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Unformatted text preview: Using the Gomory cutting plane algorithm, solve the original IP problem. Do not add more than two cutting planes. (14 marks) AMA405 2 Continued AMA405 3 (i) With reference to the following linear programming problem: maximise z = x 1 + 2 x 2 subject to x 1 + x 2 10 2 x 1 + 3 x 2 10 3 x 1 + 2 x 2 = 10 x 1 ; x 2 give reasons for the need for arti cial variables and indicate how the Simplex method can be modi ed to take account of them. (4 marks) (ii) Use the TwoPhase Simplex method to solve the problem. (21 marks) AMA405 3 Turn Over AMA405 4 (i) Using the usual notation, the tableau for the problem maximise z = c T 1 x 1 + c T 2 x 2 subject to A x 1 + I x 2 = b x 1 ; x 2 ; at a stage in the Simplex algorithm when the basis matrix is B , is repre sented by " 1 c T B B 1 A c T 1 c T B B 1 c T 2 B 1 A B 1 # z x 1 x 2 = " c T B B 1 b B 1 b # where x 2 is the starting basis. (a) Write down the conditions necessary for this tableau to be optimal. (3 marks) (b) Write down an expression for the optimal dual variables in terms of the notation above and verify that the dual constraints are satis ed. (7 marks) (ii) In the context of deriving the dual of a maximisation linear programming problem, describe the transformations necessary to convert the following into the Standard Form: (a) 4 x 1 + 3 x 2 = 10 (b) x 3 unrestricted (c) minimise z = 10 x 1 + 3 x 2 (d) 2 x 1 x...
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 Spring '13
 405
 Operations Research, Statistics, Linear Programming, Optimization, Simplex algorithm

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