New Approaches to Improving the Robustness of Airline Schedules

1
New Approaches to Improving the Robustness of
Airline Schedules

• Prof. John-Paul ClarkeDepartment of Aeronautics
  AstronauticsMassachusetts Institute of
  Technology

2
Outline

• Background Motivation
• Robust Maintenance Routing
• Flight Schedule Re-Timing
• Degradable Airline Schedule
• Conclusions

3
Airline Schedule Planning Process

• Most existing airline schedule planning methods
  assume that aircraft, crews, and passengers will
  operate as planned

4
Airline Operations

• Bad weather reduces airport capacity
• Airlines cancel or delay flights to reduce demand
• Delays propagate through the network
• Airlines must reschedule aircraft/crew and
  re-accommodate passengers
• Passengers are not satisfied
• They are delayed
• They have no control over their delay
• All passengers on a given aircraft are delayed
  equally regardless of fare class

5
Delays Cancellations

• Trend (1995-1999)
• Significant increase (100) in flights delayed
  more than 45 min
• Significant increase (500) in the number of
  cancelled flights
• Year 2000
• 30 of flights delayed
• 3.5 of flights (approx. 140,000) cancelled
• Future
• Delays and cancellations may increase
  dramatically ? more frequent and serious schedule
  disruptions and revenue loss

6
Passenger Disruptions

• Flight delays and cancellations often cause
  passenger schedule disruptions
• 26 million passengers (4 of passengers)
  disrupted
• 65 of disruptions caused by missed connections
• Very long delays for disrupted passengers
• Average delay for disrupted passengers is approx.
  4 hours (versus 15 min delay for non-disrupted
  passengers)
• Significant revenue loss - approx. 4 Billion
  /year

7
Robustness

• Need schedules that are robust (insensitive) to
  delays and cancellations
• Definitions of robustness
• Minimize cost (expected/worst case deviations
  from optimal)
• Minimize aircraft/passenger delays and
  disruptions
• Easy to recover (aircraft, crew, passenger)
• Isolate disruptions and reduce the downstream
  impact
• Two ways to provide robustness
• Re-optimize schedule after disruptions occur
  (operation stage)
• Build robustness into the schedules (planning
  stage)

8
Outline

• Background Motivation
• Robust Maintenance Routing
• Graduate Student Shan Lan
• Joint work with Prof. Cindy Barnhart
• Flight Schedule Re-Timing
• Degradable Airline Schedule
• Conclusions

9
Robust Maintenance Routing

• Objective
• Reduce the propagation of delays by combining
  flight segments in optimal (from the point of
  view of follow-on delays) maintenance routings
• Total delay for a route is uniquely determined by
  routing
• Solution Approach
• Derive distributions from historical data for
  delay introduced into a route by an airport
• Formulate and solve maintenance routing model
  that minimizes the propagation of delays subject
  to maintenance feasibility

10
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
11
Propagated v. 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
• Appropriately allocated slack can reduce
  propagated delay
• Add slack where advantageous
• Reduce slack where less needed

12
Definitions
TDD
i
i
i
PD
IDD
PDT
ADT
Slack
Min Turn Time
j
Planned Turn Time
j
PAT
AAT
PD
IAD
TAD
13
Illustration of the Idea
MTT
f1
f2
MTT
f3
f4
Original routing
14
Modeling Issues

• Difficult to use leg-based models to track the
  delay propagation
• One variable (string) for each aircraft route
  between two maintenance events (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 havent been realized
  yet
• PD and TAD are a function of routing
• PD and TAD for these routes cant 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

15
String Based Formulation
16
Solution Approach

• Random variables (PD) can be replaced by their
  mean
• Distribution of Total Arrival Delay
• Possible distributions analyzed Normal,
  Exponential, Gamma, Weibull, Lognormal, etc.
  ?lognormal distribution is the best fit
• A closed form of expected value function
• 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)

