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Canadian Mathematical Society Montréal, December 11 -13, 1999

Canadian Mathematical Society Montréal, December 11 -13, 1999. 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)

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Canadian Mathematical Society Montréal, December 11 -13, 1999

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  1. Canadian Mathematical SocietyMontréal, December 11 -13, 1999

  2. 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 SUCCESSFUL APPLICATIONS

  3. CREW-OPT BUS-OPT ALTITUDE-Pairings ALTITUDE-Rosters ALTITUDE-PBS RAIL-WAYS The GENCOL Optimizer … at the Core of Various Software Systems 60 installations around the world

  4. 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 RESEARCH TRENDS

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

  6. 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 integervariables ! Acceleration Techniques ...

  7. Primal - Dual Stabilization Restricted Dual Perturbed Primal Stabilized Primal

  8. Primal - Dual Stabilization ... Dual Solution Primal Solution Primal Solution Dual Solution Approximate Primal & Dual Primal & DualSolutions

  9. 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

  10. 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

  11. 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

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

  13. 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 SPDP for Constrained Paths B&B(Y) with MIP(X, y) at each node

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

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

  16. 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

  17. 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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