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

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.

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

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

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 1986 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.

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

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

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

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)

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}

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}

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

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

Solution of LP Relaxation - Trade Off Curve

* 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

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

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*.

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

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

Structure of Solutions

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