Benders Decomposition for the Uncapacitated Multicommodity Network Design Problem

Transcription

1 Benders Decomposition for the Uncapacitated Multicommodity Network Design Problem 1 Carlos Armando Zetina, 1 Ivan Contreras, 2 Jean-François Cordeau 1 Concordia University and CIRRELT, Montréal, Canada 2 HEC Montréal and CIRRELT, Montréal, Canada Journées de l Optimisation 2017, Montréal, Canada 1/34

2 Agenda 1 Uncapacitated Multicommodity Network Design 2 Benders Decomposition for Uncapacitated Multicommodity Network Design 3 Enhancements to Benders decomposition for UMNDP 4 Computational Experiments 5 Concluding Remarks 2/34

3 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 3/34

4 Uncapacitated Multicommodity Network Design Literature Review Simplest fundamental problem in the Network Design class. It s NP-hard. Table: Exact Algorithms used to solve UMNDP Authors Magnanti et al. (1986) Balakrishnan et al. (1988) Holmberg and Hellstrand (1998) Methodology Benders Decomposition Dual Ascent w/ heuristic Lagrangian Branch and Bound undirected design arcs Table: Heuristics used to solve UMNDP Authors Billheimer and Gray (1973) Los and Lardinois (1980) Kratica et al. (2002) Methodology Add-Drop Heuristics Add-Drop Heuristics Genetic Algorithm 4/34

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 portion of commodity k s demand routed through arc (i, j). 5/34

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. Motivation to use Benders Decomposition Favourable results in the undirected design variable case [4]. Improvement in computational power compared to the 1980s. Recent increase in literature of enhancements to improve Benders performance. Can be combined with additional LP bound strengthening techniques. 6/34

7 Benders Decomposition for UMNDP (by the book) Obtaining the Slave Problem Assume design variables y ij are fixed. Dualize the remaining LP (referred to as DSP) Observations Feasible region of DSP is independent of y ij. Objective function estimates transportation cost. 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. 7/34

8 Benders Decomposition for UMNDP (by the book) The original MIP can be reformulated as follows. Benders Master Problem (MP) 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) Iteratively solve (MP) and (DSP) obtaining lower and upper bounds respectively. 8/34

9 Enhancements to Benders decomposition for UMNDP Cuts to ensure feasibility Extreme Rays of DSP from simplex method (as in classic Benders) Optimization over recession cone of DSP Minimal Infeasible Subsytem [1] Cutset inequalities Pareto Optimal Cuts Magnanti and Wong, 1981 [3] Papadakos, 2008 [5] Embedding Benders in a Branch-and-Cut [2] For the Uncapacitated Multicommodity Network Design Problem there are additional modifications that can be done to Benders Decomposition Algorithm. Decomposable dual subproblem leading to single or multiple cut MP Efficient algorithm to obtain Pareto Optimal cuts. 9/34

10 Finding the right combination 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 and updated in the same way for all versions. 10/34

11 Network Design test instances These algorithms were tested on the C instances used by T.G. Crainic, A. Frangioni, B. Gendron, 2001 with a time limit of 1 hour. Table: C Instances Class I Class II (N,A,K) No. (N,A,K) No. 20,230, ,230, ,300, ,300, ,520, ,520, ,700, ,700, /34

12 Finding the right combination Table: Instances not solved in 1 hour 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 /34

13 Finding the right combination 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, /34

14 The Enhancements Sweet Spot Our preliminary experiments show the best combination of enhancements to Benders decomposition for UMNDP are: Embedded in a Branch-and-cut Cuts added at some fractional and all integer solutions. Feasibility of the solution network is checked first followed by optimality cuts. Cutset inequalities are used to ensure feasibility Breadth first search used to separate at integer solutions Edmonds-Karp algorithm to obtain max flow and separate fractional solutions K minimum cost flow problems are solved with CPLEX s network optimizer to obtain Pareto optimal cuts as in Magnanti and Wong [4]. 14/34

15 Fine-tuning Branch-and-Cut parameters Leaving root node with LP bound. MP is a reformulation of the initial MIP. No more violated cuts at root node=lp of MIP. Weak root bound = large search tree. Frequency of adding cuts. Too few = poor trans cost estimation = weaker bounds. Too many = larger LPs. 15/34

