Crew Pairing Optimization with Genetic Algorithms

Harry Kornilakis and Panagiotis StamatopoulosDepartment of Informatics and Telecommunications

University of Athens {harryk,takis}@di.uoa.gr

The Crew Pairing ProblemSolution to the Crew Pairing Problem

Pairing GenerationOptimization Using Genetic Algorithms

Experimental ResultsConclusions

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.

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.

Constraints of the CPP

Temporal constraintsSpatial constraintsConstraints 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.)

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 crew’s 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

Two Phase Procedure1. 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.

Solving the Crew Pairing Problem (1/2)

Solving the Crew Pairing Problem (2/2)

Timetablewith Flight

Legs

Duty Generation

Applicationof

Constraints

Pairing Generationfrom Duties

Duties

Pairings

Optimization

Set CoveringProblem

FinalSolution

Pairing Generation

Applicationof

Constraints

Pairing generation is also divided in two phases:We generate a large set of legal duties from the

flight legsWe generate the set of pairings by using the

duties previously found

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.

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 dutiesWorks similarly to the first phase but instead of

duties it generates pairings.

Pairing Generation (2/2)

Optimization (1/2)

Modeled as a set covering problem:Given a set M with m elements and n subsets Μj 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)

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)

Optimization (2/2)

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).

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.

Genetic Algorithms (2/2)

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:

Σi 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

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.

Genetic Algorithm for Crew Pairing Optimization (2/4)

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.

Genetic Algorithm for Crew Pairing Optimization (3/4)

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.

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.

Progress of the Optimization (1/2)

Generation 100 1000 3000 5000 8000

Cost 1247776 1121057 1064647 1036596 1008276

Pairings 635 592 567 551 537

Deadheading 71 10 6 6 4

Generation 10000 13000 16000 20000

Cost 996994 988418 980344 976004

Pairings 531 529 526 525

Deadheading 4 2 0 0

Progress of the Optimization (2/2)

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%).

Comparison of our Method and Parachute (2/2)

Total Cost No. ofPairings

DeadheadedLegs

Our Method 976004 525 0

PARACHUTE 1170152 611 25 (1.1%)

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.

Conclusions