Post on 26-Mar-2015
Personnel and VehicleScheduling
History and Future Trends
25th Anniversary of GERAD
May 13, 2005 GERAD
SummaryHistory
1. A GENERIC PROBLEM WITH MANY APPLICATIONDifficult to solve and large market
2. MATHEMATIC FORMULATIONComplex constraints and huge size
3. DANTZIG-WOLFE REFORMULATIONTo eliminate complex constraints
4. Column GENERATIONTo reduce member of variables
5. HEURISTIC ACCELERATIONS
6. RESULTS: AIR, BUS, RAU Transportation
7. COMMERCIAL PRODUCTS
On Going Research
8. ANALYTIC CENTER AND STABILIZATIONReduce number of column generation iterations
9. OBTAIN INTEGER SOLUTIONS FASTER
10. TASK AGGREGATIONReduce number of constraints
11. REPLACE SEQUENTIAL PLANNING BY INTEGRATED OPTIMIZATION
GENERIC PROBLEM
COMMODITY
TASKTASK
COVER AT MINIMUM COST A SET OF TASKS WITH FEASIBLE PATHS
EXAMPLEBUS DRIVER SCHEDULING
WORK SHIFT CONSTRAINTS MAX 8 HOURS MIN 6 HOURS1 HOUR LUNCH TIME
……
GLOBAL CONSTRAINTS80% OF SHIFTS ≥ 7 HOURS
TASK
BUS ROUTE RELIEF POINT
TIME
EXAMPLEBUS DRIVER SCHEDULING
WORK SHIFT CONSTRAINTS MAX 8 HOURS MIN 6 HOURS1 HOUR LUNCH TIME
………
GLOBAL CONSTRAINTS80% OF SHIFTS ≥ 7 HOURS
TASK
BUS ROUTE RELIEF POINT
TIME
SHIFT
1 2 3 4 ... 31 1 ─ ─ ─ ─ 2 ─ ─ ─ ─ ...
TRIP
BUSROUTE
ROSTERING
DRIVERSHIFT
STATIONS
GARAGE
GARAGE ?
DRIVERS
TRIP TRIP ...
TRIPS
RELIEF POINT
ROUTE 1
DAYS
DAY-OFF SHIFT
ROUTE 2
1 2 3 ...1 7:00 7:30 7:402 7:05 7:35 7:45...
URBAN BUS MANAGEMENTSCHEDULING DIVIDED IN 3 STEPS
AIR SCHEDULING PROCESS
1 2 3 4 5 ... 31 1 2 ..
FLIGHT
AIRCRAFT
CREWROSTERING
CREWPAIRING
PLANNING
A 320
DC-9
CREWMEMBERS
BASE
FLIGHT
DAYS
DAY-OFF PAIRING
MTL TOR7:00 8:008:00 9:00
...DUTY
DUTY
REST PERIODDUTY
FLIGHT
AIRCRAFT
CREW
OPERATION REPAIR
AIRCRAFT ROUTES
PERSONALIZED PAIRINGS AND BLOCKS
AIR SCHEDULING PROCESS
COVERING OF EACH OPERATIONAL FLIGHTEXACTLY ONCE; 1000
SET OF GLOBAL CONSTRAINTS; 10
100,000 ARCS x 20 RESOURCES
PROBLEM STRUCTURE(CREW PAIRING: 1000 FLIGHTS)
SEPARABLE CREW COST FUNCTIONS
...
PATH STRUCTUREFOR EACH CREW;
LOCAL FLOW ANDRESOURCE COMPATIBILITIES;
NETWORK WITH50,000 NODES,100,000 ARCS{
...
