Department of Electrical and Electronic Engineering EIE373 Microcontroller Systems and Interface Laboratory Exercise 4: AVR Serial Port Programming Objective: To develop C programs with serial port communication under the Arduino platform. Introduction: This experiment introduces an application for serial port communication inside the AVR microcontroller. Equipment: Atmel Studio The Arduino Starter Kit Procedure: Section A: Write a C program to echo a character Write a C program to complete the following task: Receive a character from the serial port by using the polling method and then send it to a PC terminal through the serial port. The baud rate should be 4800. To view the serial port communication, you should use the freeware “Tera Term,” a PC terminal. You can locate this software on your computer using the search function. Select “Serial” and an appropriate port for the serial port communication. The port number should be the same as the port the Arduino connects to your computer. Select “Setup” from the main menu and then “Serial port” . Set the baud rate to 4800. Important information: Note that you must close Tera Term before you burn your program into Arduino; otherwise, Tera Term holds the serial port, and you cannot burn your program successfully. A character can be sent to the Arduino by typing it (i.e., using the keyboard). When you type a character, Tera Term gets the input and sends it to the Arduino through the serial port. When the Arduino receives a character, it will send it to the serial port, and you can read it through Tera Term. Section B: Write a C program to send and receive strings Write a C program to complete the following tasks: 1. (Do it once at the beginning) Send “We Are Ready” to the serial port using polling. Note that the clock frequency of the Arduino Start Kit is 16 MHz. 2. (Do it repeatedly) Receivedata from the serial port by polling. If the received string is “Hi”, your program should send “Bye” to the serial port. (Hint: To get a string from the serial port, your program should receive it character by character. For example, if your program gets a string “abc”, the characters received from the serial port should be “a” first, and then “b”, and finally “c”.) The string “Hi” can be sent to the Arduino by typing it (i.e., using the keyboard) in Tera Term. When you type “Hi”, Tera Term gets the input and sends it to the Arduino through the serial port. When the Arduino receives the string, it will send a message “Bye!” to the serial port, and you can read it through Tera Term (see the figure below): Section C: Write a C program to keep sending and receiving characters Write a C program to complete the following tasks using the polling method: 1. Before you press any keys, the character ‘a’ is printed continuously. 2. When you press a key (say ‘b’), ten characters of this key (i.e., ‘b’) are printed out and then stop. 3. After that, nothing happens when you press a key other than the first key (i.e., ‘b’). 4. When you press the key again (i.e., ‘b’), the character ‘a’ is printed continuously (i.e., resume). Set the baud rate of the PC terminal (i.e., Tera Term) to 9600. If your program runs successfully and the setting of Tera Term is correct, you should see the following output: Section D: Write C programs by using the serial port interrupt You should repeat Sections B and C, but this time, you should use the serial port interrupt (not the polling method).
7SSGN110 Environmental Data Analysis | Practical 4 | Correlation & Regression 1. About this practical The aim of this practical is to explore correlation and regression techniques that are common to environmental research. The practical is composed of four sections that consider (1) Pearson’s r correlation coefficient, (2) non-parametric correlations coefficients, (3) simple linear regression and (4) multiple linear regression. The practical will use R but offer suggestions for how to do some things in Excel. There are three data files used in this practical, all available on KEATS: ‘PlantAbundance.csv’, ‘GQA.csv’ and ‘UKJanClimate.csv’. These data sets have been obtained from Moggridge (2006) [Plant Abundance and GQA] and MetOffice (2013) [Climate Data]. 1.1. Getting started in R Before you start the data analysis In R, ensure that you have: · Created a folder on your computer for this week’s practical · Set the working directory for the practical work in R Studio · Loaded the required packages. We will be using the packages car, ggExtra, GGally, Hmisc. · Read the ‘PlantAbundance.csv’, ‘GQA.csv’ and ‘UKJanClimate.csv’ files into R. 2. Pearson’s r correlation coefficient We are interested in whether the abundance of Juncus effusus (J. effusus) in a riverine wetland is related to the particle size and the organic content of the soil. To investigate this, 103 quadrats were placed in random locations. In each quadrat, the percentage cover of J. effusus was recorded and a soil sample was taken. The soil samples were taken back to the laboratory, where they were analysed for particle size and organic content. In this data set, the particle size is represented as % sand and the organic content is expressed as a percentage of the total volume of the soil sample. The abundance of J. effusus is also recorded as a percentage. All data follow a normal distribution, so parametric tests will be used (you can test this yourself in R using the standard normality tests). 2.1. Data visualisation 2.1.1. Scatterplot types The 1st thing we are going to do is visualize the possible relationship between J. effusus abundance, % sand and % organic content. The simplest way to do this is by creating a simple scatterplot. We can use the base R function plot() to do this: PA.data
Faculty of Arts & Science Fall 2024 Quiz 6 - V2 CSC 110 Y1F Question 1. Object Mutation [3 marks] Part (a) [1 mark] Consider the code below. >>> lst = [3, 1, 2] >>> lst.sort() What will lst refer to after executing every line above? Circle the correct option below. a) [1, 2, 3] b) [3, 1, 2] c) [3, 2, 1] d) This will raise an Error since there is no method called list .sort. Part (b) [1 mark] Consider the following code: >>> lst1 = [1, 2, 3] >>> lst2 = [1, 2, 3] Which of the following expressions would evaluate to True? Select all that apply. a) type(lst1) == type(lst2) b) lst1 is lst2 c) lst1 == lst2 d) id(lst1) == id(lst2) Part (c) [1 mark] Consider the following code: >>> s = ' CSC110 ' >>> z = s >>> s = s + ' Y ' >>> m = s Which of the following is true after all the above code is executed? Select all that apply. a) z refers to ' CSC110Y ' b) m refers to ' CSC110Y ' c) z and m have the same id after all the above code is executed d) An error is raised when the above code is executed. Question 2. Object Mutation [5 marks] Consider the function f1: def f1(lst: list, s: set) -> set: """ """ ... new_set = set() new_list = lst for element in sorted(s): new_list.append(element) new_set.add(element) return new set And suppose we execute the following in the console: >>> my_list = [0, 1, 2] >>> my_set = {3, 2, 1} >>> my_other_set = f1(my_list, my_set) Part (a) [2 marks] What value will my_list refer to? a) [0, 1, 2] b) [0, 1, 2, 1, 2, 3] c) [0, 1, 2, 3, 2, 1] d) [0, 1, 1, 2, 2, 3] Part (b) [3 marks] Which expressions below evaluate to True? Select all that apply. a) my_set == {0, 1, 2, 3} b) my_set == {1, 2, 3} c) my_set == {3, 2, 1} d) my_other_set is my_list e) my_other_set is my_set f) my_other_set == my_set Question 3. Memory Model [4 marks] Below is an object-based memory model diagram showing the state of the memory model at the moment immediately before a program ends. Part (a) [1 mark] If the last thing the program did was print(stuff), what would be the output? Part (b) [3 marks] Complete the following code so it will generate the memory model diagram shown above by putting an expression or statement in each of the three boxes. Do not add in any extra lines or change anything else. def bar(num: float, lst: list) -> None: def foo(lst: list, num: int) -> None: bar(12.5, lst) bar(100.0, lst) if __name__ == ' __main__ ' : stuff = i = 1 foo(stuff, i)
Linear Algebra - Fall 2023 Exam 1 1. (a) Determine if A is invertible, and if so determine its inverse. (b) Suppose B, C are 3⇥3 invertible matrices such that BC = A. Calculate the product C−1B−1. 2. The row reduced echelon form. of the augmented matrix of a linear system is State whether the system is consistent or inconsistent and why; find the solution of the system if it has solutions, and state which variables are leading variables and which are free variables. 3. Four matrices are given below, each being the row reduced echelon form. of a matrix of a system of linear equations. For each of these, determine the number of solutions of the original system; briefly justify: 0 (zero, or inconsistent system), 1 (unique) or 1 (infinitely many solutions). (a) (b) (c) (d) 4. Write the matrix of the linear transformation that represents each of the following: (a) A rotation about the origin of angle 2/π counterclockwise (you MUST simplify your answer). (b) A reflection about the line x1 = x2. (c) The transformation that is obtained by doing a rotation about the origin of angle 2/π counter-clockwise FIRST, followed by a reflection about the line x1 = x2. (that is, doing (a) first, then (b)). 5. (a) Let Calculate det(A), determine if A is an invertible matrix and find its inverse if it is. (b) For what values of k is the matrix invertible? 6. A middle school student was trying to solve a linear system. The pencil written notebook took some water damage, and some writing was no longer visible. The system was of the form. where a and b represent numbers that are no longer visible. Below the system, the student wrote several solutions to the system (but which now can no longer be clearly distinguished). Reconstruct this part of the notebook and find the numbers a and b. JUSTIFY your answer. 7. Consider the matrix (a) Interpret the linear transformation T(x) = Ax geometrically. (Write a sentence that explains its effect, and/or draw a picture.) (b) Possibly using (a), calculate A20.