17
Computational Results

• Test Networks

• Model Building and Validation

• Propagated delays (August 2000)

18
Results - Delays

• Total delays and on-time performance

• Passenger misconnects

19
Outline

• Background Motivation
• Robust Maintenance Routing
• Flight Schedule Re-Timing
• Graduate Student Shan Lan
• Joint work with Prof. Cindy Barnhart
• Degradable Airline Schedule
• Conclusions

20
Flight Schedule Re-Timing

• Objective
• Reduce the number of passenger misconnections by
  adjusting departure times so that passenger
  connection times are correlated with the
  likelihood of a missed connection (disruption)
• Add connection slack where it is need most
• Solution Approach
• Derive distributions from historical data for
  number of passengers disrupted for each
  connection
• Formulate and solve re-timing model that
  minimizes the number of disrupted passengers

21
Definitions

• AAT Actual Arrival Time
• ACT Actual Connection Time
• ADT Actual Departure Time
• MCT Minimum Connection Time
• PAT Planned Arrival Time
• PCT Planned Connection Time
• PDT Planned Departure Time

22
Illustration of the Idea
Suppose 100 passengers in flight f2 will connect
to f3
Airport A
Airport B
Airport C
Airport D
? Expected disrupted passengers reduced 20
23
Implementation Options

• Passenger disruption depends on flight delays, a
  function of fleeting and routing
• Before maintenance routing problem
• Delay propagation not considered
• New fleeting and routing solution may cause delay
  to propagate in a different way ? change the
  number of disrupted passengers
• After maintenance routing problem
• Delay propagation considered
• Need to enable the current fleeting and routing
  solution

Schedule Design
Fleet Assignment
Maintenance Routing
Crew Scheduling
24
Connection-Based Formulation

• Objective
• minimize the expected total number of passenger
  misconnects
• 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.

25
Connection-Based Formulations

• Theorem 1
• The second set of constraints are redundant and
  can be relaxed
• Theorem 2
• The integrality of the connection variables can
  be relaxed

• Formulation I CFSR

26
Alternative Formulations

• Formulation II ACFSR

• Formulation III DCFSR

27
More Model Properties

• Theorem 3
• ACFSR model is equivalent to CFSR model
• Theorem 4
• DCFSR model is equivalent to CFSR model
• Theorem 5
• The LP relaxation of CFSR model is at least as
  strong as that of ACFSR, and can be strictly
  stronger.
• Theorem 6
• The LP relaxation of CFSR model is at least as
  strong as that of DCFSR, and can be strictly
  stronger.

28
Solution Approach

• Random variables can be replaced by their mean
• Distribution of
• Branch-and-Price

29
Computational Results

• Network
• We use the same four networks, but add all
  flights together and form one network with total
  278 flights.
• Model Building and Validation
• Strength of the formulations

30
Computational Results

• Assume 30 minute minimum connecting time
• Assume 25 minute minimum connecting time
• Assume 20 minute minimum connecting time

31
Computational Results

• Number of copies
• Estimated reduction in total passenger delays
  (30 minutes MCT)
• 20 (30 minute time window), 16 (20 minute time
  window), 10 (10 minute time window)

32
Outline

• Background Motivation
• Robust Maintenance Routing
• Flight Schedule Re-Timing
• Degradable Airline Schedule
• Graduate Student Laura Kang
• Conclusions

33
Degradable Airline Schedule

• Objective
• Develop airline schedule that is robust, i.e.
  delays are isolated
• Provide priority (and thus reliability) for each
  flight
• Improve customer satisfaction by giving
  passengers an accurate expectation of the level
  of service
• Provide basis for revenue management and ATC
  auctions
• Solution Approach
• Partition schedule into smaller independent
  prioritized schedules (layers) subject to
  operational feasibility

