Airline Schedule Optimization (Fleet Assignment II) - PowerPoint Presentation

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Airline Schedule Optimization (Fleet Assignment II)

Saba NeyshabouriSlide2

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.Slide3

Modified Time-Space Network

By incorporating mentioned modifications our network will change to the following network:

Wraparound arcSlide4

Basic Fleet Assignment Model (FAM)

Based on the assumptions of the model, here is the list of model parameters, data :

Decision Variables:Slide5

Basic FAM Model

Having defined the parameters and decision variables, the Basic FAM model is as the following:Slide6

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.Slide7

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.Slide8

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.Slide9

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.Slide10

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 requirementsGate restrictionsCrew considerationsSlide11

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:Slide12

FAM and Impacts

Here are some examples of the impacts of using Fleet Assignment Models (FAM) :Slide13

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 airline’s 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!Slide14

Example: Problems of Basic FAM

Consider the data for the FAM : (X-Z :connecting through Y)Slide15

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=31875Slide16

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.Slide17

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 225Slide18

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.Slide19

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).Slide20

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 airline’s flight schedule

Itinerary-based passenger demand

Aircraft operating cost dataSlide21

Extended FAM Modeling

To include the network effects in FAM we need to keep track of number of passengers assigned to each itinerary in airline’s network.Slide22

Extended FAM Modeling

Data and variables:Slide23

Extended FAM (IFAM) ModelSlide24

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 modelSlide25

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.Slide26

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.Slide27

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.