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MATH 4113/5113 - Operations Research 1 Take Home EXAM 1 - Take Home Due: Monday October 11, 2021 by 11:59pm (CDT) INSTRUCTIONS: • This is an open-book, open-note examination with the following restrictions: You may use any printed textbook, your notes from this class, any notes that I have made publicly available, and Wikipedia. This excludes almost all information found on the internet. Additionally, you may NOT discuss the questions with anyone, share work with anyone, or receive assistance from anyone except the instructor. Any violation of this policy will be reported to the Office of Student Conduct and will result in (at minimum) a grade of zero for this exam. • All written work MUST be done on plain Letter (8.5"x11") paper and submitted as a PDF file. Start each part on its own page. Please print your name clearly at the top of every page. Number the pages sequentially. Include this page as the first page of your exam. • I have tried to make the questions as clear as possible. If you are unsure what a question is asking, contact me and ask. Until I reply, make a reasonable assumption and state it as part of your answer. • You can contact your instructor at any time by calling/texting 405-753-0191. I (printed name) hereby certify that this exam is all my own work. I received help from no person other than the instructor of this course, Dr. Paynter. Signed: Fall 2021 Page 1 of 5 MATH 4113/5113 - Operations Research 1 Take Home Part 1: KJ Investment Partners is starting a new derivative fund containing a mix of bonds, home loans, auto loans, and personal loans. The estimated rate of return on each of these derivatives is 10% for bonds, 16% for home loans, 13% for auto loans, and 25% for personal loans. In order to manage risk, the following stipulations have been put on the fund: • There must be at least $1.20 of bonds for each $1 of personal loans. • There must be at least 75c of auto loans for each $1 of home loans. • The amount invested in personal loans cannot exceed 20% of the fund’s value. Formulate a linear program to determine the mix that maximizes estimated return while satisfying the conditions. You should submit: (a) A GNU MathProg file named “username-exam1.1.mod” containing: i. A model that runs in MathProg and generates a solution to this problem. [5 points] ii. A model that gives the correct optimal solution and objective value. [6 points] iii. A model that is written with future readability in mind (comments, variable names, whitespace, etc.). [4 points] iv. A model that is correctly written in set notation. [3 points] (b) Included in your submitted PDF: i. A written version of your linear model using correct mathematical notation. [3 points] ii. A written model that correctly uses set notation. [2 points] iii. A written report of the optimal solution. [2 points] Optimal Objective Value: 15.64% Part 2: ProChem Inc. runs a complex chemical plant that runs 24-7. One section of the plant deals with the processing of two raw materials, I and II into four sellable products and waste. • Process A combines the raw materials in a 2:3 ratio to produce Product 1, Product 3, and waste in a 3:5:2 ratio. This process takes 9 worker-hours of labor per ton of raw material processed. The machinery required for Process A can handle 0.75 tons of raw material per hour and costs $50 per hour to run. • Process B is run in batches requiring exactly 1.5 tons of Raw Material I, 0.8 tons of Raw Material II, 3 hours on the appropriate machinery, and 14.5 worker-hours of labor. Process B produces 0.5 tons of Product 2 and 1.4 tons of Product 3, leaving 0.4 tons of waste. Process B’s machinery costs $25 per hour to run. • Process C produces only Product 4. It consumes 0.6 tons of Product 1 and 1.2 tons of Product 2 for each ton of Product 4, generating 0.8 tons of waste in the process. • Process D turns Product 3 and waste into Products 2 and 4. Equal amounts of each input are required. The process produces 1.5 tons of Product 4 for each ton of Product 2. • Processes C and D run on the same machinery which can process 0.5 tons of input material per hour, costs $38 per hour to run, and requires 8 workers to constantly monitor it. Additionally, if Process C and D are both produced, the machinery needs to be shut down to be converted from one process to the other. The conversion process requires 15 hours (of clock time) and 20 worker-hours. Fall 2021 Page 2 of 5 MATH 4113/5113 - Operations Research 1 Take Home • Our existing contracts with our suppliers requires them to deliver 90 tons of each raw material each week for a fixed charge. Another contractor disposes of remaining waste at a cost of $120 per ton. They can dispose of up to 5 tons of waste per week. • Our contracts with our customers oblige us to deliver 25 tons of Product 1 and 30 tons of Product 2 