Two Major Problems Problems Crew Pairing Problem (CPP) Find a set of legal pairin Find gs (each pairing

Transcription

1 Solving Airline s Pilot-Copilot Rostering Problem by Successive Bipartite Weighted Matching by Xugang Ye Applied Mathematics and Statistics, The Johns Hopkins University

2 Motivation Crew-related related cost is the second largest to fuel. The planning and scheduling hdli of aircrews is highly complex. In today s highly competitive markets, optimization-integrated integrated decision support systems have become a critical success factor of the airlines.

3 Two Major Problems Crew Pairing Problem (CPP) hub Find a set of legal pairings (each pairing is an itinerary that starts from and returns to the same h b) that hub) h covers all ll the h fli flight h llegs with i h minimum i i expected cost.

4 Two Major Problems Crew Rostering Problem (CRP) Dispatch Home base (hub) Return Find crew team for each pairing such that the schedule Legality, cost issue, and the crew satisfaction are respected. Task set

5 Data Flow of the Crew Management System Task execution ecut Crew data Flight data Crew pairing optimization Crew rostering optimization

6 Mathematical Modeling of Crew Rostering Problem

7 Problem Description Suppose in a period of time, we plan to assign a set of pairings to a group of crews in same fleet at same hub. A pilot and a copilot are teamed together to accomplish a task. Some crews cannot be teamed together. Some crews are not available sometime. We want to find a roster to cover all the pairings while respecting schedule legality and cost issue, and also minimize the discrepancy of cumulative flying time of pilots minimize the discrepancy of cumulative flying time of copilots

8 Formulation by Linear Programming (LP)

9 Formulation by Linear Programming (LP)

10 Formulation by Linear Programming (LP)

11 Formulation by Linear Programming (LP)

12 Formulation by Linear Programming (LP)

13 Model Analysis Intractability Muti-objectives; NP-hardness Scale Tens of thousand variables Millions of constraints Data Uncertainty Non-deterministic dynamic decision environment

14 Strategy Heuristic Algorithm Sequential decomposition Successive partial rostering (day-by-day) optimization Submodel extraction Common formulation of fbi bipartite i weighted matching for different subproblems Subalgorithm Network flow optimization

15 Fundamental Submodel Construction

16 Fundamental Submodel Construction

17 Fundamental Submodel Construction

18 Fundamental Submodel Construction

19 Submodels for Subcases Submodel 1: match pairings with pilots Bipartitite weighted matching problem 1 Submodel 2: match pairings with copilots Bipartitite weighted matching problem 2 Submodel 3: copilot vacancy filling Bipartitite weighted matching problem 3

20 Algorithm for solving bipartite weighted matching problem Minimum-cost maximum flow algorithm Overall complexity = O(R S) S is the complexity of the Bellman-Ford algorithm R is the number of daily pairings.

21 Global Effectiveness L = max k 2 T 2 k Where T k is the flying time of the kth pairing ii (k=1,2,,n).

22 Numerical Simulations Scale Flight tasks table that contains 17 pairings each day Maximum m flying time of a pairing is 805 minutes L=284.61minutes i

23 Data Simulations

24 Data Simulations

25 Data Simulations Simulation of pilot-copilot teaming infeasibility Simulation of pilot-pilot teaming infeasibility ID No. of copilots f pilots ID No. of ID No. of pilots ID No. of pilots

26 Data Simulations Day-off simulation for pilots Day-off simulation for copilots of pilots 30 of copilots 30 ID No 40 ID No Calendar date in considered period Calendar date in considered period

27 Data Simulations 7 Simulation of task perturbation 6 5 State of task ID N o. of Tas k

28 Data Simulations Cum mulative flying tim me ID No. of pilots Cum mulative flying tim me ID No. of copilots 2 2 tatus of pilots St 1 0-1: In day-off -1 0: Waiting 1: In task ID No. of pilots Sta atus of copilots 1 0-1: In day-off -1 0: Waiting 1: In task ID No. of copilots

29 Numerical Results mulative flying time of pilots Cu 2 x ID No. of pilots Cum mulative flying time of co opilots 2 x ID No. of copilots 2 2 Stdv. of cumulative flying time among pilots of pilots Status 1 0-1: In day-off -1 0: Waiting 1: In task ID No. of pilots Stdv. of cumulative flying time Upper bound Date Stdv. of cu umulative flying time am mong copilots Status of copilots 1 0-1: In day-off -1 0: Waiting 1: In task ID No. of copilots Stdv. of cumulative flying time Upper bound Date

30 Results from Multiple Simulations ng pilots Stdv. of cum mulative flying time amo 1400 Stdv. of cum ulative flying tim e 1200 U pper b ound Date Stdv. of cumu ulative flying time among copilots Stdv. of cum ulative flying tim e Upper bound Date Multiple simulations under the circumstance that both the pilots and the copilots are enough many (left: for pilot group; right: for copilot group) flying time among pilots Stdv. of cumulative S tdv.of c um ulative fly ing tim e Upper bound Date s lying time among copilots Stdv. of cumulative fl S tdv.of c um ulative fly ing tim e Upper bound Date Multiple simulations under the circumstance that the pilots are enough many, there are not enough copilots, some pilots are selected to fill the copilot vacancies on some days (left: for pilot group; right: for copilot group).

31 Some Statistics on Computational Efficiency Implemented with Matlab 7.0 and tested in Dell Dimension 4600 with P G CPU and 1.0 G DDR MEM.

32 Sample Output

33 Conclusion The heuristic algorithm is efficient, effective and applicable to real large-scale aircrew rostering problems. 1) Time efficient. 2) Heuristically accurate.

34 Possible Extension More frequent updates of pairing plan within one day Insertion of newly hired aircrews Aircrew shift among multiple homebases and among multiple fleets

35 Questions? Thanks very much!