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Cut selection strategies for Bendersdecomposition in uncapacitated multicommodity

network design

1 Carlos Armando Zetina, 1Ivan Contreras, 2Jean-François Cordeau

1Concordia University and CIRRELT, Montreal, Canada2HEC Montréal and CIRRELT, Montreal, Canada

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1 Uncapacitated Multicommodity Network Design

2 Benders Decomposition for Uncapacitated Multicommodity NetworkDesign

3 Computational Experiments

4 Conclusions and Future Work

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Uncapacitated Multicommodity Network Design

Applications[2]

-Roads-Railways-Transportation Services-Telecommunications

Characteristics

Given a set of commodities each with an origin, destination and demandquantity, and a network of potential arcs each with a fixed installationand unit transportation cost, construct a network that minimizes theinvestment and operational cost.

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Uncapacitated Multicommodity Network Design

Problem Definition

Let G = (N,A) be a directed graph. N denotes the nodes and A the setof arcs, each with a fixed cost fij and unit transportation cost cij .

Let K be a set of commodities with origin,destination and demandquantity (o(k), d(k), dk).

select Ā ⊂ A to be installed,route all commodities k ∈ K from their origins to destinations usingonly the arcs in Ā.

Arc Selection

𝑑1𝑂1

𝑂2

𝑑2

𝑂3

𝑑3

𝑂4

𝑑4

Commodity Routing

𝑑1𝑂1

𝑂2

𝑑2

𝑂3

𝑑3

𝑂4

𝑑4

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Mathematical Model

The following Mixed Integer Program models the Uncapacitated MulticommodityNetwork Design problem.

min∑

(i,j)∈A

fijyij +∑k∈K

∑(i,j)∈A

cijxkij

∑j∈N:(j,i)∈A

xkji −∑

j∈N:(i,j)∈A

xkij =

−dk if i = o(k)0 if i /∈ {o(k), d(k)}dk if i = d(k)

∀i ∈ N,∀k ∈ K

xkij ≤ yij ∀(i , j) ∈ A, k ∈ Kxkij ≥ 0 ∀(i , j) ∈ A, k ∈ Kyij ∈ {0, 1} ∀(i , j) ∈ A

fij is the fixed cost of installing arc (i , j) ∈ A.cij denotes unit transportation cost of arc (i , j) ∈ A.dk is the demand of commodity k.yij is a binary variable modelling whether arc (i , j) is installed.xkij models the amount of commodity k routed through arc (i , j).

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Benders Decomposition

1962 Benders Decomposition

Work in the space of the discrete variables y and a continuousartificial variable z (Master Problem).

Solve a special linear programming problem (DSP) to obtain cuts forthe projected problem.

Process is done iteratively until convergence.

Enhancements to Benders decomposition

Pareto Optimal Cuts

Magnanti and Wong, 1981Papadakos, 2008

Cut Covering inequalities

Primal Heuristics (Local Branching)

Embedding Benders in a Branch-and-Cut

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Benders Decomposition for Uncapacitated MulticommodityNetwork Design

For the Uncapacitated Network Design Problem there are additionalmodifications that can be done to our Benders Decomposition Algorithm

Decomposable Subproblem leading to Single or Multiple Cut MP

Efficient algorithm to solve the Magnanti and Wong Subproblem

Efficient algorithm to solve the Papadakos Subproblem

Let ȳ denote the solution obtained from MP and y0 thecorepoint/Magnanti Wong point being used.

Choosing Pareto-Optimal Cuts

Magnanti and Wong- Solve an additional subproblem using y0 in theobjective function but guarantee that the obtained solution is alsooptimal for ȳ .

Papadakos-Substitute the point ȳ for y0 in the subproblem andsolve.

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Benders Decomposition for Uncapacitated MulticommodityNetwork Design

Efficient vs LP method to solve the slave problem.

Magnanti, Mireault and Wong (86) show that for UMNDP one onlyneeds to solve |K | MinCostFlow problems to get Pareto optimal cuts.One can also solve the Papadakos subproblem by solving |K |MinCostFlow problems (different from Magnanti and Wong).

For our computational study we take into account:

Classic Iterative Benders vs Benders Branch-and-Cut

Single Cut vs Multiple Cut Master Problem

Use of LP vs efficient algorithms to solve the slave problem

Magnanti-Wong vs Papadakos Pareto Optimal Cut selection

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Preliminary Computational Experiments

Table: Algorithm Versions

Iterative Benders Benders Branch-and-CutSingle Cut Multiple Cuts Single Cut Multiple CutsBenders- LP Benders- LP Benders- LP Benders- LPMW- LP MW- LP MW- LP MW- LPPapadakos- LP Papadakos- LP Papadakos- LP Papadakos- LPEff. M-W Eff. M-W Eff. M-W Eff. M-WEff. Papadakos Eff. Papadakos Eff. Papadakos Eff. Papadakos

All algorithms written in C using CPLEX and run on Intel XeonProcessors at 3.10 GHz

Parameters of Branch-and-Cut are fine tuned individually.

Corepoints are chosen in the same way for all versions.

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CANAD Network Design instances

These algorithms were tested on the CANAD problems used by T.G.Crainic, A. Frangioni, B. Gendron, 2001 with a time limit of 3 hours.

Table: CANAD Instances

Class I Class II

(N,A,K) No. (N,A,K) No.

