New Approaches to Add Robustness into Airline Schedules: New Approaches to Add Robustness into Airline Schedules Shan Lan, Cindy Barnhart and John-Paul Clarke
Center for Transportation and Logistics
Massachusetts Institute of Technology
May 5 , 2002 Courtesy of Shan Lan, Cindy Barnhart and John-Paul Clarke. Used with permission
Outline: Outline Background, Motivation and Our Contributions
Overview of Robust Airline Schedule Planning
Robust Aircraft Maintenance Routing – reduce delay propagation
Flight Schedule Retiming – reduce passenger missed connections
Summary and Future Research Directions
Airline Schedule Planning Process: Airline Schedule Planning Process
Most existing planning models assume that aircraft, crew, and passengers will operate as planned
Airline Operations: Airline Operations Many reasons can cause delays
Severe weather conditions, unexpected aircraft and personnel failures, congested traffic, etc.
Delays may propagate through the network
Long delays and cancellations cause schedule disruptions
Airlines must reschedule aircraft/crew and re-accommodate passengers
Huge revenue loss:
Delays cost consumers and airlines about $6.5 billion in 2000 (Air Transport Association)
Flight Delays & Cancellations: Flight Delays andamp; Cancellations Trend (1995-1999) (Bratu and Barnhart, 2002)
Significant increase (80%) in flights delayed more than 45 min
Significant increase (500%) in the number of cancelled flights
Year 2000 (Bratu and Barnhart, 2002)
30% of flights delayed
3.5% of flights cancelled
Future:
Air traffic in US is expected to double in the next 10-15 years (Schaefer et al. (2001))
Each 1% increase in air traffic a 5% increase in delays (Schaefer et al. (2001))
Lead to more frequent and serious delay and schedule disruptions
Passenger Disruptions: Passenger Disruptions Passengers are disrupted if their planned itineraries are infeasible because
flights cancellation
Insufficient time to connect
4% of passengers disrupted in 2000 (Bratu and Barnhart, 2002)
Half of them are connecting passengers
Very long delays for disrupted passengers
Average delay for disrupted passengers is approx. 419 minutes (versus 14 min delay for non-disrupted passengers) (Bratu and Barnhart, 2002)
Significant revenue loss
Our Contributions: Our Contributions Provide alternative definitions for robustness in the context of airline schedule planning
Develop an optimization model and solution approach that can generate aircraft maintenance routes to minimize delay propagation
Develop optimization models and solution approach to minimize the expected total number of passengers missing connection, and analyze the model properties
Proof-of-concept results show that these approaches are promising
Develop integrated models for more robustness
Outline: Outline Background, Motivation and Our Contributions
Overview of Robust Airline Schedule Planning
How to deal with schedule disruptions
Challenges of building robust airline schedules
Definitions of robustness
Robust airline schedule planning approaches
Robust Aircraft Maintenance Routing -- reduce delay propagation
Flight Schedule Retiming – reduce passenger missed connections
Summary and Future Research Directions
How to Deal with Schedule Disruptions: How to Deal with Schedule Disruptions Two ways to deal with schedule disruptions
Re-optimize schedule after disruptions occur (operation stage)
Build robustness into the schedules (planning stage)
Existing planning systems do not have effective methods to manage disruptions
A more robust plan can reduce the effect of disruptions on the operations reduce operation costs and improve quality of service
Robust airline schedule planning methods are needed
Challenges of Building Robust Plans: Challenges of Building Robust Plans Lack of a systematic way to define robustness in the context of airline schedule planning
Aircraft, crew and passenger flows interact in the hub-and-spoke network
Huge problem size tractability issue
Difficult to balance robustness and costs
Definitions of Robustness: Definitions of Robustness Minimize cost
Minimize aircraft/passenger/crew delays and disruptions
Easy to recover (aircraft, crew, passengers)
Isolate disruptions and reduce the downstream impact
Robust Airline Schedule Planning: Robust Airline Schedule Planning Min
Cost Ease of
recovery Min delays/
disruptions Isolation of
disruptions
Where Should We Start?: Where Should We Start? Difficult to balance cost that airlines are willing to pay for robustness versus cost of operation
Looking for robust solution without significant added costs
Aircraft maintenance routing problem: The financial impact is relatively small It is more a feasibility problem
How to route aircraft has impacts on flight delays and cancellations, passengers, crews
Question:
What robustness can be achieved for the maintenance routing problem?
