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

2. 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

3. 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

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

5. 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

6. 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

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

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

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

10. PROBLEM STRUCTURE (CREW PAIRING: 1000 FLIGHTS)
SEPARABLE CREW COST FUNCTIONS
...
COVERING OF EACH OPERATIONAL FLIGHT
EXACTLY ONCE;
SET OF GLOBAL CONSTRAINTS;
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;

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

12. 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

13. 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

14. 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

15. 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

16. 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

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. 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 %
 10 MIN T.W %
 10 MIN T.W.
+ FLEET OPTIMIZATION %
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 HRS 4
OPTIMIZER
39 HRS 3
SAVINGS
20 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
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

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, ...
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 %

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
TASKS, NETWORKS
PATHS
GENCOL OPTIMIZER

26. 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

27. 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

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
BUS ROUTE
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 REOPTIMIZING A GOOD INITIAL SOLUTION
- AGGREGATES ↔ DRIVER ROUTES
- REOPTIMIZATION KEEP MANY SEQUENCES OF TASKS
 BUS
BUS ROUTE
RELIEF POINTS
DRIVERS

32. TASKS AGGREGATION … ….. MASTER PROBLEM AGGREGATED PROBLEM 1/2 0 =11/ =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

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

35. 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)

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

37. 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