PROGRAMMING IT (COMPSCI 4039) December 2023 Important note: throughout this exam, whenever you are asked to write Java code, do not worry about whether your code compiles. You should attempt to produce ‘correct’ code but the markers will never test any submitted code from this exam. 1. (a) Define a class named Book with the following attributes: title, author, and publicationYear. The publicationYear is stored as a number and the other two attributes are textual. Choose appropriate data types to store the attributes and include a parameterised constructor that initialises these attributes. [5] (b) Override the constructor for the Book class to allow for default values of the attributes. The default value of title should be ”Unknown Title”, the default value of author should be ”Unknown Author” and the default value for publicationYear should be 0. You should define four additional constructors, one each for the cases where individual attributes are missing as well as a version where no attributes are known. [4] (c) Create a method within the Book class that will allow the title, author and publicationYear for each book to be printed out using System.out.println(b), where b is a Book object. The book title, author and publication year should be printed out in that order, in the fixed width order given below (note that the periods should be printed as spaces – periods are just being used in this diagram so you can count them easily): The Hobbit..........;....JRR Tolkien;.1937 The width of these 3 elements is respectively 20, 15 and 5 characters. [4] (d) The method you created for (c) prints the attributes to specific lengths: what would be the result of applying this method to a (Book) object that has a very long title, for instance “Harry Potter and the Prisoner of Azkaban”? [2] 2. (a) Consider a class hierarchy for representing different shapes. Define the base class called Shape, which has an attribute numSides, for storing the number of sides of the shape, a single size attribute, size, and a method calculateArea, which returns the area of the shape. You should assume that there are no restrictions on the value that size can take, and therefore that calculateArea can return. When designining your base class you should use appropriate data types for all attributes and methods but assume that all values, including the return value of calculateArea, are zero. Include a basic constructor method as well as a calculateArea method in your base Shape class; [5] (b) Create subclasses Circle and Rectangle, which inherit from Shape. For each, you should include any extra attributes needed to describe the shape as well as an appropriate calculateArea method. You should assume that a circle is a one-sided shape and note that the area of a circle is A = πr 2 where π = 3.14159 and r is the radius of the circle. [6] (c) Override the constructor of Rectangle to allow for the situation where the rectangle being declared is in fact a square, i.e. a rectangle where all sides are the same length. [2] (d) An oval can be considered to be like a circle with two different radiuses, a squashed one and a stretched one and an area that can be calculated as A = πr1r2. Describe the changes required to your Circle class that would allow it to also describe ovals. [2] 3. (a) The Book class you defined in 1a) is to be used by the university library to store the details of the books in their collection. Define an array javaTextbooks of Book objects that will store the details of the 47 Java textbooks in the collection. [3] (b) You are given a comma separated file holding the details of these 47 textbooks. In this file the books are stored as Title, Author, Year with each line of the file representing a single book. Write a method, readBooksFromFile, that will open the file and read the contents into the array declared in 3a). This method should be passed the name of the file and the empty array to be populated. Your method should be passed the filename, as a String named filename, and the empty Book array and return the array once it has been populated. Your method should be reusable for other sections of the collection so you should not assume the array is of any particular size but you can assume the array is always the correct size to store all the books in the file. Your method should also handle any exception raised if filename does not contain the name of a valid file. [10] (c) Since the first version of Java was released in 1996 we know that the publicationYear of any of the Java textbooks cannot be before that year. Declare an exception, JavaBookYearException, to be thrown if a book appears to be unreasonably old and explain how it would be thrown if such a book is found. [2] 4. (a) Below is some Java code that generates an array of the Book objects we have been working with, having asked the user to enter the number of books that will be stored. It then prompts the user to enter the title, author and publication year, while performing checks on the values entered. It checks if the author entered is a specific name and if it is, prints a message to screen. No books published earlier than 1700 should be entered and the year is checked to ensure that it is valid. This code contains a number of errors. Find up to 6 errors and describe in detail how they could be repaired: there may be more than 6 errors but you will only be given credit for 6. The errors are a mix of logical errors (‘bugs’ that mean the code will not do what is expected) and syntax errors (such as spelling and punctuation errors or incorrect labels). Line numbers are given to help you report where the errors are. (1) import java.util.Scanner; (2) public class BookInputWithValidation { (3) public static main(String[] args) { (4) Scanner scanner = new Scanner(System.in); (5) System.out.print("Enter the number of books: "); (6) int numBooks = scanner.nextInt(); (7) Book bookArray = new bookArray[numBooks]; (8) for (int i = 0; i < numBooks; i++) { (9) System.out.println("Enter details for Book " + (i + 1)); (10) String title = scanner.next(); (11) System.out.print(Enter author: ); (12) String author = scanner.next(); (13) if (author == "JRR Tolkien") { (14) System.out.println("Here be Hobbits!"); (15) } (16) int publicationYear = scanner.nextInt(); (17) int publicationYear; (18) do { (19) System.out.print("Enter publication year " + "between 1700 and 2023):"); (20) publicationYear = scanner.nextInt(); (21) if (publicationYear < 1700 || publicationYear > 2023) { (22) System.out.println("Invalid year! Please enter a year between " + "1700 and 2023."); (23) } while (publicationYear < 1700 && publicationYear > 2023); (24) bookArray[i] = Book(author, title, publicationYear); (25) } (26) System.out.println("Entered Book Details:"); (27) for (i=0;i
Applied Stats and Data Analysis EN.553.413-613, Fall 2024 Oct 3rd, 2024 Exam 1 Question 2 (20 pts). The following TRUE/FALSE questions concern the Simple Linear Regression model Yi = β0 + β1Xi + εi , E(εi) = 0, V ar(εi) = σ 2 , cov(εi , εj ) = 0, for i ≠ j. (a) TRUE or FALSE. Yi and εj are uncorrelated for i ≠ j. (b) TRUE or FALSE. Denote b0 as the least squares estimator for β0 and βˆ 0 as some unbiased estimator for β0. The variance of b0 is smaller or equal than the variance of βˆ 0. (c) TRUE or FALSE. If the least squares estimator for β1 is 0, X and Y are not linearly related. (d) TRUE or FALSE. Semi-studentized residual vs Xi plot can be used to find outliers. (e) TRUE or FALSE. P n i=1 εi = 0. (f) TRUE or FALSE. Variance stabilizing transform. can be used to transform. X. (g) TRUE or FALSE. P n i=1 Yˆ i(Yi − Yˆ i) = 0. (h) TRUE or FALSE. Using Working-Hotelling procedure to find joint confidence interval for mean response at Xh1 = 3 with three other points will give the same confidence interval for EYh1 comparing to joint confidence interval for mean response at Xh1 = 3 with ten other points. (i) TRUE or FALSE. In the Correlation model of regression predictor Xi ’s and the response variable Yi are both normal random variables. (j) TRUE or FALSE. Var(ei) = σ 2 for all i. Question 3 (15 pts). Let X1, X2, X3 ∼ iid N(0, 1), i.e. they are independent, identically distributed standard normal random variables. Let Y ∼ N(2, 1) and Y is independent on X1, X2, X3. For the following random variables state whether they follow a normal distribution, a t- distribution, a χ 2 distribution, an F distribution, or none of the above. State relevant parameters (e.g. degrees of freedom, and means and variances for normal RVs) Question 4 (20 pts). Suppose a data set {(Xi , Yi) : 1 ≤ i ≤ n} is fit to a linear model of the form. Yi = β0 + β1xi + εi where εi are independent, mean zero, and normal with common variance σ 2 . Here we treat Y as the response variable and X as the predictor variable. The output of the lm function is given. Some values are hidden by ‘XXXXX’. We provide you with additional value: X¯ = 1.11. Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.9088 0.4791 1.897 0.0724 x 1.7480 0.7385 XXXXX 0.0281 --- Residual standard error: 1.088 on 20 degrees of freedom Multiple R-squared: XXXXXX,Adjusted R-squared: XXXXXX F-statistic: 5.602 on XX and XX DF, p-value: XXXXXX (a) How many data points are there (what is n, the sample size)? What is the estimated mean of the response variable Y at Xh = 1 for this dataset? (b) Based on all of this output, do you reject H0 : β1 = 0 in favour of Ha : β1 = 0 at level α = 0.01 significance? Why? What does the test tell us about the relationship between X and Y ? (c) Based on all of this output, do you reject the H0 : β1 = 0 vs Ha : β1 > 0 at level α = 0.01 significance? Briefly explain why, or why not. (d) The degrees of freedom, and the p-value of the F statistic are hidden. Is it possible to reconstruct all of them based on the data shown? Recover as many values as you can. (e) Find MSE and Sxx = P n i=1(Xi − X¯) 2 based on the data above (f) Is it possible to find the coefficient of simple determination from the data shown? If yes, do it. If not, briefly explain why. Question 5 (15 pts). Consider the following diagnostic plots for Model 1 and Model 2. Two simple linear regression models Y = β0 + β1X + ε are fitted to the two different datasets (X, Y ). For each model 3 diagnostic plots are shown: plot of Yi vs Xi , plot of semi-studentized residuals e*i versus fitted values Yˆ i , QQ-plot of the semi-studentized residuals e*i . (a) What is the main issue do you diagnose with the Model 1, if any? Why? Which plot was the most useful in diagnosing this problem? Be as specific in describing the issue as you can. (b) What is the main issue do you diagnose with the Model 2, if any? Why? Which plot was the most useful in diagnosing this problem? Be as specific in describing the issue as you can. (c) This question is unrelated to the above plots. Assume that you conclude that in the model the errors are not normally distributed. How does this affect your inference on b1? Briefly justify. Question 6 (30 points). For the dataset of n = 100 observations a simple linear regression model Yi = β0 + β1Xi + εi is fit. The following estimates are obtained. Above, ni , ki are some functions of Xi . We have listed additional information here (a) Find ni in terms of Xi . Show your work. You may use the least-squares solution formulas for b0, b1. (b) Prove Var Hint: You may use the fact that (c) What is the estimated variance s 2 of the error term based on the data above? (d) Find a 99% confidence interval for β0. Write it in the form. A ± B · t(C, D), compute values of A, B, C, D if possible. (e) Find the joint confidence intervals with confidence at least 99% for β0, β1 in the form. Ai±Bi ·t(Ci , Di). Compute values of Ai , Bi , Ci , Di if possible. Without any computation how does the interval for β0 for this part compare to the one in part (d)? (f) This question is unrelated to the data above. Derive the expression for cov(b0, b1) in terms of some, or all, Xi , Yi , β0, β1, σ2 . Simplify as much as you can. Question 7 (15 points). Suppose we are given n data points {(X1, Y1, Z1),(X2, Y2, Z2), . . . ,(Xn, Yn, Zn)}. We are interested in fitting the linear regression model Yi = α + βXi + εi and Zi = γ + βXi + ηi for i = 1, 2, . . . , n, where εi and ηi are independent random variables N(0, σ2 ). Note that the slope β in both equations is the same. (a) Derive the least squares estimates of α, β and γ. Hint: Write a single least squares objective function as a function of α, β, γ and proceed, i.e. Q(α, β, γ) = . . . (b) What do you think should be an estimator for σ 2 for this model?
