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Crew Pairing Optimization with Genetic Algorithms

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Crew Pairing Optimization with Genetic Algorithms Harry Kornilakis and Panagiotis Stamatopoulos Department of Informatics and Telecommunications – PowerPoint PPT presentation

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Title: Crew Pairing Optimization with Genetic Algorithms


1
Crew Pairing Optimization with Genetic Algorithms
  • Harry Kornilakis and Panagiotis Stamatopoulos
  • Department of Informatics and Telecommunications
  • University of Athens
  • harryk,takis_at_di.uoa.gr
  • The Crew Pairing Problem
  • Solution to the Crew Pairing Problem
  • Pairing Generation
  • Optimization Using Genetic Algorithms
  • Experimental Results
  • Conclusions

2
The Crew Pairing Problem (1/2)
  • The Crew Pairing Problem (CPP) is an important
    optimization problem that is part of the airline
    crew scheduling procedure.
  • Crew scheduling Crew pairing Crew assignment
  • Given a timetable containing all the flight legs
    that an airline company must carry out, the
    objective of the crew pairing problem is to
    generate leg combinations (pairings) of minimum
    cost that cover all given flight legs.

3
The Crew Pairing Problem (2/2)
  • A pairing is a round trip, consisting of a
    sequence of legs, which starts and ends at the
    crew's home base.
  • Each pairing is composed of a number of legal
    workdays, called duties, which are separated by
    rest periods.
  • The CPP is the problem of finding a set of
    pairings, covering all legs that the airline has
    to carry out, with a minimal cost.

4
Constraints of the CPP
  • Temporal constraints
  • Spatial constraints
  • Constraints concerning the type of airplane used
    (fleet constraints)
  • Constraints due to laws and regulations (e.g.
    maximum duration of a duty, minimum rest period
    between two duties etc.)

5
Further Considerations on the CPP
  • Deadheading Crew members travelling as
    passengers, in order to move to a specific
    airport and continue a round trip.
  • Cost Function Usually depends on the crews
    wages and can be quite complicated.
  • Modeling of the Constraints Highly non-linear
    and hard to model constraints. Linear programming
    inadequate.
  • Huge size of the search space

6
Solving the Crew Pairing Problem (1/2)
  • Two Phase Procedure
  • 1. Pairing Generation A large number of legal
    pairings, composed of the legs in the timetable,
    is generated.
  • 2. Optimization A subset of the generated
    pairings is selected, so that every flight leg in
    the schedule is included in at least one pairing
    and the total cost is minimized.

7
Solving the Crew Pairing Problem (2/2)
  • Pairing generation is also divided in two phases
  • We generate a large set of legal duties from the
    flight legs
  • We generate the set of pairings by using the
    duties previously found

8
Pairing Generation (1/2)
  • L Set of all the flight legs
  • Constraints defined as a function C 2L ?
    0,1, where 2L is the powerset of L, C (p)1 if
    p is a legal pairing and C (p)0, if not.
  • We want to generate a set of pairings
  • P p ? 2L C (p)1, i.e. the set of
    pairings that satisfy all the constraints.
  • Checking for the satisfaction of the constraints
    is independent of the optimization phase.

9
Pairing Generation (2/2)
  • First Phase Generation of duties from legs
  • Implemented as a depth-first search in the space
    of all possible subsets of the set of all flight
    legs.
  • The number of legal duties found might be too
    large, so we also perform an algorithm that
    selects the best in order to use them in the next
    phase.
  • Second Phase Generation of pairings from duties
  • Works similarly to the first phase but instead of
    duties it generates pairings.

10
Optimization (1/2)
  • Modeled as a set covering problem
  • Given a set M with m elements and n subsets ?j of
    M with associated costs cj , j 1,2 n, select a
    number of these subsets such that every element
    of M is contained in a least one selected subset
    and the total cost of all selected subsets is
    minimum.
  • Set covering has been proven to be NP-complete
    (Garey Johnson, 1979)

11
Optimization (2/2)
  • Existing Methods for the set covering problem
  • Exact Methods For average sized problems
  • Approximate Methods Give approximate solution to
    large sized problems
  • Our Method is a modification of an algorithm by
    Beasley Chu (1996)

12
Genetic Algorithms (1/2)
  • Genetic algorithms (GA) Methods for random
    search based on the evolutionary process of
    species, found in nature.
  • Each possible solution is encoded as a string
    (chromosome). Each character of the chromosome is
    called a gene.
  • The quality of each solution is represented by a
    real valued function defined over the set of all
    chromosomes (fitness function).

