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CREW SCHEDULING Past and Future Jacques Desrosiers HEC & GERAD Montréal, Canada

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Voir la présentation Power Point Crew Sched - Complements.ppt pour deux autres transparents

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The Mathematics behind Vehicle Routing and Crew Scheduling Jacques Desrosiers HEC & GERAD Montréal, Canada Canadian Mathematical Society Montréal, December 11 -13, 1999

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OUTLINE Problem Structure Solution Approaches Successful Applications Research Trends Conclusions

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PROBLEM STRUCTURE Time-Space Networks Local Schedule Restrictions Task Covering Schedule Composition Non Linear Cost Functions

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Time-Space Networks

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SOLUTION APPROACHES Branch & Cut involving... Lagrangean Relaxation Dantzig-Wolfe Decomposition Kelley’s Cutting Plane Method Column Generation

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Resource Constrained Shortest Path Problem on G=(N,A) P(N, A) :

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Integer Multi-Commodity Network Flow Structure

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Crew Scheduling Problems Sub-Problem is Strongly NP- hard Does not possess the Integrality Property Master Problem : Set Partitioning/Covering

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Branching & Cutting Decisions Branch-and-Cut Decomposition Process applied at all decision nodes

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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

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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

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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

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Acceleration Techniques Column Generator Master Problem Global Formulation Heuristics Re-Optimizers Pre-Processors …to obtain Primal & Dual Solutions

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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 !

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Primal - Dual Stabilization Restricted Dual Perturbed Primal Stabilized Primal

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Dual SolutionPrimal Solution Primal SolutionDual Solution Approximate Primal & Dual Primal & Dual Solutions Primal - Dual Stabilization...

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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

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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

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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

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Two-Level Problems Benders Decomposition Algorithm for Simultaneous Assignment of Buses and Drivers Aircraft and Pilots Pairings and Rosters Locomotives and Cars

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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

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Benders MP Benders SP B & B IP LP DP CG MP CG SP

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Column Generation with Interior Point Algorithm ACCPM Algorithm (Goffin & Vial) Applications Linear Programming Non-Linear Programming Stochastic Programming Variational Inequalities

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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

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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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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

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