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Expert help when you need it- Q1:An electronics repair shop has 3 technicians, and can handle 4 types of repairs (jobs) only. Not all the technicians have the skills to perform all repairs, and their wages are different. However, repairs can be split and finished independently by those capable of performing them. Data are given in the tables below. 1. The objective is to minimize the cost of all incoming repairs (60 items). Formulate the problem as an optimization problem, identifying all variables and constraints properly and clearly. Explain as necessary . 2. Write the set of AMPL statements which model this problem (by hand or otherwise)See Answer
- Q2:The following data are obtained during an experiment involving 2 independent variables, y being the measured response. It is assumed that the intercept 𝛽0 = 0 and the relationship is linear with 2 parameters y = 𝛽1x1 + 𝛽2x2. 1. What is the associated least squares optimization problem? 2. Determine the model parameters. 3. What is the predicted value for x₁ = 3, x₂ = 1? 4. How would the problem be modified to include an interaction between the 2 predictors?See Answer
- Q3:Q1: Consider the function f: R2 → R given by f(x1, x₂) (a) Find all stationary points of f. mum/maximum or a saddle point. x²₁x₁x₂ + x² + x₁-x2. Classify each point, i.e., determine if it is a local mini- (b) Starting at x = (0,0), find the local minimum of f using Newton's method (algorithm below). Stop when ||Vf(x, ak)|| <0.01. For the optimal step size use Matlab or similar to find the minimum of g(a) = f(x*+adk). You may also use Matlab or similar to work with the gradient and Hessian of f. [15 marks] Newton's Method 1: Start from an initial point º, set k=0. 2: If (Vf(x) < ) then exit. If V2f(a) is positive definite then dk-[V2 f(x)]-¹Vf(x) else dk = -f(x). 3: Calculate ak = minazo f(x + adk). 4: Set æk+1=k+ad and k=k+1 then return to step 2.See Answer
- Q4:Q2: Consider the minimisation of the following function of two variables: f(x1, x2)=-In(1+x₁) = x₂. Subject to the linear constraints: 2x1 + x₂ ≤3; 21, x₂ ≥ 0. (a) Prove that this is a convex minimisation problem. (b) Write down the Karush-Kuhn-Tucker conditions for this problem. (c) Find all solutions of the above KKT conditions. (d) Are the solutions you found a local or a global minimum (maximum)? Justify your answer. [15 marks]See Answer
- Q5:Q3: Choose one of the following two problems: 1 (a) Show that the following equality is true: d² dt2 m A¹h(x(t)) = x(t)¹A₁V²h₂(x(t)) [EXX³h (@(0))] # i=1 (t) + A¹Vh(x(t))ä(t) [15 marks] Where A Rmx1, h: Rnx1 → Rmx1, x R → Rx1 and t E R. The bold notation Vh denotes the Jacobian matrix of h. This calculation is used in week 10 lectures notes (along with some similar versions). (b) Independently investigate non-linear least squares. Provide a concise derivation and con- clude by showing that each iteration of the Gauss-Newton algorithm is equivalent to solving a linear system. [20 marks]See Answer
- Q6:1. The Simple Linear Regression model can be written as Y₁ = B₂+ B₁2, +4 i=1,2,..., n where 3, and 3, are unknown parameters and e,' s are independent and identically dis- tributed random variables with E(e) = 0 and Var(e) = 0². Also, B, and 3₁ denotes the least-squares estimates for the parameters 3, and 3₁. (a) Derive the formulae for 8, and 3, directly from the sum of squared error (SSE) formula. [3 Marks] (b) Derive the formulae for the variances of the estimators , and 3₁. Express the results in terms of o2. [4 Marks] 2. Data on horse racing were collected. The following table gives the attendance (in thousands of people) at a racetrack and the amount of money (in millions) that was bet on n = 20 randomly selected days. The data is also available in the file Gambling.xlsx. (See Test4_supplementary.txt for instructions to import the Excel data into R.) Attendance Amount Attendance Amount (thousands (millions) (thousands) (millions) 36.5 149.8 50.6 200.2 234.0 174.3 216.1 250.1 201.8 229.4 257.7 311.3 301.7 308.9 339.4 317.4 51.4 50.9 48.8 58.2 61.5 68.5 66.7 71.2 77.0 307.8 349.2 404.5 Page 2 43.9 57.9 54.5 63.2 64.4 67.9 70.9 83.8 79.7 340.4 455.6 410.3See Answer
