Optimization of Airport Ground Operations

8/10/2019 Optimization of Airport Ground Operations

1/23

1

Optimization of Airport Ground Operations

Integrating Genetic and Dynamic FlowManagement Algorithms

Jes us Garca a , Antonio Berlanga a ,Jose M. Molina a and Jose R. Casar ba Dpto. Informatica Univ. Carlos III de Madrid e-mails: { jgherrer@inf, aberlan@ia,molina@ia }.uc3m.es b Dpto. Senyales, Sistemas y Radiocomunicaciones Univ. Politecnica de Madrid e-mail: [email protected]

This work * presents a new method to automaticallysearch the best routes and schedules for airport groundoperations, within a decision support system for towercontrollers, a hard real-world application. It exploresthe potential advantages of hybridizing two comple-mentary types of algorithmic approaches to nd solu-tions with minimum delay: a genetic algorithm and atime-space dynamic ow management algorithm. Anintegration scheme to combine the strengths of eachone and exploit their complementary nature has beenanalyzed. The proposed ow-management algorithmdeterministically optimizes an over-simplied problem,while the genetic algorithm is able to search within amore realistic representation of the real problem, butsuccess is not always guaranteed if search space grows.The performance of this combination is illustrated withsimulated samples of a real-world scenario: ground op-erations in the Madrid-Barajas International Airport.

Keywords: Network Flow Management, Hybrid Ge-netic Algorithms, Scheduling, Routing, AirportGround Operations

1. Introduction

The automatic planning function for airportground operations will assist tower controllers

* Funded by CICYT (TIC2002-04491-C02-02) and CAM(07T/0034/2003 1)

to improve surface operations and safety in theprocess of traffic ow management. During peakperiods of traffic ow or during capacity drops dueto weather deterioration, demand temporarily ex-ceeds the available operational capacity, and se-vere congestion appears resulting in expensive de-lays for users and airlines. Research on new surfaceprocedures has been carried out to increase effi-ciency in the usage of current resources, in order toimprove the current methods based on xed rout-ing schemes and pure mental processes [2]. Thisset of new technologies and procedures support-ing the future airport traffic management consti-tute the A-SMGCS (Advanced Surface Movement,Guidance and Control Systems, [9],[10],[29]) con-cept, whose development will increase safety andefficiency in operations. Planning, the least matureof A-SMGCS functions, is conceived as a support-ing tool to help in the selection of appropriate se-

quences and ground routes for demanded opera-tions. The goal is the integration of the availableinformation in A-SMGCS to automatically providecontrollers with appropriate suggestions for com-plex situations.

A prototype for A-SMGCS system is currentlybeing implemented at Barajas-Madrid interna-tional airport, the busiest airport in Spain [5][14].There, the current mode of operation is a segre-gated scheme, with one runway exclusively usedfor landings and the other for takeoffs, because of the simple advantage of direct management andxed congurations. However, a mixed exploita-

tion, with runways used both for landings andtakeoffs may potentially increase available capac-ity, since the en-route separation between aircraftimposes lower utilization of runways when they areonly devoted to landing. Besides, the ight-planscurrently have pre-assigned gates and runways,with xed routes from gates to runways depend-

AI CommunicationsISSN 0921-7126, IOS Press. All rights reserved


2/23

2 Garca et al. / Optimization of Airport Ground Operations Integrating GA and D-MCMF

ing on airport conguration, so that ground con-troller selects the starting time to meet the timeslots delivered from European Central Flow Man-agement Unit (CFMU). An automatic scheme dy-namically selecting appropriate routes and sched-ules for demanded departures to obtain minimumground delay is currently open to medium-termresearch, considering also the analysis of poten-tial advantages with respect to current modes of operation. Some previous techniques have beenstudied in the authors research group, developedand integrated in a prototype system, IPAGO [13](Intelligent Planning for Airport Ground Opera-tions). A full description of planning problem de-tails and solving strategies is presented in [15], to-gether with the implications to the real problemof tower controllers handling dynamic situations.Several techniques were analyzed to improve theperformance currently achieved with conventionalprocedures (a pure mental process performed byhuman controllers). The continuation developed inthis current work explores the advantages of hy-bridizing complementary algorithmic approachesto nd better solutions, combining their strengthsand complementary nature. The information pro-vided by a deterministic ow-management algo-rithm, referred to an over-simplied problem, isapplied here to guide a genetic algorithm, poten-tially able to search within a more realistic repre-sentation of the real problem.

