A Branch and Price Approach for an Airport Vehicle Routing Problem

1. A Branch and Price Approach for an Airport Vehicle Routing Problem Michaël SchynsQuantOM Research Center HEC-Management School of theUniversity of Liege

2. Agenda • Problem statement • Goals • Branch-and-Cut-and-Price for VRPTW • Impact of the parameters • Conclusions

3. Problem Statement • Project initiated by a main Europeanfreightairport • Set of planes (TW) • Set of trucks (Ci) • Demandevolves… • Goals: min Km (and satisfy the TW)

4. Project goals (work in progress) • Model and solve the problem • (Scheduling or) VRPTW • Exact method (and/or heuristic: ant colonies) • "Survey" of the exact methods for the VRPTW • 30 last years of research: lots of "things" • Selection of the methods, tricks and parameters: impact! • Improvements? • Open Source Java Code

5. Exact methods for the VRPTW Main papers • Baldacci R., Mingozzi A. and R. Roberti, 2012, Recent exact algorithms for solving the vehicle routing problem under capacity and time windows constraints, European Journal of Operational Research, 218, 1-6. • Cordeau JF., Desaulniers G., Desrosiers J., Solomon M. and F. Soumis, 2002, Vehicle Routing Problems with Time Windows, In: Toth P. and D. Vigo (Editors), "The Vehicle Routing Problem", Siam monographs on Discrete Mathematics and Applications, 157-195. • Desaulniers G., Lessard F. and A. Hadjar, 2008, Tabu Search, Partial Elementary, and Generalized k-Path Inequalitites for the Vehicle Routing Problem with Time Windows, "Transportation Science", 42(3),387-404. • Desrochers M., Desrosiers J. and M. Solomon, 1992, A new optimization algorithm for the vehicle routing problem with time windows, "Operations Research", 40, March-April, 342-354 • Desrosiers J. and M.E. Lübbecke, 2005, A primer in column generation, In: Desrosiers, Desaulniers, Solomon (Editors), "Column Generation", Springer, (GERAD, 25th anniversary) • Feillet D., 2010, A tutorial on column generation and branch-and-price for vehicle routing problems, "4OR-Q J Oper Res", 8, 407-424 • Feillet D., Dejax P., Gendreau M. and C. Gueguen, 2004,An Exact Algorithm for the Elementary Shortest Path Problem with Resource Constraints: Application to some Vehicle Routing Problems, Networks, 44, 216-229 • Feillet D., Gendreau M. and LM Rousseau, ?after 2007?, New Refinements for the Solution of Vehicle Routing Problems with Branch and Price, Gerad, http://... • Gambardella L., Taillard E. and G. Agazzi, 1999, MACS-VRPTW: a multiple ant colony system for Vehicle Routing Problems with time windows, In: Corne D., Dorigo M. and F. Glover (Editors), "New Ideas in Optimization", McGraw-Hill. • Gutiérrez-Jarpa G., Desaulniers G., Laporte G. and V. Marianov, 2010, A branch-and-price algorithm for the Vehicle Routing Problem with Deliveries, Selective Pickups and Time Windows", "European Journal of Operational Research", 206, 341-349. • Irnich, S. and G. Desaulniers, 2005, Shortest path with resource constraints, In: Desrosiers, Desaulniers, Solomon (Editors), "Column Generation", Springer, (GERAD, 25th anniversary) • Jepsen M., Petersen B., Spoorendonk S., D. Pisinger, 2008, Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows, Operations Research, 56(2), pp497-511. • Kallehauge B., Larsen J., Madsen O. and M. Solomon, 2005, Vehicle routing problem with time windows, In: Desrosiers, Desaulniers, Solomon (Editors), "Column Generation", Springer, (GERAD, 25th anniversary) • Laporte G., 1992, The Vehicle Routing Problem: An Overview of exact and approximate algorithms, "European Journal of Operational Research, 59, 345-358. • Lozano, L., Medaglia, A. L., 2012, An Exact Algorithm for the Elementary Shortest Path Problem with Resource Constraints, Tech. Rep. COPA-2012-2, Universidad de los Andes. • Righini G. and M. Salani, 2006, Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints, "Discrete Optimization", 3, 255-273. • Righini G. and M. Salani, 2008, New Dynamic Programming Algorithms for the Resource Constrained Elementary Shortest Path Problem, "Networks", 51(3), 155-170. • Toth P. and D. Vigo (Editors), 2002, "The Vehicle Routing Problem", Siam monographs on Discrete Mathematics and Applications

6. Exact methods for the VRPTW • Twomodels and approaches: • Initially: Branch and cut (arc flows model) • Mainly: Branch and price (route formulation) (one) Route model • … 1 D 2 3 Route: feasible sequence of clients; from and to D

7. Branch and (Cut and) Price • Integer Problem  Branch and Bound • Node: simplexEach column of the table corresponds to one route • Let's first assume that all (usefull) feasible routes could be incorporated  Ok • But it is rarely the case… Problem • Good news: we don't need all the routes but just a subset for the basis • Start with a few routes and construct new promising ones dynamically (those that would have a negative reduced cost) • Column generation • B&B+CG Branch & Price • B&B+CG+cuts  Branch and Cut and Prices • CG+cuts  Column and Cuts Branch & Bound (Integer solution) Node optimisation Relaxation (simplex)

8. Branch and (Cut and) Price Strategies, parameters, tricks • Var selection: max impact… • Branching strat: 0, 1 … • Node selection: dive, broad… • Branching on "arcs"!!! Closest one Ants Diving Frequency, best… Theta in {0,1} FP! Min cost, short, long… 1nf first of nb best Time limits Routes selection … Branch & Bound (Integer solution) • Very 1st routes: elementary, heuristics… • 1st routes node: all previous ones • Cleaning: remove "unused" routes • Stabilisation of the dual variables? • Model: set partitioning, set covering • Theta in IN • Cuts: k-cycle, SRC, ng-routes… • SRC: #, threshold… ColumnGeneration Master Problem (simplex) • Algo: SPPRC (+k-cycle), ESPPRC, Pulse? • ESPPRC: forward, bidir, DSSR • Pulse: delta, horizon • Heuristics: specific, relax dom… • Stronger resource vector (Feillet) • Label selection: cyclic, cyclic&sorted • 1 nb best routes • 1 nf first routes • Strengthening Time Windows • Cleaning • Code optimisation, good practices, hardware improvements,luck ColumnGeneration Pricing (ShortestPathProblem)

9. Simulations • Java: memory 4GB,Eclipse development env. • IBM ILOG CPLEX 12.5 (simplex Master P) • Ordinary laptop, windows 7 • Airport instances and Solomon's benchmark • More than 20 "parameters" with lots of different possible values • http://www.mschyns.be/demonstration/vrp • More than 500 simulations up to now • Tools to compare the results • Bibliography • Summary of results found in the literature

10. Results and conclusions • I'm crazy! Work in progress • Parameters: does-it matter? YESA slight change in some parameters and no more solution!  Limited robustness!  Best choicesNew question: for hard instances, is it more complex to solve the problem or to find the right values of the parameters? • Network topology and data: the airport instances are hard! Large TW and overlaps, parkings on a same line (symmetry)… • New trend: improve the lower bound with cuts • Old abandoned recipes could be reconsidered! • First results for the airport: • From 5 to 3 trucks for a classical shift • No more TW violation • Distance: 50% of reduction!

11. Thank you Project: http://www.mschyns.be/demonstration/vrp QuantOM: http://www.quantom.hec.ulg.ac.be