Content :Content :1. Introduction2. Traffic forecasting and traffic allocation

◦ 2.1. Traffic forecasting◦ 2.2. Traffic allocation

3. Fleeting problems◦ 3.1. The daily fleet assignment model

3.1.1. Minimizing number of aircraft 3.1.2. The flight network for the daily fleet assignment model 3.1.3. Integer programming formulation

◦ 3.2. Swapping equipment types in a daily fleet assignment

◦ 3.3. Weekly fleet assignment

Mathematical models in airline schedule planning: A survey (Gopalan,1998)

1. Introduction1. Introduction

What’s the airline schedule?What’s flight leg?What period of time ?

The column labeled frequency indicates the days on which this flight is offered

Two different flights may share a flight number. Typically, this is the case for one-stop or two-stop flights.

Flt. No.FromToDep.Arr.Frequency

547BOSPIT525p711p12345

1753BOSPHL730p851p1234567

............…..…..…..…

The schedule planning The schedule planning paradigmparadigm

Operational difficulties

1- traffic forecasts for the month2- tactical and strategic initiatives 3- seasonal demand variations4- …

2. Traffic forecasting and 2. Traffic forecasting and traffic allocationtraffic allocationTraffic forecasting:

◦ Airline demand that tends to be seasonalTraffic allocation:

◦ determines how the demand will be allocated across the various available itineraries competing for it. Thus, an allocation model determine the approximate market share of each competing airline based upon the schedule offerings

Traffic allocationTraffic allocation:: the traffic share of an airline:

MSi : the market share of airline iFSi : frequency share of airline i in a particular marketThe factor β is determined from prior history by

regressionNote:

◦ ignores the fact that passengers have a stronger preference for traveling at certain times of the day

UnconstrainedUnconstrained demand:demand:ascribe a “desirability” factor to each itinerary. the market share of an airline is the sum of the

desirabilities of all itineraries run by the airline divided by the sum of all desirabilities of all itineraries serving the market.

we can always estimate the demand for a particular itinerary by multiplying the total market demand by the itinerary’s desirability fraction

in practice, heavily traveled itineraries “spill” passengers

flight (leg) based spill model:flight (leg) based spill model:

f (x): probability density of demand for a flight (Normal with a given mean µ and variance σ)

C: capacity of the aircraft assigned to the flight ES: expected number of spilled passengers

◦ evaluate the expected revenues from assigning different equipment types to a particular flight

◦ the expected opportunity cost of spilling passengers because of insufficient capacity

3. Fleeting problems (3. Fleeting problems (daily fleet daily fleet assignment)assignment)Advantages:widest application in practice Its not too restrictive in practice as most airlines fly the

same schedule, at least during weekdayseasier to schedule gates, crew, and maintenanceMore benefits

◦ intangible benefits like ease of scheduling maintenance, crew, and gates.

◦ it is not clear if it is possible to capture more demand by going to a variable fleeting

3. Fleeting problems (3. Fleeting problems (daily daily fleet assignment)fleet assignment)Disadvantages:Airline demand varies by day of the week (it is typically

higher on Mondays and Fridays, lower during the middle of the week and lowest on Saturdays)

It is a common practice among airlines to solve the daily fleet assignment problem and modify it for weekend flying (i.e. most US domestic carriers)

3.1. The daily fleet assignment (model) :

Goal: ◦ matching capacity to demand as much as possible ◦ The objective function used is combination of the

operating cost and the opportunity cost of spilling passengers

Constraints: ◦ rigid constraints:

the right aircraft should be present at the right place at the right time

to ensure that we do not assign more aircraft of each type than present in our fleet

every flight leg in the schedule has to be assigned exactly one equipment type

◦ constraints relating to crew and maintenance

Case studyCase study::American, Delta, and USAir have reported routinely

solving the problem to near-optimality in a few hours on a workstation, a feat unthinkable just a few years ago

A simple schedule of two flightsA simple schedule of two flights

a daily fleet assignment is impossible.

3.1.1. Minimizing number of aircraft

polynomial-time algorithms for the problem

3.1.2. The flight network for the daily fleet assignment model

3.1.3. Integer programming formulation

The basic constraint classes

◦ specify that each flight leg gets assigned exactly one equipment type

◦ equipment balance is maintained at every node

◦ number of aircraft used in any equipment type do not exceed the number of aircraft of that equipment in the fleet

3.2. Swapping equipment types in a daily fleet assignment

Introduction: changing the equipment type on a specific flight leg from the assigned equipment type to another equipment type.

The solution of integer programming method is a set of flight legs of an equipment type that form disconnected components, a situation (called a locked rotation) that may be unacceptable for maintenance routing purposes

Use when:◦ equipment failure◦ schedule disruptions◦ unexpected demand◦ to connect up the components and unlock locked rotations◦ rebalance a fleeting

Same-Day Swap Same-Day Swap AlgorithmAlgorithmThe Same-Day Algorithm finds a swap

opportunity between two fleets, if one exists, that involves the fewest number of changes to the existing assignment

Same-Day Swap Algorithm (steps) :

Same-Day Swap Algorithm ◦ Step 1. Remove all the overnight arcs

from the flight network.

◦ Step 2. Reverse the direction of all arcs in the remaining flight network that have the equipment type B assigned to them. Call this network the swap network. Let s be the head of f (the flight whose equipment type has to be swapped) and t its tail.

◦ Step 3. Find a path from s to t in the swap network that uses the fewest number of arcs. This can be done by running a shortest-path algorithm from s to t with all arc costs equal to one. The arcs involved in the path represent a same-day swap opportunity that involves the fewest number of changes to the existing assignment.

3.3. Weekly fleet assignment

A heuristic solution:◦ first solve the daily fleet assignment problem for a representative

day of the week

◦ solve a two-day fleet assignment problem for only Saturday and Sunday and require the “overnight flow” to match the Friday evening count as suggested by the daily fleet assignment problem

Case study:◦ for most domestic airlines in the US, weekend schedules do

differ from weekday schedules. Also, airlines with a strong presence in international flying change their schedules significantly even during the regular week