Crew Scheduling Problems Sub-Problem is Strongly NP- hard Does not possess the Integrality Property Master Problem : Set Partitioning/Covering
Branching & Cutting Decisions Branch-and-Cut Decomposition Process applied at all decision nodes
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 & Multiple Branching & Cutting: quickly gets local optima Branching & Cutting: on integer variables !
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
Sub-Problem Speed-up Resource Constrained Shortest Path Labels at each node : cost, time, load, … Resource Projection Adjust A dynamically Generalized Lagrangian Relaxation Results on Sub-Problem cpu time divided by 5 to 10
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 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 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 Larger Problems to Solve Mixing of Decomposition Methods Strong Exact and Heuristic Algorithms Faster Computers Parallel Implementations Still a lot of work to do !!
Some Collaborators Jean-Louis Goffin Pierre Hansen Gilles Savard Marius Solomon François Soumis Jean-Philippe Vial Jean-François Cordeau Michel Denault Guy Desaulniers Daniel Villeneuve