Outline: Outline Background, Motivation and Our Contributions
Overview of Robust Airline Schedule Planning
Robust Aircraft Maintenance Routing – reduce delay propagation
Delay Propagation
Modeling Idea
String based formulation
Solution approach
Proof-of-concept results
Flight Schedule Retiming – reduce passenger missed connections
Summary and Future Research Directions
Delay Propagation: Delay Propagation Arrival delay may cause departure delay for the next flight that is using the same aircraft if there is not enough slack between these two flights
Delay propagation may cause schedule, passenger and crew disruptions for downstream flights (especially at hubs) f1 MTT f2
Propagated Delay vs. Independent Delay: Propagated Delay vs. Independent Delay Flight delay may be divided into two categories:
Propagated delay
Caused by inbound aircraft delay – function of routing
20-30% of total delay (Continental Airlines)
Independent delay
Caused by other factors – not a function of routing
Definitions: Definitions i j Slack Min Turn Time PDT PAT ADT AAT PD IAD TAD j’ i’ i’’ PD IDD TDD Planned Turn Time
Modeling Idea: Modeling Idea Delays propagate along aircraft routes
Only limited slack can be added
Appropriately located slack can prevent delay propagation
Routing aircraft intelligently better allocated slack
Essentially add slack where advantageous, reducing slack where less needed
Illustration of the Idea: Illustration of the Idea f1 MTT f2 f3 f4 MTT Original routing
Modeling Issues: Modeling Issues Difficult to use leg-based models to track the delay propagation
One variable (string) for each aircraft route between two maintenances (Barnhart, et al. 1998)
A string: a sequence of connected flights that begins and ends at maintenance stations
Delay propagation for each route can be determined
Need to determine delays for each feasible route
Most of the feasible routes haven’t been realized yet
PD and TAD are a function of routing
PD and TAD for these routes can’t be found in the historical data
IAD is not a function of routing and can be calculated by tracking the route of each individual aircraft in the historical data
Generating Flight Delays for Any Feasible Route: Generating Flight Delays for Any Feasible Route Step1: Determine propagated delays from historical data:
PDij = max (TADi – slackij,0)
Step 2: Determine Independent Arrival Delays (IAD) from historical data:
IADj= TADj – PDij
Step 3: Determine TAD and PD for feasible routes:
For the first flight on each string, New_TAD = IAD
New_PDij =max (New_TADi – slackij,0)
New_TADj= IADj+ New_PDij
String Based Formulation: String Based Formulation
Objective Function Coefficient: Objective Function Coefficient Random variables (PD) can be replaced by their mean
Distribution of Total Arrival Delay
Possible distributions analyzed: Normal, Exponential, Gamma, Weibull, Lognormal, etc.
Our statistical analysis shows that lognormal distribution is the best fit
A closed form of expected value function can be obtained
Solution Approach: Solution Approach This formulation is a deterministic mixed-integer program with a huge number of 0-1 variables
Branch-and-price
Branch-and-Bound with a linear programming relaxation solved at each node of the branch-and-bound tree using column generation
IP solution
A special branching strategy: branching on follow-ons (Ryan and Foster 1981, Barnhart et al. 1998)
Computational Results: Computational Results Test Networks
Data divided into two sets:
First data set (Jul 2000) used to build model and generate routes
Second data set (Aug 2000) used to test these new routes
Results - Delays: Results - Delays July 2000 data
August 2000 data
Results - Delay Distribution: Results - Delay Distribution Total delays for existing and new routings
Results - Passenger Disruptions: Results - Passenger Disruptions Disruptions calculated at the flight level
If a flight was cancelled, all passengers on that flight is disrupted
If actual departure time of flight B – actual arrival time of flight A andlt; minimum connecting time all passengers connecting from A to B are disrupted
