Exam_Solution - IOE 310 Winter 2007 Final Exam Name...

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Unformatted text preview: IOE 310 Winter 2007 Final Exam April 13, 2007 Name: Instructions • This is a closed book, closed notes exam. • You may use one double sided 8 × 11 inch note sheet and calculator during the exam. Note that not all problems carry the same weight. • There are 5 problems in the exam and 12 pages (including this one). You can use the back of each sheet for rough calculations. • Please write your solutions to each problem in the space provided. Show all work and write legibly. Your grade will depend in part on the completeness and clarity of your answers. • Sign your name below after reading the Honor Pledge. Instructors are not required to grade exams in which the Honor Pledge is not signed. Good luck! Honor Pledge: “I have neither given nor received aid on this examination.” Signed: 1 Problem 1 (10 points) For each of the following statements, determine whether it is true or false and explain why. a) The total float of an activity in a project is f . Then this means the activity can be delayed by f time units without delaying any of the successors of that activity. Answer: T/F False Reason: That is the definition of free float not total float. b) Suppose we have a transportation problem with the following objective function: min z = X i ∈ S X j ∈ D c ij x ij Then the following is the correct AMPL structure for the objective function: minimize z { i in S } : sum { j in D } c[i,j]*x[i,j]; Answer: T/F False Reason: The correct syntax is the following. minimize z: sum { i in S, j in D } c[i,j]*x[i,j]; c) Suppose we are solving an integer program using branch and bound. As we branch further down the tree, the objective function at each of the nodes can only increase. Answer: T/F False Reason: It depends on whether we are minimizing or maximizing. d) Consider the following problems: (1) min c T x (2) max b T y s.t. s.t. Ax ≥ b A T y ≤ c x ≥ y ≥ Assume ¯ x and ¯ y are feasible solutions to problems (1) and (2) respectively. Then by the weak duality theorem: c T ¯ x ≤ b T ¯ y Answer: T/F False Reason: Weak duality states that: c T ¯ x ≥ b T ¯ y 2 Write your explanations for the following question. e) Consider the following problems: (1) max c T x (2) max c T y s.t. s.t. Ax = b Ax = b x ∈ { , 1 } ≤ x ≤ 1 Assume ¯ x and ˜ x are feasible solutions to problems (1) and (2) respectively. Argue that c T ¯ x ≤ c T ˜ x Answer: Problem (2) is the LP relaxation of Problem (1), therefore it has a better objective function value. In this case we are maximizing so the optimal objective function value of Problem (2) is going to be larger....
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This homework help was uploaded on 04/02/2008 for the course IOE 310 taught by Professor Saigal during the Spring '08 term at University of Michigan.

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Exam_Solution - IOE 310 Winter 2007 Final Exam Name...

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