Optimal Crane Scheduling

Transcription

1 Opimal Crane Scheduling Samid Hoda, John Hooker Laife Genc Kaya, Ben Peerson Carnegie Mellon Universiy Iiro Harjunkoski ABB Corporae Research EWO - 13 November /16

2 Problem Track-mouned cranes move maerials & equipmen. We focus on longiudinal moves (lef-righ). Cranes mus no cross pahs. 3/16

3 Problem Example crane rajecories longiude Schedule cranes o ransfer maerial beween locaions jobs wih 1-8 asks each. Origin/desinaion and sop imes for each ask. Time windows and prioriies. Precedence consrains. Goal: Follow producion schedule as closely as possible. ime 4/16

4 Problem Three problem levels. Assign jobs o cranes. 9 Crane Crane Sequence jobs on each crane. Crane Crane Plan crane rajecories. Time granulariy 10 sec over 24 hrs Crane 1 Crane 2 6/16

5 Algorihm 1: Two-phase search For 2-crane problems. Phase 1 Assign and sequence wih local search. Crane Crane Crane Crane Phase 2 Compue opimal rajecory wih dynamic programming.

6 Compues penaly ( ) + = c c c c s u x S s u x s s g s u x f s u x f 1, 1, 1, 1, 1, min 1, 1, 1, Basic DP recursion: Transiions ha saisfy consrains Posiion Loading/ unloading ime Segmen Algorihm 1: Two-phase search

7 Sae Space Reducion Only cerain rajecories mus be considered. Canonical rajecory for he lef crane: Wai as long as possible Move as soon as possible Follow lefmos rajecory ha never moves backward (away from desinaion)

8 Sae Space Reducion Depar from canonical rajecory A each momen, follow canonical rajecory or righ crane s rajecory, whichever is furher o he lef Lef crane Righ crane

9 Sae Space Reducion Properies of he minimal rajecory: Depar from canonical rajecory The lef crane never sops en roue unless i is adjacen o he righ crane. The lef crane never moves backward unless i is adjacen o he righ crane. Lef crane Righ crane

10 Sae Space Reducion Theorem: Some opimal pair of rajecories are minimal wih respec o each oher.

11 Sae Space Reducion Represen processing-ime saes as an inerval of saes. Exploi he fac ha cranes are processing (and herefore a res) mos of he ime. Sore hese saes in a 2-dimensional circular queue daa srucure ha persiss hrough ime.

12 Sae Space Reducion Compue hese coss when a ask pair for which boh asks are a heir sop locaions appears in he sae space. c 41 c 31 c 21 c 11 c 12 c 13 c 14 c 15 c ij = cos-o-go when he lef crane has been processing i ime unis and he righ crane has been processing j ime unis.

13 Sae Space Reducion Compue hese coss in he nex period. No need o updae exising coss. Daa srucure is a 2-dimensional circular queue. c 12 c 13 c 42 c 11 c 32 c 41 c 22 c 31 c 23 c 14 c 24 c 15 c 25 c 21 c ij = cos-o-go when he lef crane has been processing i ime unis and he righ crane has been processing j ime unis.

14 Sae Space Reducion And similarly in he nex period. c 23 c 24 c 14 c 12 c 43 c 41 c 42 c 33 c 34 c 25 c 13 c 22 c 15 c 35 c 21 c 11 c 31 c 32 c ij = cos-o-go when he lef crane has been processing i ime unis and he righ crane has been processing j ime unis.

15 Sae Space Reducion These coss do no correspond o separae saes. Afer his poin he able is no longer needed and memory can be released. c 34 c 35 c 25 c 14 c 23 c 15 c 44 c 13 c 45 c 31 c 24 c 33 c 21 c 11 c 41 c 32 c 22 c 12 c 42 c 43 c ij = cos-o-go when he lef crane has been processing i ime unis and he righ crane has been processing j ime unis.

16 Compuaional Resuls New daa srucure reduced compuaion ime an order of magniude. Compuaion ime is sensiive o widh of ime windows. Can now solve 60-job problem wih wide ime windows ( 1 hour) Wide windows are necessary for feasible soluion.

17 Effec of New Daa Srucure

18 Soluion of 20-job Problem

19 Sae Space Size

20 Soluion of 40-job Problem

21 Sae Space Size

22 Soluion of 60-job Problem

23 Sae Space Size

24 Compuaion Time for one DP

25 Algorihm 2: Inexac DP Sae Space Crane simulaion. Splis a decision poins. Sae rimming: Eliminaes inferior saes Applies o > 2 cranes. Simulaion Time crane 2 new job: 2,3,none crane yield: 1,2 crane 1 new job: 1,2,none Trim Saes Trim Saes Trim Saes 12/16

26 Algorihm 2: Inexac DP Discree-even simulaion Coninuous ime 4 crane acions Sae ransiion only when an acion is compleed Cranes will work and move unil: Time o pick up new job. Pah splis for each job choice. Cranes inerfere. Pah splis for each yielding choice. WORKING WAITING MOVING YIELDING 13/16

27 Algorihm 2: Inexac DP C1 C2 C3 Exac sae rimming by dominaion. Sae D dominaes sae S if Obj fcn value of D Obj fcn value of S {jobs compleed in D} {jobs compleed in S} For all cranes c, c working on same job in D & S or c is waiing in D, and progress on c s job in D progress on c s job in S Inexac sae rimming by size limi Drop saes wih higher obj fcn values S D S D S D 14/1

28 Compuaional Resuls Schedules hundreds of asks per minue under ypical condiions Runime is essenially linear in number of jobs Sae explosions caused by specific schedule feaures crane, 72-job schedule es Uncu Saes Cu Saes /16

29 Compuaional Resuls Inexac DP algorihm: Mees runime goals. Can accommodae muliples cranes. Sae space size: Sill sensiive o ime window widh. Inexac pruning required for 1 hour windows. Unfinished business: Compare soluion qualiy wih Algorihm 1. Solve addiional problems. 16/16