Outline: Outline Background, Motivation and Our Contributions
Overview of Robust Airline Schedule Planning
Robust Aircraft Maintenance Routing
Flight Schedule Retiming – reduce passenger missed connections
Passenger delays and disruptions
Modeling Idea
Formulations and their properties
Solution approach
Proof-of-concept results
Summary and Future Research Directions
Passenger Delays and Disruptions: Passenger Delays and Disruptions Flight delay and passenger delay (Bratu and Barnhart, 2002)
Passenger delay caused by disruptions is the most critical part
Minimize number of disrupted passengers
A good proxy for passenger delays
Definitions Related to Passenger Disruption : Definitions Related to Passenger Disruption If ACT – MCT andlt; 0, passengers are disrupted
Minimize Passenger Missed Connections: Minimize Passenger Missed Connections If the slack is 'eaten' by flight delay, passengers are disrupted
Adding more slack can be good for connecting passengers, but can result in reduced productivity
Appropriately located slack can prevent passenger disruptions
Moving flight departure times in a small time window can lead to better allocated slack
Illustration of the Idea: Illustration of the Idea Airport A Airport B Airport C Airport D Suppose 100 passengers in flight f2 will connect to f3 Expected disrupted passengers reduced: 10
Where to Apply: Where to Apply Whether a passenger will be disrupted depends on flight delays, a function of fleeting and routing
Before solving maintenance routing
Impact of the propagation of flight delays won’t be considered
New fleeting and routing solution may cause delay propagate in a different way may eventually change the number of disrupted passengers
After solving fleeting and routing problem
Delay propagation has been considered
Need to maintain the current fleeting and routing solution
Schedule Design Crew Scheduling Fleet Assignment Maintenance Routing
Connection-Based Formulation: Connection-Based Formulation Objective
minimize the expected total number of passengers missing connection
Constraints:
For each flight, exactly one copy will be selected.
For each connection, exactly one copy will be selected and this selected copy must connect the selected flight-leg copies.
The current fleeting and routing solution cannot be altered.
Connection-Based Formulation: Connection-Based Formulation Theorem 1:
The second set of constraints are redundant and can be relaxed
Theorem 2:
The integrality of the connection variables can be relaxed
Alternative Connection-based Formulations: Alternative Connection-based Formulations Formulation II Formulation III
Model Properties: Model Properties Theorems on constraints:
The second set of constraints are redundant and can be relaxed in formulations two and three
The integrality constraints of the connection variables can be relaxed in formulations two and three
Theorem on LP relaxations
The LP relaxation of formulation one is at least as strong as those of formulations two and three
Problem Size: Problem Size A network from a major US airline used by Barnhart et al. (2001)
2,044 flights and 76,641 itineraries.
Suppose 7 copies will be generated for each flight (if 5 minutes interval is used, 7 copies correspond to a 30 minute time window)
Assume on average every flight connects to 12 flights with connecting passengers.
How to Maintain Current Fleeting and Routing Solution: How to Maintain Current Fleeting and Routing Solution For an aircraft maintenance route: the planned turn time andgt;= minimum turn time
Force , if the time between the arrival of flight copy and the departure of flight copy is less than the minimum turn time.
The upper bounds will be set to zero for these x variables
Solution Approach: Solution Approach Random variables can be replaced by their mean
Deterministic Problem
Distribution of
Branch-and-Price
Computational Results: Computational Results Network
We use the same four networks, but add all flights together and form one network with total 278 flights.
