Gamst, Mette

Lusby, Richard Martin

Røpke, Stefan

Spoorendonk, Simon

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Mar 29, 2019Simultaneously Exploiting Two Formulations: An Exact Benders DecompositionApproachLusby, Richard Martin ; Røpke, Stefan; Gamst, Mette; Spoorendonk, SimonPublication date:2018Document VersionPeer reviewed versionLink back to DTU OrbitCitation (APA):Lusby, R. M. (Author), Røpke, S. (Author), Gamst, M. (Author), & Spoorendonk, S. (Author). (2018).Simultaneously Exploiting Two Formulations: An Exact Benders Decomposition Approach. Sound/Visualproduction (digital)Simultaneously Exploiting Two Formulations:An Exact Benders Decomposition ApproachRichard M. Lusby Stefan Røpke Mette Gamst Simon SpoorendonkDTU Management EnginneringThe Technical University of DenmarkEURO 2018Agenda1 Combining Two MIP Formulations• Reformulations and Relaxations• Benders Decomposition2 Application One• The Bin Packing Problem3 Application Two• The Split Delivery Vehicle Routing Problem (SDVRP)4 Summary2 DTU Management Engineering Exploiting Two Formulations 09/07/2018Motivation• Multiple ways to model a problem using mixed integer linear programming• Each formulation may have its own advantages and disadvantages→ Branching in one is easier than the other→ One provides a tighter relaxation than the other• Can we couple two separate formulations together to exploit the benefits of each?• Solve the resulting formulation using Benders Decomposition• We consider two different cases:→ One formulation is a reformulation of the other→ One formulation is a relaxation of the other3 DTU Management Engineering Exploiting Two Formulations 09/07/2018Methodology (1)(P1) : min cTxs.t. A1x ≥ b1x ≥ 0x ∈ Zn1(P2) : min fTys.t. A2y ≥ b2y ≥ 0y ∈ Zn2(P3) : min fTy,s.t. A1x ≥ b1A2y ≥ b2Dx = Wyx,y ≥ 0x ∈ Zn1y ∈ Zn24 DTU Management Engineering Exploiting Two Formulations 09/07/2018Methodology (1)(P1) : min cTxs.t. A1x ≥ b1x ≥ 0x ∈ Zn1(P2) : min fTys.t. A2y ≥ b2y ≥ 0y ∈ Zn2(P3) : min fTy,s.t. A1x ≥ b1A2y ≥ b2Dx = Wyx,y ≥ 0x ∈ Zn1y ∈ Zn24 DTU Management Engineering Exploiting Two Formulations 09/07/2018Methodology (1)(P1) : min cTxs.t. A1x ≥ b1x ≥ 0x ∈ Zn1(P2) : min fTys.t. A2y ≥ b2y ≥ 0y ∈ Zn2(P3) : min fTy,s.t. A1x ≥ b1A2y ≥ b2Dx = Wyx,y ≥ 0x ∈ Zn1y ∈ Zn24 DTU Management Engineering Exploiting Two Formulations 09/07/2018Methodology (2)Branch-and-Benders-Cut• Benders Decomposition is viewed as a Branch-and-Cut Algorithm(MP) : min fTys.t. A2y ≥ b2y ≥ 0y ∈ Rn2(SP) : min 0Txs.t. A1x ≥ b1Dx = Wy¯x ≥ 0x ∈ Rn1• Feasibility cuts are generated at each node of the Branch-and-Bound treeuT1 b1 + uT2Wy ≤ 0• Branching can be implemented in either problem• Branching on the x variables could be appealing→ Implications on (MP) enforced through feasibility cuts5 DTU Management Engineering Exploiting Two Formulations 09/07/2018Application One: Bin Packing ProblemReformulation• The Cutting Stock Problem is a well known problem in the OR literature• Involves determining the minimum number of rolls (or stocks) of a given lengththat must be cut in order to meet the demand for a set of items, of shorter,specified lengths.• The Bin Packing Problem is a special case of this where the cost of using a givenstock is equal to one.