The planning problem of searching optimal

time-space assignments has a NP-hard complex-ity [6], due to the complexity induced by dynamictraffic assignment when route and time of depar-tures are simultaneously optimized, and usuallyhave several optima. The search has a double na-ture of combinatorial space (the selection of theset of routes for all individual demands) and con-tinuous adjustment (time intervals for operationskept in holding areas), with constraints to sat-isfy the safety requirements. Due to this complex-ity, the problem, represented with an appropriateformulation, is suitable to apply Articial Intel-ligence techniques such as planning or stochastic

optimization [31].This particular problem of airport traffic man-

agement can be regarded as a special case of TrafficFlow Management, appearing both in communica-tion and transport networks elds. In the case of airspace traffic, we can mention approaches basedon temporal and spatial Operations Research tech-

niques complemented with heuristics [41], [35],[32], [43], dynamic programming [33], and evolu-tionary algorithms for different levels of Air Traf-c Control, such as traffic assignment [6], [36], de-sign of airspace sectors [16] or en-route conictresolution [8]. In the special case of traffic owmanagement at airports, there has been a stronginterest to improve the use of available capacity.Simulation tools modeling airport operations, suchas TAAM 1 [39], SIMMOD2 (FAA) or TARMAC 3

(DLR) [30] have been applied to analyze alter-native congurations and bottlenecks in airportssuch as Schiphol [39], Orly, C. De Gaulle [24] orSt. Louis [20]. Simulation has been complementedwith data analysis to study the capacity enhance-ment derived from expansions or recongurationsin airports such as DWF [27] or Newark [11].

With respect to specic techniques for planningairport operations, several problems with differ-ent levels of detail have been addressed. Most ap-proaches are oriented to optimize the global use of runways or minimize congestion at destination air-ports. So, there are techniques aimed at computingappropriate landing sequences [4] and schedulingto assign multiple runways to landings (segregatedmode)[44]. Integer programming has been also ap-plied to on-line optimize the mixed assignment of takeoffs and landings to runways depending on de-mand [18], [19], and recently expanded to includecollaborative decision-making paradigms [17], [20].Other approaches decide the delays on ground to

solve future problems at arrival on destination air-ports [21], [34], [1]. The major challenge is to ad-dress the details of planning ground operations,considering alternative surface routes for taxiing.Some suboptimal approaches search for solutionsby considering individual operations one by oneand the previously assigned traffic as constraints,[38], while only a few works address the search of global solutions [38], [24], [25]. They are based onheuristics and genetic algorithms to explore appro-priate decisions.

In this work, a new approach is applied to theairport-surface planning problem in the highest de-

tail. Two types of previous approaches for solvingtime-space planning (presented in [15]) are nowhybridized to improve efficiency in the search. Amodied version of the Minimum-Cost Maximum-

1 Total Airspace and Airport Modeller2 Simulation Model3 Taxi and Ramp Management and Control


3/23


4/23


Find routes for operations with minimum de-lays (taxiing length).

Find a sequence of operations and time sched-ule (assignation of time delays) to achieve op-timal use of capacity.

The selected representation of airport resources(runways, taxiways and aprons), is a directedgraph containing constrained capacity arcs, tran-sit nodes and ow-source nodes. All the con-straints to be considered for each interval plannedshould be collected and reected in this graph.Constraints cover the runways state, consideringtime slots previously allocated for other opera-tions, the current and predicted surface-traffic sit-uation, safety alerts and modications placed bycontroller. Source nodes in the graph represent theorigin of demanded operations, basically airportpassenger terminals for take-offs and landings xes

from close airspace. The rest of nodes in the graphare the reference places in the airport layout wherean aircraft can be located, representing both way-points in trajectory, generally junctions betweenrunways and taxiways, or hold-on areas before ac-cessing runways. An aircraft path, or route, is de-ned as a sequence of nodes, each one associatedbesides with an estimated time of arrival. Nodes inthe graph are linked by means of arcs. An arc hasthree attributes: direction, cost, representing thetime needed to cover it, and capacity. The availablecapacity of each arc represents the free space whereow units (operations) may be assigned along time

for each planned interval, and so represents thereal resources to be managed by the system. Theyhave been represented as capacity vectors for eacharc, which will reect the difference between maxi-mum capacity and the already planned operations.When a planned route occupies a certain segment,its capacity is decreased one unit for those inter-vals while the aircraft is supposed to be traversingthat specic segment.

