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# Assessment 2. Sem2.2017 Statistical and Optimization Methods for Engineers IMPORTANT: β’ Your submission should be entirely reproducible by using R Markdown. β’ 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 ππ(π₯π₯) = 4π₯π₯3 β 18π₯π₯2 + 27π₯π₯ β 7 (5 marks) b) Determine the convexity of the following function using the second derivative test. ππ(π₯π₯) = π₯π₯4 + π₯π₯2 (10 marks) (Note: You are encouraged to use functions provided in R to check whether your solution is correct or not) Q2: Minimize ππ(π₯π₯) = βπ₯π₯3 β 2π₯π₯ + 2π₯π₯2 + 0.25π₯π₯4 (1) Apply the bisection method with initial bounds π₯π₯ = 0 and π₯π₯ = 2.4 and with an error tolerance ππ = 0.04. Please present your search procedure in a tabular form. (10 marks) (2) Apply Newtonβs method with an error tolerance ππ = 0.001 and π₯π₯1 = 1.2. Please present your search procedures in a tabular form. (10 marks) Q3: The attached dataset (please download it separately from the blackboard) π₯π₯ was generated using the model: π¦π¦ = ππ1exp (ππ2). You are asked to use the dataset to estimate the two parameters ππ1 and ππ2 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 ππ = β18π₯π₯1 β 15π₯π₯2+20 Subject to π₯π₯1 + π₯π₯2 β€ 5 3π₯π₯1 + 2π₯π₯2 β€ 12 π₯π₯1 β₯ 0 π₯π₯2 β₯ 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: ππ(π₯π₯, π¦π¦, π§π§) = 2π₯π₯2 + π¦π¦2 + 3π§π§2 subject to 2π₯π₯ β 3π¦π¦ β 4π§π§ = 49 (10 marks) The End

Assessment 2. Sem2.2017

Statistical and Optimization Methods for Engineers

IMPORTANT:Β

Β

• Your submission should be entirely reproducible by using R

Markdown.

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

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

Β Β Β Β Β Β Β  generated by using your R Markdown file.Β

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

ππ(π₯π₯) = 4π₯π₯3 β 18π₯π₯2 + 27π₯π₯ β 7Β  (5 marks)

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

ππ(π₯π₯) = π₯π₯4 + π₯π₯2Β Β  (10 marks)

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

Q2:Β  MinimizeΒ  ππ(π₯π₯) = βπ₯π₯3 β 2π₯π₯ + 2π₯π₯2 + 0.25π₯π₯4

• Apply the bisection method with initial bounds π₯π₯ = 0 and π₯π₯ = 2.4 and with an error tolerance ππ = 0.04. Please present your search procedure in a tabular form.
• marks)

• Apply Newtonβs method with an error tolerance ππ = 0.001 and π₯π₯1 = 1.2. Please present your search procedures in a tabular form.
• marks)

π₯π₯ was generated using the model: π¦π¦ = ππ1exp (ππ2). You are asked to use the dataset to estimate the two parameters ππ1 and ππ2 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.Β Β  (20 marks)

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

MinimizeΒ Β Β Β Β Β Β Β  ππ = β18π₯π₯1 β 15π₯π₯2+20 Subject toΒ Β Β Β Β Β Β Β  π₯π₯1 + π₯π₯2 β€ 5

3π₯π₯1 + 2π₯π₯2 β€ 12

π₯π₯1 β₯ 0

π₯π₯2 β₯ 0

• The graphical solution; (5 marks)

• The Simplex method; (10 marks)

Queensland University of Technology Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β  Assessment 2.Sem2.2017

ENN542 Statistical and Optimization Methods for Engineers

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

ππ(π₯π₯, π¦π¦, π§π§) = 2π₯π₯2 + π¦π¦2 + 3π§π§2

subject to 2π₯π₯ β 3π¦π¦ β 4π§π§ = 49

(10 marks)

The End

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