Math Stats Practice Final 1. Suppose X0 = 0 and X1, · · · , Xn are given by the following Gaussian autoregressive process i.e. the AR(1) model Xi = θXi−1 + εi with θ ∈ [0, 1) unknown, and the errors εi ∼ N (0, σ2) i.i.d. with σ2 > 0 known. (a) Write down the log-likelihood function for the unknown parameter θ. (b) Find θ ˆMLE. (c) Find the least squares estimate θ 2. Suppose you have a true model y = αeβx+ϵ where ϵ ∼ N (0, σ2) and the following data x y 2 4 3 7 5 8 (a) Find the least squares estimates ˆα and β (b) Construct a 95% Confidence Interval for β. 3. X1 . . . Xn ∼ exp( θ/1 ) IID. (a) Calculate the MLE θˆ (b) Calculate the bias, variance, MSE, and efficiency of θˆ (c) Show that ˆθ → p θ/1, ie. that is it consistent. (d) Calculate the exact and asymptotic sampling distribution of θˆ 4. A coin is thrown independently 10 times to test the hypothesis that the probability of heads is 2/1 versus the alternative that the probability is 10/1 . The test rejects if either 0 or 10 heads are observed. (a) What is the significance level of the test? (b) What is the power of the test? 5. In a famous sociological study called Middletown, Lynd and Lynd (1956) administered questionnaires to 784 white high school students. The students were asked which two of ten given attributes were most desirable in their fathers. The following table shows how the desirability of the attribute ”being a college graduate” was rated by male and female students. Did the males and females value this attribute differently? Male Female Mentioned 86 55 Not Mentioned 283 360 6. Suppose that in the model yi = β0 + β1xi + ei , the errors have mean zero and are independent, but Var (ei) = w1 2σ 2 , where the ρi are known constants, so the errors do not have equal variance. This situation arises when the yi are averages of several observations at xi ; in this case, if yi is an average of ni independent observations, wi 2 = 1/ni (why?). Because the variances are not equal, the theory developed in this chapter does not apply; intuitively, it seems that the observations with large variability should influence the estimates of β0 and β1 less than the observations with small variability. The problem may be transformed as follows: w−1iyi = wi −1 β0 + wi −1 β1xi + wi −1 ei Find the least squares estimate. 7. n1 people are given treatment 1 and n2 people are given treatment 2 . Let X1 be the number of people on treatment 1 who respond favorably to the treatment and let X2 be the number of people on treatment 2 who respond favorably. Assume that X1 ∼ Binomial (n1, p1) X2 ∼ Binomial (n2, p2). Let ψ = p1 − p2. (a) Find the MLE ψˆ for ψ. (b) Find the standard error of ψˆ. (c) Suppose that n1 = n2 = 200, X1 = 160 and X2 = 148. Find ψˆ. Find an approximate 90 percent confidence interval for ψ. (d) Can we say one treatment is more effective than the other? (e) Using the prior f (p1, p2) = 1, find a posterior given X1, X2. (f) What if n1 = n2 = 200 and X1 = 20, X2 = 180? 8. Phillips and Smith (1990) conducted a study to investigate whether people could briefly postpone their deaths until after the occurrence of a significant occasion. The senior woman of the household plays a central ceremonial role in the Chinese Harvest Moon Festival. Phillips and Smith compared the mortality patterns of old Jewish women and old Chinese women who died of natural causes for the weeks immediately preceding and following the festival, using records from California for the years 1960-1984. Conduct to different tests to compare the mortality patterns shown in the table. (Week -1 is the week preceding the festival, week 1 is the week following...) Week Chinese Jewish -2 55 141 -1 33 145 1 70 139 2 49 161 9. Example B from Chapter 9 Section 5 in the textbook. Consider the following data Value 0 1 2 3 4 5 6 7 8 9 10 19 Frequency 56 104 80 62 42 27 9 9 5 3 2 1 Use three different test statistics to see if this data follows a Poisson distribution. 10. You collect data from one population and you want to test if the data follows a particular distribution. What test would you run in the following scenarios: (a) The data is categorical (or discrete) and can only take on two values. (b) The data is categorical (or discrete) and can take on many values. i. You wish to test for the distribution (ie. nonparametric test). ii. You wish to test for a parameter estimate (with asymptotically normal sampling distribution). (c) The data is (continuous) numerical. i. You wish to test for the distribution (ie. nonparametric test). ii. You wish to test for a parameter estimate (with asymptotically normal sampling distribution). 11. You collect data from two populations and want to compare the distributions for both populations: (a) The data is categorical and both populations can only take on two values. i. Data is unpaired. ii. Data is paired (b) The data is continuous: • Data is unpaired – Populations follow normal distribution. – We want to compare the unknown distributions. • Data is paired – Populations follow normal distribution. – We want to compare the unknown distributions. (c) The data is categorical and both populations can take on multiple values. (d) What test would you run if you wanted to compare the distributions of multiple populations? 12. You collect data from one population and want to test if two characteristics of the population affect each other: (a) Both features can only take on two values. (b) One feature is numeric (and continuous), one feature can only take on two values. (c) Both features are numerical (and continuous).
MLF1002 Economics Assessment [30%] Trimester 3.2024 Sanna CANDLES The scented candle market has a low entry barrier because of its simple production process – molten wax [infused with fragrance oils & colouring] poured into a jar with a wick. The market is fast growing and very fragmented – ranging from: 1. Mass produced low-quality candles ($10+) sold in discount stores & supermarkets . . . . 2. and mass produced high-end, premium-brand candles ($40+) sold online, and in specialty homeware outlets & department stores . . . . 3. to hand poured artisan candles ($50+) by small-scale manufacturing enterprise that are usually family-run. Sold online and in specialty homeware outlets School teacher Sanna has been making hand poured artisan candles for the past 10 years as a hobby and side “business” – selling limited quantities to loyal customers in her neighbourhood. At the end of 2022 Sanna left her $100 000/year high-stress teaching job, obtained a business loan of $200 000 to take her business to the next level. As a sole proprietor Sanna manages her business with 2 full-time production employees from a tiny, converted warehouse in Perth. Her hand poured artisan candles tick all the boxes for “premium quality” but retail for only $35 each [online] – attributed to her self-taught production method and relatively low operating costs. Sanna’s production facility (including labour) has a maximum supply capacity of 15 000 units/year [60 units per day x 5 days x 50 weeks]. Sanna CANDLES’s income statement for the 2023 is as follows: Income 10 000 units @ $35 350 000 Expenses/costs Direct candle cost = 10 000 x $6 60 000 Wages 120 000 Interest on business loan 10% 20 000 Rent & utilities 20 000 General expenses 10 000 Net Profit (before tax) 120 000 --------------------------------- Question 1 Draw a Demand & Supply diagram showing the starting Demand Curve (D1) and Supply Curve (S1) of hand poured artisan scented candles in general. a. Briefly explain your diagram’s Price Elasticity of Demand (PED), i.e., D1. b. Briefly explain your diagram’s Price Elasticity of Supply (PES), i.e., S1. c. Briefly identify & explain the factors that can shift the Demand curve of these candles. d. Briefly identify & explain the factors that can shift the Supply curve of these candles. e. On the same diagram, show where the Demand curve shifts right by more than the Supply curve shifts left, and the new equilibrium Price. [You must clearly state all your assumptions throughout] [Your diagram must be accurate, and completely & correctly labelled] [Quality of explanation, about 400 words: sub-total 8 marks] [Accuracy of final diagram: sub-total 2 marks] Total: 10 Marks Question 2 Explain how high annual inflation (say, of 5%) can affect various aspects of Sanna CANDLES’s business especially on her income, costs and profit. o Should address both short term and longer-term impact. o Should address both cost-push and demand-pull inflation. [You must clearly state all your assumptions (if any)] [Your answer must be specific to Sanna CANDLES – otherwise, no marks!] [Please use sub-headings, bullet-points, paragraphs, etc. to separate each point.] [Quality of explanation, about 400-600 words: 10 marks] Aria Foods Co. Self-made millionaire Judy Smith started Aria Foods, a packaged-food business based in Perth 20 years ago. After several setbacks because of failed products, it is now renowned for its innovative products developed through its well-funded research & development (R & D) department. The private business has most employees on a bonus-pay system where top performing workers are given shares in the company. As a market-leader in many food categories in Western Australia, Aria Foods has expansion plans the next 5 years - doubling its production capacity. The company recently developed a ready-to-drink “liquid meal” made from soybean, dairy products, and rice; and is high in fibre, vitamins & nutrients. It has the consistency of a McDonald's thick milk shake and expands 100% in volume when ingested. The ‘meal’ part of the drink consists of chewy tapioca balls similar to those found in bubble tea (“boba”). Therefore, it is more filling than any existing food drinks and the fullness lasts as long as a full solid meal – some 5 hours! Convenient and portable, the beverage comes in patented plastic bottle that allows the drink to last up to 6 hours once opened and unrefrigerated. Both nutritional and great tasting in a wide variety of flavours, the “meal-on-the-go” beverage represents an exciting and innovative opportunity in the “liquid meal” market. The new product was developed as a low-calorie food to address the obesity problem in Australia. Almost two-thirds of Australian adults are overweight or obese with most citing overeating as the primary cause, followed by poor food choice. Question 3 Explain IN YOUR OWN WORDS at least 5 key characteristics (or features) of Capitalism – within the context of Aria Foods. Also, briefly contrast these characteristics with Socialism. [Your explanation must be specific (or relate) to Aria Foods, otherwise no marks!] [Hint: examples of two characteristics are private property and free market . . . . ] [Quality of explanation, about 400-600 words: 10 marks]
