2D/3D Packing based on LFF (Less Flexibility First) principle

Problem packing a list of (2D) rectangles or (3D) boxes of various sizes which can be rotated and placed by any orientation into one container to maximize the volume utilization of the container packing a list of (2D) rectangles or (3D) boxes of various sizes which can be rotated and placed by any orientation into one container to maximize the volume utilization of the container (Our 2D LFF packing produces today ’ s highest density results) (Our 2D LFF packing produces today ’ s highest density results)

2D Packing

3D Packing Flexibility Flexibility of empty space Pf Flexibility of empty space Pf Flexibility of a box to be packed Rf Flexibility of a box to be packed Rf cor ner side holl ow corner ： x ； side ： y ； hollow ： z R i f = ….

Structure k-d tree 2D 2D 4-d tree 3D 3D 6-d tree (x1, y1, z1)  (x2, y2, z2) A(x1,y1,z1) B(x2,y2,z2)

CCPS ( Candidate Corner Packing Step ) CCPS ( Candidate Corner Packing Step ) Greedy Greedy An simple Example An simple Example -->

Test result LFF is better than LL in most cases especially when the boxes are complex (more kinds, great contrast), but not very effective when large number and few kinds of boxes LFF is better than LL in most cases especially when the boxes are complex (more kinds, great contrast), but not very effective when large number and few kinds of boxes Test bench LL Algorithm LFF Algorithm In In2 、 In In In In

improvement Heuristic 1 Heuristic 1 Considering the weight of boxes Considering the weight of boxes Bottom First strategy Bottom First strategy Flexibility of box: R i f = {r 1 * (1 – B i / P a ) + r 2 * ( 1 – max( l i, w i, h i )/ min( L, W, H )) + r3 * ( 1 – Q i /Q t )} Q i - weight of box i ， Q t – weight of all boxes; r 1 + r 2 + r3 = 1 Heuristic 1 Original