MAGS

An Aco-based Model to Solve the Schedule Generation and Fleet Assignment

Integrated Problem

Daniel J. Caetano and Nicolau D. F. Gualda

Universidade de São Paulo, Escola Politécnica, Departamento de Engenharia de Transportes

Av. Professor Almeida Prado, Trav. 2, n

83, Cidade Universitária, CEP 05508-900, São Paulo, SP, Brazil

Keywords: Air transportation, Schedule generation, Fleet assignment, Metaheuristic, Ant system.

Abstract: Schedule Generation and Fleet Assignment problems usually are solved separately. The integrated solution

for both problems, although desirable, leads to large scale models of the NP-Hard class. This article presents

a mathematical formulation of this integrated problem along with a new heuristical approach, called MAGS,

based on the ACO metaheuristic. Both the exact solution and the one provided by MAGS are obtained and

compared for the case of a Brazilian airline. The results have shown the applicability of MAGS to real

world cases.

1 INTRODUCTION

This paper presents a heuristic model that

incorporates the Ant Colony Optimization

metaheuristic to solve schedule generation and fleet

assignment integrated problems, avoiding model

simplifications that limit its application to real world

problems (Caetano, 2011); (Caetano and Gualda,

2010); (Rabeanety et al., 2006).

Initially, a brief review of a linear programming

model is presented, followed by the Multiple Ant

Colony Group System heuristic (MAGS), based on

the traditional Ant Colony Optimization (Dorigo and

Stützle, 2004). Finally, a comparison between the

results obtained through the metaheuristic and the

optimal results obtained with the linear

programming model is presented, followed by the

conclusions of the study.

2 AIRLINE OPERATIONAL

PLANNING

The traditional approach to solving the fleet

assignment models are based on a space-time

network, where arrival or departure airports are

represented by nodes. There are two basic types of

arcs on this representation: flight leg arcs –

connecting nodes that represent different airports –

or waiting time arcs – connecting nodes that

represent different times at the same airport (Berge

and Hoperstead, 1993 apud Sherali et al., 2006);

(Hane et al. 1995).

These classical models assume that the flight

schedule is previously defined, with every flight

being covered. Traditionally, they do not include

operational restrictions at airports. To overcome

these limitations, it is necessary to define a more

comprehensive model. The model presented in this

paper is based on the concept of space-time

modelling (Berge and Hoperstead, 1993 apud

Sherali et al. 2006); (Hane et al. 1995), extended to

cope with landing and departure slots by the addition

of landing arcs that connect an arrival node to the

first viable departure node, as shown in Figure 1.

The fleet assignment model can be integrated to

schedule generation with the addition of a penalty

for non served demand and relaxing the cover

constraint so that not all flights must be assigned.

The following sets, parameters and decision

variables are defined to describe the model:

 Sets

M: set of all markets, indexed by m; each market

defines a demand and a time window which limits

which flights can serve this demand.

: set of all nodes for aircraft f, indexed by i, j, o, d

or k, representing an airport at a specific time.

Nrd: set of nodes with departure restrictions.

227

J. Caetano D. and D. F. Gualda N..

MAGS - An Aco-based Model to Solve the Schedule Generation and Fleet Assignment Integrated Problem.

DOI: 10.5220/0003673502270232

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 227-232

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Space-Time Network.

Nra: set of nodes with landing restrictions.

F: set of all types of aircraft, indexed by f.

L: set of arcs that represent the movement of

aircraft, maintenance, waiting on the ground or

wrap, indexed by (i, j), being i the source node and j

the destination node of the movement.

Lv: set of arcs that represent flight movements.

Lvd: set of arcs representing flights assigned to a

market.

: set of arcs whose origin time is equal to or less

than t and destination time is after t. The time t is set

to a valid time according to the problem.

: set of arcs associated to market m.

 Parameters

: unrestricted passenger demand on market m.

: number of seats of aircraft of type f.

: unitary revenue for a passenger on the flight

from node i to node j. Since (i,j) represent an

specific flight – including day and time – each flight

may be associated with a specific unitary revenue.

: number of aircraft of type f available.

 Decision Variables

: number of aircraft of type f flowing through arc

(i, j).

: number of passengers associated to the flight

from node i to node j.

: number of potential passengers (demand)

associated to the flight from node i to node j.

The objective function (expression 1) seeks to

minimize the sum of lost revenues. The first term

represents the difference between maximum revenue

for the assigned aircraft and the revenue received

from assigned passengers. The second term is

associated to the lost revenue due to lost demand.



(, )

[] .. .