CBS Corporate Finance Clarkson Lumber Co – Part 2 Valuing a Firm as a Going Concern Due: October 3rd on or before 11:30 AM Instructions: Your group should upload one electronic copy of your Excel solution on Canvas Canvas (go to “Assignments”, chose the appropriate assignment and click on “submit” on the bar on the right hand side of the screen). You must upload your solution no later than 10 minutes before the start of the lecture on the day in which they are due. Ensure your solution contains your cluster and each group member’s name. You should submit a solution to all “Assignment Questions” . “Discussion Questions” are designed to provoke thought and will be discussed in class but you do not need to address these in your assignment. Your solution should be formatted and set out so that it is easy to read, clearly show your working, the logic behind your answers, and any assumptions you made in the analysis. This assignment uses the same case text you read for lecture 8 (Clarkson 1). Keith Clarkson, sole owner and president of the Clarkson Lumber Company, has received a number of informal inquiries from large, nationally‐recognized building materials distributors about purchasing his company. Although, Clarkson relishes operating his own business he would be interested if an attractive offer were received. Unfortunately, Mr. Clarkson is unsure how much his company is worth so he turns to you for guidance. Value Clarkson Lumber at December 31, 1995 assuming the firm will obtain a credit line at Northrup National Bank sufficiently large to take advantage of discounts on purchases for paying within 10 days of invoice, thus increasing operating profit margins. With higher mark‐ups and continued operating expense controls, Clarkson projects a steady operating profit margin of 6% by 2000 after which he expects the sales growth rate to drop to a fairly steady 7% per year. Margins and investment requirements will also stabilize in relation to sales growth. Make your forecasts using the “Projection Assumptions” in the table below. The discount rate is 12.5%. Note that the forecast ratios already include the benefit of the 2% trade discounts. For simplicity, use a tax rate of 35% throughout the projections, i.e., disregard the tax schedule in note (c) to Exhibit 1. All notes payable on Clarkson’s balance sheet are interest‐bearing liabilities. Make whatever other reasonable assumptions are necessary to complete your analysis and explain the rationale for each. Assignment Questions 1. Project free cash flows for the next five years, 1996‐2000 and estimate the residual value at the end of year 5 using the perpetuity with growth formula. To project cash flows beyond 2000, use the 7% sales growth rate and assume all other FCF components will maintain the same ratios you used for the year of 2000. 2. Value the entire company. What is Mr. Clarkson’s equity interest worth? 3. Using your valuation for Clarkson at the end of 1995, compute the following multiples: a. V(Operations)/Sales (based on estimated enterprise value). Compute this as a forward multiple. b. V(Operations)/EBIT (based on estimated enterprise value). Compute this as a forward multiple. c. In principle, how might you use these multiples to test the reasonableness of your valuation? What other data would you need? Discussion Questions (No submission needed) As you are doing the analysis please consider the following questions. 1. What is the effect of slower growth on free cash flow? What about increased operating margins? 2. When will Clarkson start generating a positive free cash flow? 3. What is the Clarkson Lumber Company worth as a going concern? Projection Assumptions Historical Average 93- Forecast Assumptions FCF Component 95 1996 1997 1998 1999 2000 Sales 5,500 Sales growth rate 24.5% 21.7% 20% 15% 10% 7% CGS/Sales 75.6% 75.0% 74.0% 74.0% 74.0% 74.0% Op Exp/Sales 20.9% 20.4% 20.0% 20.0% 20.0% 20.0% OPM 3.5% 4.6% 6.0% 6.0% 6.0% 6.0% Tax rate 35% 35% 35% 35% 35% AR DOH 43.4 43.4 43.4 43.4 43.4 43.4 Inv DOH 59.4 59.4 59.4 59.4 59.4 59.4 NFATO (@Sales) 12.5 12.5 12.5 12.5 12.5 12.5 AP DOH 39.7 10 10 10 10 10 Acc Exp/Sales 1.46% 1.46% 1.46% 1.46% 1.46% 1.46% Ratio Definitions: OPM (Operating Profit Margin): EBIT/Sales AR DOH (Accounts Receivable Days on Hand): AR/ Daily Sales Inv DOH (Inventory Days on Hand): Inventory/ Daily COGS NFATO (Net Fixed Asset Turnover): Sales/PPE(net) AP DOH (Accounts Payable Days on Hand): AP/Daily Purchases
ECON1020 Country Report Instruction (total number of pages: 10) CONTEXT A multinational company is considering investing in the country that you have been allocated. If the macroeconomic environment of the country is deemed favourable compared to alternative destinations, then the company will proceed to look for specific investment projects. However, you are not provided with any information regarding potential investment projects for commercial confidentiality reasons. YOUR TASKS As independent an economic consultant, you are hired by this company to: a) provide an economic profile for the country; and b) benchmark its performance against Australia, which is the destination recommended by another independent consultant. In doing so, you will deliver a structured Country Report, which includes the following illustrative features: Up to three (3) figures (diagrams, graphs or tables) in total. Here, you can use a mixture of diagrams, graphs and tables provided the total number is no more than 3. You can include more than one variable in a graph or table. You must produce your figures, rather than cut and paste from some sources. You must keep a copy of the file that you use to create your figures as you could be asked to provide it as proof that those figures are your work. You can use direct quotes provided the total number of quoted words sums to 20 words or less in the whole of the report. This rule is intended to prevent students from over-relying on direct quotes without critically analysing the content of the source. COUNTRY ALLOCATION The country you must use for this report is based on the last digit of your student number: For example, if your student number is 44194356, then you need to work on country number 6 - New Zealand. Last digit Country 0 Canada 1 United Kingdom 2 South Korea 3 Switzerland 4 Japan 5 Norway 6 New Zealand 7 Sweden 8 Italy 9 Singapore REPORT STRUCTURE 1. Macroeconomic Performance (~550 words) a. Assess the macroeconomic performance of your country based on macroeconomic indicators covered in Topics 1-4 of this course, e.g., GDP per capita, inflation, unemployment etc., for 5-10 years up to 2018. b. Explain how these indicators are related to each other in the context of your country using theories and concepts covered in the course. c. Explain what factors, including economic policies, are driving these indicators. 2. Benchmarking (~450 words) a. Use Australia as a benchmark to evaluate your country’s macroeconomic performance. b. How does your country’s macroeconomy perform compared to the benchmark country? Explain the reasons for any differences. c. Which country will be a more preferred investment destination and why? 3. Reflection (~200 words) a. Which aspect(s) of the Country Report did you find most difficult and why? b. How did you overcome those difficulties? c. What advice and suggestion(s) can you give to your future fellow students about completing the Country Report? 4. References References and citations must be in the APA 7th style.
5CCE2NMS: Numerical and Statistical Methods Coursework 2 1. In a factory quality control process, two machines are responsible for producing parts. Ma- chine 1, represented by the random variable X, produces parts with quality scores ranging from 3 to 6, depending on various factors like precision and material used. Machine 2, represented by the random variable Y, produces parts with quality scores ranging from 3 to 5. These quality scores are crucial for maintaining the overall quality of the production line, as poor scores from either machine can lead to defects or product failures. Your task is to model the joint distribution of the quality scores for these two machines, considering both their individual performance and their relationship to each other. This model can be used to understand how one machine’s performance affects the other and to predict outcomes in different conditions. Let X and Y be two discrete random variables, where X has the possible outcomes {3, 4, 5, 6} (the quality scores of Machine 1) and Y has the possible outcomes {3, 4, 5} (the quality scores of Machine 2). a. Construct a bivariate probability mass function pX,Y : {3, 4, 5, 6} × {3, 4, 5} → R that satisfies both of the following conditions: [15 marks] (i) The expected quality score of Machine 1 is E[X] = 4.5. (This represents the average performance of Machine 1, giving an idea of how it performs over time.) (ii) The expected quality score of Machine 2, given that Machine 1’s score is 4, is E[Y | X = 4] = 4.6, i.e., the conditional expectation of Y given that x = 4. (This showshow Machine 2 performs when Machine 1 produces apart with a score of 4. It reveals any dependencies between the machines.) b. Calculate the covariance and correlation for your chosen distribution constructed for part (a). (These measures show whether and how the performance of Machine 1 and Machine 2 are related, and how strong that relationship is.) [10 marks] [TOTAL: 25 marks] 2. Imagine you are a data analyst working on a project for a logistics company that manages deliveries across multiple regions. Each day, the company collects data on various aspects of the delivery process to optimise efficiency and reduce costs. Three key random variables, X , Y and Z, represent different metrics: • X represents the number of successful deliveries made out of 10 attempts on a particular route, where each delivery has a 50% chance of success. This follows a binomial distribution. • Y represents the number of delivery attempts needed before the first successful delivery occurs on a challenging route, where each attempt has a 40% chance of success. This follows a geometric distribution. • Z represents the total number of customer orders received in a specific region on a given day, with an average of 4 orders per day. This follows a Poisson distribution. Let X , Y , and Z be independent random variables whose distributions are given by: X ~ Bin(10, 0.5), Y ~ Geo(0.4), Z ~ Po(4). a. Plot the probability mass functions (PMFs) of the distributions for X , Y and Z in ONE figure with three subplots, using the same x-axis range for all subplots. (This helps you visualise the behaviour of the three different metrics and compare their likelihoods.) [3 marks] b. Find a combination of X , Y and Z whose expectation equals √e (where e is Euler’s number) and whose variance is π 2 . (This could be useful for modelling a new metric that meets specific statistical require- ments for predicting future deliveries.) [10 marks] c. Find a combination of X , Y and Z whose