• Various formulations exist• Simultaneously consider an arc-flow formulation and a path based formulation6 DTU Management Engineering Exploiting Two Formulations 09/07/2018FormulationsThe network - Carvalho (1999)0 1 2 3 4 5 6 7Arc flow formulation(BP1) min z∑(j,i)∈Axj,i −∑(i,j)∈Axi,j ={z j = 00 j = 1, . . . ,W − 1−z j =W∀j ∈ N∑(h,h+wi)∈Axh,h+wi ≥ bi ∀i ∈ Ixij ∈ Z+ ∀(i, j) ∈ Az ∈ Z+7 DTU Management Engineering Exploiting Two Formulations 09/07/2018FormulationsThe network - Carvalho (1999)0 1 2 3 4 5 6 7Arc flow formulation(BP1) min z∑(j,i)∈Axj,i −∑(i,j)∈Axi,j ={z j = 00 j = 1, . . . ,W − 1−z j =W∀j ∈ N∑(h,h+wi)∈Axh,h+wi ≥ bi ∀i ∈ Ixij ∈ Z+ ∀(i, j) ∈ Az ∈ Z+7 DTU Management Engineering Exploiting Two Formulations 09/07/2018FormulationsThe network - Carvalho (1999)0 1 2 3 4 5 6 7Pattern based(BP2) min∑p∈Pλp∑p∈Paipλp ≥ bi ∀i ∈ Iλp ∈ Z+ ∀p ∈ P7 DTU Management Engineering Exploiting Two Formulations 09/07/2018Methodology• Combine both formulations and augment with a linking constraint(BP3) minimize∑p∈Pλp,s.t.∑p∈Paipλp ≥ bi ∀i ∈ I∑p∈Paipλp =∑(h,h+wi)∈Axh,h+wi ∀i ∈ I∑(j,i)∈Axji −∑(i,j)∈Axi,j =∑p∈P λp j = 00 j = 1, . . . ,W − 1−∑p∈P λp j = W ∀j ∈ Nxij ∈ Z+ ∀(i, j) ∈ Aλp ∈ Z+ ∀p ∈ P• Solve using Benders Branch-and-Cut• Compare with the Branch-Price-and-Cut algorithm of Alves & Carvalho (2008)8 DTU Management Engineering Exploiting Two Formulations 09/07/2018Methodology• Combine both formulations and augment with a linking constraint(BP3) minimize∑p∈Pλp,s.t.∑p∈Paipλp ≥ bi ∀i ∈ I∑p∈Paipλp =∑(h,h+wi)∈Axh,h+wi ∀i ∈ I∑(j,i)∈Axji −∑(i,j)∈Axi,j =∑p∈P λp j = 00 j = 1, . . . ,W − 1−∑p∈P λp j = W ∀j ∈ Nxij ∈ Z+ ∀(i, j) ∈ Aλp ∈ Z+ ∀p ∈ P• Solve using Benders Branch-and-Cut• Compare with the Branch-Price-and-Cut algorithm of Alves & Carvalho (2008)8 DTU Management Engineering Exploiting Two Formulations 09/07/2018Methodology• Combine both formulations and augment with a linking constraint(BP3) minimize∑p∈Pλp,s.t.∑p∈Paipλp ≥ bi ∀i ∈ I∑p∈Paipλp =∑(h,h+wi)∈Axh,h+wi ∀i ∈ I∑(j,i)∈Axji −∑(i,j)∈Axi,j =∑p∈P λp j = 00 j = 1, . . . ,W − 1−∑p∈P λp j = W ∀j ∈ Nxij ∈ R+ ∀(i, j) ∈ Aλp ∈ R+ ∀p ∈ P• Solve using Benders Branch-and-Cut• Compare with the Branch-Price-and-Cut algorithm of Alves & Carvalho (2008)8 DTU Management Engineering Exploiting Two Formulations 09/07/2018InstancesCategory1 |I| W Instancesbpp1 120 150 8bpp2 250 150 14bpp3 500 150 11bpp4 1000 150 11bpp5 60 100 7bpp6 120 100 20bpp7 249 100 20bpp8 501 100 20Total 111Algorithm coded in C++, uses COIN-BCP, Cplex 12.6 & Gurobi 7.5 for LPsMax run time of 3 hours, Intel Xeon 2.6GHz processor, Linux1https://www.math.u-bordeaux.fr/~fvanderb/publications.html9 DTU Management Engineering Exploiting Two Formulations 09/07/2018ResultsAlves & Carvalho (2008) Benders ApproachClass Nodes Price (s) Solve (s) Nodes Cuts Price (s) Solve(s)bpp1 73.00 0.62 4.04 24.34 0.00 0.49 1.60bpp2 60.27 1.00 6.15 20.07 0.00 0.54 1.77bpp3 200.50 7.48 88.56 48.82 0.00 0.49 4.56bpp4 399.91 13.10 143.86 150.00 0.00 1.91 13.22bpp5 26.00 1.75 3.08 29.20 0.00 0.29 5.44bpp6 111.12 15.63 41.20 192.90 1.25 1.26 50.56bpp7 423.60 187.88 933.07 1058.90 9.00 6.43 505.59bpp8 778.80 769.35 3990.16 3873.50 67.75 30.56 3313.9610 DTU Management Engineering Exploiting Two Formulations 09/07/2018ResultsAlves & Carvalho (2008) Benders ApproachClass Nodes Price (s) Solve (s) Nodes Cuts Price (s) Solve(s)bpp1 73.00 0.62 4.04 24.34 0.00 0.49 1.60bpp2 60.27 