...100,000 ARCS
BINARY FLOWS;
30 COMMODITIES
REFORMULATION
ADVANTAGES
- SIMPLER CONSTRAINTS
- FEW CONSTRAINTS
DIFFICULTY
- MILLIONS OF MILLIONS OF VARIABLES
= 1 TASKS
PATH
{
COLUMN GENERATION
= 1
BASE UNKNOWN COLUMNS
REDUCEDPROBLEM
SUB-PROBLEM
REDUCED COST
NEW COLUMNS
DUALVARIABLES
REDUCEDCOST = 0
OPTIMALADD NEWCOLUMNS
NO YES
1- SOLVE THE REDUCED PROBLEM2- GENERATE NEW COLUMNS BY SOLVING THE SUB-PROBLEM
(MINIMIZING REDUCED COST)
SUB-PROBLEMSSHORTEST 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 DURATION3.5
∑ MAX ( , ∑ MAX (4, WORK TIME)) – DUAL COST
PAIRING DAY
10 TO 20CONSTRAINTS
GENCOL FEATURESCOVER TASKS
1, =1, bi GLOBAL CONSTRAINTS
– FLEET / CREW COMPOSITIONSUB-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
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
INTEGERSOLUTIONS
EXAMPLESTASK PATH
BUS
BUS ROUTING BUS TRIP ROUTE
DRIVER SCHEDULING TRIP SEGMENT SHIFT
ROSTERING SHIFT ROSTER
AIRLINE
AIRCRAFT ROUTING FLIGHT ROUTE
CREW PAIRING FLIGHT PAIRING
ROSTERING PAIRING ROSTER
RAIL
LOCO. ROUTING TRAIN ROUTE
PRODUCTION
JOB-SHOP OPERATION SEQUENCE ON A MACHINE
SUBWAY DRIVERSTOKYO
• 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, ...
AIR CANADA91 AIRCRAFTS, 9 TYPES, 33 STATIONS
• FLEET REDUCTION WITH TIME WINDOWS ON FLIGHT SCHEDULE
AIR FRANCE51 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 %
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 PAIRINGAIR CANADA
• FLIGHT – ATTENDANT
• A 320 + DC-9
• MONTHLY PROBLEM
• 12,000 FLIGHTS
• 5 BASES (MAX TIMES)
RESULTSFLIGHT ATTENDANTS
DC-9 + A 320
FLIGHTS % FAT
DAILY 430 .47
WEEKLY 2425 1.39
MONTHLY 11914 2.03
SAVINGS VS A.C. SOLUTION7.8 % 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 ARCHITECTUREUSER
GRAPHICAL USER INTERFACE
DATA BASE
MODELING MODULE
GENCOL OPTIMIZER
TASKS, NETWORKS PATHS
FAMILY OF PRODUCTS
SCHOOLCITY
BU
S
DR
IVE
RS
HA
ND
ICA
PE
D
PE
OP
LE
RA
IL
CR
EW
R
OS
TE
RIN
G
CR
EW
P
AIR
ING
BUS AIRCRAFTS
CIVIL and
MILITAIRYS
DAY-OFF
AIRCRAFT CREW
GIRO AD OPT
GENCOL
+100 INSTALLATIONS
SH
IFT
SC
HE
DU
LIN
G
On Going Research
8. ANALYTIC CENTER AND STABILIZATIONReduce number of column generation iterations
9. OBTAIN INTEGER SOLUTIONS FASTER
10. TASK AGGREGATIONReduce number of constraints
11. 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 FASTERVARIABLE 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 ROUTE BUSRELIEF 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 ROUTE BUSRELIEF POINTS
DRIVERS
FAST PIVOTS
PIVOTS NEEDING DESAGGREGATION
TASKS AGGREGATION• MASTER PROBLEM
• AGGREGATED PROBLEM
110011000011001110101010
… …..
1/2 0
=1=1=1=1=1=1
TASKS
BASE NON BASE
110011000011001110101010
… …..
110000111010
NON BASIC COMPATIBLE COLUMNS
INCOMPATIBLECOLUMN
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
PAIRING
ROSTERING
INTEGRATEDOPTIMIZATION
COVER FLIGHTS WITH PAIRING
COVER PAIRING WITH ROSTERS
INTEGRATED PLANNING
COVER FLIGHTS WITH ROSTERS(10 TO 30 000 FLIGHTS / MONTH)
• 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
WE CAN SOLVE HUGE PROBLEMS
CONCLUSION
MILLIONS OF MILLIONS OF VARIABLES
30 000CONSTRAINTS
WE CAN SOLVE HUGE PROBLEMS
CONCLUSION
MILLIONS OF MILLIONS OF VARIABLES
30 000CONSTRAINTS
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