A Tree Search Algorithm for Solving the Two-Dimensional Knapsack Problem

1. A Tree Search Algorithm for Solving the Two-Dimensional Knapsack Problem T. Fanslau and A. Bortfeldt University in Hagen, Germany Contents: 1 Introduction 2 The Tree Search Algorithm 3 Numerical Test 4 Conclusions

2. 1 Introduction 2D-Knapsack Problem (2D-KP) Given: - a large rectangle (container) - a set of small rectangles (pieces); each piece has a value (area, …) Determine: feasible arrangement of subset of pieces in the container so that total value of the packed pieces is maximized and relevant constraints are met Special case: value = area  total covered area to maximize Feasibility conditions: • any two packed pieces do not overlap • each packed piece lies entirely inside the container • each packed piece is placed parallel to the container sides

3. Usual constraints: (C1) Orientation constraint: orientation of all pieces fixed; 90° rotating not allowed (C2) Guillotine constraint: all pieces of a pattern can be cut by guillotine cuts (edge-to-edge-cuts, parallel to container edges)Subtypes of 2D-KP (Lodi et al. 1999):

4. Literature overview Common approaches (2D-KP - NP-hard ):tree search (TRS), dynamic optimization (DO), metaheuristics (GA, TS, …)

5. The Tree Search Algorithm 2.1 TRS Algorithm for non-guillotine case A) Overall algorithm initialize: search_effort := 1; repeat generate complete solution s; // using search_effort! update best solution if necessary: sbest := s; update parameter: search_effort := 2 * search_effort; until time_limit exceeded; • multiple solutions generated one by one • internal parameter search_effort . range of tree search . doubled per solution • current best solution - upper bound • stop after time limit exceeded

6. B) Generating one solution Residual space (RS): - rectangular free space within container- data: 2 dimensions, reference corner RS_set:- generated, yet unprocessed RS - first RS to be filled: empty container - not disjoint in general Procedure:- RS filled one after another- each RS filled by best rectangle placement, one rectangle placed in reference corner- if no rectangle fits – RS only removed - RS_set updated after placement- new solution complete if RS_set empty- chosen placement – no more changed!

7. // initialize set of residual rectangles Rres := {all rectangles}; solution s := ; set of residual spaces RS_set := {container}; // fill residual spaces successively while RS_set   do rs := RS nearest to origin; // Manhattan dist.: x+y ->min!remove rs from RS_set;determine best rect. placement rpl* = (r,o,x,y) for rs; // R!if placement foundthen extend solution s := s  {rpl*}; Rres := Rres \ {r}; endif;update RS_set; // R! endwhile; // “R!” – parts to be refined Generating one solution – pseudocode

8. C) Updating the set of residual spaces • always to be done after RS processed (also if no placement) - updating includes: .processed RS removed .new RS created .existing RS reduced wC, y legend: - RS - rectangle - ref. corner of RS RS0 IC, x

9. C) Updating the set of residual spaces • always to be done after RS processed (also if no placement) - updating includes: .processed RS removed .new RS created .existing RS reduced wC, y RS1 2 new overlapping RS RS2 rect1 IC, x

10. C) Updating the set of residual spaces • always to be done after RS processed (also if no placement) - updating includes: .processed RS removed .new RS created .existing RS reduced wC, y new RS RS3 rect2 reduced RS RS2 rect1 IC, x

11. D) Evaluating of placements • multiple possible placements per RS in general- evaluation criteria: 1) specific value := value / area 2) smoothness index (0, 1 or 2) 3) area- all poss. placements ordered by evaluation criteria 1) – 3); first ns placements tried in this order (ns = curr. no. of succ.) wC, y current RS rect2 rect1 IC, x

12. D) Evaluating of placements • multiple possible placements per RS in general- evaluation criteria: 1) specific value := value / area 2) smoothness index (0, 1 or 2) 3) area- all poss. placements ordered by evaluation criteria 1) – 3); first ns placements tried in this order (ns = curr. no. of succ.) wC, y xN – nearby edge on the right of xC yN – nearby edge above yC no coincidence: x‘ < xN y‘ > yN smoothness index = 0 y‘ yN rect2 yC rect1 xC x‘ xN IC, x

13. D) Evaluating of placements • multiple possible placements per RS in general- evaluation criteria: 1) specific value := value / area 2) smoothness index (0, 1 or 2) 3) area- all poss. placements ordered by evaluation criteria 1) – 3); first ns placements tried in this order (ns = curr. no. of succ.) wC, y xN – nearby edge on the right of xC yN – nearby edge above yC one coincidence case: x‘ < xN y‘ = yN smoothness index = 1 y‘, yN rect2 yC rect1 xC x‘ xN IC, x

14. D) Evaluating of placements • multiple possible placements per RS in general- evaluation criteria: 1) specific value := value / area 2) smoothness index (0, 1 or 2) 3) area- all poss. placements ordered by evaluation criteria 1) – 3); first ns placements tried in this order (ns = curr. no. of succ.) wC, y xN – nearby edge on the right of xC yN – nearby edge above yC two coincidence cases: x‘ = xN y‘ = yN smoothness index = 2 y‘, yN rect2 yC rect1 xC x‘, xN IC, x

15. E) Determining the best placement for a RS Task:- given:partial solution s with n - 1 rectangles (n ≥ 1), set Rres - request: best nth placement rpl* for current residual space rs Approach: Partition Controlled Tree Search (PCTRS)1) multiple complete solutions sc generated – all extending s; best placement rpl* = nth placement of best solution sc2) additional search depth sd specified - RS of levels n,…,n+sd–1: multiple placements per RS tried RS of levels n+sd,…: only first placement tried (evaluation!) ith search level ↔ ith placement - sd increases with parameter search_effort3) complete solutions generated by different extension strategies - each strategy calculates its own best solution - best solution sc over all strategies determines best placement rpl* - each extension strategy defined by a partition of sd4) advantages of PCTRS - only small fraction of complete solutions tried compared to const. no. of succ. - mix of strategies cares for suitable balance between 2 factors: . width of seach – no of succ. per extension . degree of foresight – no. of simultaneous considered search levels.

