Soumitra Pal [05305015]

1. M Tech Project – First Stage Improving Branch-And-Price Algorithms For Solving 1D Cutting Stock Problem Soumitra Pal [05305015]

2. Agenda • Cutting stock problem & formulation • Generic Branch-and-price algorithm • Implementation by Vance [1994], etc • Work in next stage • Conclusion

3. Cutting stock problem • Bigger rolls (raw) available • Orders of smaller rolls are to be cut (items) • Minimize no of raws used

4. IP formulation • λj denotes the no of rolls cut in pattern j • aij denote no of times item i is cut from pattern j • bi is order for item i

5. Solution using branch-and-price • Branch-and-bound technique of IP • The bound is calculated using LP relaxation solved by column generation

6. Generic branch-and-price algorithm Generate Initial heuristic. Set it as incumbent. Make root node of BB tree and enter into Q Y Unexamined node exists in Q? N Undiscovered? Y N Calculate LB (column generation)Set the node as discovered Set the node as examined LB >= incumbent? LB = some feasible sol X and X< incumbent? Y N N Y Fathom(do nothing) Update incumbent to X Branch (i.e. make two nodes and make them undiscovered and unexamined and enter in Q) Stop; incumbentis the solution

7. 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

8. BAP implementation variations • Initial heuristic solution • Branching rule • Node selection rule • Bounds

9. Branching rules • Excludes fractional solution • Guarantees feasible solution after a finite number of steps • Must encode branching info in subproblem • Creates trees of ~ equal size • Keeps master & subproblem tractable

10. Conventional branching • Excludes fractional solution • Finite no of steps since finite no of variables • Encoding of branch info is explained next λp = α λp ≥ ceil(α) λp ≤ floor(α)

11. Subproblem modification • On the right branch, it is equivalent to reducing the demand vector & solve the residual • No need to modify the subproblem • On the left branch, the column should not be regenerated • This is done by keeping forbidden list in subproblem

12. Problem in conventional branching • Solution space is not equally divided on both the branches • Here, solution space is equivalent to all possible columns • On the right branch, no of possible new columns is reduced • Left branch reduces only one column • Subproblem may become difficult

13. Branching for BCS (demand=1) • Branching rule by Vance [1994]

14. Branching in BCS (2) • Left branch should include both items l and m together • This can be solved by replacing the two items by one item of combined width • On the right branch, at most one of them should get included • This is solved by including edge constraints in the subproblem • The subspaces are of ~ equal size

15. Solving the subproblem • When no edge constraint, solve using Horowitz-Sahni algorithm • When no overlapping edge constraints, solve using modified HS using Jhonson-Padberg bounds • When overlapping edge constraints, use general IP solver

16. Horowitz-Sahni algorithm • Backtracking with bounds • Items ordered in decreasing profit density (profit/weight) • In forward move, try inserting one item • Bound • If U>current best, move forward • Otherwise backtrack (remove last inserted item) & repeat • Update current best when last item is considered • Stop when no more backtracking is possible

17. Jhonson-Padberg algorithm • Solves Knapsack Problem with SOS • SOS is a set of variables, at most one can be set to 1

18. Jhonson-Padberg algorithm (2) • Order SOSs according to max p/w • If w1≥W, set corresponding x1=W/w1 and all other xi=0 • Otherwise, RECORD xi for the item with minimum weight • Remove from S1, all items having profit less than wi; update remaining items in S1, pj-=pi, wj-=wi and W-=wi • Repeat • If multiple items in same SOSs is RECORDED, set variables as shown in the example

19. JP algorithm Example

20. Modified HS • Use SOS ordering • In forward move at most one from each SOS can be inserted • Bound is calculated using JP • In backtrack, other item in the same SOS is considered next

21. Branching for general CSP • Branch on a set of variables • Need to explore more

22. Bounding • Bounding can be used to avoid tailing-off effect • When the following condition is satisfied, column generation can be stopped

23. Initial heuristic solution • First Fit Decreasing (FFD) • Items in decreasing size is fit in existing roll • If can not be fit, use a new roll

24. Work in next stage • Complete literature survey • Vance [1998] • Vanderbeck [1999] • Degraeve and Peeters [2003] • Carvalho [1999] • Comparative study of them • Different set of experimental instances • Time quoted are on different machines • Need to gather absolute numbers such as no of nodes, no of sub-problems etc

25. Knapsack with forbidden list • A dynamic programming algorithm • KPFL(I,W,S) := max{KPFL(I-1,W,S), KPFL(I-1,W-wI,S uinon I)} • O(n x W x 2n x L) • Need to improve that • Core algorithms

26. Conclusion • We explored solution of 1D CSP using branch-and-price • Need to improve the solution to subproblem for overall improvement • Dynamic programming etc to solve subproblem instead of using IP solvers