The maximum available capacity for each arc,dened as the number of operations that can enterby time unit, depends basically on the safe min-imum longitudinal separation between operations

and on the aircraft groundspeeds. Specic valuesfor costs and capacities of the arcs in the graph rep-resenting Madrid-Barajas airport, depending onthese characteristics, will be detailed in section 4.

The two basic requirements to the solutionssearched, in order to be useful, are the generationfrom a global and dynamic point of view:

Generation of global solutions imply the con-sideration at the same time of all operations tobe served and the state of all resources, avoid-ing the generation of particular solutions use-ful only for individual interests. So, a globalcost function such as the sum of all delays re-sulting from a certain solution must be eval-uated to decide the most protable actions.

The scheme must integrate dynamically theinformation obtained about the current stateof traffic, operations served, and other eventssuch as indications from controllers or con-icts. So, it should be reactive to the evolutionof global state and select the most adequatesolution at each time. In the case that anom-alous or hazardous situations are detected, ormodications are introduced by controllers,the ow management system should dynami-cally adapt and nd the most adequate solu-tion.

A simple and incremental illustration can be use-ful for explaining what means global planning,compared with other approaches oriented to one-to-one assignment, searching the shortest not-constrained path for each individual operation. Inthe case that a single departure operation in thequeue has to be assigned, considering an emptyairport, the system would obviously provide theshortest path to the closest runway, directly ob-tained with a shortest-path algorithm such asDijkstras or A* algorithms [25]. However, if this

shortest path currently is already occupied withother operation, considering a non-empty but pre-assigned airport situation, the system must decidenow between two basic alternatives: delaying thestarting time until the resources get free or select-ing an alternative route to follow at the same timeby both operations. Finally, situations will involveseveral competing operations simultaneously de-manded for the same planning interval while at thesame time resources will be shared by other op-erations in progress. The system will have to de-cide now their sequence, scheduled timetable, androutes assigned to each, in order to achieve the

nal goal of global minimum delay.So, airport traffic ow management is a planning

problem with particular features. Decisions mustbe taken about the details of a set of operations toserve, being the control tower a centralized posi-tion. It must take into account constraints amongoperations, such as aircraft separation to guaran-


5/23

Garca et al. / Optimization of Airport Ground Operations Integrating GA and D-MCMF 5

Fig. 1. Graph representation of Madrid-Barajas International Airport

tee safety or minimum time intervals in the useof runways, and constraints on available resources,since they can be occupied by other pre-assignedoperations. As an example of constraint, landingsdelivered by Air Traffic Control (ATC) centers at

close airspace centers will have higher priority thatdepartures authorized on surface. If all possiblemaneuvers for individual trajectories were consid-ered, the decision variables for planning would becertainly complex. As a simplication, the systemwill decide only about routes and initial delays,supposing than each operation spends all neces-sary waiting time to meet its assigned slot stoppedin the gate, situation preferred from the points of view of safety and manageability.

3. Solvers for Ground Planning

Once the airport-planning problem has beenrepresented with a directed-graph format in theprevious section, here we propose a hybrid ap-proach, accordingly to the desired goals of mini-mizing times for demanded operations, satisfyingthe constraints.

Firstly, we detail a direct GA approach perform-ing an explicit search using an Articial Intelli-gence technique based on stochastic optimization.Routes and time schedules for all demanded op-erations are represented as decision variables in

a constrained space, where the minimum separa-tions between aircraft are explicitly modeled andthe optimum solution is searched. The search isperformed within the whole space of routes andtime schedules.

In the second place, a hybrid strategy to inte-grate more information in the GA is presented, as asequential application of two phases. First, a deter-ministic ow algorithm provides an initial ow dis-tribution optimized for a simplied problem. Then,a renement is left as a task to perform by meansof GA with specic tness function, taking the ini-tial solution as starting point to search better so-

lutions considering the individual constraints anddetailed scheduling. The ow algorithm handles asimplied airport-planning problem represented asa ow-management one, in order to apply networkow algorithms extended to consider also time as-signments. It represents the demanded operationsas ow units, and determine the paths and time


6/23


intervals in the graph to achieve a maximum owwith minimum transit delays. The required min-imum separations and assumed aircrafts ground-speeds have been translated to time-varying ca-pacities of arcs.