expectation is equal to its variance. (This helps identify cases where the variability matches the average performance for indicating consistency. For instance, in a delivery system, if the expectation equals the variance, it suggests the process is stable and predictable. A balanced route with steady deliveries means fewer fluctuations that make the system easier to manage and less prone to disruptions.)) [5 marks] d. Compute the covariance of the resulting variable in part (c) with Z. (This allows you to understand how the new metric from part (c) relates to the number of customer orders Z. It gives insight into how deliveries depend on demand.) [7 marks] [TOTAL: 25 marks] 3. A company operates a customer service system where a customer request is handled by four different departments: A (Initial Support), B (Technical Team), C (Billing), and D (Man- ager Escalation). The probabilities of a customer issue being handled by these departments stabilise over time and the long-term behaviour is described by the following probabilities: p∞ (A → A) = 0.1, p∞ (A → B) = 0.4, p∞ (A → C) = 0.2, p∞ (A → D) = 0.3. This scenario can be modelled using a Markov chain, where the customer request moves between departments according to a set of transition probabilities. You are tasked with constructing and analysing this Markov chain. a. Construct a Markov chain with four states {A,B,C, D} representing the departments, such that the long-term transition probabilities from state A match those given above. Present the transition matrix and draw the diagram of the resulting Markov chain. In the transition matrix, the value in Column i and Row j represents the probability of transitioning from state si to state sj . (The transition matrix should show the probabilities of a customer request being han- dled by each department, and the diagram will help visualise how the request moves through the system.) [13 marks] b. What is(are) the most likely path(s) for a customer request to move from Initial Support (A) to Billing (C) within 4 time steps or fewer in your proposed Markov chain? (This analysis helps determine the most probable route the request will take through the departments within a limited number of steps.) [4 marks] c. Modify the Markov chain by introducing a fifth state E to represent an external consul- tant that sometimes handles customer requests. In this case, the long-term behaviour is such that p∞ (E → E) = 0, but at any given time step k > 0, the probability of staying in state E is non-zero: pk (E → E) 0. Draw the updated diagram of the Markov chain. (This addition reflects situations where a request is temporarily passed to an external consultant but will eventually return to the internal team.) [8 marks] [TOTAL: 25 marks] 4. a. In a board game, success may depend on rolling a specific number on a die multiple times. You are rolling a fair six-sided die and need to determine how many trials it will take to roll the number ‘1’ exactly s times, with the restriction that the first roll is not a ‘1’ . What is the probability that it takes exactly n rolls to achieve this outcome? (This scenario helps analyse the likelihood of reaching a specific number of successes in a game with the consideration of a restriction on the first trial.) [15 marks] b. In a network management setting, server reliability is crucial. Suppose a server has a probability of going down once every 10 days, modelled by a Poisson distribution, with the probability of a single downtime being 0.1. What is the probability that the server will experience exactly 5 downtimes in a 365-day period? (This situation models server reliability over time, helping plan for maintenance sched- ules and system downtime management.) [10 marks] [TOTAL: 25 marks]
Applied Stats and Data Analysis EN.553.413-613, Fall 2024 Oct 31st, 2024 Midterm 2 Question 1 (22 points). Consider the linear model Y = Xβ + ε, where Y is a n-by-1 vector of response variables, X is an n-by-p design matrix, β is a p-by-1 vector of coefficients (we assume n > p), ε is a multivariate normal Nn(0, σ2 In). Denote by b the least-squares estimate of β. (a) TRUE or FALSE. Y ∼ Nn(Xβ, σ2 I) (b) TRUE or FALSE. Cook’s distance measures the influence of one predictor on all fitted values. (c) TRUE or FALSE. Hβ = b. (d) TRUE or FALSE. Studentized deleted residuals are the same as Internally Studentized Residuals. (e) TRUE or FALSE. Polynomial regression is an example of Multiple Linear Regression. (f) TRUE or FALSE. rank(X) = rank(XTX). (g) TRUE or FALSE. SSR = ∥Y − Yb∥ 2 . (h) TRUE or FALSE. Externally studentized residuals are better than internally studentized residuals in determining outliers with respect to Y . (i) TRUE or FALSE. Variance Inflation Factor helps us determine influential observations. (j) TRUE or FALSE. Outliers with large leverage are always influential. (k) TRUE or FALSE. As we add more predictors, the adjusted coefficient of multiple deter-mination Radj 2 always increases. Question 2 (16 points). In this problem we deal with the same multiple linear regression model as in Question 1. (a) Write the least-squares objective function Q(β) that we need to minimize to fit a multiple linear regression. Write it in terms of Y, X, β. (b) Take the derivative ∂β/∂Q and equate it to 0. How are these equations called? (c) From part (b) find the formula for the least-squares estimate for the regression vector b in terms of matrices X, Y. What are conditions needed for the unique solution to exist? (d) State the distribution of b. Make sure to specify mean and variance. You don’t need to derive anything here. (e) This part could be solved independently from other parts. Compute the covariance ma-trix Cov(e, Y), where e is a vector of residuals, Y is a vector of observed response vari-ables. Simplify as much as you can. What are the dimensions of the matrix Cov(e, Y)? Question 3 (16 points). Below is the sketch of a column space Col(X) of an n-by-2 design matrix X. The vector Y is an n-by-1 vector of values Yi , 1 is an n-by-1 vector of 1’s, X1 is an n-by-1 vector of predictor values Xi1. Let b be the least squares estimate of β. • Vector (iv) is the orthogonal projection of Y onto Col(X). • Vector (v) is the orthogonal projection of Y onto the vector of 1’s. (a) Express vectors (i),(ii),(iii) in terms of Y, X, b, Y. Here, Y is an n-by-1 vector of Y ’s. (b) Express vectors (i),(ii),(iii) in terms of Y, H, I, J, n, where H is the hat matrix, I is an n-by-n identity matrix, J is an n-by-n matrix of 1’s. (c) Write the coefficient of multiple determination R2 in terms of the norms of some or all vectors (i),(ii),(iii),(iv),(v), e.g. ∥(i)∥ is the norm of the vector (i). (d) (BONUS Q) Prove that SSR = YT (H − n 1J)Y. Show your steps. Question 4 (20 points). Consider the following regression model: Yi = β0 + β1X1i + β2X2i + εi where the εi are iid N(0, σ2 ). We provide the following information: We use lm in R to fit a linear regression model, obtaining the output below: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.8436 0.4591 1.837 0.1256 x1 0.4052 0.1346 3.011 0.0297 x2 0.8340 XXXXXX XXXXX 0.0150 --- Residual standard error: 0.5353 on 5 degrees of freedom Multiple R-squared: 0.8327,Adjusted R-squared: 0.7657 F-statistic: 12.44 on 2 and 5 DF, p-value: 0.01145 (a) State the null and the alternative hypothesis for the model utility test for this model. What is your conclusion at the significance level α = 0.02? Briefly explain. (b) Is β1 significant at the significance level α = 0.02? Is β2 significant at the significance level α = 0.02? Briefly explain. (c) Rewrite the hypothesis test H0 : β1 = β2 in the form. of the General Linear Test: H0 : Cβ = γ, Ha : Cβ ≠ γ. Identify C and γ and state their dimensions. What is going to be the degrees of freedom for the computed F ∗ statistic? You do not need to compute any statistic here. (d) The plot below shows the 98% Confidence Intervals for β1, β2 individually (along the axes) and the joint 98% Confidence Region (the ellipse). Shade, or indicate otherwise, the areas where (0, 0) might lie in this plot consistent with the R output above. (e) Standard error for the slope β2 is hidden by XXXXX. Recover it using the data provided above. Question 5 (20 points). Consider a fitness study tracking weekly calorie burn among par-ticipants. Let X1 represent the total distance run in kilometers, X2 the average intensity of workout sessions as a percentage (ranging from 0 to 100), and X3 an indicator variable that is 1 if the participant had a recovery day in that week and 0 otherwise. The outcome variable Y denotes the total weekly calorie burn. We fit the multiple linear regression model: Yi = β0 + β1Xi1 + β2Xi2 + β3Xi3 + β4Xi1Xi3 + β5Xi2Xi3 + εi where εi are i.i.d. N(0, σ2 ). The regression output from fitting this model is shown below: Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2799.28178 23.47759 119.232 < 2e-16 *** x1 0.13253 1.51817 0.087 0.930835 x2 -1.00306 0.59743 -1.679 0.100247 x3 120.25577 33.25913 3.616 0.000767 *** x1:x3 0.34690 2.15538 0.161 0.872873 x2:x3 0.07701 0.84579 0.091 0.927869 --- Residual standard error: 3.225 on 44 degrees of freedom Multiple R-squared: 0.9979,Adjusted R-squared: 0.9977 F-statistic: 4257 on 5 and 44 DF, p-value: < 2.2e-16 The first 3 rows of the dataset are the following Y X1 X2 X3 2746 7 56 0 2709 20 90 0 2858 13 73 1 (a) How many observations are in the full dataset? (b) Write only the first three rows of the design matrix X for this model. What are the dimensions of the full design matrix X for this model? (c) Do you notice any issues with this model? If you do, how would you fix the issue? (d) Write the mean calorie burn for a week that includes a recovery day, expressed in terms of all, or some, X1, X2, and X3. (e) Should the model above include only X3 based on the results above? Briefly justify. (f) Should both interaction terms be included? Write the Reduced and Full model, specify how the F ∗ statistic needs to be computed. Do not compute it. What are the degrees of freedom for that F statistic? Can we make a conclusion about this based on the results shown above? Briefly explain. Question 6 (16 points). The table below contains leverages hii, residuals ei , internally studentized residuals ri , externally studentized residuals ti for the model with 2 predictors and 8 observations. Table 1: Values of hii, ei , ri , and ti . (a) Which points seem to be outliers with respect to X? Briefly justify. (b) What should be the decision rule to identify observations outlying with respect to Y at the significance level α = 0.10? Write it in the form.|A| ≥ t(B, C) Identify A, B, C. Be as specific as you can. (c) Below is the Cook’s distance plot for this dataset. Based on this plot, which two ob-servations appear the most influential here? Determine approximately the value of the Cook’s distance for these observations based on the plot below.