- Q7:Use R for computations in this question. The linear model function 1m() is ***NOT*** allowed. • Question 2 ***MUST*** be answered properly in a .pdf file. (That is, the PDF file should include answers to Question 1 and Question 2) • In addition, all computations in Question 2 ***MUST*** be properly recorded in the file Test4_supplementary. txt and ***MUST*** be reproducible by simply copying a block of annotated R codes and running them in the R Console. (a) Fit a simple linear regression model E(Y) = 3, + Biz to the data (amount of money as a function of attendance). (b) What is the expected attendance if the amount bet is 250 millions? (c) Conduct a test of hypothesis [2 marks] * END ** [1 Marks] H.: 3₁0 versus Ha: 31 0. Do the data present sufficient evidence to indicate that the slope 3₁ differs from 0 at a significance level a = 0.05? Describe the process leading to the findings. [2 marks] (d) If appropriate (justify your answer), calculate a 95% confidence interval for the slope 3₁ of the regression line. Provide a proper conclusion. [3 Marks]See Answer
- Q8:12:25 PM Fri Jun 30 SIE340 SUM23 Lecture14 Bieten item 1 2 3 4 5 volume, cubic feet 0.2 0.5 0.2 0.2 0.3 A hiker's knapsack will hold no more than 0.8 cubic feet of canned goods. Some food items that are being considered for inclusion in the knapsack are shown in the table. The hiker wants to maximize the protein content of the items that are selected for inclusion in the knapsack. Quiz grams of protein 12 40 15 20 10 [... O (i) If the hiker choose item 1, he must choose item 2. (ii) If the hiker choose item 3, he cannot choose item 4. (a) Write the decision variables and give the meaning of each decision variable. (b) Write the whole LP model. (objective function, constraints, sign restrictions) (c) Add a constraint to satisfy each of the following cases: 76% Done 25/25See Answer
- Q9:1.1 Identify degrees of freedom for static and dynamic analysis of this frame. Figure P1.1 A gable frame for Problem 1.1See Answer
- Q10:1.2 Identify degree of freedom and derive the relationship between structural resisting force and displacement of this structure. ΕΙ m L Figure P1.2 A simple beam for Problems 1.2 to 1.4See Answer
- Q11:1.3 Formulate equation of motion of the structure in Figure P1.2 when the mass is subjected to externally applied force p(t).See Answer
- Q12:1.4 Formulate the equation of motion of structure in Figure P1.2 when the structure is subjected to vertical ground excitation (1).See Answer
- Q13:3.1 - Sketch the phase portrait for the linear system below and justify its trajectories and the stability and type of its critical point. You need to deduce the equations describing the system to justify your answers. dy₁-2y₁-3y₂ dt dy₂ dt = 3y₁-2y₂See Answer
- Q14:3.2 -Use the plane TrA/DetA to identify the Type and Stability of the critical points for the linear systems d = A X, in each of the cases of matrix A, as given below; not to be done using the direct calculation of eigenvalues and eigenvectors of the matrices. dt 21 2 1 1-4 0 2 -18 (@)^= { ? ! )}; (4=( 7 _! }(4=( 12 + )-0^4 = [ 2₂ 4 )³ 4 = ( 2 } }); { A= A= 12 1-3 -1 -3 -1 -27 Justify your answers based on the values of Det A and TrA that you obtain directly from the matrices A and hence describe the eigenvalues for each system. (Help: Remember that when Det A<0, you don't need any other calculation as the CP is identified in the lower part of the plane TrA/DetA....)See Answer
