A Branch and Price Approach for an Airport Vehicle Routing Problem

Michaël SchynsQuantOM Research Center

HEC-Management School of theUniversity of Liege

Agenda• Problem statement

• Goals

• Branch-and-Cut-and-Price for VRPTW

• Impact of the parameters

• Conclusions

Problem Statement• Project initiated by a main European

freight airport• Set of planes (TW)

• Set of trucks (Ci)

• Demand evolves…

• Goals: min Km (and satisfy the TW)

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

Exact methods for the VRPTWMain papers

1.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.

2.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.

3.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.

4.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

5.Desrosiers J. and M.E. Lübbecke, 2005, A primer in column generation, In: Desrosiers, Desaulniers, Solomon (Editors), "Column Generation", Springer, (GERAD, 25th anniversary)

6.Feillet D., 2010, A tutorial on column generation and branch-and-price for vehicle routing problems, "4OR-Q J Oper Res", 8, 407-424

7.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

8.Feillet D., Gendreau M. and LM Rousseau, ?after 2007?, New Refinements for the Solution of Vehicle Routing Problems with Branch and Price, Gerad, http://...

9.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.

10.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.

11.Irnich, S. and G. Desaulniers, 2005, Shortest path with resource constraints, In: Desrosiers, Desaulniers, Solomon (Editors), "Column Generation", Springer, (GERAD, 25th anniversary)

12.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.

13.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)

14.Laporte G., 1992, The Vehicle Routing Problem: An Overview of exact and approximate algorithms, "European Journal of Operational Research, 59, 345-358.

15.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.

16.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.

17.Righini G. and M. Salani, 2008, New Dynamic Programming Algorithms for the Resource Constrained Elementary Shortest Path Problem, "Networks", 51(3), 155-170.

18.Toth P. and D. Vigo (Editors), 2002, "The Vehicle Routing Problem", Siam monographs on Discrete Mathematics and Applications

Exact methods for the VRPTW• Two models and approaches:

– Initially: Branch and cut (arc flows model)– Mainly: Branch and price (route formulation) (one)

Route model– …

D

3

2

1

Route: feasible sequence of clients; from and to D

Branch and (Cut and) Price

Branch & Bound(Integer solution)

Node optimisationRelaxation(simplex)

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 basisStart with a few routes and construct new promising ones dynamically (those that would have a negative reduced cost)Column generation

B&B+CG Branch & PriceB&B+CG+cuts Branch and Cut and Prices CG+cuts Column and Cuts

Branch and (Cut and) Price

Branch & Bound(Integer solution)

Column Generation

Master Problem(simplex)

Column Generation

Pricing(Shortest Path

Problem)

• Var selection: max impact…• Branching strat: 0, 1 …• Node selection: dive, broad…• Branching on "arcs"!!!

• 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…

• 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

Strategies, parameters, tricksClosest oneClosest one

AntsAntsDivingDivingFrequency, best…Frequency, best…

Theta in {0,1}Theta in {0,1}

FP!FP!

Min cost, short, long…Min cost, short, long…11nf first of nb bestnf first of nb bestTime limitsTime limitsRoutes selectionRoutes selection……

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

Results and conclusions• I'm crazy! Work in progress• Parameters: does-it matter? YES

A 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!

Thank you

Project:

http://www.mschyns.be/demonstration/vrp

QuantOM:

http://www.quantom.hec.ulg.ac.be