For this assignment, you are required to 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 (45 marks)
You have a certain type of chicken wire to build a temporary enclosure for holding a chicken in your backyard. You have 100m chicken wire and want to maximise the enclosure area for your given materials.
(a) You plan to build a triangular enclosure (the lengths of three sides are x, y, and z,respectively. See Figure 1)
Suppose x is 10m longer than y, find the lengths of the three sides x, y, and z.
– Construct a one-dimensional optimisation model and solve it using the Golden Section Search method. Solution details and MATLAB codes are required. (15 marks)
– Construct a two-dimensional optimisation model and solve it using the Lagrange multiplier method. Solution details are required (no MATLAB code required). (10 marks)
(b) If it is a right-angled triangular enclosure, the longest side is known to be 41.4m, please find the two sides (x, y) using the Linear Programming (LP) method.
– Please construct an optimisation model and solve it using MATLAB function linprog(). Solution details and MATLAB codes are required. (20 marks)
Question 2 (55 marks)
Suppose a linear equation is to be fit predicting raw material price as a linear function of the number of product A and product B (made of the same raw material) sold given the following data:
Assume the prediction equation is yi=c0+czx1i+c2x2i where are the prediction parameters on the quantity of products A and B sold, respectively, and is the intercept.
Define x1i，x2i as the observations on the quantity of products A and B sold, respectively, and as the observed price. identifies the observation.
(a) 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 construct an LP model to minimise the largest absolute deviation.
– Write down the tabular form of the formed LP problem. Explain the steps to get to the final tabular form. (20 marks)
– Solve the formed LP problem using the MATLAB function-linprog(); explain the result. (10 marks)
(b) 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)