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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: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
Q14: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
Q15:Let x, y and z be the sides of a rectangular body. 1. Formulate the problem of finding the tetrahedral body of maximum volume with diagonal of unit length. 2. Write the KKT conditions for this problem. 3. Is the cube an optimal solution? Justify your answer.See Answer
Q16: 1. You have been hired as a consultant by the company "Bird Baths R Us” to help determine why one of their products is frequently returned by customers.The CEO expects you to justify your conclusions using both graphical and numerical data. You should aim to be as precise as possible in your analysis.Please upload screenshots of any graphical material as well as any Excel files (not screenshots) you use. The product in question is a hemispherical bird bath known as the "Avosphere."It is ten inches deep and features six perches for visiting birds. The height x, in inches, of the water as the Avosphere is being filled is modeled by the differential equation: \frac{d x}{d t}=\frac{60\left(1-20 k x+k x^{2}\right)}{20 x-x^{2}} where the time t is measured in hours and k is a constant that measures how quickly water evaporates. If there were no evaporation, k would be zero. Your preliminary tests have determined k to be .02. We will assume the bird bath initially has 1 inch of water. • In the "DFIELD Direction Field" window menu bar, select Options → Delay Time Per Point → 10 Milliseconds Options → Solution Direction → Forward ● In the “DFIELD Equation" window, you can change the values in the"Display Window." Use Min t = Min x = 0. You will need to decide what you want the maximum values to be. (a) Using your evaporation constant, what is the height of the water after 2hours? What would the height be after 2 hours if there were no evaporation? (b) Assuming no evaporation, how long until the bird bath is full of water? (d) Based on your answers above, why do you think customers are dissatisfied with this product? (e) Approximately what value would Bird Baths R Us need to reduce theevaporation constant to so that customers can get the Avosphere at least70% full? (c) With your evaporation constant, what is the maximum depth of water acustomer can achieve in their Avosphere?See Answer
Q17: 1. Consider the model defined by x^{\prime}(t)=x(2-0.4 x-0.3 y) y^{\prime}(t)=y(1-0.1 y-0.3 x) nd and classify the equilibrium points of this system. You can use alculator/computer to help with the computations! (b) Find the x and y nullclines for this system. You can use a calcula-tor/computer to help with the computations! (c) Now, use "pplane.jar" and print the phase plane with the nullclines. Inorder to see the nullclines, you might have to change the values of x-max,y-max, x-min, y-min in the “PPlane Equation Window”, based on your answers to the previous part. To show the nullclines, go up to "Solution"then "Show Nullclines". Label then equilibrium points and print the phase plane with nullclines. (d) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0)1.5 and y(0) = 3.5.= (e) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0) = 1 and y(0) = 1.See Answer
Q18: 2. Consider the model defined by x^{\prime}(t)=x(1-0.1 x-0.05 y) y^{\prime}(t)=y(1.7-0.1 y-0.15 x) (a) Find and classify the equilibrium points of this system. You can use a calculator/computer to help with the computations! (b) Find the x and y nullclines for this system. You can use a calculator/computer to help with the computations! (c) Now, use "pplane.jar” and print the phase plane with the nullclines. In-order to see the nullclines, you might have to change the values of x-max,y-max, x-min, y-min in the “PPlane Equation Window", based on your answers to the previous part. To show the nullclines, go up to "Solution"then “Show Nullclines". Label then equilibrium points and print the phase plane with nullclines. (d) Using your phase plane, determine what happens to these two species in-the long run under the initial conditions x(0) = 1 and y(0) = 1. (e) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0) = 4 and y(0) = 10.See Answer
Q19: 2. You've been hired by the New York City Office of Public Sanitation to analyze the pests in the sewers. The Chief of Public Sanitation expects you to justify your conclusions using both graphical and numerical data. You should aim to be as precise as possible in your analysis. Please upload screenshots of any graphical material as well as any Excel files (not screenshots) you use. The following system of differential equations describes the populations of cock-roaches and rats living in the sewers. x^{\prime}(t)=.05 x-.001 x y y^{\prime}(t)=-.01 y^{2}-.1 y+.02 x y Both species are measured in millions and time is measured in months. • In the "PPLANE Phase Plane" window menu bar, select Options → Delay Time Per Point → 10 Milliseconds Options → Solution Direction → Forward ● In the “PPLANE Equation” window, you can change the values in the"Display Window." Your minimum values should be zero since populations can't be negative. You will need to decide what you want the maximum values to be. • Remember that the command Graph → Both x-t & y-t allows you to seea graph of the solution functions! (a) Which of x or y is the population of cockroaches? Which of x or y is the population of rats? Completely justify your answer. (b) In the absence of one of these two species, describe the growth of the other. (c) Starting with a population of 6 million rats and 4 million cockroaches,when does the number of cockroaches overtake the number of rats? Then,how long does it take for the number of rats to catch back up and overtake the number of cockroaches? What is the minimum rat population during this period of time? (d) Find and classify the equilibrium points of the system. What does this tell you about the long term behavior of the two species? The Chief of Public Sanitation wants to know whether it is more effectiveto poison the cockroaches or rats. What recommendation do you make?Completely justify your answer.See Answer

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