U NIT C REWING FOR R OBUST A IRLINE C REW S CHEDULING Bassy Tam Optimisation Approaches for Robust Airline Crew Scheduling Bassy Tam Professor Matthias Ehrgott Professor David Ryan Dr. Golbon Zakeri

I RREGULAR O PERATIONS AND R OBUST S CHEDULING A cost minimal solution usually has high resource utilisation Crew or aircraft spend minimal amount of time on the ground between arrival and departure of flights I.e. minimise any idle time incurred Once disruption occurs Little slack between flights to compensate for delays A single initial delay might propagate to later flights Airlines are interested in robust schedules Optimisation Approaches for Robust Airline Crew Scheduling

R OBUST S CHEDULE P LANNING Robust schedule Optimal planned cost is not necessary Aim for low operational cost Robustness of a schedule Not well defined Two broad categories Improve flexibility of the schedule To allow easy recovery of schedule by switching aircraft and/or crew Depends on the recovery procedure Minimise impacts from disruption To reduce delay propagation Schedule is operationally robust Optimisation Approaches for Robust Airline Crew Scheduling

R OBUST S CHEDULE P LANNING Robust scheduling can be considered in almost all stages of the airline scheduling process Four approaches for crew pairing Expected crew cost by simulation Schaefer et al. (2001) Bi-criteria Ehrgott and Ryan (2002) Stochastic programming Yen and Birge (2006) Move-up crew Shebalov and Klabjan (2006) Optimisation Approaches for Robust Airline Crew Scheduling

C OMPARISON OF T WO R OBUSTNESS A PPROACHES Similar robustness measure Both look for operationally robust crew schedules Improve robustness by Reducing the number of switching aircraft connections when ground time is short Lengthen ground time when switching aircraft is necessary Compared robustness indicators Simulation to compare on-time performance Studied some historical data to ensure the same delay distribution is used for both approaches Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING It is usual to solve the crew pairing problem separately for different crew ranks and crew types Different resources allocated in each crew base Different operational rules Crew schedules are unlikely to be unit crewed Crew members perform different activities after operating a flight However, it is more robust to keep the crew working together as a unit Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING Little consideration of the flight connections of other crew ranks when constructing a crew schedule What is unit crewing? An approach to construct crew schedules so that members of a crew perform the same sequence of flights within their duty periods as much as possible A unit-crewed schedule is considered to be more operationally robust in the sense that it is less likely to delay other flights due to waiting for a member of the crew from a disrupted upstream flight Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY S EQUENTIAL S CHEDULING Obtain a cost minimal crew schedule for one crew rank Bi-criteria optimisation problem for the other crew rank Minimise crew cost Maximise unit crewing connection To solve the bi-criteria optimisation problem Penalty method Weighted sum of two objectives Elastic constraint method Reformulated the crew cost objective as a constraint Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY S EQUENTIAL S CHEDULING Penalty method No interpretation of the sum of two objectives Specification of the penalty is difficult Relatively few Pareto optimal solutions may be found Elastic constraint method More difficult to solve compared with the penalty method Might not be able to obtain good solutions within reasonable time Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY S EQUENTIAL S CHEDULING Need to experience the order of crew pairing problems to be solved The final set of crew schedules might not be optimal in terms of total crew cost and number of unit crewing connections Solution of the second crew rank (the bi-criteria optimisation problem) is limited by the solution of the first crew rank (the cost minimal problem) Optimality or feasibility of the second crew rank is not considered when solving the crew pairing problem for the first rank Might result in higher cost to obtain unit crewed solution Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY P ARALLEL S CHEDULING Optimisation Approaches for Robust Airline Crew Scheduling Unit Crewing Constraints One constraint represents one possible unit crewing connection If x kj contains connection i, u kij = 1 and 0 otherwise The i th connection will be unit crewed if u 1ij x 1j – u 2ik x 2k = 0 Non-unit crewed connection will have a slack or surplus value of 1 To minimise total number of non-unit crewing connections

U NIT C REWING BY P ARALLEL S CHEDULING Problem is difficult to solve due to the size of the resulting problem Flight and crew base balancing constraints are doubled Number of unit crewing constraints is 5.7 times of the number of flight constraints (for single crew rank) in our test data Branching method Use special branch selection method to increase number of unit crewing connections Limiting number of constraints Include a subset of unit crewing constraints into the problem Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY P ARALLEL S CHEDULING Branching method All unit crewing constraints are removed Optimisation Approaches for Robust Airline Crew Scheduling Total crew cost is always minimised, i.e. no choice for making trade-offs between unit crewing objective and cost objective

U NIT C REWING BY P ARALLEL S CHEDULING Branching with elastic constraint Following aircraft objective introduced Cost objective is reformulated as a constraint Has problem to find a quality crew schedule that has high number of unit crewing connections Number of unit crewing connections is not included in the objective Unit crewing is only considered as long as the cost and following aircraft objectives are optimised Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY P ARALLEL S CHEDULING Two methods to define unit crewing constraints Define unit crewing constraints before the optimisation process Difficulties arise from the number of possible connections Size of the problem Define unit crewing constraints during the optimisation process Size of the problem increases dynamically Add constraints when needed Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY P ARALLEL S CHEDULING Observations Difficult to obtain an integral solution Connections are unit crewed in a “fractional” way Sum of fractional values spill over on many connections A lot of constraints need to be constructed Makes branch and bound difficult Sometimes makes originally unit crewed connections no longer unit crewed Impose 1-branch on unit crewed connections that are robust (following aircraft connections) Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY P ARALLEL S CHEDULING Advantages Avoid the sum of fractional values to spill over on too many connections Limit number of possible connections in the flight network, i.e. column generation process is faster Limit number of variables in the problem, i.e. reduced cost calculation is faster Disadvantage Optimality cannot be guaranteed Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY P ARALLEL S CHEDULING Observations If not many unit crewed following aircraft connections at the end of the first LP relaxation, problems of spill over remains The spill over of the sum of fractional values is caused by the unit crewing constraints The more unit crewing constraints, the worse the problems Limit number of unit crewing constraints Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY P ARALLEL S CHEDULING Disadvantage Order the priority of connections Decide maximum number of unit crewing constraints allowed in the problem If too many unit crewing constraints allowed Problems of the spill over remains If the limit on number of unit crewing constraints is too small Good quality unit crewed schedules cannot be found Optimisation Approaches for Robust Airline Crew Scheduling

U NIT C REWING BY P ARALLEL S CHEDULING Dantzig-Wolfe decomposition Block diagonal structure Use Dantzig-Wolfe decomposition to break the combined model into two problems Master problem contains the unit crewing constraints Sub-problems are the two crew pairing problems Structure of the two crew pairing problems remain unchanged Longest computational time Optimisation Approaches for Robust Airline Crew Scheduling