Canadian Mathematical Society Montral, December 11 -13, 1999 Jacques Desrosiers cole des Hautes tudes Commerciales & GERAD Montral, 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
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 CREW-OPT BUS-OPT ALTITUDE-Pairings
ALTITUDE-Rosters ALTITUDE-PBS RAIL-WAYS 60 installations around the world 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 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 Ax y1 y2 b x 0 0 y1 1 ,0 y2 2 Perturbed Primal
min cx Ax b x 0 max b A c max b A c d1 d 2 Restricted Dual min cx d1 y1 d 2 y 2
Ax y1 y 2 b 0 y1 1 , x 0 0 y 2 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, n A Resource Projection R R m
Adjust A dynamically Generalized Lagrangian Relaxation 4 Results on R R 2 Sub-Problem cpu time divided by 5 to 10 mn 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 Montral 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 !!
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