Data divided into two sets:
First data set (Jul 2000) used to build model and generate schedule
Second data set (Aug 2000) used to test the new schedule
Strength of the formulations
Computational Results: Computational Results Assume 30 minute minimum connecting time
For July 2000 data
For August 2000 data
Computational Results: Computational Results August 2000 data
Assume 25 minute minimum connecting time
Assume 20 minute minimum connecting time
Computational Results: Computational Results How many copies to generate
Outline: Outline Background, Motivation and Our Contributions
Overview of Robust Airline Schedule Planning
Robust Maintenance Routing
Flight Schedule Retiming
Summary and Future Research Directions
Summary of Contributions
Future Research Directions
Summary of Contributions: Summary of Contributions Provide alternative definitions for robustness in the context of airline schedule planning
Develop an optimization model and solution approach that can generate aircraft maintenance routes to minimize delay propagation
Develop optimization models and solution approach to minimize the expected total number of passengers missing connections, and analyze the model properties
Proof-of-concept results show that these approaches are promising
Develop integrated models for more robustness
Future Research Directions: Future Research Directions Integrated Models
Integrated robust aircraft maintenance routing with fleet assignment
Robust aircraft maintenance routing with time window
Integrated flight schedule re-timing with FAMTW
Other approaches
Fleet assignment with minimal expected cost
Fleet assignment under demand uncertainty
Aircraft routes with swap opportunities
Aircraft routes with short cycles
Computational Results: Computational Results July 2000 data
Assume 25 minute minimum connecting time
Assume 20 minute minimum connecting time
Impact on Passengers: Impact on Passengers Disruptions calculated at the flight level
If a flight was cancelled, all passengers on that flight is disrupted
If actual departure time of flight B – actual arrival time of flight A andlt; minimum connecting time all passengers connecting from A to B are disrupted
Number of disrupted passengers only calculated for connections between flights that both have ASQP records
ASQP has records only for domestic flights flown by jet airplanes and major airlines
Actual departure and arrival times for flights without ASQP records are unknown Assume no disruptions for these flights
Passengers only counted as disrupted once
If passenger is disrupted on any flight leg of itinerary, passenger not counted as disrupted on the following flight legs
Passenger Delays and Disruptions: Passenger Delays and Disruptions Passenger delays
the difference between scheduled and actual arrival time at passengers’ destination
Passengers are disrupted if their planned itineraries are infeasible
Flight delay and passenger delay (Bratu and Barnhart, 2002)
Passenger Disruption: Passenger Disruption Disrupted passengers
Significant numbers: 4% 20-30 million in U.S.
Experience very long delay
Contribute to more than half of the total passenger delay
Cause huge revenue loss
Destroy airlines’ image
Reduce disrupted passengers
Passenger delay caused by disruption is the most critical part
Hard to determine the delays for each disrupted passengers
Minimize number of disrupted passengers
LP Solution: LP Solution Algorithm for LP relaxation
Step 0: Create initial feasible solution
Step 1: Solve the restricted master problem (RMP)
Find optimal solution to RMP with a subset of all strings
Step 2: Solve the pricing problem
Generate strings with negative reduced cost
If no string is generated, stop: the LP is solved
Step 3: Construct a new restricted master problem
Add the strings generated
Go to step 1
Notation: Notation S: set of feasible strings
F: set of flights
G: set of ground variables
:set of strings ending (starting) with flight i
: binary decision variable for each feasible string s
y: integer variable to count number of aircraft on the ground at maintenance stations
: number of aircraft on the ground before (after) flight i departs at the maintenance station from which flight i departs
: number of aircraft on the ground before (after) flight i arrives at the maintenance station from which flight i arrives
Notation (Cont.): Notation (Cont.) : propagated delay from flight i to flight j if flight i and flight j are in string s
: indicator variable, equals 1 if flight i is in string s, and equals 0 otherwise
: number of times string s crosses the count time, a single point time at which to count aircraft
: number of times ground arc g crosses the count time
N : number of planes available.
Data: Data Airline Service Quality Performance (ASQP) provides good source of delay information
ASQP provides flight operation information:
For all domestic flights served by jet aircraft by major airlines in U.S.
Planned departure time and arrival time, actual departure time and arrival time (including wheels-off and wheels-on time, taxi-out and taxi-in time, airborne time)
Aircraft tail number for each flight
Cancelled flights (reasons for cancellation, and aircraft tail number are not available)
Effect of Cancellations: Effect of Cancellations For cancelled flights in the historical data
we don’t know which aircraft supposed to fly them
We don’t have the delay information
We assume the propagated delays for these flights are zero
Lower cancellation rates
Less passengers disrupted because of cancellation
More passengers disrupted because of flight delays
7 days in Aug 2000 with very few cancellations (cancellation rate = 0.19%)
For Aug 2000, 65% of disrupted passengers are disrupted because of flight delays
For 7 selected days in Aug 2000, 92% of disrupted passengers are disrupted because of flight delays
Results - Low Cancellation Days: Results - Low Cancellation Days Passenger disruptions for 7 selected days in Aug 2000 with very few cancellations
Reduction in number of disrupted passengers per non-cancelled flights is same as that for entire month
Extensions: Extensions Combine with scheduling
More slacks may be added further reduce delay propagation
Combine with fleet assignment
Need to determine cost for propagated delay
More feasible strings better solution
Minimum turn time is a function of fleet type
Integrate with fleet assignment and schedule generation