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
• 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

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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.

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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.

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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.)

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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

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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.

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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

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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.

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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.

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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)

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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)

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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).

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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.

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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

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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.

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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.

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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.

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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.

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Progress of the Optimization (1/2)
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Progress of the Optimization (2/2)
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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).

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Comparison of our Method and Parachute (2/2)
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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.