20,230,40 3 20,230,200 420,300,40 4 20,300,200 4

30,520,100 4 30,520,400 430,700,100 4 30,700,400 4

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Preliminary Computational Results

Table: Instances not solved in 3 hours of CPU time

Iterative Benders Benders B&C

Class Algorithm Single Cut Multi Cut Single Cut Multi Cut

I (15)

Benders-LP 15 3 8 0M-W LP 12 3 7 0Eff. M-W 12 3 8 0Papadakos- LP 8 3 3 0Eff. Papadakos 9 3 4 0

II (16)

Benders-LP 16 9 16 2M-W LP 16 2 16 1Eff. M-W 16 2 15 1Papadakos- LP 16 2 13 1Eff. Papadakos 16 2 11 2

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Preliminary Computational Results

Table: Average Gap % for instances not solved

Iterative Benders Benders B&C

Class Algorithm Single Cut Multi Cut Single Cut Multi Cut

I

Benders-LP 98.657 28.554 44.769 0M-W LP 90.969 7.771 81.764 0Eff. M-W 98.741 5.176 71.201 0Papadakos- LP 8.571 6.217 7.804 0Eff. Papadakos 8.131 5.945 7.880 0

II

Benders-LP 95.667 11.388 47.081 1.246M-W LP 89.97 3.068 47.437 1.356Eff. M-W 93.192 4.498 56.278 1.576Papadakos- LP 11.464 2.907 3.809 1.432Eff. Papadakos 11.656 2.084 4.723 0.989

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Preliminary Computational Results

Table: Average times (seconds) of solved problems Benders Branch and Cut-Multiple Cuts

|N|, |A|, |K | Benders-LP M-W LP Pap.- LP Eff. M-W Eff. Pap.

20,230,40 0.677 1.01 0.137 0.097 0.09320,300,40 0.753 1.36 0.318 0.113 0.125

30,520,100 1366.595 976.275 1596.73 660.06 1602.67530,700,100 105.993 69.1 37.248 11.793 25.64820,230,200 143.313 96.543 74.81 56.335 97.86520,300,200 133.645 72.393 138.193 36.34 115.2830,520,400 108.888 213.923 213.968 7.305 31.72330,700,400 216.045 1305.077 859.893 174.007 155.545

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Preliminary Computational Results

Table: Average times (seconds) of solved problems Benders Branch and Cut-Single Cut

|N|, |A|, |K | Benders-LP M-W LP Pap.- LP Eff. M-W Eff. Pap.

20,230,40 35.86 8.727 0.877 1.453 0.36720,300,40 144.09 123.338 4.565 4.878 1.563

30,520,100 N/A N/A 379.785 N/A 59.0530,700,100 N/A N/A 970.5 N/A 902.25520,230,200 N/A N/A 3598.77 N/A 1544.320,300,200 N/A N/A 355.37 215.35 5257.1530,520,400 N/A N/A 2532.29 N/A 954.66

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Preliminary Computational Results

Table: Comparison of average time (seconds) of best Benders approach vsCplex 12.6.3

|N|, |A|, |K | Best Benders Approach Cplex Defaults

20,230,200 52.295 180.0120,230,40 0.09 0.077

20,300,200 36.34 210.7220,300,40 0.095 0.138

30,520,100 660.06 526.32530,520,400 7.305 88.55830,700,100 11.793 55.21330,700,400 174.007 2260.553

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Conclusion

Based on our preliminary computational experiments we see that:

It is advantageous to embed Benders in a Branch-and-cut ratherthan the iterative form.

If possible it is preferable to use a multiple cut version of the MasterProblem.

For Multi cut Benders M-W’s method to obtain Pareto-OptimalCuts leads to shorter CPU time.

For a Single cut Benders Papadakos’ method to obtainPareto-Optimal Cuts leads to shorter CPU time.

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Future Research

Test the algorithms on larger instances.

Propose a method to select the “best” corepoint for each algorithm.

Make additional enhancements to the algorithm so as to outperformCplex in all instances for a given version of the algorithm.

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J.F. Benders.Partitioning procedures for solving mixed-variables programmingproblems.Numerische Mathematik, 4:238–252, 1962.

Alysson M. Costa.A survey on benders decomposition applied to fixed-charge networkdesign problems.Computers & Operations Research, 32(6):1429–1450, 6 2005.

Teodor Gabriel Crainic, Antonio Frangioni, and Bernard Gendron.Bundle-based relaxation methods for multicommodity capacitatedfixed charge network design.Discrete Applied Mathematics, 112(13):73 – 99, 2001.Combinatorial Optimization Symposium, Selected Papers.

T.L. Magnanti, P. Mireault, and R.T. Wong.Tailoring benders decomposition for uncapacitated network design.In Giorgio Gallo and Claudio Sandi, editors, Netflow at Pisa,volume 26 of Mathematical Programming Studies, pages 112–154.Springer Berlin Heidelberg, 1986.

S. Orlowski, M. Pióro, A. Tomaszewski, and R. Wessäly.SNDlib 1.0–Survivable Network Design Library.In Proceedings of the 3rd International Network OptimizationConference (INOC 2007), Spa, Belgium, April 2007.http://sndlib.zib.de, extended version accepted in Networks, 2009.

Uncapacitated Multicommodity Network DesignBenders Decomposition for Uncapacitated Multicommodity Network DesignComputational ExperimentsConclusions and Future Work