Pisinger, David

Trivella, Alessio

Online Research Database In Technology

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 18, 2017Integrating load-balancing into multi-dimensional bin-packing problemsTrivella, Alessio; Pisinger, DavidPublication date:2016Document VersionPeer reviewed versionLink back to DTU OrbitCitation (APA):Trivella, A., & Pisinger, D. (2016). Integrating load-balancing into multi-dimensional bin-packing problems[Sound/Visual production (digital)]. 28th European Conference on Operational Research, Poznan, Poland,03/07/2016Integrating load-balancing intomulti-dimensional bin-packing problemsTECHNICAL UNIVERSITY OF DENMARKALESSIO TRIVELLADAVID PISINGERPOZNAN˜, 06.07.2016Contents1 Recall the bin-packing problem2 Integration of load-balancing: problem definition3 MIP model for balancing a single bin4 MIP model for joint packing and balancing5 A multi-level local-search heuristic6 Computational resultsIntegrating load-balancing into multi-dimensional bin-packing problems 1/20Multi-dimensional Bin-Packing problem (MBP)InstanceSet of D-dimensional rectangular-shaped boxesV = {1, . . . ,n}. Box i has width = wi,d in dimension dIdentical bins with width = Wd in dimension dProblemOrthogonally insert all boxes into the bins avoiding overlappingand using as few bins as possible. Rotations are not allowedApplicationsShipping and transportation industry, filling up containers,loading trucks etc. Most real-world problems have D ≤ 3,but all results hold for any dimensionIntegrating load-balancing into multi-dimensional bin-packing problems 2/20MIP model for the MBPmin Ns.t.∑d∈D(lijd + ljid)+ pij + pji ≥ 1 ∀ i < j ∈ Vxid − xjd + Wd lijd ≤Wd − wid ∀ i 6= j ∈ V , d ∈ Dxid ≤Wd − wid ∀ i ∈ V , d ∈ Dai − aj + n pij ≤ n − 1 ∀ i 6= j ∈ V1 ≤ ai ≤ N ∀ i ∈ Vvar : aj , N ∈ N, xid ∈ R+, lijd , pij ∈ {0,1} ∀ i , j ∈ V , d ∈ DDifficult to solve in practice due to many:symmetriesbig-M constraintsIntegrating load-balancing into multi-dimensional bin-packing problems 3/20Load-Balanced MBP (LB-MBP)InstanceMBP instance + density of items ρi (or mass)ProblemArrange items into the miminum number of bins, in such a waythat the barycenters of the loaded bins fall as close as possibleto an ideal point (e.g. the center of the bin or center of its base)ApplicationsTransport (ship, truck, aircraft’s cargo): a good position of thecenter of mass increases the safety and effciency of the travel,minimizing the waste of fuelIntegrating load-balancing into multi-dimensional bin-packing problems 4/20Objective functionMinimize the total imbalance over:used binsdimensions10 10 1010 10 10211.51Optimal barycenterActual barycenterimbalance = 3 imbalance = 0 imbalance = 2.5total imbalance = 5.5Integrating load-balancing into multi-dimensional bin-packing problems 5/20Balancing a single binAssume to have a set V of items which fit into a single bin:min∑d∈Dkd (rd + sd )s.t. : rd − sd = Boptd −1M(∑imi(xid +wid2))∀d ∈ D∑d∈D(lijd + ljid) ≥ 1 ∀ i < j ∈ Vxid − xjd + Wd lijd ≤Wd − wid ∀ i 6= j ∈ V , d ∈ Dxid ≤Wd − wid ∀ i ∈ V , d ∈ Dvar : xid , rd , sd ∈ R+ lijd ∈ {0,1} ∀ i , j ∈ V , d ∈ DTo solve the LB-MBP we could:1) Find the smallest number of bins2) Balance each bin to optimality...but the packing and balancing phases are not linked together!Integrating load-balancing into multi-dimensional bin-packing problems 6/20MIP model for the LB-MBPmin N C +D∑d=1N∑j=1Kd(rjd + sjd)s.t. :D∑d=1(lijd + ljid)+ pij + pji ≥ 1 ∀ i < jxid − xjd + Wd lijd ≤ Wd − wid ∀ i, j, ∀ dai − aj + n pij ≤ n − 1 ∀ i, jxid ≤ Wd − wid ∀ i1 ≤ ai ≤ N ∀ in (cij − 1) ≤ ai − j ≤ n (1− cij ) ∀ i, j1− (n + 1) (1− δij ) ≤ ai − j ≤ −1 + (n + 1) (1− γij ) ∀ i, jcij + γij + δij = 1 ∀ i, jmi Wd(cij − 1)≤ eijd − mi (xid + wid/2) ≤ mi Wd(1− cij)∀ i, j, ∀ dmi Wd(cij − 1)≤ αijd − mi(Woptd − rjd + sjd)≤ mi Wd(1− cij)∀ i, j, ∀ deijd ≤ cij Wd mi ∀ i, j, ∀ dαijd ≤ cij Wd mi ∀ i, j, ∀ dN∑i=1eijd =N∑i=1αijd ∀ j, ∀ dvar : aj , N ∈ N, xid , rjd , sjd , eijd , αijd ∈ R+, lijd , pij , cij , γij , δij ∈ {0, 1}Integrating load-balancing into multi-dimensional bin-packing problems 7/20Sequential vs. joint problem3D instance with 18 items, ρi = 1, Bopt = (5,5,0)Sequential problembin 1 1 2 3 4 5 7 8 10bin 2 6 11bin 3 9 12 14 17bin 4 13 15 16 18Optimal 3DBPP: uses 4 binsbin Bx By Bz fbinbin 1 5.00 5.00 4.18 4.18bin 2 6.12 5.00 4.44 5.56bin 3 5.00 5.00 4.38 4.38bin 4 5.00 5.00 3.32 3.32fcoord 1.12 0.00 16.32 17.44Joint problembin 1 1 2 3 5bin 2 4 6 7 8 10 12bin 3 9 11 14 17bin 4 13 15 16 18Different 3DBPP solutionbin Bx By Bz fbinbin 1 5.00 5.00 4.22 4.22bin 2 5.00 5.00 3.64 3.64bin 3 5.00 5.00 4.59 4.59bin 4 5.00 5.00 3.32 3.32fcoord 0.00 0.00 15.77 15.7710% improvement!But running time is 4 vs. 132 seconds. In general the jointmodel cannot be solved for instances larger than 15-20 itemsIntegrating load-balancing into multi-dimensional bin-packing problems 8/20Heuristic load balancingWe now develop a heuristic algorithm to solve large instancesIt is possible to characterize feasible packings by means of aset of Interval Graphs (Fekete-Schepers)GxGyxyIntegrating load-balancing into multi-dimensional bin-packing problems 9/20Properties (1/2)Theorem 1If D graphs Gd , d ∈ D, are obtained from a packing, then thefollowing conditions are fulfilled:P1 : Each Gd is an interval graphP2 : ∩d Gd = ∅P3 : The stable sets of Gd have total weight less than thed-dimension of the binDefinitionLet G be an undirected graph. An orientation Φ of G is calledtransitive orientation (TRO) if:(a,b) ∈ Φ ∧ (b, c) ∈ Φ =⇒ (a, c) ∈ ΦIntegrating load-balancing into multi-dimensional bin-packing problems 10/20Properties (2/2)Theorem 2If G is an interval graph, then its complement G is transitivelyorientableTheorem 3Let Gd , d ∈ D be D graphs satisfying P1, P2, P3, and callΦ = (Φd)d∈D, where Φd is a transitive orientation of Gd .The function pΦ : V −→ R+D0 defined by:pΦd (v) ={0 if @u ∈ V : (u, v) ∈ Φdmax{pΦd (u) + wd (u) | (u, v) ∈ Φd } otherwiseproduces a packingIntegrating load-balancing into multi-dimensional bin-packing problems 11/20How many transitive