Before explaining the algorithms details, it isimportant to state here that they will deal witha simplied model of airport conditions and air-craft motion on the ground. The main assump-tions are (i) aircraft move on ground with uniformmotion between the gates and runways; (ii) oncethe ground movement plan is delivered for an air-craft, there is no uncertainty about the trajectoryit will follow; (iii) all the delay suffered by an oper-ation is translated to the initial waiting time at thegate, which is the preferred situation under normalconditions. The planning function will decide thestarting time for an operation and, once it startsto move, it will not stop until it arrives at its des-tination. Therefore, all the information about air-port surface occupation and demanded operationsfor allocation is known in advance for a centralizedfunction to decide plans, and information is con-tinuously renewed in time to allow dynamic searchof best decisions against time.

Under these conditions, the algorithms suggestthe appropriate solutions, which can be helpful tothe ground controllers to select the alternatives. Tobe a completely useful tool in operational condi-tions, some further steps should be addressed. Themost important aspect would be a exible interface

to continuously re-dene the problem, taking intoaccount modications made by human controllers.Deviations in operations with respect to allocatedplans could be considered to correctly model theavailable resources and then decide appropriateplans for reaction. Changes in plans should, as faras possible, rule out jumps. To do this, onlythe deviated operations should be considered asvariable decisions, not moving the other alreadyassigned plans. The exception is when, due to adeviation, some allocated operations violate con-straints, in which case they must be also consid-ered for re-planning. Finally, more details about

the operations could be included in the models,such as holding positions at taxiing or before take-off, variations in groundspeed, variable time sepa-rations in runways depending on weight categories,and uncertainty in maneuvers and speed through-out the planned time could be considered in thegenerated plans.

3.1. A Genetic Algorithm for Operation Planning

Network problems are one of the earliest appli-cations of a kind of stochastic global optimizationtechniques labeled as Evolutionary Computation

[6]. Genetic Algorithms search in the space of com-binations of input parameters, providing fast andaccurate solutions. In the eld of transportationmanagement, in the particular case of Air TrafficManagement (ATM), the work developed by [6],[36], [16], [25], [4], [37] proposed GA solutions inorder to improve some aspects of ATM.

We have developed a GA inspired in the pre-vious mentioned works, incorporating ad hoc themutation operator and tness function to sched-ule the demanded operations. The algorithm de-scribed in [22], namely Canonical Genetic Algo-rithm (CGA), was applied in order to obtain thesurface movements plan. We dened a plan as thedeparture schedule and the paths that a set of air-craft follows from gates to takeoff runways. Theobjective is to nd the plan that reduces the aver-age delay per operation, subject to the restrictionof no conict between operations.

J. Holland formally introduced the Genetic Al-gorithms (GA) [28] and, since then, their char-acteristics have made them widely applied to op-timization problems, especially of combinatorialtype. Their main characteristics are robustnessand parallelism in the search, although optimal-

ity is not always guaranteed. The airport surfaceoperation planning, formulated as a combinatorialproblem, has a very large search space and an ap-proximate solution could be satisfactory. Thus, theuse of GA paradigm is justied since the trade-off between qualities of solutions and processing timesis advantageous.

The three most important aspects of using GAsare the denition and implementation of:

Genetic representation. Each solution is codedas an instance of the vector with the deci-sion variables. This codication is called the

genotype of a solution. Genetic operators. The exploration and ex-ploitation of the search space are performedapplying three operators that produce newsolutions from preexisting ones: selection,crossover (genes re-combination) and muta-tion.


7/23


Objective function. The criteria to measurethe goodness of a solution are typically im-plemented by means of an evaluation func-tion, generally referred to as tness func-tion. Using the biological analogy, the geno-type of a solution is expressed as the pheno-type of one individual, and over this individ-ual is applied the tness function.

3.1.1. Genetic representation For the case of air traffic ground plans encod-

ing in a GA, they are codied with two sequencesof numbers with length equals to the demandeddepartures, d.

r = ( r 1 , r 2 ,...,r d ) t = ( t 1 , t 2 ,...,t d ) (1)

For each i-th operation, a plan assigns a route,r i , selected from a predened set with all possi-ble routes, and the time, t i , that the aircraft will

delay its departure from the gate. This specialcodication allow us an easily implementation of the crossover and mutation operators, adapted toproblem characteristic. Restriction of infeasible so-lutions, such as those which have operations as-signed to the same route at the same time, cannotbe taken into account in the codication so tnessfunction will penalize solutions that violate the re-strictions. The codication only restricts r i and t ito be valid values; r i must be within the rangeof possible operations from a set of xed routes,and t i must be an integer value between 0 and themaximum allowed delay.