- Q15:dt 3.3 - For each of the linear dynamical systems AX, with matrix A given below, find the differential equations describing the systems and describe the trajectories, stability, and type of their expected phase portraits and critical points. Sketch their phase portraits. It is crucial that you justify your answers: just drawing a phase portrait without justifying its features attracts no marks. (a) A= :) (c) A= Below you can find relevant output from Mathematica to use in your solution. 9 4 -9 -3 A = {{9,4}, {-9, -3}} Eigensystem[A] {{3, 3}, {{-2,3}, {0,0}}} 1 = { (b) A= -2 1 02 A = {{-2, 1}, {0,2}} Eigensystem[A] {{-2, 2}, {{1,0}, {1,4}}} -12 4 -26 -8 A = {{-12,4}, {-26, -8}} Eigensystem[A] {{-10+10 i, -10-10 i }, {{1-5i, 13}, {1+5i, 13}}}See Answer
- Q16: MT 328 - Non-Linear dynamical systems: Routes to Chaos Problem Sheet 3 Your solutions are to be submitted online via Moodle by Thurday 8.02.2024(up to 11pm). Please write your name clearly at the top of each page, scan your work as a single PDF file and upload it onto Moodle, as per instructions you can find on this MT328 Moodle page. 3.1 _ - Sketch the phase portrait for the linear system below and justify its trajectories and the stability and type of its critical point. You need to deduce the equations describing the system to justify your answers. dy=-2y-3y2 dt dy, =3y₁-2y2 dt 3.2 — Use the plane TrA/DetA to identify the Type and Stability of the critical points for the linear dx dt systems = AX, in each of the cases of matrix A, as given below; not to be done using the direct calculation of eigenvalues and eigenvectors of the matrices. 2 1 1 -4 2 1 1 2 (a)4={ } } };16)4={ } } });(0)4=| | 13 1): 004-(2, 3):10 - (223) ):(e)4={ 2 -3 2 8 -2 7 Justify your answers based on the values of Det A and TrA that you obtain directly from the matrices A and hence describe the eigenvalues for each system. (Help: Remember that when Det A<0, you don't need any other calculation as the CP is identified in the lower part of the plane TrA/DetA....) 3.3 For each of the linear dynamical systems - dx = A✗, with matrix A given below, find the dt differential equations describing the systems and describe the trajectories, stability, and type of their expected phase portraits and critical points. Sketch their phase portraits. It is crucial that you justify your answers: just drawing a phase portrait without justifying its features attracts no marks. 9 (a) A= 4 -9-3 (b) A= -21 02 (c) A= -12 4 -26-8 Below you can find relevant output from Mathematica to use in your solution. A = {{9,4}, {-9, -3}} Eigensystem[A] {{3,3}, {{-2, 3}, {0,0}}} A = {{-2, 1}, {0, 2}} Eigensystem[A] {{-2,2}, {{1,0}, {1,4}}} A = {{-12, 4}, {-26, -8}} Eigensystem[A] {{-10+10 i, -10-10 i }, {{1-5i, 13}, {1+5i, 13}}}See Answer
- Q17:(60pts) (40) Instructions: (100 points) Assignments should be neatly-written, well-organized, correct and concise. To receive full credit please show all your work. All class assignments and other announcements will be posted on the class website. 1. IKEA is opening a special factory to manufacture dinner tables and chairs in Kuwait. A dinner table is estimated to be sold for a profit of KD30. The chairs are estimated to be sold for a profit of KD10 each. It takes six hours to make a dinner table and three hours to make a chair. The factory will be able to crunch wood for up to 40 hours per week. Demand in Kuwait requires that the plant makes at least three times as many chairs as dinner tables. Tables take up four times as much storage space as chairs. Finally, there is room for at most four tables each week. Use the linear programming tool in Matlab (linprog) to determine the number of tables and chairs manufactured that will maximize the profit. 2. Use linprog to solve the following problem, Minimize 4x+5y+6z s. t. x+y≥ 11 x-y≤5 2-x-y=0 7x≥35-12y and 20, y≥0, z≥0 reducing the problem by eliminating one of the constraints.See Answer