orientations?varying  Φxvarying  ΦyDifferent transitive orientations produce different packingsIntegrating load-balancing into multi-dimensional bin-packing problems 12/20Local search among TROsHow many TROs?From graph theory: number of TROs of a graph is∏ki=1 ri !where ri is the number of vertices of particular substructuresIs it possible to find TROs?From graph theory: we can characterize them all TROs of agraph (it’s complicated though)Local SearchWe define a best-improvement local search exploring aquadratic neighborhood of TROs. For each TRO:go back to the corresponding packingevaluate the load balancingIntegrating load-balancing into multi-dimensional bin-packing problems 13/20Example for a 2D caseItems have different densities, bin= 5x5, Bopt = (2.5,2.5)ΦxΦyB = ( 2.32, 2.97 )ΨxΦyB = ( 2.50, 2.97 )ΨxΨyB = ( 2.50, 2.50 )Integrating load-balancing into multi-dimensional bin-packing problems 14/20Local search at graph levelPurpose: improve cases where the number of TROs is limitedHow: modifying the structure itself of the graphs:Consider interval graphs GdAdd or remove edges using specific rules (Crainic et al.)0 2 4 6 80246 Initial packing1 2 340 2 4 6 80246 New packingInterval graph Ey(3,4) is added to Ey (1,4) and (2,4) are also added to Ey1 2 34xyxyIf new graphs correspond to a packing, then start localsearch on TROsIntegrating load-balancing into multi-dimensional bin-packing problems 15/20Local search at bin-packing level (1/2)PurposeExploit the balancing potential of having a large number ofbin-packings solutions with the same number of binsHowIteratively repack and rebalance n-tuples of weakly balancedbins using variable-depth neighborhood search (VDNS)Integrating load-balancing into multi-dimensional bin-packing problems 16/20Local search at bin-packing level (2/2)Define a k-neighborhood as the set of all bin-packing solutionsobtained by repacking at most k binsVDNS algorithm1 Assign imbalance scores to the bins2 Select k bins using roulette wheel selection3 Repack the bins using a heuristic for MBP4 If k bins are still used, balance them5 If balancing is improved: save solution and update scoresk is dynamically adjusted:if no solutions are found after n iterations: k = k + 1Integrating load-balancing into multi-dimensional bin-packing problems 17/20Results for 3D Bin-Packing instances (1/2)Optimal barycenter(W12 ,W22 ,W32): center of the binρi = 1 ρi ∼ U(1, 2) ρi ∼ U(1, 6)cl. size LB bins Init. Bin VDNS Init. Bin VDNS Init. Bin VDNS1 100 24.12 25.64 16.48 0.91 0.011 17.03 0.68 0.054 19.22 2.26 0.2012 100 24.64 26.12 16.19 0.85 0.011 16.69 0.68 0.042 19.01 2.28 0.1983 100 24.48 26.08 16.34 0.87 0.006 16.78 0.66 0.030 19.16 2.21 0.2454 100 57.44 60.60 31.74 0.58 0.011 30.64 0.29 0.030 30.91 0.93 0.0905 100 13.60 14.60 15.83 0.92 0.003 15.87 0.71 0.068 17.65 2.70 0.4256 100 18.20 20.08 9.83 1.45 0.190 10.30 1.70 0.396 13.49 4.54 1.2107 100 11.12 12.36 16.89 1.19 0.012 16.53 0.79 0.050 18.37 3.11 0.5598 100 15.52 17.08 15.66 0.78 0.008 15.34 0.64 0.056 17.35 