1. Constructing Robust Crew Schedules with Bicriteria Optimisation
My name is David Ryan; I am a Professor of OR at the University of A.
In this presentation on “Optimised Crew Scheduling at Air NZ”, I will be joined by Rod Butchers, Crew Resource Manager responsible for the total crew scheduling process for pilots and flight attendants at Air NZ.
Our fellow authors are all graduates in OR from the Dept of Engineering Science at the U of A. I have been responsible for the supervision of their graduate research into apsects of this overall project. Stephen Miller, Amanda Scott and Chris Wallace are now all employees of AirNZ while Paul Day, Andrew Goldie and Jeff Meyer have formed a hi-tech consulting company called Optimal Decision Technologies Ltd and are consultants at Air NZ.
Matthias Ehrgott and David M. Ryan
Department of Engineering Science
University of Auckland

2. Outline of Presentation
Tour of Duty (Pairings) Optimisation
Measuring Robustness of Crew Schedules
Bicriteria Optimisation in Tour of Duty Planning
Method of Elastic Constraints
Conclusions
In this presentation I will first introduce Air NZ, NZ’s national airline and then describe the important characteristics of the aircrew scheduling problems which all airlines must solve.
After outlining the underlying mathematical model and solutions methods we have developed for these problems, I will describe in some detail features of the eight optimisation based systems that we have implemented to solve all aspects of AirNZs crew scheduling problems.
Rod Butchers will then discuss aspects of the implementation of the systems and their impact within the company.

3. The airline crew scheduling problems
Airline scheduling process
Schedules Planning
ToD
Planning
Rostering
Delivery
(On the Day)
today
+12 weeks
4-6 weeks
1-2 weeks
Airline crew scheduling involves two distinct phases:
Tour of Duty (ToD) planning
Involves the construction of ToDs (or pairings)
Rostering
Allocation of ToDs to individual crew members to create rosters

4. The ToD Planning process
A ToD definition:
Alternating sequences of duty periods and rest periods
Each duty period can contain one or more flight sectors
Duty periods may include passengering
Each ToD begins and ends at a crew base

5. Tour of Duty Planning Optimisation
Generalised Set Partitioning model
Variables - all legal ToDs
Constraints ensure coverage of flights
Base constraints with inequalities or non-unit right-hand-sides
Objective is to minimise total dollar costs
Solved via LP relaxation and Branch and Bound
Dynamic column generation to accommodate the very large number of possible variables
Constraint branching based on successive flight pairs to resolve fractions
To solve the ToD Planning problem, we use a GSP model which is a form of zero-one integer linear programme with a specially structured zero-one constraint matrix. The variables correspond to all legal ToDs and the constraints ensure that all flights are included in one ToD and that limits on the number of ToDs at each crew base are satisfied. The objective of the optimisation is to minimise total dollar costs.
We solve the GSPP using LP relaxation and Branch and Bound to resolve fractional solutions.
Because the model has a very large number of variables, we use column generation based on a resource constrained shortest path algorithm to dynamically generate variables as required during the LP optimisation.
We also use a novel branching scheme called constraint branching which we have developed to resolve fractional solutions that occur in the LP relaxation. The constraint branch used in ToD Planning is based on successive flight pairs. This branching scheme is now used in many other applications.
AirNZ’s first optimised ToD system was implemented almost 15 years ago in but nowdays this is probably the most commonly solved problem in aircrew scheduling.
In 1991, a previous Edelman finalist described an approach used by American Airlines to solve this aspect of crew scheduling.

6. Measuring Robustness of Tours of Duty
Delays propagate through the schedule
Data: 46,000 flights over 18 months of domestic operations
Result: Delays get longer during day, linear increase

7. Examples of Tours of Duty (Pairings)
Crew follows aircraft: most robust solution
Aircraft change: delays propagate through schedule, not robust

8. Non-Robustness Measures
A simple non-robustness measure for each flight in aTOD:
penalty = expected delay of flight
ground time not yet considered
Non-robustness measure for each ToD:
total accumulated penalty over consecutive flight pairs of a ToD
no penalty if the ToD follows the aircraft
Non-robustness measure for a ToD solution (a schedule)
total accumulated penalty over all ToDs

9. A better robustness masure
Resulting from discussions with Air NZ } -{
Penalty = ,0
Max[ ]
Ground time(between consecutive sectors on different aircraft)
- Expected delay of incoming sector
- Ground duty time (e.g. meal break)

10. Bicriteria Optimisation in Tour of Duty Planning
Minimise penalty for non-robust solutions, minimise cost
The model
min rtx
min ctx
s.t. A1x = e flight constraints
A2x = b base constraints
x {0,1}

11. Solution Approach Use cost objective as a constraint
Allow increase in cost (percent of optimal IP solution cost)
min rtx
s.t. ctx  (1+objch/100) CIP
A1x = e
A2x = b
x  {0,1}

12. Illustration (Solution of LP Relaxation)
Robustness
All ToD solutions
(LP relaxation)
objch2
objch1
Cost

13. Illustration (Solution of Integer Problem)
Robustness
objch2
All ToD solutions
(IP solution)
objch1
Cost

14. Solution of LP Relaxation - Trade Off Curve

15. * Best solution after 1000 nodes ($ constraint still active)
IP Solution
>20,000 sec*
* Best solution after 1000 nodes ($ constraint still active)
2,716 sec
88 sec

16. IP Solution - Making the $ Constraint Elastic
min rtx + Penalty *elastic-surplus
ctx - elastic-surplus  (1+objch/100)CIP;
ctx  (1+objch/100)CIP
min rtx
8,800.38
2,716.24
14.28
870.44
88.15
15.56
73.74
135.3
69.14
Penalty 100
Penalty 10
Penalty 0.01

17. The Method of Elastic Constraints
General Model
min rtx + p sp
s.t. ctx + sl - sp = 
A1x = e
A2x = b
x  {0,1}
Theorem:
If p > 0, optimal solution is Pareto optimal.
If x* Pareto optimal there are  and p* such that x* is optimal for all p  p*.

18. Example min (x1, x2) Parameters p,  for Pareto optimality
x1, x2  0, integer
Parameters p,  for
Pareto optimality
min x2 + p1sp1
x1 + sl1 - sp1 = 
x  X

19. Penalty Parameters (Boundgap of 2%)
Choice of Decision Maker
Robustness gain per dollar violation
“True” value = trade off = slope of LP curve
Decreasing value of additional gains

20. Structure of Solutions

21. Conclusions and Future Rresearch
Results indicate robustness can be gained by increasing cost
Considerable robustness gains by small increase in cost
System easy to use as cost increase can be specified by user
Improve cost AND robustness by combining crew scheduling and fleet assignment
Bicriteria rostering: max crew satisfaction and minimize fatigue