{

[]

[] }

ij ij ij ij

ij Lv f F

ij ij ij

Min R C x R pa

Rd pa









(1)

1 ( , )

ij Lvd







(2)

|( , ) |( , )

0 ,

ok kd f

ook L d kd L

xkNfF





 



(3)

(, )

ij f

ij Lt

AfF









(4)

|( , )

fFdid Lv

iNrd







(5)

|( , )

fFooj Lv

jNra







(6)

. 0 ( , )

ij ij

Cx pa ij Lv







(7)

0 ( , )

ij ij

dpa ijLv

(8)

(, )

ij m

ij Lm

dD mM



 



(9)

Binaries:

x {0,1} for (i,j) Lvd

(10)

Integers:

x 0 for (i,j) L\Lvd

(11)

d 0 for (i,j) Lv

(12)

a 0 for (i,j) Lv

(13)

Expressions 2 to 4 represent the traditional cover,

balance and number of aircraft restrictions (Berge

and Hoperstead, 1993 apud Sherali et al. 2006);

(Hane et al. 1995).

Expressions 5 and 6 represents slot constraints,

assuring that only one aircraft will depart or land on

those nodes, respectively. Expressions 7 to 9 assure

that each market demand will be associated to each

flight and that the passengers of a flight will never

be greater than the associated aircraft capacity.

The variables representing demanded flight arcs

are binary, and are specified on expression 10. All

other arc variables are integers greater than or equal

to zero, as stated on expression 11, 12 and 13.

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

228

3 ANT COLONY MODELING

Flight scheduling and fleet assignment are

traditionally solved using integer linear

programming techniques such as node clustering and

constraint relaxation. However, practical instances,

representing the operation of major airlines, remain

a challenge, given the computational complexity

involved. On the other hand, there are many

heuristics that are capable of finding very good

solutions to several types of combinatorial problems

(Rayward-Smith et al., 1996 apud Abrahão, 2005),

suggesting the search for heuristics that can provide

appropriate solutions for the problem in lower

processing times. The successful application to

problems like Vehicle Routing Problem (VRP) and

Aircraft Rotation Problem (ARP) draws attention to

the metaheuristic known as Ant Colony

Optimization (ACO), one of the many swarm

intelligence metaheuristics (Teodorovic, 2008).

3.1 ACO Applied to the Schedule

Generation and Fleet Assignment

Integrated Problem

Although it was possible to adapt the basic ACO

metaheuristic to solve the integrated flight schedule

and fleet assignment problem, the results obtained

through such approach were not satisfactory. Since

the basic ACO leads to a single shortest path, it must

be executed several times, assigning one aircraft at a

time and removing the selected arcs from the list,

leading to suboptimal solutions, with objective

function values almost three times the optimal ones.

However, the problem has specific

characteristics that can be used to improve the

overall solution and thus an alternative heuristic is

proposed, called Multiple Ant Group System

(MAGS), incorporating elements of Multiple Ant

Colony Optimization (MACO)(Vrancx and Nowé,

2006), Multiple Ant Colony System (MACS) and

Elitist Ant System (EAS)(Dorigo and Stützle, 2004),

as well as new elements not present on other ACO

metaheuristic variants.

3.2 Multiple Ant Group System –

MAGS Heuristic

MAGS is a multiple ant colony heuristic, such as

MACS and MACO. As in MACO, a solution is

represented by multiple ants; on the other hand, the

number of ants that build a specific solution is

previously known: there must be one ant per aircraft.

The ants that compose a solution are called an ant

group. A group may be composed of ants from

different colonies and, similar to what is presented in

MACS, each colony has a different objective

function. This means that ants from each colony

make decisions based on different criteria. In

MACS, however, pheromones are identical for all

colonies, which means it is substantially different

from MAGS.

The proposed solution construction process is

substantially different from the classical ACO, to

reduce the number of invalid and unrealistic

solutions. During the construction of a solution, the

ants of a group will alternately choose graph arcs.

The group’s ant which will take the next step is

randomly selected and whenever a flight arc is

associated to an ant, this arc will be no longer

available to other ants in the same group, ensuring

the construction of solutions in which two or more

aircraft do not share flights.

Additionally, when a flight is selected by an ant,

part of the flight’s market demand is also allocated,

reducing the demand available for other flights that

share the same market. This strategy avoids the

association of ants to flight arcs for which the

demand is no longer available in that solution. As

the demand for each arc becomes dynamic during

the construction of the solution, the problem

presents similar characteristics to the dynamic

routing in communication networks, as solved by the

AntNet heuristic (Dorigo and Stützle, 2004). The

exclusion of arcs and the demand allocation during

the solution construction have relevant effects on the

results, which are complementary to that provided

by the repellent pheromones proposed on MACO,

which continues to affect the selection probability of

each arc.

Considering the described construction process,

each ant group has the same role of a single ant in

the basic ACO: the group of ants represents the

complete objective function, each ant associated

with a different term of it. The MAGS basic logic is

presented in Figure 2.