1.00 6.15 20.07 0.00 0.54 1.77bpp3 200.50 7.48 88.56 48.82 0.00 0.49 4.56bpp4 399.91 13.10 143.86 150.00 0.00 1.91 13.22bpp5 26.00 1.75 3.08 29.20 0.00 0.29 5.44bpp6 111.12 15.63 41.20 192.90 1.25 1.26 50.56bpp7 423.60 187.88 933.07 1058.90 9.00 6.43 505.59bpp8 778.80 769.35 3990.16 3873.50 67.75 30.56 3313.9610 DTU Management Engineering Exploiting Two Formulations 09/07/2018Comments• Proposed methodology is competitive• Benders Decomposition evaluates more nodes• Feasibility cuts needed for some instances only11 DTU Management Engineering Exploiting Two Formulations 09/07/2018Application Two: SDVRPRelaxation• Variant of the classical Capacitated Vehicle Routing Problem (CVRP)• Given a set of Customer demands, a depot, and a fleet of vehicles with a certaincapacity, partition the customers into vehicle tours that start and end at thedepot, satisfy the vehicle capacities, and which minimize the transportation cost.Formulated on a Graph G = (V,E)• In the SDVRP a customer can be visited by multiple vehicles• Notoriously difficult problem• Simultaneously consider an exact formulation and a relaxation12 DTU Management Engineering Exploiting Two Formulations 09/07/2018Formulations• Use the exact three index formulation of Li et al. (2006)• xkij ∈ {0, 1} determines with vehicle k travels between nodes i and j• yik ∈ [0, 1] states the portion of node i’s demand serviced by vehicle k• tik ≥ 0 is the time at which vehicle k visits node i• Combine with the two index formulation of Archetti at al. (2014)• xij ∈ {0, 1} counts the number of vehicle’s travelling between nodes i and j• zi ≥ 0 states the number of times node i is visited13 DTU Management Engineering Exploiting Two Formulations 09/07/2018Formulationsmin∑i,j∈V∑k∈Kcijxkijs.t.∑j∈Vxk0j ≤ 1 ∀k ∈ K∑j∈Vxkji −∑j∈Vxkij = 0 ∀i ∈ V \ {0}, ∀k ∈ Ktik − tjk + (|V |+ 1)xkij ≤ |V | ∀i, j ∈ V \ {0}, ∀k ∈ K∑k∈Kyik = 1 ∀i ∈ V \ {0}∑i∈Idiyik ≤ Q ∀k ∈ Kyik −∑j∈Vxkji ≤ 0 ∀i ∈ V \ {0}, ∀k ∈ Kxkij ∈ {0, 1} ∀(ij) ∈ E, ∀k ∈ K0 ≤ yik ≤ 1 ∀i ∈ V, ∀k ∈ Ktik ≥ 0 ∀i ∈ V, ∀k ∈ K13 DTU Management Engineering Exploiting Two Formulations 09/07/2018Formulationsmin∑i,j∈Vcijxijs.t.∑i∈δ(i)xij = 2zi ∀i ∈ V∑(i,j)∈δ(S)xij ≥ 2 ·⌈∑i∈S diQ⌉∀S ⊂ N, |S| ≥ 2xij ∈ {0, 1} ∀(i, j) ∈ E(N)kN ≤ xij ≤ 2 · |K| ∀(i, j) ∈ E \ E(N)kN ≤ z0 ≤ |K| and integer1 ≤ zi ≤ |K| and integer ∀i ∈ N13 DTU Management Engineering Exploiting Two Formulations 09/07/2018Methodology• Couple both formulations together & augment the resulting model withxij =∑k∈Kxkij ∀(i, j) ∈ E• Apply Branch-and-Benders-Cut• Use the two index formulation as the master problem• Branch in the master problem• Separate feasibility cuts on finding integer solutions• Compare with the Branch-and-Cut Algorithm of Archetti et al. (2014)• Dynamically, and heuristically, separate capacity cuts, connectivity cuts14 DTU Management Engineering Exploiting Two Formulations 09/07/2018Results: Proposed MethodologyTested on 25 instances from Belenguer et al. (2000) (Integer distances)Instance Ben. Cap. Con. Subs LP IP Nodes Time (s)eil22 0 47 0 1 375.00 375.00 1 0.11eil23 0 20 1 1 569.00 569.00 1 0.06eil30 1,216 3,055 1,430 2,021 508.00 510.00 4,679 109.71eil33 0 163 0 1 833.50 835.00 3 1.46eil51 0 5,221 81 12 514.10 521.00 237 17.35S51D1 0 3,59 0 20 454.33 458.00 73 9.13S76D1 0 10,750 80 16 586.16 592.00 399 63.66Algorithm