16 Impact of Corepoint selection Corepoint is a point in the interior of the discrete variable space in the master problem used to obtain Pareto optimal cuts for MP. N.B. Corepoint must be a feasible network. We use the solution obtained from ignoring fixed costs (Routing solution) to make the following observations. > 85% of arcs in optimal solution are in Routing solution. Routing solution contains < 50% of potential arcs. Routing solution provides good candidate arcs for optimal solution and base for corepoint selection. 16/34

17 Computational Experiments Cplex- General MIP solver with default settings. Cplex 12.7 was used. Ben1 Ben2 Add cuts at every node (including root node) only if there has been at least 5% improvement in last 10 iterations. Update corepoint at each call to separation by taking the mid point between current MP solution and previous corepoint. Leave root node with LP solution. Check for cuts at every depth of 7 and add at most 5 cuts. Keep corepoint fixed at 0.7 if arc is in routing solution 0.5 otherwise. All use of CPLEX is limited to one thread and the use of the traditional MIP search. 17/34

18 Computational Experiments Table: Comparison of solution methods for UMNDP (averages per description) Cplex Ben1 Ben2 Class Description LP Gap Nodes CPU time Nodes CPU time Nodes CPU time 20,230, ,300, I 30,520, , , ,700, II 20,230, ,300, ,520, ,700, , , , , , Average , , , /34

19 More Difficult Instances We use the instances from Frangioni and Gorgone (2013). Table: Computational performance with difficult instances- Time limit 24 hours Solved Instances Time (sec)of solved instances Gap of instances not solved Class Description Ben2 Cplex Ben2 Cplex Ben2 Cplex I 20,300,100 4/4 4/ ,300,200 2/4 2/4 1, , ,600,100 2/4 2/ ,600,200 0/4 0/ ,1200,100 0/4 0/ ,1200,200 1/4 0/4 5, II 20,300,400 0/4 0/ ,300,800 0/4 0/ ,600,400 0/4 0/ ,600,800 0/4 0/ ,1200,400 0/4 0/ ,1200,800 0/4 0/ Grand Total 9/48 8/ /34

20 Summing it up.. Conclusions Selection of adequate enhancements of Benders Decomposition saves an order of magnitude of CPU vs CPLEX. Fine tuning of the Branch-and-Cut parameters has a significant impact. Proper corepoint selection plays an important role. Underlying network topology defines the instance s difficulty. 20/34

21 But wait, there s more... 21/34

22 But wait, there s more... 21/34

23 MIPs to solve MIPs 22/34

24 MIPs to solve MIPs 23/34

25 MIPs to solve MIPs 24/34

26 MIPs to solve MIPs 25/34

27 MIPs to solve MIPs 26/34

28 MIPs to solve MIPs 27/34

29 MIPs to solve MIPs 28/34

30 MIPs to solve MIPs 29/34

31 MIP for MIPs Advantage of embedding MP in this framework Benders Cuts are valid for any MIP t. Benders LP is calculated faster than Cplex LP. Other refinements can be added to solve the final MIP faster. Use a better starting solution. Selecting different types of cuts to add. Use LP to further restrict the final solution space to search. 30/34

32 Computational Experiments Table: Comparison of methods for UMNDP Cplex Benders MIP for MIPs w/benders Class Description LP Gap CPU time CPU time Iterations CPU time 20,230, ,300, I 30,520, ,700, II 20,230, ,300, ,520, ,700, Testbed Average /34

33 Conclusions Summary: A fine-tuned Benders with adequate enhancements goes a long way. A general methodology similar to local branching and cut and solve has been proposed. More fine-tuning of the latter could lead to solving the difficult instances. Future Work Compare with CPLEX 12.7 s Benders. Include new features to solve difficult instances. Primal Heuristic Lower bound improvement 32/34

34 Conclusions Summary: A fine-tuned Benders with adequate enhancements goes a long way. A general methodology similar to local branching and cut and solve has been proposed. More fine-tuning of the latter could lead to solving the difficult instances. Future Work Compare with CPLEX 12.7 s Benders. Include new features to solve difficult instances. Primal Heuristic Lower bound improvement Questions? 32/34

35 References I Matteo Fischetti, Domenico Salvagnin, and Arrigo Zanette. A note on the selection of benders cuts. Mathematical Programming, 124(1-2): , 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 , /34

36 References II 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 Springer Berlin Heidelberg, Nikolaos Papadakos. Practical enhancements to the magnantiwong method. Operations Research Letters, 36(4): , /34