School of Information Technology & Electrical Engineering

Engg7302 Advanced Computational Techniques in Engineering

Assignment 2: Optimisation Question

Due date:  23/10/2024 3:00 pm

Where to submit:  Submit the  assignment  report  via  the  Blackboard  Turnitin  submission system.

Instruction:

For this assignment, you must attempt 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 10% of the total marks for the course.

Question On Optimisation     (This question is worth 10% of the total marks for the course)

A furniture company will sell three types of chairs to UQ (types A, B, and C), all made from the same wood material. Suppose a linear equation is to befit to predict the wood material price as a function of the quantity of these three types of chairs sold, given the following data:

Table 1

Assume  the  prediction  equation  is  p(n) = ax(n) + by(n) + CZ (n),  where  a, b, C  are   the prediction parameters for the quantities of chair types A, B and C to be sold, respectively. Define x(n), y(n), Z(n)  as  the  observations  on  the  number  of  types  A,  B,  and  C  sold, respectively (see columns 1, 2, 3 in Table 1), and p(n) as the observed price (see column 4 in Table 1). 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 wood material price and its prediction.

Please develop a Linear Programming (LP) model to minimise the sum of the absolute deviations and solve the formed LP problem using the MATLAB function-linprog.

(2)   Suppose the desired criterion for equation fit is that the fitted data exhibit a minimum of the largest absolute deviation between the wood material price and its prediction.

Please develop an LP model to minimise the largest absolute deviation and solve the formed LP problem using the MATLAB function-linprog.

(3)

Suppose the desired criterion for the equation fit is that the fitted data exhibit a minimum sum of the squared deviations between the wood material price and its prediction. In Table 1, the first dataset is supposed to be emphasised 10 times more than the remaining four datasets. You are then asked to solve the resulting least squares (LS) problem.

-    Write down the linear system equation (Ax=B) of the LS problem.

-    Solve the LS problem using the normal equations approach.