TRI201 Translation: Theory and Practice COURSE WORK II Academic year 2024/2025 Semester 1 INSTRUCTIONS 1. This course work takes the form. of a piece of critical essay, and is 1,500-word in length, accounting for 60% of the final mark. 2. Your essay should be clearly structured with an introduction, a conclusion and a body part. 3. Your essay should discuss a translation phenomenon with theories you have learned in class. The topic of your essay should come from translation practice and aim at providing guidance for it. It may, for instance, be: Translation-related issues such as translation quality, equivalence and subtitling; Language-related issues such as the use of gender-neutral language, the translation of modifiers, and translationese; Social and cultural issues such as patronage, visibility of the translator that lead to changes to the ST. 4. Your essay should address ONE or TWO research questions regarding the text or context. DO NOT attempt to address more than two research questions in a single research essay. The research essay should have a thesis or central argument supported by original ideas and detailed analysis using illustrative examples from the text. 5. Your essay should give an original and critical reading of the translation process and/or translation product using illustrative examples and references to ELABORATE on ideas in your central argument. Discussions that only describe the translation phenomena without critical reflection or organizing the ideas into a persuasive analysis will receive a satisfactory grade only. The paper will be evaluated on the quality of your writing, your ability to analyse texts and apply theories of translation to practice, your independent and critical reflection, and your skills in crafting a coherent and compelling argument. Outstanding papers demonstrate depth of understanding, original thought, and a fresh perspective on the topic. 6. The research paper must be fully referenced and include a bibliography in line with the APA referencing system. Each essay must have a word count at the end of the essay (excluding all references). Pages should be numbered. The text must use Times New Roman, size 12 and double spacing. Make sure to run a grammar and spelling check after finishing the essay. 7. A soft copy of your essay has to be uploaded onto LMO by 5 PM Monday, 30 December (subject to change). The file name of the essay should follow the format: Full name [Student ID]. Full name is First name + Surname, with first name your registered name on LMO. For example, Xiaofeng Zhang [123456]. 8. Submissions without following exactly the instructions given herein may not reach the instructor, and marks will be deducted for course works that require re-submission. 9. For late submission, penalties will be given in accordance with the University policy. 10. The use of generative AI and machine translation is NOT allowed.
PROGRAMMING IT (COMPSCI 4039) Monday 12 December 2022 Important note: throughout this exam, whenever you are asked to write Java code, do not worry about whether your code compiles. The markers will never test any submitted code from this exam. 1. (a) The International Olympic committee are designing an IT system, built in Java, that will record the medal tables in their summer and winter games. The following class will be used to represent a team: public class Team { private String name; private int numGold; private int numSilver; private int numBronze; } Here, name is the team name (for instance, “Ireland”), and numGold, numSilver, and numBronze are the total numbers of gold, silver and bronze medals that the team has accumulated so far in the competition. (i) Write a constructor for the Team class that is called bypassing only the team name. This should set name to be the passed string and initialise numGold, numSilver and numBronze to 0. [3] (ii) Write a second constructor that takes four parameters to add a team to the table after they have already been awarded medals. The four parameters should be the team name String as well as three integers for the numbers of medals already won by that team. [4] (b) Create getter and setter methods for the name string and the three integer attributes (i.e. the medal counts) of the Team class. [8] 2. (a) Since each team can only win at most 1 medal of each kind for any given event, overload the setter methods for the medal counts so that the new version is passed no parameters and simply increments the count by 1. [3] (b) Create a method within the Team class that will allow the team name and medal counts for each team to be printed out using a System .out .println(t), where tis a Team object. The team name, number of golds, number of silver and number of bronze should be printed out in that order, in the fixed width order given below (note that the periods should be printed as spaces – periods are just being used in this diagram so you can count them easily): Australia . . . . . . | . . . 17 | . . . .7 | . . .22 [7] (c) A class in now required to hold the leaderboard for the games, in an array of Team objects. Part of this class is as follows: public class Leaderboard { private int numTeams; private Team[] teams; } Write a constructor for this leaderboard class. The constructor should take a single parameter, an integer representing the number of Teams competing at the games. The number of teams should be stored in numTeams and this value should be used to instantiate teams, the array of Team objects. [3] (d) Give the command that could have been used to initialise a leaderboard for this year’s Beijing Winter Olympics, in which 91 teams took part. [2] 3. (a) The code shown below is an attempt by the college in printing a simple welcoming message for the new students of this semester. Unfortunately, the code will not work. Identify 4 mistakes that can be found in the code. public class Welcome New Students { static void main(String[] args) { System .out .println(Welcome students!) } } [4] (b) The college is asking developers to find a way of sorting out the student’s records. (i) In the first part the developers need to identify the highest grade. You are asked to create a methods that can be used to identify and print out the highest grade. Assume that the marks are in an integer format and the records are saved as an array. For the array use the values 25, 89, 95, 60, 45,78. [7] (ii) Write another function in the same class you created for Q3bi in order to identify the lowest grade. The lowest grade should also be printed. All the details are the same as presented in 3bi. [4] 4. (a) One student’s surname was entered wrongly in the students’ text record. His surname was “Ferguson” but the entry was “Farguson” . (i) Write a method that can easily check if the wrongly typed student’s surname is in the text file. This means that the method should return true if the line is found or false if it is not. Assume that the path of the file is just “filepath”, the name of the text file is “Srecord” and no newline characters are included. Error handling for FileNotFound should be considered. [4] (ii) After producing code for reading the text file (4ai) include additional code that should be included in the previous method, for replacing the wrong surname (Farguson) with the correct one (Ferguson). Make sure to include a printout of the updated surname. [2] (iii) What would be the difference if we wanted to replace any entries having the surname “Farguson” with an empty line? Write how the previous code from 4aii would be modified so the entry ”Farguson” would be replaced with an empty line instead. Write the relevant code. [1] (b) For the same course a string array was created called “Sname” which produces as an output the names of the students in an alphabetical order. The following names were included (Thomas,Amaya, Gustavo, Melina, Reirse, Bell). This is a bad coding example and there are several mistakes in the code. You are asked to identify 8 mistakes and point them out, explaining them. Class AlphName{ public static void main{ int n = 0; String Sname[] = {"Thomas, Amaya, Gustavo, Melina, Reirse, Bell"}; String temp for (int i = 0; i < n;) { for ( k = 100; k < n; k++) { if (Sname[i] .compareTo(Sname[k]) >0{ temp = Sname[i]; Sname[i] = Sname[k]; Sname[k] = temp; } } } System .println("Names in alphabetical order:"); for (int i = 0; i < n; i++) { System .out .println(); } } } [8]
Linear Algebra - Fall 2023 Exam 2 1. (a) Find a basis in ker(A), where (b) Find a basis in Im(A), where 2. Let A be the matrix (a) Find the rank and nullity of this matrix. (b) Find a matrix C whose image is the kernel of the linear transformation whose matrix is A. 3. Let P2 = {a + bx + cx2|a, b, c 2 R} be the vector space of two polynomial functions of degree less or equal to two. Consider the map T : P2 ! P2 given by T(f) = f + f' + f'' where f' represents the derivative of the polynomial f. (a) Verify, using the definition (or otherwise) that T is a linear transformation. (b) Find the matrix A of the linear transformation T with respect to the basis U = (1, x, x2). PART II of 3. (c) Find the change of basis matrix S from B to U, where B is the basis B = (1,(x + 1),(x + 1)2) (you don’t have to verify here that B is a basis). (d) Using any method, find the matrix B of the linear transformation T with respect to the basis B. 4. Find an orthonormal basis in the subspace V of R4, where 5. Let v1, v2 2 R4 be the vectors (a) Check that (v1, v2) is an orthonormal basis in V = Span{v1, v2}. (b) Find the projection projV (w) of the vector w = onto V . PART II of 5. (c) What are the coordinates of the vector z = projV (w) with respect to the basis (v1, v2)? (d) Determine the matrix of the linear transformation T(x) = projV (x). 6. A 4 x 3 matrix has rank 2. Answer the following: (a) What is the nullity of this matrix? (b) Are the columns of A linearly independent? (c) If the system Ax = b is consistent, how many free variables does it have, if any? (d) What is dim(ker(A)) and dim(Im(A))? BONUS A+ Let A be a 2 x 3 matrix and B be a 3 ⇥ 2 matrix, such that AB = I2, the 2 ⇥ 2 identity matrix. Consider the linear transformation T : R3 ! R3 defined by T(x) = BAx Show that T is a projection onto a plane V and describe V in terms of A and B.