3.1.2. Genetic operators The main idea behind the GA performance is

the cumulative selection, although this term isnot enough to provide the denite answer to thequestion of why it works. Cumulative selection isnot a new concept at all, appears in stochasticoptimization and other methods like descent gra-dient. The GA innovation is the incorporation of characteristics inheritance and variation trials inthe search. These features, in a simplied form, re-semble to the biological natural selection and areimplemented through the genetic operators: selec-

tion, crossover and mutation.Because the operator must be adapted to a par-

ticular problem, there are many genetic operatorsreported in the literature. In this work, the tour-naments selection [22] was the selection schemechosen for selecting the parent individuals in thepopulation to generate following offspring. The

crossover operator produces new solutions recom-bining the existing ones. In this work, a single-point crossover has been used. The crossover oper-ator must be modied to consider the codicationof plans as two sequences of numbers. The samecrossover point is applied, to both part of two par-ent plans, in order to obtain the offspring.

Two mutation operators are used in this work.One is the traditional mutation described in CGAand the other one is a new operator included tospeed up the appearance of small delay-time solu-tions. A random variation of the delay time, uni-formly distributed in the range [-8, 2], is appliedwith probability 5%.

Figure 2 describes the main steps of the algo-rithm in order to obtain a new population of solu-tions.

P o p u l a t i o n : g e n e r a t i o n ( k )

P o p u l a t i o n : g e n e r a t i o n ( k + 1 )

T o u r n a m e n t s

S e l e c t i o n

C r o s s o v e r

R o u t e a n d D e l a y

T i m e M u t a t i o n

D e l a y T i m e

D e c r e m e n t M u t a t i o n

Fig. 2. Steps of GA algorithm

3.1.3. Objective function Finally, the tness function, which measures

the goodness of solutions, is generically de-scribed here. The description is general to allowboth a direct GA application and the proposed hy-brid approach detailed in next section, where ex-tra information obtained with a ow-assignment

algorithm is included. The tness value measureshow a solution (a plan of operations) solves aproblem, in this case represented as a series of demanded departures registered in a list of two-dimensional vectors, (G, R), containing the depar-ture gates and the takeoff runways. These speci-cations may be included for each individual op-


8/23


eration in the problem formulation, they may beomitted, or they might be the result of applyingthe ow-assignment algorithm.

In order to calculate the tness values of a plan,f p , a monitoring of the surface movements is per-formed with the assumptions enumerated before.The following quality measures are assessed, usedas parameters of the following tness function:

f p = oop G

i

li + wop R

i

li + t plan +50 c 50k+ r (2)

The terms are dened as follows:

When a plan contains an operation using agate G (source) or runway R (sink) differentto those specied, this plan is penalized witha value equal to the time that takes the wrongoperation, li , multiplied by a weight. For the

i-th operation, the time elapsed since the en-gine starts at the gate until the takeoff at run-way, is represented as li . The weights, o andw, are zero when the gate and the runwayare right, respectively. And both weights havebeen xed to 1 for the case of operations witherrors. The reason to enable errors in the as-signment of gates and takeoff runway is justi-ed as an improvement of the search in ampli-tude. Is a consequence of that for this prob-lem, the valid solutions are connected throughsolutions that violate some restriction, andthe genetic search must be produced by means

of little modications over initially feasible so-lutions. Time to carry out the whole plan, t plan . The

simulation nishes when the last aircraft hastaken off. This time, similar to the makespanof workow problems, represents the totalamount of time needed to carry out all re-quired operations on surface, and it is so aglobal target to be reduced as much as possi-ble.

Number of conicts, c. When two aircrafts vi-olate the security distance, a conict is re-ported. Obviously, a plan containing only a

conict is unacceptable; therefore, this pa-rameter is strongly weighted. In the experi-ments, the best plan never has any conictafter the 20th generation.

Number of take-offs, k. The objective is to ob-tain plans that process all demanded depar-tures.

r is the sum of the operations delays, normal-ized by the number of operations that effec-tively nalize. The delay of an operation, ti,explicitly codied for each operation as indi-cated in section 3.1, is measured as the timeelapsed since the beginning of operating planuntil the operation is authorized to departfrom the gate. The maximum time of simu-lation has been xed equal to 5000 seconds.This means that all operations should havenished within this time interval.

Regarding the coefficients in tness function(50x factors for c and k), they were adjusted toobtain appropriate tradeoffs among convergencetime, quality and feasibility of solutions, after re-peating some experiments with different seeds inthe process. The design of the tness function toachieve the best solutions followed a quite heuris-tic approach. Initial values were derived from pre-liminary runs and then they were experimentallytuned.

