problem in standard form to create a minimum cost pizza which satisfies both the
nutritional requirements of Table 1 and bounds on item quantities given in Table 2. Use the nutritional
data of Table 3 and the cost data of Table 4 in your model.
Use the MATLAB linear optimization routines linprog to solve the problem. Interpret your results.
Table 1
Nutrient Requirement Units
Calcium 750.0 mg
Iron 12.0 mg
Protein 48.5 gram
Vitamin A 4500.0 IU
Thiamine 1.3 mg
Niacin 16.0 mg
Riboflavin 1.6 mg
Vitamin C 30.0 mg
1
Table 2
UPPER AND LOWER BOUNDS ON PIZZA ITEMS
Item Upper Bounds* Lower Bound*
Sauce 1.986 1.140
dough 5.249 4.266
cheese 2.270 1.703
pepperoni 0.983 N/A
ham 1.135 N/A
bacon 0.993 N/A
g.pepper 1.561 N/A
onion 0.993 N/A
celery 1.561 N/A
mushroom 1.135 N/A
tomato 1.703 N/A
pineapple 1.703 N/A
meat N/A 0.993
veg. N/A 0.993
fungi N/A 0.922
* Amount in hundreds of grams.
Table 3
NUTRITIONAL DECOMPOSITION OF PIZZA ITEMS*
Item Calc Iron Prot Vit A Thia Niac Ribo Vit C
cheese 517.700 .222 20.000 3000.000 .022 6.000 .244 -
Sauce 14.000 1.800 2.000 800.000 0.100 1.400 .060 6.000
dough 18.233 3.826 14.224 - .586 8.852 .628 -
pepperoni 10.000 2.500 15.000 - - 2.000 - -
ham 9.031 2.291 14.692 - .740 4.009 0.178 -
bacon 13.000 1.189 8.392 - .361 1.828 .114 -
g.pepper 9.459 .675 1.351 209.460 .081 .540 .081 127.030
onion 27.273 .545 1.818 18.182 .363 .545 .036 10.000
celery 40.000 .250 - 125.000 .025 .500 .025 10.000
mushroom 6.000 .800 3.000 - .100 4.300 .460 3.000
tomato 13.333 .533 1.333 450.000 .066 .800 .04 22.667
pineapple 12.016 .310 .387 25.194 .081 .193 .019 6.977
* Units as in Table 1
2
Table 4
COSTS OF PIZZA ITEMS
Item Cost in cents/100 grams
cheese 93.20
sauce 70.48
dough 19.26
pepperoni 88.22
ham 88.10
bacon 88.54
green pepper 50.20
onion 10.66
celery 28.28
mushrooms 62.40
tomatoes 44.06
pineapple 51.96
2. Portfolio Selection Problem. An individual with $50,000 to invest has identified three mutual
funds as attractive opportunities. Over the last five years, dividend payments (in cents per dollar
invested) have been as shown in Table 5, and the individual assumes that these payments are indicative
of what can be expected in the future. This particular individual has two requirements: (1) the
combined expected yearly return from his/her investments must be no less than $4,000 (the amount
$50,000 would earn at 8 percent interest) and (2) the variance in future, yearly, dividend payments
should be as small as possible. How much should this individual invest in each fund to achieve these
requirements?
Table 5
Years
1 2 3 4 5
Investment 1 10 4 12 13 6
Investment 2 6 9 6 5 9
Investment 3 17 1 11 19 2
[Hint: Let xi(i = 1, 2, 3) designate the amount of funds to be allocated to investment i, and let
xik denote the return per dollar invested from investment i during the kth time period in the past
(k = 1, 2, . . . , 5). If the past history of payments is indicative of future performance, the expected
return per dollar from investment i is
Ei =
1
5
5∑
k=1
xik.
The variance in future payments can be expressed as
f(x1, x2, x3) =
3∑
i=1
3∑
j=1
σ2ijxixj = x
TCx,
where the covariances σ2ij are given by
σ2ij =
1
5
5∑
k=1
xikxjk − 1
52
(
5∑
k=1
xik
)(
5∑
k=1
xjk
)
. ]
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(a) Using the following table, calculate the covariance matrix C = [σ2ij ].
Table 6
k x1k x2k x3k x
2
1k x
2
2k x
2
3k x1kx2 x1kx3 x2kx3
k k k
1 10 6 17 100 36 289 60 170 102
2 4 9 1 16 81 1 36 4 9
3 12 6 11 144 36 121 72 132 66
4 13 5 19 169 25 361 65 247 95
5 6 9 2 36 81 4 54 12 18
total 45 35 50 465 259 776 287 565 290
(b) Set up a standard form optimization problem (i.e quadratic optimization problem) that will
determine the best investment mix.
(c) Solve the problem using the MATLAB quadratic programming routine quadprog. Interpret your
results.
NOTES: Essential information for accessing files from the MATH3161/MATH5165 Course Web page
and for using Matlab.
ˆ Matlab can be accessed from your own laptop using the myAccess service. (see the link on the
Course Web-page, UNSW Moodle, Computing facilities (labs, virtual apps, software).
ˆ Matlab M-files can be obtained from Matlab Worksheets in Class Resources at the Course Web-
page, UNSW Moodle. The Matlab files for Q1, Problem Sheet-1 (ss21.m) and for Q5, Problem
Sheet-6 (qp21.m) are available at this page in the assignment folder.
ˆ Matlab is run by typing
matlab
at the UNIX prompt. Inside Matlab use ‘help command’ to get help,
e.g.
help optim
help linprog
help quadprog
ˆ To run a Matlab .m file from within Matlab simply type the name of the file:
ss21
This assumes the file ss21.m is in the current directory (use the UNIX command ‘ls’ to see what
files you have; if it is not there get a copy of the file from Matlab worksheets page at the Course
Web page and save it as ss21.m).
ˆ An entire Matlab session, or a part of one, can be recorded in a user-editable file, by means of
the diary command. The recording is terminated by the command diary off. A copy of the
output produced by Matlab can be stored in the file ‘ss21.out’ by typing diary ss21.out For
example
diary ss21.txt
ss21
diary off
will save a copy of all output in the file ss21.txt
ˆ The file ss21.out may be viewed using ‘more’ or any text editor (xedit, vi) or printed using the
‘lpr’ command.
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