34
Implementation Options
schedule design
fleet assignment
aircraft routing
crew scheduling
35
IP Model

• Prioritize layers based on revenue (e.g. group
  highest revenue flights together in most reliable
  layer)
• Revenue is protected if all flight legs in an
  itinerary are in a protected layer
• IP model maximizes the total protected revenue
  subject to feasibility constraints
• Prototype 2 layers implementation
• Layer 1 60 (protected layer)
• Layer 2 40

36
Model Statistics

• 1,134 flight legs
• 274 aircraft
• 1,744 itineraries (8 of total)
• Single flight leg 1,130
• 2 flight legs 613
• 3 flight legs 1
• 53,091 passengers (80 of total)
• 10,839,340 revenue (84 of total)

37
Notation

• Indices
• r route
• f itinerary
• ij flight
• k layer (k1 K)
• ?ijf 1 if flight ij is in itinerary f, 0
  otherwise
• Decision variables
• yrk 1 if route r is in layer k, 0 otherwise
• zfk 1 if itinerary f is in layer k, 0 otherwise
• xijk 1 if flight ij is in layer k, 0 otherwise
• Parameters
• vfk revenue for itinerary f is placed in layer k
• Ch capacity at hub h in bad weather
• Sk fraction of layer k
• ar number of flights in route r
• arh number of flights departing at hub h in route
  r
• ACN number of aircraft

38
Flight-based Formulation
39
Route-based Formulation
40
Greedy Flight-Leg Pairing

• STEP 0 Fix connections for non-hub to non-hub
  flights
• STEP 1 Pair flight segments at spoke airports
  using the revenue paring with aircraft
  utilization heuristic
• STEP 2 Combine paired flight segments from step
  1 at hub airports using the revenue paring with
  aircraft utilization heuristic
• STEP 3 Partition very long routes into several
  shorter routes

41
Greedy Flight-Leg Pairing
10
100
100
10
42
Swapping Search
route i
route j

• Check swapping feasibility
• Check constraints satisfaction
• Check objective function improvement
• Assume revenue is protected proportionally to the
  number of flight legs in the protected layer
• Swap

43
Tabu Search

• STEP 0 start with initial solution x from
  revenue paring heuristics
• WHILE( number of iteration is less than N )
• STEP 1 Swapping Search. If f(x) f(x), x ? x
• STEP 2 Update Tabu list
• If a pair was in a tabu list for Y iterations,
  remove it from the tabu list
• Set X pairs which were swapped in the search in
  the tabu list
• Tabu search is sensitive to its parameters X, Y,
  N
• State-of-art decision for X, Y, N

44
IP Objective Function Value
Upper bound for route-based DAS
D-SPM
8,667,632
D-ARM w/Heuristics
8,123,060
Lower bound for route-based DAS
Current routing
6,492,895
45
Protected Revenue
D-SPM
9,624,460
74.5
D-ARM w/Heuristics
9,057,750
70.1
Current routing
7,302,040
56.6
46
Protected Passengers
D-SPM
44,984
67.5
D-ARM w/Heuristics
43,051
64.6
Current routing
37,587
56.4
47
Simulation Results - Good Weather
Layer 1
Layer 2
Current Routing
Average delay
6 min
6 min
6 min
Pr(delay 0)
0.37
0.48
0.42
Pr(delay 15)
0.14
0.17
0.16
48
Simulation Results - Bad Weather
Layer 1
Layer 2
Current Routing
Average delay
13 min
25 min
17 min
Pr(delay 0)
0.52
0.69
0.61
Pr(delay 15)
0.30
0.48
0.37
49
Outline

• Background Motivation
• Robust Maintenance Routing
• Flight Schedule Re-Timing
• Degradable Airline Schedule
• Conclusions

50
Conclusions

• Robust Maintenance Routings provide
• Airline schedule with reduced delay propagation
• Flight Schedule Re-Timing provides
• Airline schedule with fewer passenger disruptions
  or missed connections
• Degradable Airline Schedules provide
• Airline schedule that isolates delays
• Tool for managing passengers expectation
• Potential revenue enhancement