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Spring 2010 Exam 1 Solutions

Spring 2010 Exam 1 Solutions - STOR 112 Spring 2010 Exam 1...

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STOR 112 Spring 2010 Exam 1 Solution 2010-2-25 Instructions: The exam consists of 6 questions for a total of 120 points. Work the problems in a blue book. You can write on the exam, but it will not be graded. Make sure your name and section number are on your blue book. (Section 1 = Pataki; Section 2 = Anderson). Clearly mark each problem number. Please answer the problems in order. Start each problem on a new page.(If you want to skip a problem, leave plenty of space for that problem). You may use a calculator. For LP and IP formulations, you must give a clear definition of the decision variables, an objective function, and the constraints. Give a descriptive label for the objective AND each constraint(such as ”profit”). Do not make assumptions about the solution based on the problem data (such as, product A makes much less profit than product B, so we will not make any of it, and exclude it from the formulation), even if it is possible to do so. Write all constraints in the standard form: variables on the left side and a number on the right. When turning in your exam, please show your UNC OneCard to one of the in- structors. Please return the exam paper as well. Question 1. (10 points) Consider the linear programming problem max 10 x 1 + 3 x 2 st. - x 1 - x 2 ≤ - 6 (1) 3 x 1 - 2 x 2 3 (2) - 2 x 1 + 3 x 2 13 (3) x 1 , x 2 0 . Graph the feasible region. List all the corner points, and show how to find their coordinates. List the objective value of the corner points. What is the optimal value of the LP? 1
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Solution. Graph the feasible region. List all the corner points, and show how to find their coordinates. List the objective value of the corner points. Corner point A: at the meeting of constraints (1) and (2). We need to solve - x 1 - x 2 = - 6 3 x 1 - 2 x 2 = 3 to get A = (3 , 3) . Objective value is 39. Corner point B: at the meeting of constraints (1) and (3). We need to solve - x 1 - x 2 = - 6 - 2 x 1 + 3 x 2 = 13 to get B = (1 , 5). Objective value is 25. Corner point C: at the meeting of constraints (2) and (3). We need to solve 3 x 1 - 2 x 2 = 3 - 2 x 1 + 3 x 2 = 13 to get C = (7 , 9). Objective value is 97. 2
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What is the optimal value of the LP? It is 97, attained at the point C. Question 2. (15 points) A farmer has 10 acres to plant in wheat and corn. He has to plant at least 3 acres of wheat. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of corn. The profit is $300 per acre of wheat and $500 per acre of corn. Formulate a linear programming problem to find out how many acres of each should be planted to maximize total profit.
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