HW3 - Ad = 0, d ≥ 0. 1 P3 Consider the LP (from Homework...

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3. Homework for IEOR162 Linear Programming Semester: Fall 2010 Instructor: Dr. Sarah Drewes Due to: Thu, October 7, 2010 (at the beginning of class) P1 Consider the problem (compare Homework 1, P2, problem b)) max z = 4 x 1 + x 2 s.t. 8 x 1 + 2 x 2 16 5 x 1 + 2 x 2 12 x 1 , x 2 0 a) Write down the extreme points of the feasible region in two di- mensions. (You can use the graph of the feasible region from Homework 1.) b) Reformulate the problem in standard form. c) Write down all basic solutions to this problem and determine if they are basic feasible solutions (bfs) to the LP. d) Identify for each basic feasible solution of the LP in standard form to which extreme point it corresponds. e) Is the LP degenerate? P2 Consider an LP in standard form, hence with feasible region Ax = b , x 0. Show that d is a direction of unboundedness if and only if
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Unformatted text preview: Ad = 0, d ≥ 0. 1 P3 Consider the LP (from Homework 1, P2, problem c) ): max z =-x 1 + 3 x 2 s.t. x 1-x 2 ≤ 4 x 1 + 2 x 2 ≥ 4 x 1 , x 2 ≥ a) Reformulate the problem in standard form. b) Determine a direction of unboundedness for this problem. c) Express the vector x = x 1 x 2 s 1 e 1 with coordinates x 1 = 1 , x 2 = 4 in the representation of Theorem 2, i.e., determine a direction of unboundedness d ∈ R 4 and scalars σ i ≥ 0 and ∑ k i =1 σ i = 1, such that x can be written as x = d + k X i =1 σ i b i , where b 1 , . . . b k are the basic feasible solutions of the given LP. Hint: Choose one basic feasible solution b 1 and σ 1 = 1, σ i = 0 for i = 2 , . . . k . 2...
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This note was uploaded on 01/21/2011 for the course IEOR 162 taught by Professor Zhang during the Fall '07 term at University of California, Berkeley.

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HW3 - Ad = 0, d ≥ 0. 1 P3 Consider the LP (from Homework...

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