Personnel and Vehicle Scheduling

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Personnel and VehicleScheduling

• History and Future Trends
• 25th Anniversary of GERAD

May 13, 2005 GERAD
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Summary

• History
• A GENERIC PROBLEM WITH MANY APPLICATIONDifficult
  to solve and large market
• MATHEMATIC FORMULATIONComplex constraints and
  huge size
• DANTZIG-WOLFE REFORMULATIONTo eliminate complex
  constraints
• Column GENERATIONTo reduce member of variables
• HEURISTIC ACCELERATIONS
• RESULTS AIR, BUS, RAU Transportation
• COMMERCIAL PRODUCTS

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• On Going Research
• ANALYTIC CENTER AND STABILIZATIONReduce number
  of column generation iterations
• OBTAIN INTEGER SOLUTIONS FASTER
• TASK AGGREGATIONReduce number of constraints
• REPLACE SEQUENTIAL PLANNING BY INTEGRATED
  OPTIMIZATION

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GENERIC PROBLEM
TASK
TASK
?
COMMODITY
COVER AT MINIMUM COST A SET OF TASKS WITH
FEASIBLE PATHS
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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
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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
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URBAN BUS MANAGEMENT
SCHEDULING DIVIDED IN 3 STEPS
TRIPS
1 2 3 ... 1 700
730 740 2 705 735 745 . . .
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
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AIR SCHEDULING PROCESS
PLANNING
MTL ? TOR 700 800 800 900
FLIGHT
FLIGHT
AIRCRAFT
A 320
DC-9
FLIGHT
REST PERIOD
CREW PAIRING
BASE
DUTY
DUTY
...
DUTY
DAYS
1 2 3 4 5 ... 31 1 2 . .
CREW ROSTERING
CREW MEMBERS
? DAY-OFF
?PAIRING
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AIR SCHEDULING PROCESS
REPAIR
OPERATION
AIRCRAFT
AIRCRAFT ROUTES PERSONALIZED PAIRINGS AND
BLOCKS
CREW
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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
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REFORMULATION


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TASKS
PATH
ADVANTAGES - SIMPLER CONSTRAINTS
- FEW CONSTRAINTS DIFFICULTY - MILLIONS
OF MILLIONS OF VARIABLES
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COLUMN GENERATION
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BASE
UNKNOWN COLUMNS

NEW COLUMNS

REDUCED PROBLEM
SUB-PROBLEM
DUAL VARIABLES

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
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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
• - ...

? MAX (
, ? MAX (4, WORK TIME)) DUAL COST
PAIRING DURATION 3.5
DAY
PAIRING
10 TO 20 CONSTRAINTS
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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

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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
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EXAMPLES
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SUBWAY DRIVERSTOKYO

• PROJECT CNRC GIRO GERAD
• 2000 3000 TASKS
• 1 OR 2 DAYS SHIFTS
• COMPLEX COLLECTIVE AGREEMENT
• RESULTS
• SAVINGS 15
• CONTRACT gt US 1,500,000
• CUSTOMERS TOKYO, SINGAPOUR, NEW YORK, CHICAGO,
  ...

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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

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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)

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AIRCRAFT ROUTING AND SCHEDULINGCANADIAN ARMY
(C-130)

• WEST CHALLENGE
• 750 SOLDIERS AND EQUIPMENT
• 19 CITY-PAIRS
• MAX 65 SOLDIERS PER FLIGHT
• SAVINGS

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CREW PAIRINGAIR CANADA

• FLIGHT ATTENDANT
• A 320 DC-9
• MONTHLY PROBLEM
• 12,000 FLIGHTS
• 5 BASES (MAX TIMES)

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RESULTSFLIGHT ATTENDANTSDC-9 A 320
SAVINGS VS A.C. SOLUTION 7.8 ? 2.03
CUSTOMERS TRANSAT, CAN. REGIONAL, NORTHWEST,
U.P.S. DELTA, SABENA, SWISSAIR, FEDEX
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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, ...

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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)

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PRODUCTS ARCHITECTURE
USER
GRAPHICAL USER INTERFACE
DATA BASE
MODELING MODULE
TASKS, NETWORKS
PATHS
GENCOL OPTIMIZER
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FAMILY OF PRODUCTS
GIRO
AD OPT
SCHOOL
CITY
CIVIL and MILITAIRYS
AIRCRAFT CREW
SHIFT SCHEDULING
BUS DRIVERS
HANDICAPED PEOPLE
RAIL
CREW PAIRING
CREW ROSTERING
BUS
AIRCRAFTS
DAY-OFF
GENCOL
100 INSTALLATIONS
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• On Going Research
• ANALYTIC CENTER AND STABILIZATIONReduce number
  of column generation iterations
• OBTAIN INTEGER SOLUTIONS FASTER
• TASK AGGREGATIONReduce number of constraints
• REPLACE SEQUENTIAL PLANNING BY INTEGRATED
  OPTIMIZATION

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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 ADDEDMORE 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 COMPATIB
LE WITH CPLEX AND ACCELERATION TECHNICS
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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
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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
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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
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TASKS AGGREGATION

• MASTER PROBLEM
• AGGREGATED PROBLEM

1/2
0
1 1 1 1 1 1
1100 1100
1100 1100
..
..
0011 0011
0011 0011
TASKS
1010 1010
1010 1010
BASE NON BASE
INCOMPATIBLE COLUMN
1100
NON BASIC COMPATIBLE COLUMNS
0011
1010
FAST PIVOTS PIVOTS NEEDING
DESAGGREGATION
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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

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

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