As proposed by Dorigo and Stützle (2004), the

nearest neighborhood solution can be adopted as an

initial solution. On the addressed problem, the

"nearest neighbor" was defined as the arc associated

with the minimum revenue loss, avoiding waiting

arcs whenever possible. The objective function value

for this solution is used to determine the initial

pheromone deposit on each arc.

Differently from the basic ACO, the initial

pheromone deposit is not the same for all arcs. Arcs

associated with smaller heuristic values must receive

substantially more pheromones in the initial distri-

MAGS - An Aco-based Model to Solve the Schedule Generation and Fleet Assignment Integrated Problem

229

procedure MAGS

Setup Parameters

best=Nearest Neighborhood Solution

Setup Pheromones

for s Seasons

for g Ant Groups

Create Ant Group

sol=Build Ant Group Solution

sol=LocalSearch(sol)

if( sol < 1,05*best OR

sol > 2*best)

sol=LocalSearch2(sol)

if (sol < best) best=sol

end for g Ant Groups

Evaporate Pheromones

Update Pheromome trails

Update best AntGroup trail

end for s Seasons

end Procedure

Figure 2: MAGS basic logic.

bution than those associated to higher heuristic

values. Considering the basic ACO probability

equation (expression 14), an increased pheromone

deposit value on arcs that have low heuristic value

will also increase their likelihood of being chosen, at

least in the initial heuristic stage.

|( , )

ij ij

il il

lil N



















(14)

The probability equation adopted (expression 15),

however, presents some additional parameters. The

first one is 

, which represents the amount of

pheromone of other colonies, as in MACO, with

their respective coefficient . Additionally, the

parameter 

reduces the probability of selecting a

sequence of several unprofitable flights. The 

value is always 1.0 for profitable flight,

maintenance, and waiting arcs. For unprofitable

flight arcs, its value starts as 1.0 but upon the

addition of an unprofitable arc to the solution, the

value of 

is reduced by 50% for the next

unprofitable flight arc. This value is only reset to 1.0

when a profitable flight is selected to compose the

solution.

|( , )

.. .

ij ij ij

il il il

lil N









 



























(15)

The 

value is proportional to the flight profit,

given that there is demand available in the market.

The 

is made equal to 

for repositioning flight

arcs – which have no markets associated to them –,

waiting arcs and maintenance arcs, since the

heuristic value based on lost revenue on these arcs

would be always non positive. The objective of this

measure is also to reduce the myopic heuristic

behavior, adding more emphasis on historical quality

of the solutions containing a specific arc, which is

represented by the pheromone deposit value.

After the initial pheromone is distributed, g ant

groups are generated, but no changes are made on

pheromones, as in the basic ACO. As the generation

of the ant groups is completed, pheromone

evaporation takes place, at a fixed rate, and then all

g ant groups will update their pheromone trails. As

in the EAS, the best solution will reinforce its own

pheromone trail, leading to convergence toward that

solution.

The pheromone deposit for each ant group is

proportional to the objective function value, as in the

basic ACO, but each ant of that group shall deposit

only part of the group total pheromone: the amount

of pheromone each ant of a group deposits is

proportional to the ant’s contribution to the quality

of the solution represented by that group. The

proposed distribution rule is defined by the

expression 16, where 

is the deposit of each ant,

with 

= 1/C

, where C

is the cost of the solution

represented by the group, calculated through the

objective function. R

is the revenue generated by

that ant and MR

is the maximum revenue that could

be generated by that group of ants.

.{0.5 [ / 2. ]}

fg f g

RMR





(16)

This formulation guarantees that each ant will

deposit a value not smaller than 50% of the deposit

calculated for the group and also ensures that it will

increase when the ant has a large contribution to the

group total revenue.

It is important to notice that when each group

finishes its solution construction, a local search is

performed to improve that solution. This local

search is divided into two steps: LS, which is

quicker and handles all the solutions, and LS2,

which is slower and processes only the solutions that

have a value close to the optimum one or are too far

from it. The LS is a procedure that removes

sequences of two unprofitable flight arcs. This

procedure also includes a corrective heuristic, which

adjusts the solution so that each ant’s terminal and

initial airports are the same. The LS2 supplements

LS, looking for profitable flights that could replace

waiting arcs on each ant’s path.

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

230

3.3 Application and Results

The mathematical model and MAGS were applied to

instances based on a domestic regional airline case

that carries 104 weekly flights and uses three ATR-

42/300 aircraft (for 50 passengers each). Alternative

flight networks were generated with different fleet

configurations, involving Embraer 120 (for 30

passengers each), and Embraer 170 (for 70

passengers each). Some of those instances also

include alternative flights for a new destination,

expanding the base network to 164 weekly flights,

plus thousands of potential repositioning flights with

no markets associated to them.

