1. 1 A Second Stage Network Recourse Problem in Stochastic Airline Crew Scheduling Joyce W. Yen University of Michigan John R. Birge Northwestern University INFORMS San Antonio, TX Nov. 5, 2000

2. 2 Presentation Overview u Problem Motivation: Stochastic Airline Crew Scheduling u Stochastic Programming Formulation u Branching Algorithm u Large Problem Computational Results u Conclusion and Future Directions

3. 3 What is airline crew scheduling? u Begin with a set of flights that need to be flown by airline crews u Crews have restrictions for working hours u Want to generate trip itineraries and select a minimum cost set of trip itineraries for crews to fly that will cover every flight

4. 4 Why study airline crew scheduling? u Airlines want to increase profits u Crew costs second largest cost next to fuel costs –costs run in the billions of dollars for large carriers u Key to lowering crew costs –Optimizing crew schedules and crew assignments

5. 5 General Formulation u let a i,j = 1 if pairing j covers flight i u let x k = 1 if pairing k is selected to be in the solution

6. 6 Problems with Current Strategies u Crew schedules optimized only to be “deoptimized” in the actual implementation because of changes to the schedules u Problem solved as a linear program –LP’s optimized at extreme points t push solutions to brink of infeasibility u Unable to respond to changes –Companies spending millions to find ways to better react to schedule disruptions u Pairings considered individually

7. 7 Disruptions and Schedules u Different schedules will react differently to disruptions u Schedules can perpetuate the problem u Want to find a way to identify and select schedules that are less disrupted by changes to the flight schedule

8. 8 Stochastic Crew Scheduling u Integrate information about the short range (recovery) problem into the long range (traditional crew scheduling) problem u Objective: Include expected cost of disruptions in the model formulation u Capture interaction effects of planes, crews, and schedule recovery

9. 9 Problem Formulation For each disruption , we have different recourse. The overall problem becomes: where is the expected value of future actions due to disruptions  in the original schedule

10. 10  : delay scenario j : flight k : round trip itinerary Recourse Formulation subject to: consistency constraints for flight arrival and departure times consistency constraints for plane predecessor requirements consistency constraints for crew predecessor requirements time_dep(j) - [ time_arr(crew_pred(j,k))-crew_gnd(j,k)]x k,j 0 **  delay(j)

11. 11 General form of Recourse LP u Note cross product terms u This form, in general, is not convex in x u Traditional SIP solution methods don’t work

12. 12 General Idea for Algorithm u eliminate delays caused by crews changing planes u flight pair ranks based on switch delay value –higher value means more costly delay u branch on pairings with particular plane changes –identifies costly decisions and disallows or forces such decisions u Note: branching is on a large set of variables — not just a single one u Algorithm terminates with an optimal solution u Algorithm can be augmented onto existing SPP solvers

13. 13 Large Problem Results u Air New Zealand delay data from 1997 u Create model for disruptions based on data u 3 test problem –9 planes, 61 flights –10 planes, 66 flights –11 planes, 79 flights u Disruption time distribution: lognormal and gamma with maximum value 200 minutes

14. 14 Assumptions u State-of-the-Art pairing generator and set partitioning solver available u No cancellations u Crews still able to fly schedule when flight delayed –no time restriction problems u Plane ground time constant, crew ground time constant u Static plane assignments u Maximum delay time bounded

15. 15 10 Planes Results u 100 scenarios u Penalty = 100 u Daily schedule u No limit on number of crews scheduled u Run length: 184 iterations, 367 nodes created –computational time limit

16. 16 10th best

17. 17 Observations from Example u Quick savings early in the algorithm –small changes to c T x value –larger changes to the recourse value and objective value u Can get reasonably good solution after very few iterations –current optimal obtained after 214 nodes created (total nodes created = 367) –within 5% of best after 10 nodes created –within 2% of best after 26 nodes created –within 1% of best after 109 nodes created

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19. 19 Effect of Penalty Parameter u Penalty parameter reflection of how much disruption is worth u As increase penalty parameter, track the trade-off of disruption costs and crew costs u How to determine penalty value –penalty = capacity of plane –penalty = # passengers / # crew –penalty = cost / minute of disruption –etc.

20. 20 Penalty = 0 Penalty = 1000

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22. 22 General Comments u How much value disruptions reflected in penalty value –trade-off between disruption cost and crew cost u Can stop at any time and use algorithm to solve for desired degree of accuracy or efficiency –gap between LB and UB small enough –computational budget exhausted u Can add a heuristic to pick a solution that has a value within those bounds u Reach point where impossible to eliminate more disruption costs

23. 23 Challenges u Algorithm –subproblems may be slow t However, can find good solution quickly –bounds may not be tight t higher the penalty the less effective bounds are u Data –Need for lots of data –Must carefully evaluate relationships and correlations –problem size

24. 24 Future Research u Direct column generation for pairings using delay costs to avoid costly plane changes –difficulty is delays depend on all the crew assignments jointly –traditional column generation does one trip at a time –improvement:generate collection of columns that work well together

25. 25 Future Research u Investigate alternative data models u Explore alternative recourse models u Apply method to general network design problems –telecommunications networks –supply chain –job shops