3.2. A Hybrid strategy to search for planning solutions

The previous general GA approach directly in-cluded the individual operations in the encodedproblem, so the solution is referred to each oper-ation: route assigned and time schedule. Our hy-brid approach incorporates information provided

by a dynamic ow management algorithm, Dy-namic MCMF (D-MCMF). This algorithm was ini-tially proposed by the authors in [15] as an exten-sion of classical Minimum-Cost Maximum-Flow al-gorithm (MCMF) [7], and it will be briey summa-rized later. In this algorithm, the operations to as-sign are rst abstracted as (undistinguishable) owunits to compute the optimum ow distribution. Ittakes the number of operations from each terminaland selects how and when to deliver them to theavailable runways. In this approach the safety re-quirements on separation between operations aretranslated in the ow approach as capacity con-

straints. However, this is considered only as themaximum number of operations that can be sentalong certain arcs to keep enough separations, buta ne adjustment of schedules is needed to satisfythe constraints. Besides, D-MCMF can only han-dle longitudinal separations for operations enter-ing the same arcs, but no constraints among close


9/23


operations in different arcs, which must be explic-itly checked. Other limitation of this ow approachis that some limitations refer to individual opera-tions and should be explicitly considered. For in-stance, some operations can be constrained to de-part only from a certain runway, so only routesending in that runway from the departing terminalshould be considered.

3.2.1. Integration of external ow distributions in the GA

The hybrid approach starts with the computa-tion of ow distribution from the D-MCMF algo-rithm. This solution provides the ow units, x l [k],cl [k], for all arcs in the directed graph (le E) andall time slots considered (k=1,...,N). This informa-tion is taken by the GA to search the solutions intwo aspects: initial population and tness, mod-ifying the pure approach where solutions in the

initial population would be uniformly generated.The ow distribution will be used in the initializa-tion population process of GA. The starting pop-ulation of operations plans is lled with a randomselection of operations matching with the distrib-ution of ow units in the terminals and runways,with starting times randomized within the minuteinterval indicated in this distribution.

The information of the D-MCMF ow algo-rithm, summarized below, has been incorporatedin the tness function with four variations, to ex-plore the different behaviors, all of them consid-ering the rst term in the expression above. The

rst one is labeled as Pure-GA, applying theCGA explained in section 3.1, with a random ini-tial population and the tness function not includ-ing restrictions about ows calculated with theoptimal-ow algorithm (parameters o,w dependonly on external constraints for individual oper-ations, if there exists any). The other three vari-ant hybrid algorithms take the supplied ow dis-tribution, all of them injecting the initial pop-ulation as indicated above. The rst one, labeledas GA+Teminal-Runway Flow, incorporates inthe tness function the likelihood of terminal andrunways distribution with the ows provided. So,

the term o, number of incorrect origin gates,takes into account the departures from the termi-nal gates and the differences with the proposedow distribution, and penalizes the difference. Thesame strategy is applied regarding destination run-ways, the term w. The other two variants, la-beled as GA+Runway Flow, and GA+Teminal

Flow, remove constraints on one of both distrib-utions, which is equivalent to setting constants o,w, respectively to zero in tness computation.

3.2.2. Dynamic ow management algorithms for optimal distribution

This algorithm handles a simplied representa-tion of the airport planning problem, with a di-rected graph with constrained capacity arcs. Thisfact allows it compute in a deterministic way anoptimum solution. D-MCMF extends the classicalnetwork algorithms to include dynamic variationin capacity along time, extensions in the numberof sources and sinks and constraints in nodes. Theproblem addressed under this perspective is sim-plied and open to the further search to addressall the relevant real-world constraints.

Classical algorithms for stationary conditions

Algorithms for ow management on networkscome from Operational Research eld [26,7], specif-ically from optimization techniques applied tointeger-constrained linear programming. They arewell-known methods to provide optimal routes andow distributions in networks under stationaryconditions (all ows are characterized with con-stant values or long-term statistics). A directedgraph (V, E) is the data structure handled, be-ing V a set of nodes and E a set of directedarcs or edges linking the nodes. The network isable to move some commodity along the arcs, be-ing dened the ow as the quantity of commod-

ity moved per time unit. The decision variablesare positive-valued real variables xl, containing theow distribution for all arcs in the network, l E,according to the direction dened by each arc.Each node N in the graph is classied into one of three possible types, depending on the ow bal-ance of arcs leaving the node and arcs arrivingto it, bN : source , i f b N > 0, sink , when b N

Documents

Optimization of Airport Ground Operations