The demand distribution adopted on each

instance can be of three different types:

 Fixed

: the demand associated to each flight is

fixed at 50 passengers.

 Flight

: the demand is associated to each flight

and is the average demand per flight, based on

values provided by the Brazilian Civil Aviation

regulatory agency –

ANAC (2007).

 Period

: the demand between two airports

associated to a period of day – morning or evening –

is the average demand by day period, based on

values provided by ANAC (2007).

The instances were solved by integer linear

programming techniques through Gurobi Optimizing

software version 3, on an Intel Core2 Quad

computer with 2GB of memory, using four

processing cores and 200GB of available virtual

memory. MAGS was implemented using Java SE

version 6, running on the same equipment while

using only one of the cores, since MAGS was not

implemented using parallel programming.

Table 1 shows the results obtained by both the

exact mathematical model and MAGS for some of

the instances, given a weekly schedule. The values

of the objective function represent the total lost

revenue and, thus, the lower the value, the better the

solution.

Analyzing the results, it is possible to notice that

MAGS leads to results very close to those obtained

by the exact model, with small standard deviations

and much smaller processing times – none of the

instances took more than an hour on each run. The

minimum values obtained, shown in Table 2, are

even closer to the optimal ones: while the average

values are distant by up to 6% of the optimum, the

minima are no more than 3% greater than the

optimum value for each case.

The average processing time per arc was of 0.04

seconds. The processing times, though not directly

linear to the number of arcs, are very low compared

to processing times of the mathematical model

solved with the Gurobi Optimizer, even without a

parallel implementation for MAGS.

Table 1: Exact model and MAGS average results.

Instance

Demand

Type

Fleets /

Aircraft

Arcs

Flight Arcs /

Repositioning Arcs

(Potential)

Exact Model MAGS (10 Runs Average)

O.F.

Value

Time(s)

O.F.

Value***

Std.Dev.

Avg. Time

(s)

Fixed 3 / 3 35.367 312 / 15.636 0 3 0 0 0

Fixed 4 / 4 35.367 312 / 15.636 92.200 3 92.200 0 430

Fixed 3 / 3 44.907 492 / 20.316 175.000 7 177.500 0 803

Flight 3 / 3 44.907 492 / 20.316 897.345 14 916.475 0 828

Flight 5 / 5 74.845 820 / 33.860 809.365 172.800* 855.192 23.398 1.386

Flight 3 / 5 44.907 492 / 20.316 809.365 14.249 843.317 8.468 1.108

Period 1 / 3 14.969 164 / 6.772 870.120 68 878.324 3.615 2.236

Period 2 / 3 29.938 328 / 13.544 814.150 78.133** 831.716 3.220 2.792

Period 3 / 5 29.938 328 / 13.544 788.550 172.800* 807.112 9.452 3.372

(*) Processing was interrupted after the 2-day time limit (172.800 seconds).

(**) Processing was interrupted due to insufficient memory (2GiB RAM + 200GiB harddisk virtual memory).

(***) Average values do not include the construtive heuristic results.

MAGS - An Aco-based Model to Solve the Schedule Generation and Fleet Assignment Integrated Problem

231

Table 2: MAGS minimum results.

Instance

MAGS (10 Runs Minimum)

Value* MAGS / Optimum

Avg.

Time (s)

0 100.0% 0

92.200 100.0% 430

177.500 101.4% 803

916.475 102.1% 828

832.135 102.8% 1.386

833.865 103.0% 1.108

875.470 100.6% 2.236

829.170 101.8% 2.792

801.030 101.6% 3.372

(*) Minimum values do not include the constructive heuristic

results.

4 CONCLUSIONS

This study presented and compared results of two

types of models to solve the flight schedule and the

fleet assignment problems in an integrated way.

Both models incorporate the same objective function

and constraints, including real world operational

restrictions such as slots at airports.

One of the models relies on Linear

Programming. The other, addressed to solve large

scale problems, a heuristic approach called MAGS –

Multiple Ant Group System – is based on the ACO

metaheurisic. MAGS presents a distinctive way to

determine the initial pheromone level on each arc, as

well as alternative rules for the construction of

solutions, each one being represented by multiple

ants. In addition, the multiple ant solution

representation required a new rule for pheromone

updating. The exact model could reach optimal

solutions for relatively small instances. MAGS has

reached very close results to the optimal ones in

smaller processing times, addressing the possibility

to utilize it to solve larger real world problems.

ACKNOWLEDGEMENTS

The authors acknowledge CAPES (Coordenação de

Aperfeiçoamento de Pessoal de Nível Superior) and

CNPq (Conselho Nacional de Desenvolvimento

Científico e Tecnológico) for financial support and

LPT/EPUSP (Laboratório de Planejamento e

Operação de Transportes da Escola Politécnica da

USP) for technical support.

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