Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Similar presentations


Presentation on theme: "Personnel and Vehicle Scheduling History and Future Trends 25 th Anniversary of GERAD May 13, 2005GERAD."— Presentation transcript:

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

2 Summary History 1.A GENERIC PROBLEM WITH MANY APPLICATION Difficult to solve and large market 2.MATHEMATIC FORMULATION Complex constraints and huge size 3.DANTZIG-WOLFE REFORMULATION To eliminate complex constraints 4.Column GENERATION To reduce member of variables 5.HEURISTIC ACCELERATIONS 6.RESULTS: AIR, BUS, RAU Transportation 7.COMMERCIAL PRODUCTS

3 On Going Research 8.ANALYTIC CENTER AND STABILIZATION Reduce number of column generation iterations 9.OBTAIN INTEGER SOLUTIONS FASTER 10.TASK AGGREGATION Reduce number of constraints 11.REPLACE SEQUENTIAL PLANNING BY INTEGRATED OPTIMIZATION

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

5 EXAMPLE BUS DRIVER SCHEDULING WORK SHIFT CONSTRAINTS MAX8 HOURS MIN6 HOURS 1 HOURLUNCH TIME …… GLOBAL CONSTRAINTS 80% OF SHIFTS 7 HOURS TASK BUS ROUTE RELIEF POINT TIME

6 EXAMPLE BUS DRIVER SCHEDULING WORK SHIFT CONSTRAINTS MAX8 HOURS MIN6 HOURS 1 HOURLUNCH TIME ……… GLOBAL CONSTRAINTS 80% OF SHIFTS 7 HOURS TASK BUS ROUTE RELIEF POINT TIME SHIFT

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

8 AIR SCHEDULING PROCESS FLIGHT AIRCRAFT CREW ROSTERING CREW PAIRING PLANNING A 320 DC-9 CREW MEMBERS BASE FLIGHT DAYS DAY-OFF PAIRING MTL TOR 7:00 8:00 8:00 9:00... DUTY REST PERIOD DUTY FLIGHT

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

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

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

12 COLUMN GENERATION = 1 BASE UNKNOWN COLUMNS REDUCED PROBLEM SUB-PROBLEM REDUCED COST NEW COLUMNS DUAL VARIABLES REDUCED COST = 0 OPTIMAL ADD NEW COLUMNS NOYES 1- SOLVE THE REDUCED PROBLEM 2- GENERATE NEW COLUMNS BY SOLVING THE SUB-PROBLEM (MINIMIZING REDUCED COST)

13 SUB-PROBLEMS 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 -... PAIRING DURATION 3.5 MAX (, MAX (4, WORK TIME) ) – DUAL COST PAIRING DAY 10 TO 20 CONSTRAINTS

14 GENCOL FEATURES COVER TASKS – 1, =1, b i 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

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

16 EXAMPLES TASKPATH BUS BUS ROUTINGBUS TRIPROUTE DRIVER SCHEDULINGTRIP SEGMENTSHIFT ROSTERINGSHIFTROSTER AIRLINE AIRCRAFT ROUTINGFLIGHTROUTE CREW PAIRINGFLIGHTPAIRING ROSTERINGPAIRINGROSTER RAIL LOCO. ROUTINGTRAINROUTE PRODUCTION JOB-SHOPOPERATIONSEQUENCE ON A MACHINE

17 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,...

18 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 % DAILY FLEET ASSIGNMENT AND AIRCRAFT ROUTING (Management Science 1997) T.W. REDUCTION 10 MIN 3.8 % 20 MIN 8.9 % 30 MIN 13.9 %

19 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)

20 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 HRS4 OPTIMIZER39 HRS3 SAVINGS20 HRS (34 %)1 (33 %)

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

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

23 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,... ORLYCDG PAIRINGS454 X X 5 PERSONS ORLYCDG CPU TIME35 MIN3 HRS SAVINGS7.4 %7.6 %

24 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)

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

26 FAMILY OF PRODUCTS SCHOOLCITY BUS DRIVERS HANDICAPED PEOPLE RAIL CREW ROSTERING CREW PAIRING BUSAIRCRAFTS CIVIL and MILITAIRYS DAY-OFF AIRCRAFT CREW GIROAD OPT GENCOL +100 INSTALLATIONS SHIFT SCHEDULING

27 On Going Research 8.ANALYTIC CENTER AND STABILIZATION Reduce number of column generation iterations 9.OBTAIN INTEGER SOLUTIONS FASTER 10.TASK AGGREGATION Reduce number of constraints 11.REPLACE SEQUENTIAL PLANNING BY INTEGRATED OPTIMIZATION

28 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

29 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

30 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 ROUTE BUS RELIEF POINTS DRIVERS

31 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 ROUTE BUS RELIEF POINTS DRIVERS

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

33 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

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

35 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) INTEGRATED PLANNING WITH AGGREGATION

36 WE CAN SOLVE HUGE PROBLEMS CONCLUSION MILLIONS OF MILLIONS OF VARIABLES CONSTRAINTS

37 WE CAN SOLVE HUGE PROBLEMS CONCLUSION MILLIONS OF MILLIONS OF VARIABLES CONSTRAINTS BASE 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


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

Similar presentations


Ads by Google