Answer four Linear Programming questions
MATH 392 – Introduction to Linear Programming, Spring 2024 Take-Home FINAL EXAM. DUE on May 06, 2024, at 12:00 PM EDT Directions for Take-Home Final Exam: This is not a group project. Do your own work. Submit your work in .docx, .pdf, or .jpg format. Except In extraordinary circumstances, NO OTHER FOMATS WILL BE ACCEPTED. You can either email your completed Final Exam to me in the Canvas email or in my VSU email [email protected]. 1. Solve this Linear Programming Problem by-hand using the Simplex Method as presented in our class. Show an initial feasible solution, show all logic for determining which variable is entering the basis, show the ratio calculations for determining which variable is leaving the basis, and show all elementary row calculations to pivot the simplex tableau at each iteration. If correct work details are not shown, no credit will be given. Maximize: Subject to: π = ππ + πππ + πππ ππ + πππ + ππ ≤ ππ ππ + ππ − πππ ≤ ππ ππ , ππ , ππ ≥ π 2. Set up the Linear Program generated by the following scenario and solve this Linear Program using the Graphing Method. A merchant plans to sell two models of home computers at costs of $650 and $800, respectively. The $650 model yields a profit of $45 and the $800 model yields a profit of $50. The merchant estimates that the total monthly demand will not exceed 250 units. Find the number of units of each model that should be stocked in order to maximize profit. Assume that the merchant does not want to invest more than $70,000 in computer inventory. Page 1 of 2 3. State this problem as objective function and constraints. Then solve this Linear Programming Problem using the Simplex Method as discussed in our class. You may use automated tools (LP Assistant.jar or Excel Solver) but be sure to give enough information to let me know that you know what you are doing. A small petroleum company owns two refineries. Refinery 1 costs $25,000 per day to operate, and it can produce 300 barrels of high-grade oil, 200 barrels of medium-grade oil, and 150 barrels of low-grade oil each day. Refinery 2 is newer and more modern. It costs $30,000 per day to operate, and it can produce 300 barrels of high-grade oil, 250 barrels of medium grade oil, and 400 barrels of low-grade oil each day. The company has orders totaling 35,000 barrels of high-grade oil, 30,000 barrels of mediumgrade oil, and 40,000 barrels of low-grade oil. How many days should the company run each refinery to minimize its costs and still meet its orders? a. State the shadow prices and reduced costs and interpret their meanings in the context of this scenario. b. What is the solution of the dual problem? How do you know? Explain your answer. 4. Solve the following Linear Program. If you use an automated solver, state which solver was used. Show the work details from that solver (screenshots are okay). Minimize: π = πππ + ππππ + ππππ Subject to: πππ + πππ − πππ ≥ π −πππ + πππ − ππ ≥ π ππ − ππ + πππ ≥ π ππ , ππ , ππ ≥ π a. What would be the effect on the solution if the first constraint is replaced with πππ + πππ − πππ ≥ π? b. What would be the effect on the solution if the third constraint is replaced with ππ − ππ + πππ ≥ π? Page 2 of 2
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