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Assignment 1: Optimisation
Where to submit: Submit the assignment report via the Blackboard Turnitin submission
system.
Instruction:
For this assignment, you must carry out the process of attempting to solve different
optimisation problems. For each question, you are required to report your results in detail. It
should include your best solution and its corresponding solution procedures. If you are asked
to solve those sub-questions using MATLAB, their MATLAB source code with detailed
comments is required.
Marks will be awarded based on how well your submission addresses the above points.
This assignment is worth 20% of the total marks for the course.
Question 1
Suppose a linear equation is to be fit predicting raw material price as a linear function of the
quantity of product A and produce B (made of the same raw material) sold given the
following data:
Quantity of product A sold Quantity of product B sold Price of raw material
9 1 5
13 8 2
17 3 9
8 5 10
10 9 4
15 2 6
Assume the prediction equation is 𝑧(𝑛) = 𝑎𝑥(𝑛) + 𝑏𝑦(𝑛) + 𝑐, where 𝑎, 𝑏 are the prediction
parameters on the quantity of products A and B sold, respectively, and 𝑐 is the intercept. Define
𝑥(𝑛), 𝑦(𝑛) as the observations on the quantity of products A and B sold, respectively, and
𝑧(𝑛) as the observed price. 𝑛 identifies the observation index.
(1) Suppose the desired criterion for equation fit is that the fitted data exhibit a minimum of
the sum of the absolute deviations between the raw material price and its prediction.
Please develop a Linear Programming (LP) model to minimise the sum of the absolute
deviations and write down the tabular form of the formed LP problem.
(25 marks)
2
(2) Suppose the desired criterion for equation fit is that the fitted data exhibit a minimum of
the largest absolute deviation between the raw material price and its prediction.
Please develop an LP model to minimize the largest absolute deviation and solve the
formed LP problem using the MATLAB function-linprog.
(25 marks)
(3) Suppose the desired criteria for equation fit is that the fitted data exhibit a minimum sum
of the squared deviations between the raw material price and its prediction. You are then
asked to solve the formed least square (LS) problem.
- Write down the linear system equation (Ax=B) of the LS problem.
(15 marks)
- Solve the LS problem using the normal equations approach.
(10 marks)
Question 2
You have certain types of chicken wire to build a temporary enclosure for holding chicken in
your backyard. You plan to build a triangular enclosure (the lengths of three sides are x, y and
z, respectively. See Figure 1:
Figure 1 Triangular enclosure (chicken house)
You have 100m of chicken wire, and you want to maximise the area of the enclosure for your
given materials.
If the lengths of two sides have the following relationship: x=y. Please find the lengths of three
sides x, y, and z using the Successive Parabolic Interpolation method and Newton’s method.
Please convert it to a one-dimensional optimisation problem and provide your MATLAB code.
(25 marks)
x
y
z