Airline Schedule Optimization (Fleet Assignment II) Saba
Neyshabouri
Slide 3
The Fleet Assignment Model In order to develop the mathematical
optimization model for this problem, some modifications should be
made to the underlying time-space network: Constructing a time
space network for each fleet type Adding wraparound arcs for each
airport to make the possibility of keeping and aircraft at an
airport overnight. Adding the count time to the network to be able
to keep track of the total number of aircrafts of a fleet that are
assigned.
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Modified Time-Space Network By incorporating mentioned
modifications our network will change to the following network:
Wraparound arc
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Basic Fleet Assignment Model (FAM) Based on the assumptions of
the model, here is the list of model parameters, data : Decision
Variables:
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Basic FAM Model Having defined the parameters and decision
variables, the Basic FAM model is as the following:
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Basic FAM Model (description) Objective Function Tries to
minimize the (Operational Cost- Generated revenue) of all the
assignments of fleet type k to flight leg I Constraints (assignment
constraint): States that each flight leg must be assigned exactly
to one fleet type.
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Basic FAM Model (description) Constraints (Fleet balance
constraint): This constraints states that for each node and each
fleet type : All the flights of type k that are arrived at n and
are going to stay plus all the flights originating from n that are
going to fly out should be equal to the flights on the ground that
were waiting until the time at node n plus the number of flights
that are going to fly into n. This constraint is the famous Flow
Balance constraint in network flow models. This constraint states
that all the flights that are coming to a node should be leaving
that node at some point, or to state it differently, total number
of aircrafts arriving at an airport (of each type) is equal to the
total number of departing aircrafts. Not satisfying this constraint
will cause the model to accumulate all the aircrafts at one
node.
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Basic FAM Model (description) Constraints (Fleet availability):
This constraints states that for each fleet type : The Sum of all
of flights that has been on the ground during the count time plus
all the flights that has been assigned to a flight leg (flying) on
that time should be less than or equal to the total number of
available aircrafts of that type. These constraints which are
similar to resource availability constraints, will make the
optimization model not to assign more than existing aircrafts to
flight legs.
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Basic FAM Model (description) Constraints (Variable
Definition): This constraints states that for each fleet type and
each arc : Assignment variables are 0-1 variables that shows the
decision made about that particular assignment. Non-negativity
constraints for flights on the ground which states that variable
can not assume negative values. Note that for the flights on the
ground there is no indication of the variable being integer, while
these variables are inherently integer! In some network problems,
The integer constraints can be relaxed thanks to the special
structure of the problem, which makes the problem much easier to
solve.
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Basic FAM Weaknesses Basic FAM is BASIC! It captures some of
the most important constraints of the problem. It is not covering
constraints such as: Noise restrictions Maintenance requirements
Gate restrictions Crew considerations
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Solving FAM FAM is an integer, multi-commodity network flow
problem with side constraints. It can be solved (not always easily
and not for all the problems) using off- the-shelf optimization
software packages such as: CPLEX Xpress-MP Here is an example of a
problem size and solution time needed:
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FAM and Impacts Here are some examples of the impacts of using
Fleet Assignment Models (FAM) :
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Extending Basic FAM There are several shortcomings in Basic
FAMs: Spill costs: Revenue lost when the assigned aircraft for that
flight cannot accommodate all passenger demand. Recapture costs:
When an airline spills passengers from one flight leg and then
books them on other flights in the airlines network. Most FAMs
consider only aggregate demand and average fare by flight leg or by
passenger itinerary which can compromise the accuracy of the
estimated spill costs. Most FAMs assume that demand is static over
the schedule period!
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Example: Problems of Basic FAM Consider the data for the FAM :
(X-Z :connecting through Y)
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Example: Problems of Basic FAM Given the data provided in the
tables: Maximum possible revenue is : Here is the table for each
possible fleet assignment combination: Assume fleeting I is
selected: Demand for flight 1 is 150 (75+75) and demand for flight
2 is 225 (150+75) by this fleeting, each flight leg will have
capacity of 100 (capacity of aircraft A) so 50 passengers on flight
1 and 125 (225-100) passengers on flight 2 will spill. Since the
fare for X-Z is less than the sum of fares for X-Y and Y-Z The
Revenue maximizing strategy is to spill passengers from X-Z (50)
with the cost of 15000 (50*300) and flight 1 will have enough
capacity. Because the local fare of Y-Z is less than X-Z, 75
passengers are spilled from Y-Z itinerary (with the cost of 16875)
and since 50 are already spilled from X-Z, flight 2 will be at
capacity and total spill cost for this fleeting is
15000+16875=31875
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Example: Problems of Basic FAM Following the same line of
calculations, the spill cost for each possible fleeting is shown in
this table: Considering the contributions of each fleeting, the
optimal solution for this small example is fleeting I.
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Example: Problems of Basic FAM If instead of considering
network effects, we could calculate spills on a leg-based fashion.
In this case, the objective is to minimize the spill cost for each
individual flight leg, independent of the effects on the other
flights in the network. The strategy is to spill passengers
greedily in order of increasing fare, until the number of
passengers equals the capacity. In our example, local passengers
are always spilled in favor of keeping the connecting passengers
with a higher total fare so for the fleeting I: 50 X-Y passengers
are spilled at fare 200 125 Y-Z passengers are spilled at fare of
225
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Example: Problems of Basic FAM Calculating the spill costs on a
greedy leg-based approach will yield the following table: Comparing
to the network itinerary-based calculation: The main reason for big
differences is that greedy approach does not capture flight leg
interdependencies or network effects.
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Considering Network Effects For our example, it was easy to
enumerate all the possible spill costs for each fleeting, but for
problems of real sizes (thousands of flight legs), enumeration can
be computationally very expensive if not impossible! Researchers
have developed mathematical models and optimization approaches for
large scale problems, and conclude that the benefits of modeling
network effects can be significant: Network-based fleet assignment
approach at AA has yielded annual improvements in revenue for 0.54%
to 0.77% (Jacobs et al., 1999). Increased annual contributions from
$30 to over $100 million have also been reported as achievable at
United Airlines when fleeting decisions are made using network-
enhanced FAM instead of a leg-based FAM (Barnhart et al.,
2002).
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Extended Fleet Assignment Problems To capture network effects
and extend Basic FAM to include passenger spill decisions,
following inputs to the problem should be considered: An airlines
flight schedule Itinerary-based passenger demand Aircraft operating
cost data
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Extended FAM Modeling To include the network effects in FAM we
need to keep track of number of passengers assigned to each
itinerary in airlines network.
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Extended FAM Modeling Data and variables:
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Extended FAM (IFAM) Model
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IFAM: Description Objective function: The objective is to
minimize the sum of the operating cost of flying leg I with
aircraft type k for all flight legs and fleet types and the
negative of the total revenue. Constraints: First 3 constraints are
the same as in Basic FAM model
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IFAM: Description Constraints: Passenger flow and capacity
constraints: This constraint makes sure that for each aircraft in a
fleet, the number of passengers assigned to that aircraft will not
exceed its capacity. Demand Constraint: This constraint will limit
the total number of passengers traveling on or spilled from
itinerary p to the unconstrained demand of p.
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IFAM: Description Constraints: Passenger flow and capacity
constraints: These set of constraints will bound the variables to
be positive and also fleet assignment variables should be 0-1.
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IFAM vs. FAM Using IFAM will cause the problem size to grow
which can cause computation inefficiency or tractability issues, On
the other hand it will provide significant economic benefits thanks
to considering network effects.