Linear programming model
Microsoft Word - SIT718_Assessment-Task_4-T1_2019.docx ASSESSMENT DETAILS SIT718 Real World Analytics Assessment Task 4: Problem Solving Key information Due: Weighting: Thursday 13 February 2020 by 23:30 AEDT. 25% Reference style: Harvard Learning Outcomes This assessment assesses the following Unit Learning Outcomes (ULO) and related Graduate Learning Outcomes (GLO): Unit Learning Outcome (ULO) Graduate Learning Outcome (GLO) ULO3 – assessed through student ability to apply game theory, and linear programming skills and models, to make optimal decisions. ULO4 - assessed through student ability to develop software codes to solve computational problems for real world analytics. GLO1 - Discipline knowledge and capabilities GLO4 – Critical thinking GLO5 – Problem solving Purpose Assignment 4 assesses your abilities to build linear programming models and solving them. You will demonstrate your skills in using linear programming to model real life case studies. You will consider cases with two variables and solve them using the graphical method. For problems with more than two variables, you will solve the linear programming models that you built using linear programming solvers in appropriate software, such as R. You will consider game theory – two players game – to build appropriate models that describe different game scenarios. You will demonstrate your knowledge in investigating the existence of equilibrium (stable solution). You will use mixed models to find appropriate solutions and solve the models you constructed with appropriate software such as R. Instructions The work is individual. Solutions and answers to the assignment must be explained carefully in a concise manner and presented carefully. Use of books, articles and/or online resources related to SIT718 Real World Analytics is allowed. Students are expected to refer to the suitable literature where appropriate. Students can attempt all tasks below and provide an individual written report in appropriate word processor. The detailed problem description and data set will be released to students on Monday 20th January 2020. ã Deakin University 1 ASSESSMENT DETAILS ã Deakin University 2 Submission details Your final submission should consist of: 1. A pdf file (created in any word processor), containing the solutions of the questions, labelled with your name; 2. Two codes combined in one with your R file, labelled with yourname.R, with lp models for questions 2 and 3. • No more than 8 A4 sides, including Figures, Tables, Appendices and References. The report should be typed. Use minimal font 11pt and 2.5cm side margins. If the page limit is exceeded only the first 8 pages will be marked. • Assignment (a report in pdf format, software code and/or data) must be submitted via the assignment dropbox in the unit site (accessed the Program page). • No e-mail or hardcopy submissions are accepted. Extension requests Requests for extensions should be made to Unit/Campus Chairs well in advance of the assessment due date. If you wish to seek an extension for an assignment, you will need to apply by email directly to to Dr Thanh Nguyen (
[email protected]), as soon as you become aware that you will have difficulty in meeting the scheduled deadline, but at least 3 days before the due date. When you make your request, you must include appropriate documentation (medical certificate, death notice) and a copy of your draft assignment. Conditions under which an extension will normally be approved include: Medical To cover medical conditions of a serious nature, e.g. hospitalisation, serious injury or chronic illness. Note: Temporary minor ailments such as headaches, colds and minor gastric upsets are not serious medical conditions and are unlikely to be accepted. However, serious cases of these may be considered. Compassionate e.g. death of close family member, significant family and relationship problems. Hardship/Trauma e.g. sudden loss or gain of employment, severe disruption to domestic arrangements, victim of crime. Note: Misreading the timetable, exam anxiety or returning home will not be accepted as grounds for consideration. Special consideration You may be eligible for special consideration if circumstances beyond your control prevent you from undertaking or completing an assessment task at the scheduled time. See the following link for advice on the application process: http://www.deakin.edu.au/students/studying/assessment-and-results/special-consideration Assessment feedback Students will receive written feedback to aid reflection and analysis of problem strategies and solutions for consideration in the upcoming problem-solving task. Referencing You must correctly use the Harvard method in this assessment. See the Deakin referencing guide. mailto:
