Video Airline Schedule Optimization (Fleet Assignment II)

8/13/2019 Video Airline Schedule Optimization (Fleet Assignment II)

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Airline Schedule Optimization

(Fleet Assignment II)

Saba Neyshabouri


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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 timespace 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 theassignments of fleet type kto flight leg I

Constraints (assignment constraint):

States that each flight leg must be assigned exactly to one fleet type.


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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 leavingthat 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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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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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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Given the data provided in the tables:

Maximum possible revenue is :

Here is the table for each possible fleet assignmentcombination:

Assume fleeting I is selected: Demand for flight 1 is 150 (75+75) and demand for flight 2 is 225 (150+75) by thisfleeting, 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 spillpassengers 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) andsince 50 are already spilled from X-Z, flight 2 will be at capacity and total spill cost for this fleeting is15000+16875=31875


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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 optimalsolution for this small example is fleeting I.


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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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Calculating the spill costs on a greedy leg-based approach will yieldthe following table:

Comparing to the network itinerary-based calculation:

The main reason for big differences is that greedy approach does notcapture 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 ofpassengers assigned to that aircraft will not exceed its capacity.

Demand Constraint:

This constraint will limit the total number of passengers traveling on or spilled fromitinerary 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.

Video Airline Schedule Optimization (Fleet Assignment II)

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