asst7sol - C&O 350 Linear Optimization – Winter 2009...

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Unformatted text preview: C&O 350 Linear Optimization – Winter 2009 – Solutions to Assignment 7 Question # Max. marks Part marks 1 9 2 4 3 2 10 2 6 2 3 5 4 16 5 No Credit Total 40 1. Revised Simplex Method, I 9 marks = 2+4+3 Consider the linear programming problem ( P ) : maximize c T x subject to Ax = b, x ≥ where A = - 5- 1- 4 4- 2 1 0 3 2- 1 1 0 0- 1- 1- 1- 1 1 , b = 2 1 1 , and c = [1 , 2 ,- 3 , 1 , , 2 ,- 1] T . (a) Determine the basic feasible solution associated with the basis B = { 1 , 6 , 7 } . (b) Solve ( P ) with the Revised Simplex Method, starting from the (feasible) basis B . In every iteration, use the smallest subscript rule for choosing the entering and leaving variables. Clearly show the details of each iteration. (c) If you find a feasible solution that is optimal, write down the optimal solution that you find. Give a proof of its optimality using duality theory. If you find that ( P ) is unbounded, write down a sequence of feasible solutions { x ( t ) } which shows this unboundedness, and show that the corresponding objective values become arbitrarily large as t goes to infinity. Solution : (a) A B = - 5 1 0 3 0 0- 1- 1 1 . Solve A B x B = b to get x * B = 1 / 3 11 / 3 5 . Thus, the basic feasible solution associated with B = { 1 , 6 , 7 } is x = [1 / 3 , , , , , 11 / 3 , 5] T . (b) Iteration 1: Solve A T B y = c B , that is - 5 3- 1 1 0- 1 0 0 1 y = 1 2- 1 to get y = 1 5 / 3- 1 . 1 Compute c 2 = c 2- A T 2 y = 2- [- 1 0- 1 ] y = 2 > 0. x 2 enters. Solve A B d = A 2 to get d = - 1- 2 . d ≤ 0 = ⇒ LP problem is unbounded. (c) Proof of unboundedness: For each real number t , let x 2 ( t ) = t , x 3 ( t ) = x 4 ( t ) = x 5 ( t ) = 0 and x 1 ( t ) x 6 ( t ) x 7 ( t ) = 1 / 3 11 / 3 + t 5 + 2 t . It is easy to check that x ( t ) = [ x 1 ( t ) ,...,x 7 ( t )] T is a feasible solution for all t ≥ 0 (that is, x i ( t ) ≥ and Ax ( t ) = b ). Objective value of x ( t ) = 1 / 3 + 2 t + 2(11 / 3 + t )- (5 + 2 t ) = 8 / 3 + 2 t → + ∞ as t → + ∞ . Hence the LP problem is unbounded. 2. Revised Simplex Method, II 10 marks = 2+6+2 For part (b): 4 marks for the first iteration, 2 marks for the remainder Consider the linear programming problem ( P ) : maximize c T x subject to Ax = b, x ≥ where A = 1- 2 0 1 2- 1 0 2 1 1 2 1 0 0 , b = 1 2 5 , and c = [2 , , 1 , 4 , 2] T ....
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This note was uploaded on 02/05/2010 for the course CO 350 taught by Professor S.furino,b.guenin during the Spring '07 term at Waterloo.

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asst7sol - C&O 350 Linear Optimization – Winter 2009...

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