[email protected]?subject="SIT718 T3 2019 Assessment 4" http://www.deakin.edu.au/students/studying/assessment-and-results/special-consideration https://www.deakin.edu.au/students/studying/study-support/referencing/harvard ã Deakin University 3 ASSESSMENT DETAILS Academic integrity, plagiarism and collusion Plagiarism and collusion constitute extremely serious breaches of academic integrity. They are forms of cheating, and severe penalties are associated with them, including cancellation of marks for a specific assignment, for a specific unit or even exclusion from the course. If you are ever in doubt about how to properly use and cite a source of information refer to the referencing site above. Plagiarism occurs when a student passes off as the student’s own work, or copies without acknowledgment as to its authorship, the work of any other person or resubmits their own work from a previous assessment task. Collusion occurs when a student obtains the agreement of another person for a fraudulent purpose, with the intent of obtaining an advantage in submitting an assignment or other work. Work submitted may be reproduced and/or communicated by the university for the purpose of assuring academic integrity of submissions: https://www.deakin.edu.au/students/study-support/referencing/academic-integrity SIT718 Real world Analytics Assessment Task 4 2019 T3 Total Marks = 100, Weighting - 25% Your final submission must include the following two files: 1. "name-report.pdf"A report, in pdf format (created in any word processor), covering all of the items in above (where “name” is replaced with your name -you can use your surname or first name). With plots and tables, it should be up to 8 pages. 2. "name-code.R"The R code file (that you have written to produce your results) named(where “name” is replaced with your name - you can use your surname or first name). Your assignment will not be assessed if we cannot reproduce your results with your R code. https://www.deakin.edu.au/students/study-support/referencing/academic-integrity https://www.deakin.edu.au/students/study-support/referencing/academic-integrity 1. A cheese factory is making a new cheese from mixing two products A and B, each made of three different types of milk - sheep, cow and goat milk. The compositions of A and B and prices ($/kg) are given as follows, Amount (litres) per 1000 kg of A and B Sheep Cow Goat Cost ($/kg) A 30 60 40 5 B 80 40 70 8 The recipes for the production of the new cheese require that there must be at least 45 litres Cow milk and at least 50 litres of Goat milk per 1000 kg of the cheese respectively, but no more than 60 litres of Sheep milk per 1000 kg of cheese. The factory needs to produce at least 60 kg of cheese per week. a) Explain why a linear programming model would be suitable for this case study. [5 marks] b) Formulate a Linear Programming (LP) model for the factory that minimises the total cost of producing the cheese while satisfying all constraints. [10 marks] c) Use the graphical method to find the optimal solution. Show the feasible region and the optimal solution on the graph. Annotate all lines on your graph. What is the mini- mal cost for the product? [10 marks] Note: you can use graphical solvers available online but make sure that your graph is clear, all variables involved are clearly represented and annotated, and each line is clearly marked and related to the corresponding equation. d) Is there a range for the cost ($) of A that can be changed without affecting the optimum solution obtained above? [5 marks] Hint: This question does not require unit conversion. ã Deakin University 4 Maia Angelova Turkedjieva 2. A factory makes three products called Spring, Autumn, and Winter, from three materials containing Cotton, Wool and Silk. The following table provides details on the sales price, production cost and purchase cost per ton of products and materials respectively. Sales price Production cost Purchase price Spring $60 $5 Cotton $30 Autumn $55 $4 Wool $45 Winter $60 $5 Silk $50 The maximal demand (in tons) for each product, the minimum cotton and wool propor- tion in each product is as follows: Demand min Cotton proportion min Wool proportion Spring 4200 50% 40% Autumn 3200 60% 40% Winter 3500 40% 50% a) Formulate an LP model for the factory that maximises the profit, while satisfying the demand and the cotton and wool proportion constraints. [10 Marks] b) Solve the model using R/R Studio. Find the optimal profit and optimal values of the decision variables. [10 Marks] Hints: 1. Let xij ≥ 0 be a decision variable that denotes the number of tons of products j for j ∈ {1 = Spring, 2 = Autumn, 3 = Winter} to be produced from Materials i ∈ {C=Cotton, W=Wool, S=Silk}. 2. The proportion of a particular type of Material in a particular type of Product can be calculated as: e.g., the proportion of Cotton in product Spring is given by: xC1 xC1 + xW1 + xS1 . ã Deakin University 5 3. Two mining companies, Company 1 and Company 2, bid for the right to drill a field. The possible bids are $ 10 Million, $ 20 Million, $ 30 Million, $ 40 Million and $ 50 Million. The winner is the company with the higher bid. In case of a tie (equal bids) Company 1 is the winner and will get the field. For Company 1 getting the field for more than $ 40 Million is as bad as not getting it (assume loss), except in case of a tie (assume win). (a) State reasons why/how this game can be described as a two-players-zero-sum game [5 Marks] (b) Considering all possible combinations of bids, formulate the payoff matrix for the game. [5 Marks] (c) Explain what is a saddle point. Verify: does the game have a saddle point? [5 Marks] (d) Construct a linear programming model for Company 1 in this game. [5 Marks] (e) Produce an appropriate code to solve the linear programming model in part (d). [5 Marks] (f) Solve the game for Company 1 using the linear programming model you constructed in part (e). Interpret your solution. [5 Marks] ã Deakin University 6 4. Consider two companies, Company 1 and Company 2,