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# Assessment 2. Sem2.2017 Statistical and Optimization Methods for Engineers IMPORTANT: • Your submission should be entirely reproducible by using R Markdown. • Two documents should be submitted: one is the R Markdown file and one is the final product (which can be in html, word, or PDF) generated by using your R Markdown file. Q1: a) Find the maxima and minima, if any, of the function 𝑓𝑓(𝑥𝑥) = 4𝑥𝑥3 − 18𝑥𝑥2 + 27𝑥𝑥 − 7 (5 marks) b) Determine the convexity of the following function using the second derivative test. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥4 + 𝑥𝑥2 (10 marks) (Note: You are encouraged to use functions provided in R to check whether your solution is correct or not) Q2: Minimize 𝑓𝑓(𝑥𝑥) = −𝑥𝑥3 − 2𝑥𝑥 + 2𝑥𝑥2 + 0.25𝑥𝑥4 (1) Apply the bisection method with initial bounds 𝑥𝑥 = 0 and 𝑥𝑥 = 2.4 and with an error tolerance 𝜖𝜖 = 0.04. Please present your search procedure in a tabular form. (10 marks) (2) Apply Newton’s method with an error tolerance 𝜖𝜖 = 0.001 and 𝑥𝑥1 = 1.2. Please present your search procedures in a tabular form. (10 marks) Q3: The attached dataset (please download it separately from the blackboard) 𝑥𝑥 was generated using the model: 𝑦𝑦 = 𝑎𝑎1exp (𝑎𝑎2). You are asked to use the dataset to estimate the two parameters 𝑎𝑎1 and 𝑎𝑎2 in the model, by treating it as a leastsquares problem. Hint: please do not attempt to solve it manually; you need to use optimization functions available in RStudio. (20 marks) Q4: Solve the following linear programming problem using the two methods below: Minimize 𝑓𝑓 = −18𝑥𝑥1 − 15𝑥𝑥2+20 Subject to 𝑥𝑥1 + 𝑥𝑥2 ≤ 5 3𝑥𝑥1 + 2𝑥𝑥2 ≤ 12 𝑥𝑥1 ≥ 0 𝑥𝑥2 ≥ 0 (1) The graphical solution; (5 marks) (2) The Simplex method; (10 marks) Queensland University of Technology Assessment 2.Sem2.2017 ENN542 Statistical and Optimization Methods for Engineers Q5: Find the minimum value of the following function using Lagrange Multipliers: 𝑓𝑓(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = 2𝑥𝑥2 + 𝑦𝑦2 + 3𝑧𝑧2 subject to 2𝑥𝑥 − 3𝑦𝑦 − 4𝑧𝑧 = 49 (10 marks) The End

Assessment 2. Sem2.2017

Statistical and Optimization Methods for Engineers

IMPORTANT:

• Your submission should be entirely reproducible by using R

Markdown.

• Two documents should be submitted: one is the R Markdown file

and one is the final product (which can be in html, word, or PDF)

generated by using your R Markdown file.

Q1:  a) Find the maxima and minima, if any, of the function

𝑓𝑓(𝑥𝑥) = 4𝑥𝑥3 − 18𝑥𝑥2 + 27𝑥𝑥 − 7  (5 marks)

1. b) Determine the convexity of the following function using the second derivative test.

𝑓𝑓(𝑥𝑥) = 𝑥𝑥4 + 𝑥𝑥2   (10 marks)

(Note: You are encouraged to use functions provided in R to check whether your solution is correct or not)

Q2:  Minimize  𝑓𝑓(𝑥𝑥) = −𝑥𝑥3 − 2𝑥𝑥 + 2𝑥𝑥2 + 0.25𝑥𝑥4

• Apply the bisection method with initial bounds 𝑥𝑥 = 0 and 𝑥𝑥 = 2.4 and with an error tolerance 𝜖𝜖 = 0.04. Please present your search procedure in a tabular form.
• marks)

• Apply Newton’s method with an error tolerance 𝜖𝜖 = 0.001 and 𝑥𝑥1 = 1.2. Please present your search procedures in a tabular form.
• marks)

Q3: The attached dataset (please download it separately from the blackboard)

𝑥𝑥 was generated using the model: 𝑦𝑦 = 𝑎𝑎1exp (𝑎𝑎2). You are asked to use the dataset to estimate the two parameters 𝑎𝑎1 and 𝑎𝑎2 in the model, by treating it as a leastsquares problem. Hint: please do not attempt to solve it manually; you need to use optimization functions available in RStudio.   (20 marks)

Q4:  Solve the following linear programming problem using the two methods below:

Minimize         𝑓𝑓 = −18𝑥𝑥1 − 15𝑥𝑥2+20 Subject to         𝑥𝑥1 + 𝑥𝑥2 ≤ 5

3𝑥𝑥1 + 2𝑥𝑥2 ≤ 12

𝑥𝑥1 ≥ 0

𝑥𝑥2 ≥ 0

• The graphical solution; (5 marks)

• The Simplex method; (10 marks)

Queensland University of Technology                                                       Assessment 2.Sem2.2017

ENN542 Statistical and Optimization Methods for Engineers

Q5:  Find the minimum value of the following function using Lagrange Multipliers:

𝑓𝑓(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = 2𝑥𝑥2 + 𝑦𝑦2 + 3𝑧𝑧2

subject to 2𝑥𝑥 − 3𝑦𝑦 − 4𝑧𝑧 = 49

(10 marks)

The End

Interested in a PLAGIARISM-FREE paper based on these particular instructions?...with 100% confidentiality?