Personnel and Vehicle Scheduling History and Future Trends 25th Anniversary of GERAD May 13, 2005 GERAD

Summary History A GENERIC PROBLEM WITH MANY APPLICATION Difficult to solve and large market MATHEMATIC FORMULATION Complex constraints and huge size DANTZIG-WOLFE REFORMULATION To eliminate complex constraints Column GENERATION To reduce member of variables HEURISTIC ACCELERATIONS RESULTS: AIR, BUS, RAU Transportation COMMERCIAL PRODUCTS

On Going Research ANALYTIC CENTER AND STABILIZATION Reduce number of column generation iterations OBTAIN INTEGER SOLUTIONS FASTER TASK AGGREGATION Reduce number of constraints REPLACE SEQUENTIAL PLANNING BY INTEGRATED OPTIMIZATION

GENERIC PROBLEM  TASK TASK COMMODITY COVER AT MINIMUM COST A SET OF TASKS WITH FEASIBLE PATHS

EXAMPLE BUS DRIVER SCHEDULING RELIEF POINT BUS ROUTE TASK TIME WORK SHIFT CONSTRAINTS MAX 8 HOURS MIN 6 HOURS 1 HOUR LUNCH TIME …… GLOBAL CONSTRAINTS 80% OF SHIFTS ≥ 7 HOURS

EXAMPLE BUS DRIVER SCHEDULING RELIEF POINT BUS ROUTE TASK TIME SHIFT WORK SHIFT CONSTRAINTS MAX 8 HOURS MIN 6 HOURS 1 HOUR LUNCH TIME ……… GLOBAL CONSTRAINTS 80% OF SHIFTS ≥ 7 HOURS

URBAN BUS MANAGEMENT SCHEDULING DIVIDED IN 3 STEPS 1 2 3 ... TRIPS 1 2 3 ... 1 7:00 7:30 7:40 2 7:05 7:35 7:45 . TRIPS TRIP STATIONS BUS ROUTE GARAGE  TRIP  TRIP  ... GARAGE ? RELIEF POINT DRIVER SHIFT ROUTE 1 ROUTE 2 DAYS 1 2 3 4 ... 31 1 ─ ─ ─ ─ 2 ─ ─ ─ ─ . ROSTERING DRIVERS SHIFT DAY-OFF

AIR SCHEDULING PROCESS PLANNING FLIGHT MTL  TOR 7:00 8:00 8:00 9:00 FLIGHT AIRCRAFT A 320 DC-9 FLIGHT REST PERIOD CREW PAIRING BASE DUTY ... DUTY DUTY 1 2 3 4 5 ... 31 1 2 . DAYS CREW ROSTERING CREW MEMBERS  DAY-OFF PAIRING

AIR SCHEDULING PROCESS OPERATION REPAIR AIRCRAFT AIRCRAFT ROUTES PERSONALIZED PAIRINGS AND BLOCKS CREW

PROBLEM STRUCTURE (CREW PAIRING: 1000 FLIGHTS) SEPARABLE CREW COST FUNCTIONS ... COVERING OF EACH OPERATIONAL FLIGHT EXACTLY ONCE; 1000 SET OF GLOBAL CONSTRAINTS; 10 PATH STRUCTURE FOR EACH CREW; 30 COMMODITIES NETWORK WITH 50,000 NODES, 100,000 ARCS { 100,000 ARCS x 20 RESOURCES ... LOCAL FLOW AND RESOURCE COMPATIBILITIES; 100,000 ARCS ... BINARY FLOWS;

{ REFORMULATION = 1 TASKS PATH ADVANTAGES - SIMPLER CONSTRAINTS - FEW CONSTRAINTS DIFFICULTY - MILLIONS OF MILLIONS OF VARIABLES