Section 1: Multiple Choice Questions Part1 1. In 2020, most climate scientists believed that a. the evidence for global warming remained somewhat flimsy. b. carbon dioxide was the only human made source of the greenhouse effect. c. the earth was likely to warm over the next 50 years as a consequence of greenhouse gas pollution. d. belly button lint caused cancer. e. atmospheric carbon dioxide levels were likely to stabilize naturally within 20 years. 2. Negative feedback effects on global warming a. include exposure of dark earth as polar ice caps melt. b. would accelerate the warming trend. c. would slow down the warming trend. d. would result if higher CO2 levels reduced the capacity of the ocean to absorb CO2. e. are likely if the melting of frozen tundra increases the emission of methane gas into the atmosphere. 3. If global warming does occur, economic costs include a. enhanced agricultural productivity in cold climates. b. sea-level rise. c. enhanced agricultural productivity, especially in poor countries. d. a likely increase in the diversity of natural ecosystems, as warmer climates emerge. e. b and d. 4. Benefit-cost analysis of global warming a. has proven to be relatively uncontroversial. b. suggests that on net, controlling CO2 emissions will generate higher costs than benefits. c. is a scientific process, free of ethical decisions. d. calls for at least moderate reductions in greenhouse gas emissions. e. has created a consensus view among economists as to how fast new technologies can be developed. 5. Given that government regulators operate in a world of poor information, and are subject to political influence, a. conservatives nevertheless concede that government intervention to protect the environment is generally socially beneficial. b. conservatives seek an absolute minimum of government intervention. c. progressives have faith that a laissez-faire attitude is best for the environment. d. progressives view active government as both effective and necessary. e. b and d. 6. Incentive-based regulatory approaches, such as pollution taxes a. areviewed positively by most economists-- both progressive and conservative-- as a way to control pollution. b. provide less flexibility than traditional technology-based regulation. c. tend to hurt wealthier people more than poor people. d. would be sufficient, in the eyes of progressive economists, to control global warming. e. require that the government specify certain types of pollution control technology that firms must adopt. Part2 1. Ecological economists argue that a. resource scarcity may lead to catastrophic outcomes in the long run. b. created capital can generally substitute for natural capital. c. important resource scarcities are unlikely to emerge until the second half of the 21st century. d. that population and consumption pressure may lead to near term increases in the price of key resources. e. a and d. 2. The IPAT equation tells us that to keep emissions of nitrogen oxide constant, if population doubles a. and consumption per person also doubles, technology must improve by a factor of eight. b. and consumption per person is halved, than technology must also cut emissions per unit of output by half. c. and technology cuts emissions of output by half, than consumption must stay constant. d. and consumption per person also doubles, technology must improve by a factor of 2. e. a and c. 3. To measure the sustainability of resource use, ecological economists a. have developed the measure known as NNW. b. weigh population and consumption pressure against resource stocks. c. adjust GDP to account for the costs of economic growth. d. rely on the decay rate of plutonium. e. give a generous allowance for technology to generate substitutes for the resource in question. 4. If selling timber in a country generates resource rent of $200 million in a given year, then: a. NNW will rise by $200 million that year. b. NNW fall by $200 million that year. c. IW will fall by $200 million that year. d. IW will not change that year. e. None of the above 5. Environmental footprint analysis a. concludes that we need more than 5 planets to support the current level of population. b. utilizes the acreage needed to sequester carbon to account for emissions of carbon dioxide. c. focuses largely on the acreage needed for agricultural production. d. is lower for countries high on the Human Development Index (HDI) scale. e. is a preferred tool of neoclassical economists. Part3 Plant A (MCa = 12 - 2xa) Plant B (MCb = 9 - xb) 1. In the diagram above, Plant A is currently polluting 9 units of gunk per day, while Plant B is polluting 6 units of gunk, for a total of 15. To achieve a cost-effective reduction to 9 units total a. regulators could distribute tradeable permits for 7 units per day to plant A, and 2 units per day to Plant B. b. regulators could set a tax equal to $4 per unit of gunk. c. regulators could specify that Plant A reduce emissions to 4 units per day, while plant B reduce emissions to 5 units per day. d. aor b. e. a, bor c. 2. Suppose that regulators set a tax equal to $4 per unit of gunk per day. In that case, Plant A would pay in clean-up costs and taxes a. $24. b. $20. c. $16. d. $10. e. $4. 3. Pollution taxes can be made revenue neutral by a. employing the Coase Theorem Corollary. b. rebating pollution taxes back in the form. of income tax cuts. c. improving labor market efficiency. d. using a permit give-away system. e. raising income taxes. 4. Relative to command-and-control regulation, incentive-based approaches increase the incentives for discovering new pollution control technology by a. making it costly for firms to pollute, even below the standard. b. doing away with the requirement to use a certain type of technology. c. forcing the marginal costs of reduction at each plant to be equal. d. a and b. e. a, band c. 5. Hotspots a. are more likely with marketable permit systems than pollution taxes. b. occur when pollutants are uniformly mixed. c. can be avoided under a pollution tax system if monitoring and enforcement are beefed up. d. can be controlled under a tax system by varying the level of the tax in different regions. e. area relatively minor problem with incentive-based regulation. 6. Replacing a system of command-and-control regulation with marketable permit regulation is unlikely to achieve all possible cost-savings due to each of the following except a. thin markets. b. poor information on the part of regulators about the costs of pollution reduction at each firm. c. the need for additional investment in monitoring and enforcement. d. the exercise of market power. e. the need to put some restriction on permit life. Part 4 1. The following are normative statements: a. If global warming occurs, sea-level rise is likely to pose a serious problem for some countries. b. Because of the costs associated with sea-level rise, carbon dioxide emissions should be cut to slowdown global warming. c. Survey results indicate that a majority of the population agrees with the statement: global warming should be prevented regardless of the cost. d. Ifour goal is to slowdown global warming, carbon dioxide emissions should be cut. e. b,c, and d. 2. Economists believe that normative positions on environmental issues a. area matter of opinion, and thus not amenable to study. b. can be better understood if all underlying assumptions are clearly stated. c. are only valid if based in utilitarian thought. d. area matter of opinion, and thus not relevant to real life problems. e. should not differ if all parties are rational individuals. 3. From a utilitarian perspective, preserving rainforests to protect biodiversity a. is relatively unimportant. b. should not be done if it conflicts with profits. c. always makes sense if it actually protects endangered species. d. should be pursued if it increases overall human well-being. e. is definitely a bad idea if it leads to local people losing jobs and income. 4. Equal marginal utility of consumption a. means that Joand Al receive the same level of utility from their overall consumption. b. means that Joand Al receive the same level of utility from small increases or decreases in consumption. c. means that a one dollar increase to a rich individual just offsets the decrease in social welfare of a one dollar decrease to a poor individual. d. implies that policy-makers should be quite sensitive to the distribution of costs and benefits generated by environmental protection. e. band c. 5. If Al is alive today, and Jo is not yet born, than the social welfare function SW=UAl(XAl)+UJo(w*XJo) a. reduces the value of Jo's consumption by a factor w. b. indicates that Al's consumption cannot come at Jo's expense. c. implies that an efficiency standard is called for. d. fails to take sustainability into account. e. restates Einstein's theory of relativity. 6. In the social welfare function SW=UAl(XAl, w*P)+UJo(XJo, w*P), let P stand for pollution, with Aland Jo both exposed. This SW function a. implies that a sustainability standard is called for. b. implies that an efficiency standard is called for. c. implies that a safety standard is called for. d. pays no special attention to pollution victims. e. is generally less valid than the one in question 5. Section 2: Modeling Environmental Problems 1. Two paper plants are polluting the bay outside of Lobsterville with gunk. The authorities have decided that to protect the lobster fisheries, total gunk emissions must be reduced to 9 tons per day. Plant A has marginal emission reduction costs of 10-xa dollars per day, where xa is the level of pollution, and is currently emitting 10 tons. Plant B has marginal emission reduction costs of 10-2xb dollars per day, where xb is the level of pollution, and is currently emitting 5 tons. Thus, 15 tons are currently being emitted. The situation is diagrammed below. a. Suppose the regulators decide that each plant must reduce emissions by 3 tons (xa=7, xb=2). Is this a cost-effective way to control emissions back to a level of 9? Why or why not? b. Now suppose that the regulators wish to reduce total emissions to 6 tons of gunk. What tax rate perton should they charge? In the figure below, illustrate the taxes and clean-up costs that each firm would bear under this proposal. (No need to calculate them.) c. Suppose gunk was a non-uniformly mixed pollutant. Could a uniform pollution tax be used to achieve the identical lobster health goal that the city council was shooting for in part (a)? Section 3: Bonus Problem (short answer – 3-4 paragraphs). Reflect on the negotiation exercise you did in class. What do you think are the obstacles in the way of the world coming to a climate agreement, and do you think nations will overcome these obstacles?
