assign4_5.sol

assign4_5.sol - C&O 350 Linear Optimization Winter 2010...

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1. Consider the LP problem maximize c T x subject to Ax = b, x 0, where c = - 1 1 - 1 - 1 - 1 1 , A = 1 0 0 1 0 6 3 1 - 4 0 0 2 1 0 2 0 1 2 , and b = 9 2 6 . (a) Find the tableau corresponding to the basis { 1 , 2 , 3 } and to the basis { 1 , 4 , 5 } . Which of them is feasible? (b) Beginning with a feasible tableau from part (a), solve the problem with the simplex method. Solution: (a) We start with: z + x 1 - x 2 + x 3 + x 4 + x 5 - x 6 = 0 x 1 + x 4 + 6 x 6 = 9 3 x 1 + x 2 - 4 x 3 + 2 x 6 = 2 x 1 + 2 x 3 + x 5 + 2 x 6 = 6 The tableau with basis { 1 , 2 , 3 } is: z - 9 4 x 4 + 5 2 x 5 - 29 x 6 = - 77 2 x 1 + x 4 + 6 x 6 = 9 x 2 - 5 x 4 + 2 x 5 - 24 x 6 = - 31 x 3 - 1 2 x 4 + 1 2 x 5 - 2 x 6 = - 3 2 This does not give us a feasible solution. The tableau with basis { 1 , 4 , 5 } is: z - 2 3 x 2 - 7 3 x 3 - 25 3 x 6 = - 43 3 - 1 3 x 2 + 4 3 x 3 + x 4 + 16 3 x 6 = 25 3 x 1 + 1 3 x 2 - 4 3 x 3 + 2 3 x 6 = 2 3 - 1 3 x 2 + 10 3 x 3 + x 5 + 4 3 x 6 = 16 3 This gives us the feasible solution x * = [ 2 3 , 0 , 0 , 25 3 , 16 3 , 0] T . We will use this tableau to start the simplex method in (b). (b) There are many possible solutions to this. We’ll present one that follows Dantzig’s rule (see page 78 of coursenotes). Starting with the tableau that corresponds to the basis
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assign4_5.sol - C&O 350 Linear Optimization Winter 2010...

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