13
Genetic Algorithms (2/2)
  • Two of the fittest members of the population are
    selected and combined to produce a new solution
    (crossover). Then a few random genes of the new
    solution are changed (mutation) in order to avoid
    local minima in the search.
  • New individuals replace the weaker members of the
    existing population.

14
Genetic Algorithm for Crew Pairing Optimization
(1/4)
  • Binary String Coding Each chromosome has length
    equal to number of pairings. Each gene
    corresponds to one pairing. If the gene is 1 then
    the corresponding pairing is in the solution,
    otherwise it is not.
  • Fitness Function Si ci gi
    deadheadingPenalty deadheadedFlights
  • ci is the cost if the i-th pairing
  • gi is the value of the i-th gene of the solution
  • deadheadingPenalty is a constant to penalize
    deadheaded flights
  • deadheadedFlights is the number of deadheaded
    flights

15
Genetic Algorithm for Crew Pairing Optimization
(2/4)
  • Selection of Parents based on their order in the
    population, sorted in descending order based on
    their fitness function, not on the absolute value
    of the fitness.
  • Crossover Uniform crossover, i.e. if the i-th
    gene of both parents is 1 (respectively 0) then
    the i-th gene of the offspring is set to 1
    (respectively 0). Otherwise its value is randomly
    selected.
  • Mutation A few genes of the new solution are
    randomly set to 1 or 0 with probability that
    depends on the density (the percentage of genes
    equal to 1) of the fittest individual of the
    population.

16
Genetic Algorithm for Crew Pairing Optimization
(3/4)
  • Correcting the Solution After crossover and
    mutation the new solution might be infeasible.
  • Correction Algorithm The value of some genes is
    changed to 1 until every flight leg is covered
    and the solution becomes feasible. For each
    uncovered leg in the solution, we add a pairing
    that covers that leg. To select one pairing among
    all the pairings which contain that particular
    leg, we use a heuristic that favors pairings of
    low cost that cover as many uncovered legs as
    possible and as few legs already covered as
    possible.

17
Genetic Algorithm for Crew Pairing Optimization
(4/4)
  • Randomly generate an initial population of
    chromosomes.
  • Repeat until a satisfactory solution is reached
  • Select two of the fittest members of the
    population for parents.
  • Apply the crossover operator on the two parents
    to produce a child chromosome.
  • Apply the mutation operator on the child
    chromosome.
  • Correct the child chromosome so that it becomes
    a feasible solution.
  • Select one of the weakest members of the
    population to be replaced.
  • Replace the selected individual with the child.

18
Experimental Results
  • Input data Real data from the flight schedule of
    Olympic Airways (737-200 fleet, April 1998),
    which was composed of 2100 flight legs, passing
    through 29 different airports.
  • Constraints Based on the regulations of the
    Olympic Airlines for cockpit crews.
  • Pairing Generation 22090 legal duties and then
    11981 legal parings were generated and stored.
  • Optimization After 20000 generations a solution
    of cost 976004 and of zero deadheading was found.

19
Progress of the Optimization (1/2)
20
Progress of the Optimization (2/2)
21
Comparison of our Method and Parachute (1/2)
  • We compared our result with the result found by
    the project PARACHUTE, a project that combines
    constraint programming with parallel computing.
    We ran PARACHUTE on the same data set and used
    the same constraints.
  • Our method gave a solution of cost lower by
    194148 (16.5).

22
Comparison of our Method and Parachute (2/2)
23
Conclusions
  • We separated the Crew Pairing Problem in two
    phases.
  • A depth-first search was used in the first phase
    and a genetic algorithm in the second.
  • Experimental results with real data were
    satisfactory improving on previous result and
    showing that the solution is viable.
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