2.63 0.5011 200 48.84 51.16 15.05 0.79 0.007 15.74 0.82 0.032 18.36 2.57 0.2082 200 48.48 50.80 14.81 0.77 0.006 15.57 0.85 0.044 18.24 2.58 0.2173 200 49.24 51.24 14.91 0.77 0.005 15.68 0.83 0.036 18.29 2.65 0.2624 200 117.8 122.2 31.91 0.53 0.004 30.66 0.29 0.012 30.91 0.96 0.0345 200 25.60 27.36 12.47 0.75 0.011 13.06 0.89 0.147 14.90 3.23 0.8016 200 35.84 38.24 6.74 1.30 0.241 8.17 2.19 0.523 12.09 5.44 1.5057 200 20.36 22.40 12.72 1.26 0.057 12.95 1.15 0.329 14.89 3.99 1.5678 200 29.40 32.04 12.75 0.70 0.016 12.79 0.98 0.183 14.66 3.33 0.921Results are average over 25 instances, running time is < 5-10sIntegrating load-balancing into multi-dimensional bin-packing problems 18/20Results for 3D Bin-Packing instances (2/2)Optimal barycenter(W12 ,W22 ,0): center of the baseρi = 1 ρi ∼ U(1, 2) ρi ∼ U(1, 6)cl. size LB bins Init. Bin VDNS LLB Init. Bin VDNS LLB Init. Bin VDNS LLB1 100 24.12 25.64 52.30 43.75 41.98 36.06 51.57 42.53 41.64 31.87 54.78 41.93 40.84 26.822 100 24.64 26.12 61.88 46.52 44.49 35.90 58.89 43.95 43.01 31.72 59.32 40.21 39.35 26.683 100 24.48 26.08 52.64 43.58 41.92 35.56 52.00 42.49 41.46 31.42 53.08 41.56 40.45 26.414 100 57.44 60.60 60.92 39.72 38.42 25.59 58.72 38.57 38.23 22.61 58.82 38.08 37.58 19.065 100 13.60 14.60 54.65 44.83 42.35 36.77 51.78 42.70 41.70 32.44 51.97 41.48 40.27 27.166 100 18.20 20.08 53.07 47.26 44.91 41.98 51.33 45.63 44.24 37.07 51.92 44.49 43.13 31.137 100 11.12 12.36 55.24 44.94 42.36 36.59 51.87 42.58 41.59 32.30 51.96 41.80 39.80 27.128 100 15.52 17.08 54.77 44.83 42.08 37.08 51.97 43.03 41.99 32.72 52.37 42.05 40.45 27.441 200 48.84 51.16 51.98 44.00 42.56 36.87 51.77 43.18 42.14 32.58 52.90 42.31 41.09 27.432 200 48.48 50.80 61.86 47.10 45.20 36.79 59.34 44.72 43.65 32.51 59.96 40.72 39.93 27.363 200 49.24 51.24 52.38 44.11 42.67 36.68 52.18 43.26 42.22 32.14 53.45 42.35 41.16 27.284 200 117.8 122.2 60.31 39.50 38.19 25.74 58.29 38.40 38.05 22.76 58.55 37.92 37.36 19.185 200 25.60 27.36 53.89 45.90 44.14 39.29 52.23 44.57 43.48 34.71 52.93 43.54 42.16 29.216 200 35.84 38.24 52.24 48.28 45.77 44.36 51.06 46.74 45.12 39.19 51.83 45.34 43.64 32.977 200 20.36 22.40 53.96 46.47 44.51 39.88 52.31 44.87 43.90 35.23 53.43 44.04 42.72 29.648 200 29.40 32.04 53.68 45.84 44.37 39.61 52.14 44.73 43.72 34.98 53.09 45.84 42.40 29.42Results are average over 25 instances, running time is < 5-10sLLB is a lower bound obtained from “liquifying” the itemsIntegrating load-balancing into multi-dimensional bin-packing problems 19/20Background paperAlessio Trivella and David Pisinger. “The load-balancedmulti-dimensional bin-packing problem”. Computers &Operations Research 74 (2016)152–164Integrating load-balancing into multi-dimensional bin-packing problems 20/20Thank you.

Integrating load-balancing into multi-dimensional bin-packing problems

http://orbit.dtu.dk/ws/files/125633675/TrivellaPisinger_EURO2016.pdf

Integrating load-balancing into multi-dimensional bin-packing problems

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology