Improving Revenue by System Integration and Cooperative Optimization

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Universität Paderborn
Stochastic Fleet Assignment and Disruption
Management for Airline Planning
Jan Ehrhoff, Sven Grothklags, Ulf Lorenz
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• Dr. Ulf Lorenz
• University of Paderborn
• Assistent in the research group of Prof. Dr.
  Burkhard Monien
• 
  (theoretical computer science)
• I am working "between theory and practice"
• airline fleet assignment under uncertainty
• parallel and distributed computing, also
  reconfigurable architectures
• algorithms and experiments
• game tree search
• co-author of the chess program Hydra (best in
  the world)

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Agenda

• Motivation

• Fleet Assignment in Airline Planning

• Desire for Improvement

• Modeling

• Problem Description

• What is to be optimized?

• Classification of the Problem and Solutions

• Generelly

• The Repair Game

• Simulations

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Motivation
Fleet Assignment
MarketModeling
Crew Pairing
OperationsControl
NetworkDesign
Rotation- building
RevenueManagement
Crew Rostering
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Motivation
State-of-the-Art Fleet Assignment - Mathematical
Model
xl,f1 ? leg l is assigned to subfleet f
PAD
leg l
yv,v aircraft on ground
FRA
xl,f
yv,v
MUC
v-
v
v
ground arc (v-,v)
flight event
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Motivation
long- and midterm planning profitability is
increased with the help of better utilization of
recources. consequence plans are more "narrow".
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Motivation
Operations Control realization of the plan with
the help of Ops-Control
Ops-Control fights against 'disruptions'
disruptions cause delays, fleet changes and
cancalations, which cause connections interrupts,
ad-hoc crew re-assignment and slot-problems ...
X
Cost and Trouble!
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Motivation

• Desire of planners and re-planners
• plans and repaired plans should be constructed
  such that,
• Ops-Control can quickly lead the disrupted plan
  into the old one,
• with low-cost repair-plans
• whose further repair after further disruptions
  will be inexpensive

• Problem Plan data are not exactly known at
  planning time.
• Instead Probability distribution can be
  assumed.
• Approach
• missing reliability must be incorporated in
  problem formulation
• time is made discrete
• probability distributions are made discrete
• ? multi-stage-stochastic planning problems
• ? game against Nature

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Measurement of Results Experiments in a
Simulation Environment

• Simulator acts as a communication model
• demand intuitively near to reality, and useful
• discrete time d15 minutes per interval,
  everything happening within one time-interval is
  interpreted as simultanously.
• departures are event points.
• at each event point a leg which causes an event
  can be delayed 30, 60 or 120 minutes with
  certain probabilities, or it can be cancelled.
• Simulator examines events between time T and Td
• procedure simulator informs so called 'Engine'
  about disruptions, waits for corrected solution
  and jumps ahead further d minutes.

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User View
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User View
6.54
2321
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Measurement of Results Experiments in a
Simulation Environment

• To beat The 'Myopic MIP-solver'
• Re-planning process similar to original planning
  process, with the help of a time-space networks.
• cost function differs (because changes dominate
  the costs)
• c(TIM,EQT,CNL,revenue) 50TIM 10000EQT
  100000CNL revenue
• delays and cancellations are modelled
• cancellation binary decision variable xl, for
  each leg l.
• xl, 1 if and
  only if leg l is cancelled
• delays D-many variables for subfleet f, serving
  leg l D 4 for delays 0, 30, 60,
  120 minutes
• The output is an optimal repair-plan under the
  assumtion that no further disruptions will ever
  occur.

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Complexity
Stochastic Fleet Assignment
plus additional finite probability distributions
for each leg for cancellation and delays and
optimal expected profit to be computed is PSPACE
complete
by reduction to a variant of SSAT
?x1Rx2?x3Rx4?... SAT-formula f is fullfilled
with probability gt 0.5
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Model Building What are We Aiming at?
game tree complete stochastic decision tree
value of the root
best decision
time
5.76
5.23
5.76
0.7
0.6
0.4
0.3
4.6
6.7
4.5
6.6
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1.5
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max-player
Nature / average player
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Model Building What are We Aiming at?
Def. Strategy
value at root
best decision
time
5.76
5.23
5.76
0.7
0.6
0.4
0.3
4.6
6.7
4.5
6.6
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• is a subtree
• contains root, and
• for one player one successor per node
• for the other all successors

max-player
Nature / average-player
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Model Building What are We Aiming at?
Def. Plan, Schedule
value of the root
decision corresponds to a 'emission plan'
time
5.76
5.23
5.76
0.7
0.6
0.4
0.3
4.6
6.7
4.5
6.6
5.6
1.5
4.5
4.4
4.6
6.6
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0.1
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• is a path
• contains the root, and
• the node-leaving strategy edges for the
  max-player
• the most likely edge of Nature's nodes

max-player
Nature / average-player
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Model Building, Example
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Modellbildung, Beispiel
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Possible Solution Approaches
Method 1 Enumerate (in sophisticated manner) all
strategies, take the best and
present their plan. Method 2 Enumerate all
scenarios (possibly implicitly), formulate a
deterministic planning problem and
solve it with known
BranchBound techniques. Method 3 Because
there are more strategies than plans,
the strategies can be partitioned
according to their plans.
Enumerate all plans, analyse their strategies and
present the best
plan. Remark Because of PSPACE hardness, we
cannot bound the number of scenarios for method
2 polynomial, except if PSPACE
NP.
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Contribution of Game Tree Search
1. Often error-reduction property of game trees.

• select an appropriate partial tree at the top.
  Roughly a depth-t tree, t as large as
  possible.
• put heuristically estimated values at the leaves
  and examine the selected partial tree.

Search
2. The Alphabeta Algorithm examines
2-person-minimax-trees sufficiently efficient
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Re-Planning
Heuristic values are needed.
time
5.76
5.23
5.76
0.7
0.4
0.6
0.3
4.6
6.7
4.5
6.6
5.6
6.6
1.5
4.5
4.4
1.5
4.6
6.7
0.6
0.3
0.6
0.4
0.9
0.1
0.7
0.4
0.6
0.4
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Master-plan should be the best-possible plan.
But available is 'only' an emission plan Eval
measure distance to master-plan
max-player
Nature / average-player
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The Game
nodes state of the system plus history possible
actions of max-player Choose local new
repair-plan. contains ACH (Aircraft
Change), TIM (Time Shift),
EQT (Equipment Change), CNL (Cancel) move
generator generate small list of plausible
actions. moves of Nature information update
- delays of
flights - damaged
aircrafts -
changes numbers of passengers move generator
generate plausible subset of all possible
disruption combinations. evaluation function
- penalties for changes of the old
plan (heuristisch, statisch) - distance
to master plan
(e.g. changed legs,
time
shifts,
fleet changes,...)
- penalties for lost
passengers
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The Game Tree
time
f(path)
master-plan
g(state)
goal consider the most important possible
disruptions of the future for the evaluation
of the present situation.
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Forecasting Algorithm
Search algorithm value mimav(node v, bound
alpha, bound beta) generate successors
v1,...,vb of v value val 0 if
lower-bound(v,alpha) lt alpha return alpha if
upper-bound(v, beta) gt beta return beta if b
0 return h(v) for i 1 to b if v
is MAX-node alpha max(alpha,
mimav(vi,alpha,beta) if alpha ? beta
return alpha if i b return alpha
else let w1,...,wb weights of
v1,...,vb val mimav(vi,alpha,beta)wi
if val L ?bj i1wj ? beta
return beta if val U ?bj i1wj ?
alpha return alpha if i b return
val
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Forecasting Algorithm

• move generator for optimizer
• similar to move generator of Myopic-MIP player,
  but modified BB search
• if good solution found add some cuts and go
  ahead.
• BB stops when 3 alternatives have been found.
• move generator for Nature
• possible moves possible atomic disruptions
  within the next d minutes
• probabilities of atomic disruptions are known

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Experiments
basic data Lufthansa continental plan data plus
data for generating plans Plan 20603 legs, 144
aircrafts within 6 subfleets Master-MIP 220460
columns, 99765 rows, 580793 non-zeros we simulate
14 single days, getting 14 test instances
probability for cancellation/ (120m/60m/30m
delay) 0.003/0.04/0.16/0.24
3.35 cost reduction more measure points gt
forecast significantly better that
Myopic-MIP-solver.
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thanks!