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Homework9

# Homework9 - 000001(c(10 points ³or what values of ε is...

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ENGRI 1101 Engineering Applications of OR Fall 2008 Homework 9 Linear Programming: Sensitivity and Duality Due date: 4:00pm on Monday, November 10, 2008 in the ENGRI 1101 box at the west end of the corridor on the second floor of Rhodes Hall, where it connects to Upson. Reading assignment: Chapter 3 of Winston, Chapter 2 of Chvatal and section on The Duality Theorem (included in coursepack). Make sure to put your name on the first page and staple the problems in order. 1. (60 points) (a) (20 points) Solve the following problem by the simplex method. Show all your steps. maximize 3 x 1 + 2 x 2 + 4 x 3 subject to x 1 + x 2 + 2 x 3 4 2 x 1 + 3 x 3 5 2 x 1 + x 2 + 3 x 3 7 x 1 ,x 2 ,x 3 0 (b) (20 points) Now solve the following LP, by choosing the same entering and leaving variables as in ( a ), in exactly the same order (ending with the same basis as you did in ( a )). Again, show all your steps. (If this question is unclear, please come to an office hour.) maximize 3 x 1 + 2 x 2 + 4 x 3 subject to x 1 + x 2 + 2 x 3 4 2 x 1 + 3 x 3 5 + ε 2 x 1 + x 2 + 3 x 3 7 x 1 ,x 2 ,x 3 0 (You should think of ε as a really small number, say

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Unformatted text preview: . 000001 .) (c) (10 points) ³or what values of ε is the solution you found in ( b ) feasible? (d) (10 points) ³or what values of ε is the solution you found in ( b ) optimal? 2. (40 points) In the previous question you solved the following LP: maximize 3 x 1 + 2 x 2 + 4 x 3 subject to x 1 + x 2 + 2 x 3 ≤ 4 2 x 1 + 3 x 3 ≤ 5 2 x 1 + x 2 + 3 x 3 ≤ 7 x 1 , x 2 , x 3 ≥ . The dual of this poblem is: minimize 4 y 1 + 5 y 2 + 7 y 3 subject to y 1 + 2 y 2 + 2 y 3 ≥ 3 y 1 + y 3 ≥ 2 2 y 1 + 3 y 2 + 3 y 3 ≥ 4 y 1 , y 2 , y 3 ≥ . 1 (a) (25 points) Now take the dual of the dual problem (Frst transform it into canonical form, then take the dual). Transform your answer, if necessary, into canonical form. (b) (15 points) Recall how we derived the dual, by considering how to construct upper bounds on the objective value of the primal. Can you explain your result of ( a ) in this light? 2...
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Homework9 - 000001(c(10 points ³or what values of ε is...

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