16. Expl.: extension strategy for partition “2+1” (two stages) fixed partial plan, level n-1 0 • stage 1: tree with two levels n and n+1 and ns = 2 constructed - complete all sol. s, - select best complete sol. sb, - reduce sb to n+1 placements, - pass sb to next stage • stage 2: tree with one level n+2 and ns = 8 constructed; best complete solution of 2nd stage returned at last stage 1, level n 1 add. search depth stage 1, level n + 1 2 ... stage 2, level n + 2 3 partial plan completed plan

17. 2.2 TRS-Algorithm for guillotine case Packing units: - blocks with n  1 rectangles - simple blocks (one item type), general blocks (small inner gaps) - 2 super phases: 1: simple blocks only, 2: general blocks Residual spaces (RS): - container – first RS - each RS filled by 1 block; remaining empty space then cut into three daughter spaces - unprocessed RS always disjoint - heuristic rules for cutting RS and processing order of RS Tree search: - PCTRS - largest blocks tried per RS Guillotine constraint: - observed by (single) blocks - observed by complete plans (sequences of blocks) since residual spaces generated by guillotine cuts

18. 3 Numerical Test Implementation, names: algorithm implemented in C, called: CLTRS variant for subtypes OG/RG: CLTRS–G variant for subtypes OF/RF: CLTRS–F PC/OS: AMD Athlon, 2.6 GHz, Windows XP Configuration: parameters uniformly set per CLTRS variant exception: time limitsReporting of results: - gap := relative deviation from upper bound (or known optimum) (bound – objective value) / bound (in %) - only best values per method and instance considered - only one run per instance with CLTRS (G or F)

19. A) 630 instances from Beasley (2004) Optimization criterion: packed valueInstance size: m: 40 …1000 CLTRS–F: mean computing time 22.1 Secs.

20. B) 21 instances from Hopper and Turton (2004) Optimization criterion: packed area; filling rate 100% poss. Instance size: m: 16 …197 CLTRS–F: mean computing time – OF: 129.5 Secs, – RF: 1.8 Secs.

21. C) 44 instances from Fayard et al. (1998) Piece set: constrained (C), unconstrained (U) Optimization criterion: packed area (U), packed value (W) Resulting subsets: CU, CW, UU, UW - 11 instances each Instance size: m: 25 …60 CLTRS–G: mean computing time 252 Secs.

22. 4 Conclusions Heuristic CLTRS: - excellent solution quality - time consumption: OF/RF – excellent; OG/RG – moderate - few parameters, no tuning New concepts: - problem specific: . CLTRS–F: smoothness index – heart of approach . CLTRS–G: generalized block concept - generic: Partition Controlled Tree Search (PCTRS) . new type of incomplete tree search / beam search . economical generation of solutions . balance of:search width – degree of foresight Topics of further research: - 3D-CLTRS, non-guillotine case - 2D-CLTRS, extension to strip packing - application of PCTRS to other problems.

23. Appendix: Detailed description of PCTRS

24. Search stage:- input: solution sin with m filled RS (m ≥ n-1) - procedure: subtree of partial solutions of depth d constructed (d ≤ sd), all constructed solutions of level m + d are completed - ns – no. successors per node = no. tried blocks per RS; ns = f(d, search_effort); d small ↔ ns big ! - output: best complete solution sout Extension strategy (ES):- results by chaining of search stages, i.e. input sin of stage x+1 = output sout of stage x - for given search depth sd all possible ES given by all partitions of sd example: search depth sd = 3, ordered partition p: 3; 2+1; 1+2; 1+1+1; no. of summands t: 1 2 2 3- no of summands t = no. of stages; ith summand = depth d of ith stage Partition Controlled Tree Search (PCTRS):- all possible ES used concurrently - search stages / ES differ in terms of a) search width, i.e. no. of tested placements per RS (ns) b) degree of foresight, i.e. no. of simultaneous considered RS (d)- PCTRS cares for suitable balance of both factors!

25. 1) extension strategy for partition “3” (one stage) fixed partial plan, level n-1 0 • for each RS of levels n, n+1, n+2 the first ns = 2 admissible placements are tried • each partial plan with n+2 blocks is completed; best complete solution returned stage 1, level n 1 add. search depth stage 1, level n + 1 2 stage 1, level n + 2 3 partial plan completed plan, standard completion from next level (here: n + 3)

26. 2) extension strategy for partition “1+1+1” (three stages) fixed partial plan, level n-1 0 • stage 1: for RS of level n first ns = 8 admissible placements are tried; each resulting plan is completed; s1 – best solution of stage 1 • stages 2, 3: same procedure repeated on levels n+1, n+2, yielding best solutions s2 and s3, s3 is returned • chaining: s1 is input for 2nd stage, s2 is input for 3rd stage ... stage 1, level n 1 add. search depth ... stage 2, level n + 1 2 ... stage 3, level n + 2 3 partial plan completed plan

27. 3) extension strategy for partition “2+1” (two stages) fixed partial plan, level n-1 0 • stage 1: tree with two levels n and n+1 and ns = 2 constructed • stage 2: tree with one level n+2 and ns = 8 constructed; best solution of 2nd stage returned at last • chaining: best solution of stage 1 is input for stage 2 stage 1, level n 1 add. search depth stage 1, level n + 1 2 ... stage 2, level n + 2 3 partial plan completed plan