A computational study of enhancements to Benders Decomposition in uncapacitated multicommodity network design

Transcription

1 A computational study of enhancements to Benders Decomposition in uncapacitated multicommodity network design 1 Carlos Armando Zetina, 1 Ivan Contreras, 2 Jean-François Cordeau 1 Concordia University and CIRRELT, Montreal, Canada 2 HEC Montréal and CIRRELT, Montreal, Canada CORS Annual Conference 2016, Banff, Canada 1/20

2 1 Uncapacitated Multicommodity Network Design 2 Benders Decomposition for Uncapacitated Multicommodity Network Design 3 Computational Experiments 4 Conclusions and Future Work 2/20

3 Uncapacitated Multicommodity Network Design Applications -Roads -Railways -Transportation Services -Telecommunications Characteristics Given a set of commodities each with an origin, destination and demand quantity, and a network of potential arcs each with a fixed installation and unit transportation cost, construct a network that minimizes the investment and operational cost. 2/20

4 Uncapacitated Multicommodity Network Design Problem Definition Let G = (N, A) be a directed graph. N denotes the nodes and A the set of arcs, each with a fixed cost f ij and unit transportation cost c ij. Let K be a set of commodities with origin,destination and demand quantity (o(k), d(k), d k ). select Ā A to be installed, route all commodities k K from their origins to destinations using only the arcs in Ā. Arc Selection Commodity Routing d 2 d 2 d 3 O 4 d 3 O 4 O 2 O 2 O 3 O 3 d 4 d 4 O 1 d 1 O 1 d 1 3/20

5 Mathematical Model The following Mixed Integer Program models the Uncapacitated Multicommodity Network Design problem. (P) min f ij y ij + d k c ij xij k k K (i,j) A (i,j) A j N:(j,i) A x k ji xij k = j N:(i,j) A 1 x k ij y ij xij k 0 y ij {0, 1} if i = o(k) 0 if i / {o(k), d(k)} 1 if i = d(k) i N, k K (i, j) A, k K (i, j) A, k K (i, j) A f ij is the fixed cost of installing arc (i, j) A. c ij denotes unit transportation cost of arc (i, j) A. d k is the demand of commodity k. y ij is a binary variable modelling whether arc (i, j) is installed. x k ij models the amount of commodity k routed through arc (i, j). 4/20

6 Benders Decomposition 1962 Benders Decomposition Work in the space of the discrete variables y and a continuous artificial variable z (Master Problem). Solve a special linear programming problem (DSP) to obtain cuts for the projected problem. Process is done iteratively until convergence. Enhancements to Benders decomposition Pareto Optimal Cuts Magnanti and Wong, 1981 [2] Papadakos, 2008 [3] Cut Covering Bundles [5] Maximal Non-dominated Cuts [6] Primal Heuristics (Local Branching) [4] Embedding Benders in a Branch-and-Cut [1] 5/20

7 Benders Decomposition for Uncapacitated Multicommodity Network Design Benders Master Problem (Reformulation) min z + f ij y ij s.t. (i,j) A z (λ k d(k) λk o(k) ) k K k K 0 ( λ k d(k) λ k o(k) ) k K k K (i,j) A (i,j) A µ k ijy ij ({λ k i }, {µ k ij}) Opt(DSP) µ k ijy ij ({ λ k i }, { µ k ij}) Ext(DSP) where y ij {0, 1}, (i, j) A and z R and Opt(DSP) and Ext(DSP) represent the extreme points and extreme rays respectively of the special LP (DSP) used to obtain cuts. 6/20

8 Benders Decomposition for Uncapacitated Multicommodity Network Design Dual Sub Problem (DSP) (DSP) max (λ k d(k) λk o(k) ) k K k K (i,j) A µ k ijỹ ij s.t. λ k j λ k i µ k ij d k c ij (i, j) A, k K where λ k i R i N, k K and µ k ij 0, (i, j) A, k K. For the Uncapacitated Network Design Problem there are additional modifications 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 7/20

