co227_a3_soln

co227_a3_soln - Assignment 3 Due: Wednesday October 26 at...

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Unformatted text preview: Assignment 3 Due: Wednesday October 26 at the BEGINNING of class 1. Given the following linear programs with feasible basis B , do the following: (i) solve the linear program without tableau, give an appropriate certificate (ii) solve the linear program with tableau (a) maximize 2 x 1- x 2- x 3 + 3 x 4 + x 5- x 6 subject to x 1 + x 2 + x 4 + x 5- 2 x 6 = 3- x 1 + x 2- 2 x 3 + 2 x 5 + x 6 = 2 x 1- x 2 + x 3 + x 4- x 5 = 1 x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 Basis B = { 1 , 2 , 6 } Solution: (i) We have A = 1 1 1 1- 2- 1 1- 2 0 2 1 1- 1 1 1- 1 , b = 3 2 1 , c T = (2 ,- 1 ,- 1 , 3 , 1 ,- 1). A B = 1 1- 2- 1 1 1 1- 1 A- 1 B = 1 / 2 1 3 / 2 1 / 2 1 1 / 2 1 1 y = ( A- 1 B ) T c B = 1 / 2 1 / 2 0 1 1 1 3 / 2 1 / 2 1 2- 1- 1 = 1 / 2 3 / 2 The canonical form for basis B = { 1 , 2 , 6 } is: maximize 3 + (0 , ,- 5 / 2 , 1 , 2 , 0) x subject to 1 0- 1 / 2 2 1 0 0 1- 3 / 2 1 2 0 0 0- 1 1 1 1 x = 5 4 3 x We can increase x 4 by t 0 where t = min 5 2 , 4 1 , 3 1 = 5 2 so 4 enters the the basis, 1 leaves the basis. New basis is B = { 4 , 2 , 6 } . A B = 1 1- 2 1 1 1- 1 A- 1 B = 1 / 4 1 / 2 3 / 4 1 / 4 1 / 2- 1 / 4- 1 / 4 1 / 2 1 / 4 y = ( A- 1 B ) T c B = 1 / 4 1 / 4- 1 / 4 1 / 2 1 / 2 1 / 2 3 / 4- 1 / 4 1 / 4 3- 1- 1 = 3 / 4 1 / 2 9 / 4 The canonical form for basis B = { 4 , 2 , 6 } is: maximize 11 / 2 + (- 1 / 2 , ,- 9 / 4 , , 3 / 2 , 0) x subject to 1 / 2- 1 / 4 1 1 / 2 0- 1 / 2 1- 5 / 4 0 3 / 2 0- 1 / 2 0- 3 / 4 0 1 / 2 1 x = 5 / 2 3 / 2 1 / 2 x We can increase x 5 by t 0 where t = min 5 / 2 1 / 2 , 3 / 2 3 / 2 , 1 / 2 1 / 2 = 1 so 5 enters the basis. We can choose 2 or 6 to leave the basis. If we choose 6 to leave, the new basis is B = { 4 , 2 , 5 } . A B = 1 1 1 1 2 1- 1- 1 A- 1 B = 1 / 2 1 / 2 1- 1- 1- 1 / 2 1 1 / 2 y = ( A- 1 B ) T c B = 1 / 2 1- 1 / 2- 1 1 1 / 2- 1 1 / 2 3- 1 1 = 2 3 The canonical form for basis B = { 4 , 2 , 5 } is: maximize 7 + (1 , , , , ,- 3) x subject to 1 1 / 2 1 0- 1 1 1 1 0 0- 3- 1 0- 3 / 2 0 1 2 x = 2 1 x We can increase x 1...
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co227_a3_soln - Assignment 3 Due: Wednesday October 26 at...

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