- Q18: PROBLEM 3 3. Considerthefollowingmodelfor{(xi,yi)}: Р Yi = Bo+Bjoj (xi) + ɛi for i j=1 = 1, n for some basis functions 1(x), ..., Op(x) where the parameters Bo, B₁, · Р Bo+Bjoj (xi) ≥ 0 for i j=1 = 1, ... n " To estimate ẞo, ẞ1, ..., ßp, we used constrained least squares: Minimize subject to the constraints n Р Σ Yi βο - Σ β;φ; (α;) Σβ,Φ,(;) i=1 Bo+Bjoj (xi) ≥ 0 for i = j=1 = 1, 2 ... n Bp are such that The parameter estimates can be computed using an interior point algorithm whereby we define (Bo(r), · · ·‚ßp(r)) to minimize - Bo - Р 2 n Р ½Σ ( − − £¾(x)* −Ë (+Ź 8,6,(z) j=1 ―r In (xi) i=1 j=1 for r > 0 and let r↓0. (Note that we are implicitly assuming that the unconstrained LS estimates of the parameters violate the constraints.) (a) Derive an IRLS algorithm for computing (Bo(r),···, ßp(r)) for r > 0. 3 (b) The file problem3-data.txt on Quercus contains 1000 observations (xi, yi) where 0 ≤ X1,X1000 ≤ 1. Take p 10 and define 1(x), …, 10(x) to be B-spline functions; in R, = these can be defined as follows: > library(splines) > xx <- bs(x, df=10) The object xx will be matrix with 10 columns with the number of rows equal to the length of x. Compute the constrained LS estimates for these data. (The unconstrained LS estimates violate some of the non-negativity constraints.) 4See Answer
- Q19:MT 328 - Non-Linear dynamical systems: Routes to Chaos Problem Sheet 3 3.1 -Sketch the phase portrait for the linear system below and justify its trajectories and the stability and type of its critical point. You need to deduce the equations describing the system to justify your answers. dy1 =- 2y - 3y dt 1 2 dy 2 =3y -2y 1 2 dt 3.2 -Use the plane TrA/DetA to identify the Type and Stability of the critical points for the linear dK systems -= A X, in each of the cases of matrix A, as given below; not to be done using the direct calculation of eigenvalues and eigenvectors of the matrices. (a)A=2 1);(b)A= ( 2 1 );(c)A=(1 -4 );(d)A= 0 2 ; (e)A=( -1 8 ). 1 2) 1 -3 ) ( 2 -1 ( -3 -1 ) (1-2 7) Justify your answers based on the values of Det A and TrA that you obtain directly from the matrices A and hence describe the eigenvalues for each system. (Help: Remember that when Det A<0, you don't need any other calculation as the CP is identified in the lower part of the plane TrA/DetA .... ) 3.3 - For each of the linear dynamical systems dk = A X, with matrix A given below, find the differential equations describing the systems and describe the trajectories, stability, and type of their expected phase portraits and critical points. Sketch their phase portraits. It is crucial that you justify your answers: just drawing a phase portrait without justifying its features attracts no marks. (a) A=\ ( 9 4 ) -9 -3 ) (b) A=| 0 2 ) (c) A=[ -26 -8) 1 1 ( -12 4 ) Below you can find relevant output from Mathematica to use in your solution. A = { {9, 4}, {-9, -3} } A = { {-2, 1}, {0, 2} } A = { {-12, 4}, {-26, -8} } Eigensystem[A] { {-10+10 i, -10-10 i }, { {1-5i, 13}, {1+5i, 13} } } Eigensystem[A] { {3, 3}, { {-2, 3}, {0, 0} } } Eigensystem[A] { {-2, 2}, { { 1, 0}, {1, 4} } }See Answer
- Q20:The following data are obtained during an experiment involving 2 independent variables, y being the measured response. It is assumed that the intercept 𝛽0 = 0 and the relationship is linear with 2 parameters y = 𝛽1x1 + 𝛽2x2. 1. What is the associated least squares optimization problem? 2. Determine the model parameters. 3. What is the predicted value for x₁ = 3, x₂ = 1? 4. How would the problem be modified to include an interaction between the 2 predictors?See Answer
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