9 Efficiently solving for Pareto Optimal cuts with Magnanti-Wong method (Magnanti et. al 1986) Let ȳ denote the solution obtained from MP and y 0 the corepoint being used. For a fixed k = k we obtain a Pareto-Optimal cut by solving. (P) min d k c ijx k ij R k(ȳ)x 0 (i,j) A j N:(j,i) A x ji j N:(i,j) A 1 x 0 if i = o(k) x ij = 0 if i / {o(k), d(k)} 1 + x 0 if i = d(k) x ij yij 0 + x 0 ȳ ij x ij 0,x 0 0 i N (i, j) A (i, j) A where R k(ȳ) is the cost of routing commodity k through the network defined by ȳ. Magnanti et al. argue that fixing x 0 (ij) A y 0 ij and solving the remaining Minimum Cost Flow problem gives an optimal solution. 8/20

10 Efficiently solving for Pareto Optimal cuts with Papadakos method Let ȳ denote the solution obtained from MP and y 0 the corepoint being used. For a fixed k = k we obtain a Pareto-Optimal cut by solving. (P) min j N:(j,i) A x ji j N:(i,j) A (i,j) A d k c ijx k ij 1 if i = o(k) x ij = 0 if i / {o(k), d(k)} 1 if i = d(k) x ij yij 0 x ij 0 i N (i, j) A (i, j) A Solving the K Minimum Cost Flow problems gives an optimal solution and the corresponding dual variables give a Pareto Optimal Cut. 9/20

11 Preliminary Computational Experiments Table: Algorithm Versions Iterative Benders Benders Branch-and-Cut Single Cut Multiple Cuts Single Cut Multiple Cuts Benders- LP Benders- LP Benders- LP Benders- LP MW- LP MW- LP MW- LP MW- LP Papadakos- LP Papadakos- LP Papadakos- LP Papadakos- LP Eff. M-W Eff. M-W Eff. M-W Eff. M-W Eff. Papadakos Eff. Papadakos Eff. Papadakos Eff. Papadakos All algorithms written in C using CPLEX and run on Intel Xeon Processors at 3.10 GHz Parameters of Branch-and-Cut are fine tuned individually. Corepoints are chosen in the same way for all versions. 10/20

12 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, ,230, ,300, ,300, ,520, ,520, ,700, ,700, /20

13 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 Benders-LP M-W LP I (15) Eff. M-W Papadakos- LP Eff. Papadakos II (16) Benders-LP M-W LP Eff. M-W Papadakos- LP Eff. Papadakos /20

14 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 Benders-LP M-W LP I Eff. M-W Papadakos- LP Eff. Papadakos II Benders-LP M-W LP Eff. M-W Papadakos- LP Eff. Papadakos /20

15 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, ,300, ,520, ,700, ,230, ,300, ,520, ,700, /20

16 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, ,300, ,520,100 N/A N/A N/A ,700,100 N/A N/A N/A ,230,200 N/A N/A N/A ,300,200 N/A N/A ,520,400 N/A N/A N/A /20

17 Preliminary Computational Results Table: Comparison of average time (seconds) of best Benders approach vs Cplex N, A, K Best Benders Approach Cplex Defaults 20,230, ,230, ,300, ,300, ,520, ,520, ,700, ,700, /20

18 Conclusion Based on our preliminary computational experiments we see that: It is advantageous to embed Benders in a Branch-and-cut rather than the iterative form. If possible it is preferable to use a multiple cut version of the Master Problem. For Multi cut Benders M-W s method to obtain Pareto-Optimal Cuts leads to shorter CPU time. For a Single cut Benders Papadakos method to obtain Pareto-Optimal Cuts leads to shorter CPU time. 17/20

19 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 outperform Cplex in all instances for a given version of the algorithm. Perform similar computational tests for other enhancements proposed in the literature. 18/20

20 References I B. Fortz and M. Poss. An improved benders decomposition applied to a multi-layer network design problem. Operations Research Letters, 37(5): , T. L. Magnanti and R. T. Wong. Accelerating benders decomposition: Algorithmic enhancement and model selection criteria. Operations Research, 29(3):pp , Nikolaos Papadakos. Practical enhancements to the magnantiwong method. Operations Research Letters, 36(4): , Walter Rei, Jean-François Cordeau, Michel Gendreau, and Patrick Soriano. Accelerating benders decomposition by local branching. INFORMS Journal on Computing, 21(2): , /20

21 References II Georgios K. D. Saharidis, Michel Minoux, and Marianthi G. Ierapetritou. Accelerating benders method using covering cut bundle generation. International Transactions in Operational Research, 17(2): , HanifD. Sherali and BrianJ. Lunday. On generating maximal nondominated benders cuts. Annals of Operations Research, 210(1):57 72, /20