Departments of Engineering Science and Statistics ENGSCI / STATS 255 Assignment 4 Due: 12:30pm, 2/6/2021 No extensions will be available for this assignment due to the late due date. Instructions for handing in. Please... 1. Assignments may be typed in a word processor of your choice, or handwritten neatly. 2. Set answers out in order of the questions. Do NOT jump between questions. 3. There is no need to copy the questions out in your submission. 4. If answering a question using R, all commands that you enter should be provided as part of the answer, along with all relevant output. Include your ID number on output wherever possible, e.g. as part of the title of a graph. Submission: Assignments must be submitted using Canvas as a single PDF file, before the due date and time. Handwritten assignments will need to be scanned. Prepare your assignments well in advance of the deadline in case of technical issues. Notes: • Summarising, analysing and communicating information is an important part of Operations Research. For this reason you will be expected to write answers which clearly communicate your thoughts. The mark you receive will be based on your written English as well as your technical work – review the relevant section in the Course Outline. • We encourage working together. Discussing assignments and methods of solution with other stu- dents or getting help in understanding from staff and students is acceptable and encouraged. You must write up your final assignment individually, in your own words. • By submitting this assignment, you confirm that you understand the University’s policies on cheating, plagiarism and group work; that your submission is entirely your own work and you have not allowed access to any part of the assignment to any other person. See the appropriate sections in the Course Outline for more details. • This assignment consists of THREE questions, and is marked out of 60 marks. 1 1. (20 marks) Nocturnal Aviation, Limited has possibly not chosen the best time to announce its new route between Auckland and Raoul Island. Nevertheless, the announcement has been made, and the company’s analyst is hard at work on simulations of possible pricing strategies. The analyst decides to model the arrival of inquiries from potential customers over the 10 days prior to a flight departure as a Poisson process with constant rate λ = 2 per day. A potential customer arriving t days before departure has a budget, or willingness to pay, modelled as a random variable distributed uniformly between 100 and 800 − 40t dollars, independent of other customers. A sale is made if the customer’s budget exceeds the ticket price, which has been set at $300. The aircraft has 19 passenger seats, although some of these may already be taken by passengers who booked more than 10 days in advance. (a) What is the expected number of customer inquiries occurring in the last 10 days before departure? What is the probability that there are fewer than 12 enquiries? (b) A customer enquiry comes in 2 days before the flight departs. Assuming seats are still available, what is the probability that this customer buys a ticket? (c) Perform simulations of the last 10 days before departure, with i. 16 seats still available ii. 8 seats still available at the beginning of this period. For each simulation, use 10000 runs and use the last 3 digits of your student ID number as the random seed. Make histograms showing the number of empty seats on the plane when it departs. (d) Estimate, with confidence intervals, i. the expected number of empty seats on the plane; ii. the probability that more than 5 seats are empty. (e) Has the ticket price been chosen wisely? Would it be better to price the tickets at $200, or $400 ? (f) An inhomogeneous Poisson process might be a better model for customer inquiries. Sketch a graph of a plausible-looking rate function λ(t), and explain why it is plausible. 2 2. (22 marks) This question makes use of the lake inflow dataset from Assignment 3. You’ll need the file inflows.csv as used in that assignment. (a) For each of the 26 years, obtain the mean inflow into Lake Pukaki by averaging over the 52 weeks. The range of values should be from 43.68581 (in 1992) to 98.428 (in 1998). Show the values on a histogram. (b) A simple model for the distribution of the annual mean inflow to Lake Pukaki has the following triangular probability density function. .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........ .... .... .... .... .... .... .... .... .... .... .... .... .... .... ........ ..... 40 75 100 .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .............. .......................................................................................................................................................................................................... .. .... ... .... ... .... ... .... ... What is the probability, according to this model, that an annual mean inflow is less than 60 ? (c) Another model for the distribution of the annual mean inflow to Lake Pukaki is the empirical distribution: each of the 26 values in the dataset occurs with probability 126 . What is the probability, according to this model, that an annual mean inflow is less than 60 ? (d) Devise a procedure for generating random variates according to the triangular distribution model. (e) Use your method from (2d) to generate a sample of 10000 random variates. Show them on a histogram. (f) Use your sample from (2e) to estimate, with confidence intervals, (i) the mean of the distribution; (ii) the probability that an annual mean inflow is less than 60. (g) Use the sample function in R to generate a sample of 10000 random variates according to the empirical distribution. Show them on a histogram. (h) Use your sample from (2g) to estimate, with confidence intervals, (i) the mean of the distribution; (ii) the probability that an annual mean inflow is less than 60. Do your confidence intervals contain the true values? 3 3. (18 marks) Sales transactions in the self-checkout lane at a mid-sized supermarket can be modelled as a Poisson process with a rate of 4.5 transactions per minute, and the time required for a customer to complete the checkout can be modelled as a random variable uniformly distributed between 0 and 1 minute, independently of everything else. (a) What is the minimum number of self-checkout machines required to handle sales at this rate? (b) Customers are encouraged to form themselves into separate queues, one for each self-checkout machine. Customers are assumed to join whichever queue is shortest at the time they arrive. Use the queueing-simulation web page to perform a simulation of the process under these assumptions, for a situation where there are 3 self-checkout machines operating. Use the last 3 digits of your student ID number as the random seed. Hand in a suitable summary of the simulation output, relevant to the remaining parts of this question. (c) Explain why your simulation results are (or are not) consistent with Little’s Law. (d) Construct a 95% confidence interval for the mean time spent by a customer to complete a purchase, including both time spent waiting and being served. (e) An analyst from corporate HQ suggests that it might more efficient if customers were instead organized into a single queue, with multiple servers. Modify your simulation from part (b) to represent this proposal, and so make a point estimate of how much time it would save for the average customer. 4













































































































































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