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R & D Project Accelerating Column Generation by Reducing Degeneracy of LP Solutions

R & D Project Accelerating Column Generation by Reducing Degeneracy of LP Solutions. Column Generation in brief. Take few initial feasible columns Solve the restricted master problem Use the dual solution of the master problem as the profit for the knapsack problem to get new better column

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R & D Project Accelerating Column Generation by Reducing Degeneracy of LP Solutions

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  1. R & D Project Accelerating Column Generation byReducing Degeneracy of LP Solutions

  2. Column Generation in brief • Take few initial feasible columns • Solve the restricted master problem • Use the dual solution of the master problem as the profit for the knapsack problem to get new better column • Continue as far as better columns can be found

  3. Accelerating Column Generation • The better columns are not picked early because of 0-dual (degenerate) multipliers • However, generally there are multiple dual solutions; there can be non-degenerate solutions • One could be the centroid; finding it is difficult • We used Chebyshev center

  4. Chebyshev Center

  5. Experimental data • Order size 1, integral optimum • 27% gain in # iterations • 8% degradation in time • Order size 1, fractional optimum • 14% gain in # iterations • 29% degradation in time • General order size • 23% gain in # iterations • 9% degradation in time

  6. Initial Columns • Previous data is for First Fit • We experimented with Fast Fit Maximal • % gain is reduced

  7. Another heuristic • Incase of multiple knapsack solutions, pick that which uses maximum roll space

  8. Conclusion • We tried two heuristics • Both are better than `gilmore’ in terms of number of iterations • `chebyshev’ takes more time • The combination of the two improves no of iterations most of the cases

  9. Future Work • One concrete example • Find out where ‘new-heuristic’ fails • Alternative to solve knapsack problem – some branch-and-bound • Ellipsoid method

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