Canadian Mathematical Society
Montréal, December 11 -13, 1999
Jacques Desrosiers
École des Hautes Études Commerciales & GERAD
Montréal, Canada H3T 2A7
[email protected]
The Mathematics behind Vehicle Routing and Crew Scheduling
This presentation describes the significant advances made
in time-constrained routing and scheduling. Helped by
continuously better insights into problem structures and rapid
advances in computer technology, the optimization methods
are becoming a viable tool for solving practical size problems.
SUCCESSFUL APPLICATIONS
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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
The GENCOL Optimizer
… at the Core of Various Software Systems
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CREW-OPT
BUS-OPT
ALTITUDE-Pairings
ALTITUDE-Rosters
ALTITUDE-PBS
RAIL-WAYS
60 installations
around the world
RESEARCH TRENDS
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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
Early & Multiple
Branching & Cutting:
quickly gets local optima
Partial Pricing in case of
many Sub-Problems
Branching & Cutting:
on integer variables !
• Primal - Dual Stabilization
min cx
min cx
max b
max b
Ax  y1  y2  b
Ax  b
x0
x0
A  c
A  c
d1    d 2
0  y1   1 ,0  y2   2
Restricted Dual
Perturbed Primal
min cx  d1 y1  d 2 y 2
Ax  y1  y 2  b
x0
0  y1   1 ,
0  y2   2
Stabilized Primal
Primal - Dual Stabilization ...
Dual Solution
Primal Solution
Primal Solution
Dual Solution
Approximate Primal & Dual
Primal & Dual Solutions
• Constraint Aggregation
Massive Degeneracy
on Set Partitioning Problems
A pilot covers consecutive flights on the same aircraft
A driver covers consecutive legs on the same bus line
Aggregate Identical Constraints
on Non-zero Variables
Aggregation Algorithm
• Initial Constraint Aggregation
• Consider only Compatible Variables
Solve Aggregated Master Problem
Primal & Aggregated Dual Solutions
Dual Variables Split-up
Solve Sub-Problem
• Modify Constraint Aggregation
• Sub-Problem Speed-up
Resource Constrained Shortest Path
Labels at each node : cost, time, load, …
Resource Projection R 
 R
n
A
m
Adjust A dynamically
Generalized Lagrangian Relaxation
Results on R  R
4
2
Sub-Problem cpu time divided by 5 to 10
mn
• Two-Level Problems
Benders Decomposition Algorithm
for Simultaneous Assignment of
Buses and Drivers
Aircraft and Pilots
Pairings and Rosters
Locomotives and Cars
IP(X, Y) for Two-Level Scheduling
B&B(Y) with MIP(X, y) at each node
MIP(X, y) solved using Benders Decomposition
Master
Sub-Problem
IP(X)
Simplex and B&B(X)
solved by Column Generation
MP LP(y) of Set Partitioning
SP DP for Constrained Paths
Benders
MP
CG MP
LP
CG SP
DP
B&B
Benders
SP
IP
• Column Generation with
Interior Point Algorithm
• ACCPM Algorithm (Goffin & Vial)
• Applications
Linear Programming
Non-Linear Programming
Stochastic 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
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Larger Problems to Solve
Mixing of Decomposition Methods
Strong Exact and Heuristic Algorithms
Faster Computers
Parallel Implementations
Still a lot of work to do !!