COLUMN GENERATION BASE UNKNOWN COLUMNS = 1 NEW COLUMNS REDUCED PROBLEM DUAL VARIABLES SUB-PROBLEM REDUCED COST 1- SOLVE THE REDUCED PROBLEM 2- GENERATE NEW COLUMNS BY SOLVING THE SUB-PROBLEM (MINIMIZING REDUCED COST) REDUCED COST = 0 ADD NEW COLUMNS NO YES OPTIMAL

SUB-PROBLEMS ∑ MAX ( , ∑ MAX (4, WORK TIME)) – DUAL COST SHORTEST PATH WITH CONSTRAINTS MIN REDUCED COST MIN S.T. - PATH - DAY DURATION ≤ 12 HOURS - WORK TIME / DAY ≤ 8 HOURS - WORK TIME / PAIRING ≤ MAX - NIGHT REST ≥ MIN - ... ∑ MAX ( , ∑ MAX (4, WORK TIME)) – DUAL COST PAIRING DURATION 3.5 PAIRING DAY 10 TO 20 CONSTRAINTS

GENCOL FEATURES COVER TASKS  1, =1,  bi GLOBAL CONSTRAINTS FLEET / CREW COMPOSITION SUB-PROBLEMS MULTIPLE VEHICLE / CREW TYPES MULTIPLE DEPOTS / BASES PATH STRUCTURE INITIAL / FINAL CONDITIONS CYCLIC SOLUTION PATH FEASIBILITY TIME WINDOW MAX RESOURCE UTILIZATION LINEAR, NONLINEAR, NONCONVEX CONSTRAINTS COLLECTIVE AGREEMENT

ADVANTAGES OF COLUMN GENERATION PROBLEM MIN CX AX ≤ a BX ≤ b X INTEGER ADVANTAGES - SOLVE SUB-PROBLEM AT INTEGRALITY - REDUCE INTEGRALITY GAP - EASIER BRANCH AND BOUND COST FUNCTION COL. GEN. SOLUTION OPT SOL. P. L. SOLUTION INTEGER SOLUTIONS

EXAMPLES TASK PATH BUS BUS ROUTING BUS TRIP ROUTE DRIVER SCHEDULING TRIP SEGMENT SHIFT ROSTERING ROSTER AIRLINE AIRCRAFT ROUTING FLIGHT CREW PAIRING PAIRING RAIL LOCO. ROUTING TRAIN PRODUCTION JOB-SHOP OPERATION SEQUENCE ON A MACHINE

SUBWAY DRIVERS TOKYO PROJECT: CNRC – GIRO – GERAD 2000 – 3000 TASKS 1 OR 2 DAYS SHIFTS COMPLEX COLLECTIVE AGREEMENT RESULTS SAVINGS ≈ 15% CONTRACT > US $1,500,000 CUSTOMERS: TOKYO, SINGAPOUR, NEW YORK, CHICAGO, ...

DAILY FLEET ASSIGNMENT AND AIRCRAFT ROUTING (Management Science 1997) AIR CANADA 91 AIRCRAFTS, 9 TYPES, 33 STATIONS FLEET REDUCTION WITH TIME WINDOWS ON FLIGHT SCHEDULE AIR FRANCE 51 AIRCRAFTS, 6 TYPES, 44 STATIONS PROFIT IMPROVEMENT BASIC PROBLEM 6.5 %  10 MIN T.W. 11.2 %  10 MIN T.W. + FLEET OPTIMIZATION 21.9 % T.W. REDUCTION 10 MIN 3.8 % 20 MIN 8.9 % 30 MIN 13.9 %

WEEKLY FLEET ASSIGNMENT AND AIRCRAFT ROUTING AIR CANADA 5000 FLIGHTS 1 WEEK CYCLIC 10 ARICRAFT TYPE COMPLEX CONNECTION TIME AND COST (PER CITY, PER AIRCRAFT TYPE, PAIR OF TERMINALS) MAX PROFIT AND HOMOGENITY CPU TIME: 1 HOUR (400 Mhz)

