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

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