per week at prices of $270 per ton and $310 per ton respectively. We could also sell up to 20 tons more of Product 2 per week at the same prices if it were available. • Our contract for Product 3 is to deliver 75 tons per week at a price of $120 per ton. There is a mechanism in this contract for us to deliver less, but that incurs a penalty of $30 per ton we are short each week. • Product 4 is sold on the retail market for $360 per ton. The corporate office is confident that this market could easily produce sales of up to 40 tons per week. • Our current contract with the labor union pays workers $45 per hour. Formulate a linear program that maximizes the weekly profit of this division. You should submit: (a) A GNU MathProg file named “username-exam1.2.mod” containing: i. A model that runs in MathProg and generates a solution to this problem. [5 points] ii. A model that gives the correct optimal solution and objective value. [6 points] iii. A model that is written with future readability in mind (comments, variable names, whitespace, etc.). [4 points] iv. A model that is written in set notation. [3 points] (b) Included in your submitted PDF: i. A written report of the optimal solution. [2 points] Note: • One week contains 168 hours. • In each process, the total weight of material going in is the same as the total weight of material going out. • You do not need to account for labor scheduling. That will be determined at a separate time. • Your solution will be implemented on an ongoing (cyclical) basis. As a result, you do not need to wait to start Processes C & D. The ingredients are already available from last week. • Don’t forget that, if you run both Process C and Process D, you have to shut down the machinery twice. Once to switch from C to D and once to switch back to C for the start of the next week. Optimal Objective Value: $136,754.29 Fall 2021 Page 3 of 5 MATH 4113/5113 - Operations Research 1 Take Home Part 3: LSG Manufacturing has a large, expensive piece of machinery that is vitally important to their business. Any equipment needs maintenance and repairs and those costs increase as the equipment gets older. The LSG’s machine is currently 3 years old and so they are planning the 10-year replacement schedule for the machine. Purchasing a new machine at any point costs $150,000, but the old machine can be sold to offset that cost. It costs an additional $5,000 to replace the machine in lost productivity, moving costs, etc. LSG can buy new machines at the beginning of any of the next 10 years. Machine Age Sale Price Finance Cost Maintenance Cost (years) ($) ($/year) ($/year) [0, 1) 150,000 8,278 3,500 [1, 2) 135,000 6,642 5,500 [2, 3) 121,500 4,906 8,000 [3, 4) 109,000 3,062 11,000 [4, 5) 98,000 1,105 14,500 [5, 6) 88,500 0 18,000 [6, 7) 80,000 0 22,000 [7, 8) 72,000 0 26,000 [8, 9) 64,500 0 29,000 [9, 10) 58,000 0 33,000 [10, 11) 52,000 0 36,000 [11, 12) 47,000 0 38,500 [12, 13) 42,000 0 40,500 [13, 14) 38,000 0 43,000 Formulate a linear program to determine when LSG should buy new machine(s) in order to minimize the total cost of ownership of this machine for the next 10 years. You should submit: (a) A GNU MathProg file named “username-exam1.3.mod” containing: i. A model that runs in MathProg and generates a solution to this problem. [5 points] ii. A model that gives the correct optimal solution and objective value. [6 points] iii. A model that is written with future readability in mind (comments, variable names, whitespace, etc.). [4 points] iv. A model that is correctly written in set notation. [3 points] (b) Included in your submitted PDF: i. A written version of your linear model using correct mathematical notation. [3 points] ii. A written model that correctly uses set notation. [2 points] iii. A written report of the optimal solution. [2 points] Optimal Objective Value: $265,943 NOTE: There is a template for this model posted on D2L with the model section already com- pleted. BONUS: Rewrite the parameter definitions of the model file so that the data section of your model only includes the numbers given on this page. That is, rewrite your model so that it does the calculations to determine the arc cost. [3 points] BONUS BONUS: Complete the bonus only using set notation. [3 points] Fall 2021 Page 4 of 5 MATH 4113/5113 - Operations Research 1 Take Home Part 4: A power utility has a number of power stations of three different types that they use to provide power to a specific region, 10 of type 1, 8 of type 2, and 5 of type 3. Each plant has to run between a minimum and maximum level. The hourly cost of running a plant is calculated as a base cost (for running at minimum power), an hourly cost for each additional megawatt over the minimum, plus an additional cost for starting a generator. All of this information is contained in the following table (where costs are in dollars). P ow er P la nt