AIRCRAFT ROUTING AND SCHEDULING CANADIAN ARMY (C-130) WEST CHALLENGE 750 SOLDIERS AND EQUIPMENT 19 CITY-PAIRS MAX 65 SOLDIERS PER FLIGHT SAVINGS FLIGHT TIME NUMBER OF AIRCRAFT MANUAL SOL. 59 HRS 4 OPTIMIZER 39 HRS 3 SAVINGS 20 HRS (34 %) 1 (33 %)

CREW PAIRING AIR CANADA FLIGHT – ATTENDANT A 320 + DC-9 MONTHLY PROBLEM 12,000 FLIGHTS 5 BASES (MAX TIMES)

RESULTS FLIGHT ATTENDANTS DC-9 + A 320 FLIGHTS % FAT DAILY 430 .47 WEEKLY 2425 1.39 MONTHLY 11914 2.03 SAVINGS VS A.C. SOLUTION 7.8 %  2.03 % CUSTOMERS: TRANSAT, CAN. REGIONAL, NORTHWEST, U.P.S. DELTA, SABENA, SWISSAIR, FEDEX

CREW ROSTERING (OPERATION RESEARCH 1999) AIR FRANCE FLIGHT-ATTENDANT MONTHLY PROBLEM PROBLEM SIZE RESULTS CUSTOMERS: AIR CANADA, TRANSAT, CAN REGIONAL, TWA, DELTA, SWISSAIR, SABENA, AMERICA WEST, ... ORLY CDG PAIRINGS 454 X 7 3000 X 5 PERSONS 240 840 ORLY CDG CPU TIME 35 MIN 3 HRS SAVINGS 7.4 % 7.6 %

WEEKLY LOCOMOTIVE SCHEDULING (CANADIAN NATIONAL RAIL ROAD) 2500 TRAINS, 160 LOCAL SERVICES 26 TYPES OF LOCOMOTIVE POWER CONSTRAINTS  2 TO 4 LOCO/TRAIN 18 MAINTENANCE SHOPS COMPLEX CONNECTING TIME: ( CITY, EQUIPMENT, ORIENTATION, …) SAVING OF 100 LOCO. ON 1100 AND 10% OF TRAVEL DISTANCE CPU TIME: 30 MINUTES (400Mhz)

PRODUCTS ARCHITECTURE USER GRAPHICAL USER INTERFACE DATA BASE MODELING MODULE TASKS, NETWORKS PATHS GENCOL OPTIMIZER

FAMILY OF PRODUCTS +100 INSTALLATIONS GIRO AD OPT GENCOL CITY SCHOOL CIVIL and MILITAIRYS AIRCRAFT CREW BUS DRIVERS HANDICAPED PEOPLE CREW PAIRING CREW ROSTERING RAIL SHIFT SCHEDULING BUS AIRCRAFTS DAY-OFF GENCOL +100 INSTALLATIONS

On Going Research ANALYTIC CENTER AND STABILIZATION Reduce number of column generation iterations OBTAIN INTEGER SOLUTIONS FASTER TASK AGGREGATION Reduce number of constraints REPLACE SEQUENTIAL PLANNING BY INTEGRATED OPTIMIZATION

ANALYTIC CENTER METHOD (GOFFIN, VIAL) COLUMN GENERATION WITH INTERIOR POINT ALGORITHM FOR THE MASTER PROBLEM DO NOT SOLVE THE M.P. AT OBTIMALITY AT EACH ITERATION STAY IN THE INTERIOR OF THE DUAL DOMAIN EASY RESTART WHEN COLUMN ARE ADDED MORE STABLE AND LESS ITERATIONS BUT INCOMPATIBLE WITH SOME ACCELERATION TECHNICS OF COLUMN GENERATION STABILIZATION TECHNICS USE NON-LINEAR PIECE-WISE PENALITY ON DUAL VARIABLES MORE STABLE AND LESS ITERATIONS COMPATIBLE WITH CPLEX AND ACCELERATION TECHNICS

