1. Irregular stock cutting with guillotine cuts Han Wei, Julia Bennell NanJing,China

2. Summary Problem definition Current practice Direct packing approach –Generate candidate combinations –Evaluate candidate combinations –Evolve layouts in a self adaptive forest Example solutions/demo Future work

3. Problem description Problem arises in the glass cutting industry, in this instance, specifically for conservatories (glass houses)

4. Problem description All pieces are convex Single size stock sheet (multiple sizes and off-cuts will be considered in the future) Pieces are broken out using guillotine cuts (orthogonal and non-orthogonal) No limit to the number of stages/cuts Pieces can be continuously rotated Demand met exactly

5. Current practice Nest pairs of all non-rectangular pieces into a rectangle Select best pairs according to following ratio Use standard rectangle bin packing with guillotine constraints

9. Research aim Current software encloses two pieces into rectangles then packs the rectangles to be guillotine cut-able. Our aim 1.Investigate nesting more than two pieces into rectangles 2.Removing the requirement to nest into rectangles while meeting guillotine constraint.

10. Direct packing approach overview In words ….. Evolve many layouts by recursively combining pieces, or configurations of subsets of pieces, together. Layouts evolve via a forest structure, where the size is controlled by an acceptance threshold An accepted combination of pieces becomes a single piece defined by the convex hull of the combination Pieces must be combined in such a way to result in a feasible layout

11. Direct packing approach overview 1.Generate candidate combinations of two (sets of) pieces 1.Non overlapping 2.Guillotine cut-able 2.Evaluate candidate configurations 3.Evolve solutions in self adaptive forest search

12. Generating candidate combinations For each i in P1, and j in P2, and sliding distance d, Attach Slide

13. Evaluating candidate configurations

14. Match algorithm 1.Initialise best_i, best_j, best_d, max_U 2.For each i in P1, and j in P2, Attach(P 1,P 2,i,j) If do d = 0, max{0,length(e i )-length(e j )} Slide(d) If U w f(P1,P2) > max_U set best_i, best_j, best_d, max_U 3. Attach(P 1,P 2,best_i,best_j), Slide(best_d)

15. Feasibility 1.Given all pieces are convex and the described attach procedure, the no-overlap and guillotine constraints are clearly met for two original pieces. 2.Once the best match is found the combination is defined by its convex hull, hence combining sets of pieces, defined by their convex hull is the same as 1. Guillotine cut

16. More assumptions and definitions

17. Evolve layout in self adaptive forest m = 1m = 2m = 3m = 4m = 5

18. The maximum population of D given θ and max g : At level m, the candidate set of configurations are: The mth generation of the population is described by:

19. Example results

21. Software demo

22. Future work Explore the following How scalable is the approach Sensitivity to parameters; w, θ d and θ Incorporate full constraints

23. Additional constraints Minimum angle of 30°for cuts intersecting an edge No more than two cuts can intersect Both are permitted if a 20mil gap is added  

24. Thank you Questions