Microeconomics Final Examination Fall 2024 Instructions: Follow the instructions in each part of the final examination to answer the questions on the exam. If you are timing yourself, each question should take you on average no longer than 30-minutes to answer. There is no need to rewrite each question, you may write your answers into a Word document. Be certain to include the Part and Question Number for each of your answers, and to incorporate every step instructed in the question into your answer. I will be reading your answers critically for the content. If you want to receive extra credit you may answer additional questions on the final examination after you have answered the required questions in each part of the exam. Your completed final examination must be submitted to me no later than December 21, 2024 at 11:59 p.m. via Dropbox. It may be submitted as either a Word document or a PDF. However, I highly recommend submitting your completed examination as a PDF to ensure your intended formatting remains in place. Part One: Key Principles (25 Points) Key principles provide foundation for understanding a discipline. Briefly, discuss two of the five following key principles and their role in microeconomics in not less than 250 words (double spaced in 12-point Times New Roman font). Be certain to address each of the components from the topic. Be certain your explanation includes content from the required readings, lecture notes, and digital media archive. 1.) Total Physical Product and Marginal Physical Product Total physical product is the amount of output the firm obtains in total from a given quantity of inputs. Marginal revenue product is the increase in total output that results from a one-unit increase in the input quantity. First, discuss the relationship between total physical product and marginal physical product. Then, define marginal revenue product and discuss its relationship to marginal physical product. Lastly, give one example of each from a recent news article. 2.) Input Quantities The most desirable output quantity for the firm clearly depends on how costs change as output varies. First, discuss the three types of cost curves economists use to display and analyze this information. Then, discuss the marginal product relationship. Lastly, give one example of how costs change as output varies for the firm from a recent news article. 3.) Price, Output and Profit It is a common misperception that the firm selects a price and a quantity of output that maximize profit. First, discuss why this is a common misperception. Then, discuss the impact activities of other firms in the market competing for a share of total market demand have on a firm. Lastly, give one example of the firm maximizing its profit from a recent news article. 4.) The Firm Under Perfect Competition Industries differ dramatically in the number and size of their firms. Perfect competition is a market structure in which firms are numerous and small. First, discuss the conditions for perfect competition. Then, discuss the perfectively competitive firm. Next, compare the perfectively competitive firm to monopoly power. Lastly, give one example of the perfectively competitive firms and one example of monopoly power from a recent news article. 5.) Limiting Marketing Power: Regulation and Anti-Trust To protect the public interest from monopolies, government uses anti-trust policy to prevent acquisition of monopoly power. In addition, some industries are regulated by rules that constrain firms’ pricing. First, discuss how the government uses anti-trust policy to prevent acquisition of monopoly power. Then, discuss one industry that is regulated by rules that constrain its pricing. Lastly, give an example of each from a recent news article. Part Two: Key Concepts (25 Points) Key principles develop key concepts. Briefly, discuss two of the five following key concepts and their role in microeconomics. Each discussion should be no less than 150 words. Be certain to address each of the components from the topic. Be certain your explanation includes content from the required readings, lecture notes, and digital media archive. 1.) Production, Inputs, and Cost: Building Blocks for Supply Analysis The firm can generally substitute one input for another. Whether or not it pays to substitute depends on the relative costs of labor and machinery. First, describe the alternative types of input proportions available to the firm. Then, describe the combination of inputs that represent the least costly way for the firm to produce its goods. Lastly, give one example of the firm’s substitutability from a recent news article. 2.) Production, Inputs, and Cost: Building Blocks for Supply Analysis Total profit is the net earnings of the firm during a period of time. Marginal profit is the addition to total profit resulting from one more unit of output. Total revenue is the total amount of money the firm receives from the consumers of its goods, without any deduction of costs. Marginal revenue is the addition to total revenue resulting from the addition of one unit to output. First, expand on these descriptions to completely describe their impact on the firm. Then, describe fixed cost and the profit-maximizing price. Lastly, give one example of the profit-maximizing firm from a recent news article. 3.) Output, Price, and Profit: The Importance of Marginal Analysis If the firm is losing money, in certain cases it will be better off continuing to operate until its obligations to pay the non-variable costs expire. First, describe what happens if the firm stops producing. Be certain to describe what happens to its costs that are non-variable. Then, describe in which circumstances will the firm do better by shutting down immediately and producing nothing. Lastly, give an example of each from a recent news article. 4.) Output, Price, and Profit: The Importance of Marginal Analysis When a perfectly competitive industry is in long-run equilibrium, firms maximize profits so that P = MC. First, describe the relationship between the firm and the industry under perfect competition in the long-run. Then, describe the firm supply curve under perfect competition in the short-run. Next, describe the industry supply curve under perfect competition in the short-run. Lastly, give one example of the firm under perfect competition in the long-run and one example of the industry under perfect competition in the long-run from a recent news article. 5.) Limiting Market Power: Regulation and Anti-Trust The concentration of an industry measures the share of the total sales or assets of the industry in the ownership of its largest firms. First, describe an industry that has a very low concentration ratio. Then, describe an industry that has a very concentration ratio. Next, describe what occurs when circumstances in the industry are favorable for price collusion. Lastly, give one example of the Herfindahl-Hirschman from a recent news article. Part Three: Formulas and Schedules (25 Points) Key principles and key concepts describe formulas and graphs. Briefly, discuss three of the four following formulas or schedules to discuss and their role in microeconomics. Be certain to include in your discussion how each is used as an analytic tool used by economists. Each discussion should be no less than 150 words. Be certain to address each of the components from the topic. Be certain your explanation includes content from the required readings, lecture notes, and digital media archive. 1.) Production, Inputs, and Cost: Building Blocks for Supply Analysis First, use the following schedule to calculate the marginal physical product, the marginal revenue product and the average physical product. Then, discuss the optimal point for the perfectly competitive firm. peryear percarpenter peryear percarpenter Assumptions: Price Per Garage: $15,000 Cost Per Worker: $50,000 2.) Production, Inputs, and Cost: Building Blocks for Supply Analysis First, use the following schedule to calculate the marginal revenue, the marginal cost, the total profit, and the marginal profit. Then, list the formula for total revenue and total cost. Lastly, discuss the condition under which the firm will earn a profit, incur a loss, the output that maximizes the perfectly competitive firm’s profit, and under which a firm is maximizing profit by minimizing loss. (TP)Marginal 3.) Output, Price, and Profit: The Importance of Marginal Analysis First, use the following schedule to calculate the total cost, the loss if the firm shuts down, and the loss if the firm does not shut-down. Then, identify in which case (i.e. Case A or Case B) the firm should shut-down and discuss the reasons why the firm should shut-down in one case but not the other. Finally, discuss the condition under which the firm will incur a loss and the output level at which the firm will continue to operate. ble Cost$80,000$80,000Total Cost(TC)Loss if Firm Shuts Down(short-run, non-variable -Down
ESE 5060 - Introduction to Optimization (Final Exam) Fall Semester, 2023 Instructions: You must answer any five (5) of the following six (6) problems. Do not answer more than five (5) problems. If you do, only the first 5 problems answered will be graded. Please indicate on the front of your exam booklet which problems you want graded and please be sure to cross out any problems in your exam booklet that you do not want graded. Problem #1 (20 points) - The Branch and Bound Method Suppose that an optimal tableau for the LP Problem max z = 4x1 + 6x2 + 3x3 (Objective Function) s.t. 3x1 + 2x2 + 5x3 ≤ 68 (Constraint #1) 4x1 + 6x2 + 3x3 ≤ 83 (Constraint #2) 3x1 + 4x2 + 2x3 ≤ 57 (Constraint #3) x1 ,x2 ,x3 ≥ 0 (Sign Restrictions) is Row BVs Values x1 x2 x3 s1 s2 s3 0 z 83 0 0 0 0 1 0 1 x1 5 [ 1] 0 0 0 −2 3 2 x2 13/2 0 [ 1] 0 −1/8 9/8 −11/8 3 x3 8 0 0 [ 1] 1/4 3/4 −5/4 Using the Branch and Bound Method, determine the solution to this LP if x2 is now restricted to be an integer. You must show your tableaus for full credit. Only a final answer without the tableaus yields very little or no credit. Problem #2 (20 points) - Constructing An IP Problem Consider the following non-linear programming problem max z = 5x1 + 3x2 + 7x3 (Linear Objective Function) s.t. |x1 + 2x2 + 4x3 | ≥ 20 (Non-Linear Constraint #1) 2x1 + 3x2 + x3 ≤ 30 (Linear Constraint #2) x1 ,x2 ,x3 ≥ 0 (Sign Restrictions) that contains: one (1) linear objective function, one (1) non-linear constraint, one (1) linear constraint and three (3) sign restrictions. a.) (6 points) Construct a single IP problem that can be used to solve this non-linear programming problem. Your single IP problem should contain: one (1) linear objective function, four (4) linear constraints, five (5) variables, three (3) sign restrictions and two (2) integer restrictions. Be sure to clearly state how any big M’s are computed and note that when constructing your big M’s, you need only observe that from the linear constraint 2x1 + 3x2 + x3 ≤ 30 and the sign restrictions x1 ,x2 ,x3 ≥ 0, it’s enough to note that 0 ≤ x1 ≤ 15 , 0 ≤ x2 ≤ 10 and 0 ≤ x3 ≤ 30. Of course, you need not solve the resulting IP problem. b.) (14 points) If we now want to require that at least one of the following three (3) additional constraints |4x1 − x2 + x3 | ≤ 15, |3x1 − 8x2 + 9x3 | ≥ 72 or 7x1 + 4x2 − 3x3 = 25 are also satisfied, construct a single IP problem that can be used to solve this. Your single IP problem should contain: one (1) linear objective function, nine (9) variables, eleven (11) linear constraints, three (3) sign restrictions and six (6) integer restrictions. Be sure to clearly state how any big M’s are computed. Of course, you need not solve the resulting IP problem. Problem #3 (20 points) - A Transportation Problem Steelco manufactures three types of steel at different plants. The time required to manufacture 1 ton of steel (regardless of type) and the costs to manufacture 1 ton of steel at each plant are shown in the following table. PlantSteel One Ton of Steel 1 One Ton of Steel 2 One Ton of Steel 3 Time Plant 1 $60 $40 $28 20 min Plant 2 $50 $30 $30 16 min Plant 3 $43 $20 $20 15 min Each week, 100 tons of each type of steel (1, 2, and 3) must be produced and each plant is open 40 hours per week. a.) (10 points) Formulate a balanced transportation problem to minimize the cost of meeting Steelco’s weekly requirements. Put both your answer in the form of the tableau as shown here. b.) (5 points) Use the shipping cost method to determine an initial basic feasible point and the value of w for this problem. Note that if there is a tie in shipping costs when using the shipping cost method, you must break the tie by choosing the most northwest corner results between the ties. c.) (5 points) Suppose the time required to produce 1 ton of steel now depends on the type of steel as well as on the plant at which it is produced, as summarized in the next table. PlantSteel Steel 1 (Minutes) Steel 2 (Minutes) Steel 3 (Minutes) Plant 1 15 12 15 Plant 2 15 15 20 Plant 3 10 10 15 Could a transportation problem still be formulated? Explain. Problem #4 (20 points) - Sensitivity Analysis During a sensitivity analysis on the LP min w = cTx (Objective Function) s.t. Ax = b (Constraints #1-#m) x ≥ 0 (Sign Restrictions) in which both b and c were changed, the optimal tableau is found to changed to the following tableau. Row BVs Values x1 x2 x3 x4 x5 x6 0 −w 100 0 0 0 5 −1 3 1 x1 8 [ 1] 0 0 2 −5 −2 2 x2 16 0 [ 1] 0 −4 3 1 3 x3 −4 0 0 [ 1] −1 2 0 Continue solving the LP with this tableau to arrive at a new optimal solution. You must show your tableaus for full credit. Only a final answer without the tableaus yields very little or no credit. Problem #5 (20 points) - An Interesting Cone Recall that if are n × 1 columns, we say that x