OBTAIN INTEGER SOLUTIONS FASTER VARIABLE FIXING IDENTIFY VAR. SMALLER THAN 1  FIX TO 0 AND REMOVE VAR. FROM THE PROBLEM IDENTIFY VAR. GREATER THAN 0  FIX TO 1 AND REMOVE TASK FROM THE PROBLEM CUTTING PLAN FACET COMPATIBLE WITH COLUMN GENERATION DEEP CUT IN SUB-PROBLEM NEW BRANCHING BRANCH ON MORE GLOBAL VARIABLES BRANCH MANY VARIABLES AT THE TIME (BRANCH BACK IF NECESSARY)  BRANCHING TREE LESS DEEP DEEP CUT NORMAL CUT

TASK AGGREGATION SOME TASKS WILL BE PROBABLY GROUPED IN THE SOLUTION EX. 1: CONSECUTIVE TASKS ON THE SAME BUS WILL BE PROBABLY ASSIGNED TO THE SAME DRIVER  BUS BUS ROUTE RELIEF POINTS DRIVERS

TASK AGGREGATION SOME TASKS WILL BE PROBABLY GROUPED IN THE SOLUTION EX. 1: CONSECUTIVE TASKS ON THE SAME BUS WILL BE PROBABLY ASSIGNED TO THE SAME DRIVER EX. 2 - REOPTIMIZING A GOOD INITIAL SOLUTION - AGGREGATES ↔ DRIVER ROUTES - REOPTIMIZATION KEEP MANY SEQUENCES OF TASKS  BUS BUS ROUTE RELIEF POINTS DRIVERS

TASKS AGGREGATION … ….. MASTER PROBLEM AGGREGATED PROBLEM 1/2 0 =1 1/2 0 =1 1100 1100 … ….. … ….. TASKS 0011 0011 1010 1010 BASE NON BASE INCOMPATIBLE COLUMN 1100 NON BASIC COMPATIBLE COLUMNS 0011 1010 FAST PIVOTS PIVOTS NEEDING DESAGGREGATION

TASK AGGREGATION AGGREGATION AND DESAGGREGATION TO REACH OPTIMALITY TAKE ADVANTAGE OF DEGENERACY TO REDUCE MASTER PROBLEM SIZE STRATEGIES TO CREATE MORE DEGENERACY LEES FRACTIONAL L.P. SOLUTION REDUCE SOLUTION TIME BY FACTORS OF 10 TO 20

INTEGRATED PLANNING PAIRING COVER FLIGHTS WITH PAIRING ROSTERING COVER PAIRING WITH ROSTERS INTEGRATED OPTIMIZATION COVER FLIGHTS WITH ROSTERS (10 TO 30 000 FLIGHTS / MONTH)

INTEGRATED PLANNING WITH AGGREGATION SOLVE PAIRING PROBLEM AGGREGATE FLIGHTS IN THE SAME PAIRING OPTIMIZE ROSTERS WITHOUT DESAGGREGATION  CLASSICAL ROSTERING PROBLEM REOPTIMIZE ROSTERS CHANGING AGGREGATION (REACH OPTIMAL SOLUTION BY SOLVING SMALL PROBLEMS)

CONCLUSION WE CAN SOLVE HUGE PROBLEMS MILLIONS OF MILLIONS OF VARIABLES 30 000 CONSTRAINTS

CONCLUSION WE CAN SOLVE HUGE PROBLEMS MILLIONS OF MILLIONS OF VARIABLES BASE 30 000 CONSTRAINTS SOLVING ONLY A KERNEL PROBLEM MANY TIMES REDUCE NUMBER OF VARIABLES WITH COLUMN GENERATION REDUCE NUMBER OF CONSTRAINTS WITH CONSTRAINT AGGREGATION THE KERNEL PROBLEM IS ADJUSTED DYNAMICALLY TO REACH OPTIMALITY