Canadian Mathematical Society

Montréal, December 11 -13, 1999

Jacques DesrosiersÉcole des Hautes Études Commerciales & GERADMontréal, Canada H3T 2A7Jacques.Desrosiers@hec.ca

The Mathematics behind Vehicle Routing and Crew Scheduling

This presentation describes the significant advances madein time-constrained routing and scheduling. Helped bycontinuously better insights into problem structures and rapidadvances in computer technology, the optimization methodsare becoming a viable tool for solving practical size problems.

SUCCESSFUL APPLICATIONS

• Vehicle Routing with Time Windows• Dial-a-Ride for Physically Disabled Persons • Urban Transit Crew Scheduling • Multiple Depot Vehicle Scheduling • Aircraft Routing • Crew Pairing • Crew Rostering (Pilots & Flight Attendants) • Locomotive and Car Assignment

• CREW-OPT• BUS-OPT• ALTITUDE-Pairings• ALTITUDE-Rosters• ALTITUDE-PBS• RAIL-WAYS

The GENCOL Optimizer

60 installations around the world

… at the Core of Various Software Systems

RESEARCH TRENDS

• Accelerating Techniques• Primal - Dual Stabilization• Constraint Aggregation• Sub-Problem Speed-up• Two-level Problems Solved

with Benders Decomposition

• Integer Column Generation with Interior Point Algorithm

• Acceleration Techniques

Column Generator Master Problem Global Formulation

Heuristics Re-Optimizers Pre-Processors

…to obtain Primal & Dual Solutions

Acceleration Techniques ...

Multiple Columns: selected subset close to expected optimal solution

Partial Pricing in case of many Sub-Problems

Early & MultipleBranching & Cutting: quickly gets local optima

Branching & Cutting: on integer variables !

• Primal - Dual Stabilization

0

min

xbAx

cx

cAb

max

21

max

ddcAb

Restricted Dual

2211

21

0,00

min

yyx

byyAxcx

Perturbed Primal

2211

21

2211

0,00

min

yyx

byyAxydydcx

Stabilized Primal

Dual Solution Primal Solution Primal Solution Dual Solution

Approximate Primal & Dual

Primal & Dual Solutions

Primal - Dual Stabilization ...

• Constraint Aggregation

Massive Degeneracy on Set Partitioning Problems

A pilot covers consecutive flights on the same aircraftA driver covers consecutive legs on the same bus line

Aggregate Identical Constraintson Non-zero Variables

Aggregation Algorithm

• Initial Constraint Aggregation• Consider only Compatible VariablesSolve Aggregated Master Problem

Primal & Aggregated Dual SolutionsDual Variables Split-upSolve Sub-Problem• Modify Constraint Aggregation

• Sub-Problem Speed-up

Resource Constrained Shortest PathLabels at each node : cost, time, load, …

Resource ProjectionAdjust A dynamicallyGeneralized Lagrangian Relaxation

Results on Sub-Problem cpu time divided by 5 to 10

nmRR mAn

24 RR

• Two-Level Problems

Benders Decomposition Algorithmfor Simultaneous Assignment of

Buses and DriversAircraft and Pilots

Pairings and RostersLocomotives and Cars

IP(X, Y) for Two-Level Scheduling

MIP(X, y) solved using Benders Decomposition

Master IP(X) Simplex and B&B(X)

Sub-Problem solved by Column Generation MP LP(y) of Set Partitioning SP DP for Constrained Paths

B&B(Y) with MIP(X, y) at each node

Benders MP

Benders SP

B &

B

IP

LP

DP

CG MP

CG SP

• Column Generation with Interior Point Algorithm

• ACCPM Algorithm (Goffin & Vial)• Applications

Linear ProgrammingNon-Linear ProgrammingStochastic Programming Variational Inequalities

• Integer Column Generation with Interior Point Algorithm

• Strategic Grant in Geneva– J.-P. Vial et al.

• Strategic Grant in Montréal– J.-L. Goffin et al.

Design of a Commercial Software System

CONCLUSIONS

• Larger Problems to Solve• Mixing of Decomposition Methods• Strong Exact and Heuristic Algorithms• Faster Computers • Parallel Implementations

Still a lot of work to do !!