coded in C++, uses COIN-BCP, Cplex 12.6 & Gurobi 7.5 for LPsMax run time of 3 hours, Intel Xeon 2.6GHz processor, Linux15 DTU Management Engineering Exploiting Two Formulations 09/07/2018Results: Proposed MethodologyInstance Ben. Cap. Con. Subs LP 1 LP 2 IP NodesS51D2 533 466,626 9,441 3,399 678.21 687.07 728.00 20,085S51D3 560 302,259 5,566 2,761 905.78 914.31 N/A 14,079S51D4 452 496,903 4,778 2,094 1,519.78 1,524.48 N/A 13,089S51D5 560 539,575 6,118 2,601 1,279.09 1,287.24 N/A 17,118S51D6 367 575,843 369 1,815 2,126.45 2,128.48 N/A 9,233eilA76 399 318,952 9,790 2,519 778.10 785.04 N/A 14,432eilB76 321 461,015 11,001 1,642 933.97 940.62 N/A 13,697eilC76 162 1,467,051 32,989 1,236 703.78 709.56 780.00 39,742eilD76 97 1,373,146 38,055 993 657.68 662.29 707.00 44,659eilA101 24 914,453 17,926 298 788.19 793.13 872.00 22,541eilB101 153 299,122 7,956 742 1,000.99 1,005.22 N/A 7,119S76D2 320 480,082 10,431 1,627 1,016.56 1,022.03 N/A 12,525S76D3 225 413,936 7,888 1,044 1,347.90 1,352.61 N/A 9,548S76D4 124 340,259 3,774 743 2,005.18 2,007.16 N/A 7087S101D1 58 867,619 8,785 961 702.77 708.46 726.00 24,211S101D2 87 316,617 6,454 434 1,270.25 1,276.42 N/A 4,893S101D3 77 369,606 4,668 348 1,753.37 1,755.25 N/A 4,211S101D5 35 334,711 3,078 149 2,652.75 2,652.78 N/A 3,04016 DTU Management Engineering Exploiting Two Formulations 09/07/2018Results: Proposed MethodologyInstance Ben. Cap. Con. Subs LP 1 LP 2 IP NodesS51D2 533 466,626 9,441 3,399 678.21 687.07 728.00 20,085S51D3 560 302,259 5,566 2,761 905.78 914.31 N/A 14,079S51D4 452 496,903 4,778 2,094 1,519.78 1,524.48 N/A 13,089S51D5 560 539,575 6,118 2,601 1,279.09 1,287.24 N/A 17,118S51D6 367 575,843 369 1,815 2,126.45 2,128.48 N/A 9,233eilA76 399 318,952 9,790 2,519 778.10 785.04 N/A 14,432eilB76 321 461,015 11,001 1,642 933.97 940.62 N/A 13,697eilC76 162 1,467,051 32,989 1,236 703.78 709.56 780.00 39,742eilD76 97 1,373,146 38,055 993 657.68 662.29 707.00 44,659eilA101 24 914,453 17,926 298 788.19 793.13 872.00 22,541eilB101 153 299,122 7,956 742 1,000.99 1,005.22 N/A 7,119S76D2 320 480,082 10,431 1,627 1,016.56 1,022.03 N/A 12,525S76D3 225 413,936 7,888 1,044 1,347.90 1,352.61 N/A 9,548S76D4 124 340,259 3,774 743 2,005.18 2,007.16 N/A 7087S101D1 58 867,619 8,785 961 702.77 708.46 726.00 24,211S101D2 87 316,617 6,454 434 1,270.25 1,276.42 N/A 4,893S101D3 77 369,606 4,668 348 1,753.37 1,755.25 N/A 4,211S101D5 35 334,711 3,078 149 2,652.75 2,652.78 N/A 3,04016 DTU Management Engineering Exploiting Two Formulations 09/07/2018Comments• Struggles on larger instances• Combine with an upper bound heuristic• Feasibility cuts on fractional solutions17 DTU Management Engineering Exploiting Two Formulations 09/07/2018Summary• Proposed framework couples two formulations to exploit both• Tested its performance using a reformulation and a relaxation• Shows promise• More development and testing need18 DTU Management Engineering Exploiting Two Formulations 09/07/2018Thanks for your attention!19 DTU Management Engineering Exploiting Two Formulations 09/07/2018

Simultaneously Exploiting Two Formulations: An Exact Benders Decomposition Approach

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Simultaneously Exploiting Two Formulations: An Exact Benders Decomposition Approach

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