Midterm 1 - Solutions

Midterm 1 - Solutions - ENGR62/MS&E111 Spring...

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Unformatted text preview: ENGR62/MS&E111 Spring 2005 Introduction to Optimization May 5, 2005 Prof. Ben Van Roy Midterm Exam: Due Friday May 6th, 2005, at 5:00PM Instructions Sign the honor code statement and return this hand-out when you submit your solutions. Turn in your exam to Prof. Van Roys office, Terman 315. The exam will be graded out of total of 100 points. Questions 14 are each worth 25 points. The extra credit problem is worth an additional 5 points. Partial credit will be awarded for work that is handed in that is relevant and partially solves a problem. Honor Code In recognition of and in the spirit of the Honor Code, I certify that I will neither give nor receive unpermitted aid on this examination and that I will report, to the best of my ability, all Honor Code violations observed by me. Signature: Name: 1. True or False (25 points) State whether each of the following statements is true or false. For each part, only your answer, which should be one of True or False, will be graded. Explanations will not be read. For each correct answer, you will receive five points. For each incorrect answer, you will lose two points. No points are assigned or lost for a part unanswered. a) If a linear program has at least one optimal solution and every optimal solution is a vertex of the feasible region, then it has only one optimal solution. b) For a linear program that has M variables and N constraints ( M N ), at most M constraints can be active at a vertex of the feasible region. c) If a linear program has an unbounded feasible region then it has no optimal solution. d) If a linear program has at least one optimal solution then the feasible region must have a vertex that is an optimal solution. e) If the feasible region of a linear program is nonempty and bounded then it must have a vertex that is an optimal solution. solution a) True. Suppose not, take two vertices that are optimal. Then, any convex combination of these two vertices is optimal as well, contradicting the fact that every optimal solution is a vertex. b) False. Consider an example with N = 2 and M = 3 with constraints x 1 0, x 2 0, x 1 + x 2 0; all three constraints are active at x = 0. c) False. Consider minimizing a scalar x subject to x 0. d) False. Consider minimizing a x 1 subject to x 1 = 0, for a problem with two variables: x 1 and x 2 . Then, for any x 2 , (0 , x 2 ) is optimal but not a vertex. e) True. This follows from Theorem 3.3.1. 2. Automobile Manufacturing You are the production manager for a manufacturer of three types of automobile spare parts. You must determine a monthly production plan for the coming month. The manufacture of each part requires processing on each of two machines, with the following processing times in hours: Part A B C Machine1 0.02 0.04 0.03 Machine2 0.05 0.03 0.02 Each machine is available 40 hours per month. Each part manufactured will yield a unit profit as follows: Part A B C Profit 100 90 80 The demand for each part as follows:...
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This note was uploaded on 10/12/2009 for the course ENGR 62 taught by Professor Unknown during the Spring '06 term at Stanford.

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Midterm 1 - Solutions - ENGR62/MS&E111 Spring...

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