1. A S CENARIO A GGREGATION –B ASED A PPROACH FOR D ETERMINING A R OBUST A IRLINE F LEET C OMPOSITION FOR D YNAMIC C APACITY A LLOCATION Ovidiu Listes, Rommert Dekker

2. A GENDA  Introduction  Literature Review  Fleet Composition Problem  Model  Deterministic Model  Stochastic Model  Scenario Aggregation Algorithm  Scenario Generation  Case Study  Conclusion

3. 1.I NTRODUCTION Random demand fluctuations lead to -low average load factors -a significant number of not accepted passengers Dynamic allocation of airline fleet capacity: Using most recent estimates of customers demands for accordingly updating the assignments of aircrafts to the flight schedule

4. Fleet Assignment Fleet Composition This paper focuses on creating an approach to the airline fleet composition problem that accounts explicitly for stochastic demand fluctuations

5. 2. LITERATURE REVIEW Berge&Hopperstad(1993) Hane et al.(1995) Talluri(1996) Gu et al.(1994)

6. 3. T HE F LEET -C OMPOSITION P ROBLEM Complex, upper-management decides on it. Paper adresses problem from OR perspective. Model it in relation to the basic fleet assignment. Demand is assumed to follow independent normal distribution, variability specified as the K-factor (sd/mean).

7. Each aircraft has -Fixed cost -Operational cost -Capacity for each fair class -Range capability -Family indicator o Assumptions: -Identical flying&turn around time -No recapture -Minimum number of aircrafts required is taken into account

8. 4.M ODEL Fleet composition problem can be considered as a multicommodity flow problem based on the construction of a space-time network

9. 4.1. D ETERMINISTIC MODEL NP-hard for more than three aircraft types

10. 4.2. S TOCHASTIC MODEL S representative scenarios andsolution for individual demand scenarios is same for every scenario hence,for every scenario s.

11. Because of huge number of integer second-stage variables a branch-and-bound type of procedure is not practical. For small examples: LP relaxation of SP denoted by LSP includes many integer-valued decision variables. LP relaxation gap turns out to be less than 0.5% in these cases.

12. 4.3.1 T HE S CENARIO A GGREGATION – B ASED A PPROACH Scenario aggregation is a decomposition-type of method. Main Idea: Iteratively solving individual scenario problems, perturbed in a certain sense, and to aggregate, at each iteration, these individual solutions into an overall implementable solution

13. 4.3.2. T HE S CENARIO A GGREGATION A LGORITHM Admissible solution: Feasible for each scenario s. z variables indexed over scenario s then additional constraint: : solution from previous iteration This constraint is relaxed in the Lagrangian sense using multipliers ws.

14. T HE S CENARIO A GGREGATION A LGORITHM

15. is an implementable solution not necessarily admissible w is interpreted as information prices Stopping Criteria: Variance error wrt z variables is used Stop when: Criteria Selection: -Low ρ values encourage progress in primal sequence -ε is set to 3% of minimum total number of planes

16. R OUNDING PROCEDURE fractional first stage solution with For any given fractional solution u [u] denotes integer part of u and {u} denotes fractional part of u A constant c is selected between 0 and 0.5 Rounding Procedure:

17. 4.4 S CENARIO G ENERATION Demand assumed to follow a normal distribution: Descriptive Sampling: A purposive selection of the sample values— aiming to achieve a close fit with the represented distribution—and the random permutations of these values

19. 4.5 F LEET P ERFORMANCE E VALUATION New simulated demands from demand distribution is used, size 3 to 4 times greater than number of scenarios used. Generic Fleet Flexibility Fleet Interchangibility

20. 5. C ASE S TUDY Small case validates method, Large case shows extend Nine aircraft types 40% business, 60% economy seats Small case: Large Case: -342 flight legs -18 airports -15 planes -50 scenarios -Mean Demand : 14-65 for economy class 26-48 for business class -1978 flight legs -50 airports -68 planes -25 scenarios -Mean Demand : 18-57 for economy class 21-43 for business class

21. G ENERIC F LEXIBILITY -S MALL C ASE

22. F LEET I NTERCHANGIBILITY -S MALL C ASE

23. G ENERIC F LEXIBILITY -L ARGE C ASE

24. F LEET I NTERCHANGIBILITY -L ARGE C ASE

25. 6.C ONCLUSION Increase in load factor up to 2.6% Decrease in spill up to 3.3%. Profit increase up to 14.5%. Finally, The scenario-aggregation based approach handles effects of fluctuating passenger demand on fleet-planning process and generates flexible fleet configurations that support dynamic assignments.