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ps3 - Homework 3 IE418 Integer Programming Dr Ralphs Due 1...

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Homework 3 IE418 – Integer Programming Dr. Ralphs Due March 13, 2007 1. Consider the 0-1 knapsack problem: max { n X j =1 c j x j | n X j =1 a j x j b, x B n } with a j , c j > 0 for j = 1 , . . . , n . (a) Show that if c 1 a 1 ≥ · · · ≥ c n a n > 0, r - 1 j =1 a j b and r j =1 a j > b , then the solution to the LP relaxation is x j = 1 for j = 1 , . . . , r - 1, x r = ( b - r - 1 j =1 a j ) /A r and x j = 0 for j > r . (b) Solve the instance max 17 x 1 + 10 x 2 + 25 x 3 + 17 x 4 5 x 1 + 3 x 2 + 8 x 3 + 7 x 4 12 x B 4 by branch and bound and show the result graphically. 2. Show that the stronger of the two formulation for the open pit mining problem from Homework 1 is ideal. 3. Consider an integer program max { cx | Ax b, 0 x u } Suppose the linear programming relaxation has been solved to optimality and row zero of the tableau looks like z = ¯ a 00 + X j NB 1 ¯ a 0 j x j + X j NB 2 ¯ a 0 j ( x j - u j ) where NB 1 are the nonbasic variables at 0 and NB 2 are the nonbasic variables at their upper bounds u j , so that ¯ a 0 j 0 for j NB 1 and ¯ a 0 j
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