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490 IEEE TRANSACTIONS 0 CO TROL SYSTEMS TECH OLOGY. VOL. 5. NO.5. EPTEMBER 1997
Optimizing Airport Capacity Utilization in AirTraffic Flow Management Subject to Constraints
at Arrival and Departure FixesEugene P. Gilbo
Abstract-This paper formulates a new approach for improve-ment of air traffic flow management at airports, which leads tomore efficient utilization of existing airport capacity to allevi-ate the con equences of congestion. A new model is presented,which first consider the runways and arrival and departurefixe jointly as a single system resource, and second considersarrivals and departures simultaneously as two interdependentprocesses. The model takes into account the interaction betweenrunway capacity and capacities of fixes to optimize the trafficflow through the airport system. The effects are achieved bydynamic time-dependent allocation of airport capacity and flowsbetween arrivals and departures coordinated with the operationalconstraints at runways and arrival and departure fixes a wellas with dynamics of traffic demand and weather. umericalexamples illustrate the potential benefits of the approach.
Index Terms-Airport capacity, air traffic flow management,delay, optimization, queue.
I. I TRaDUCTION
I ABILITY of airport and airspace capacity to meet thegrowing air traffic demand is a major cause of conge tion
and extremely costly delay. Severe congestion during peakperiod when traffic demand exceeds available capacity be-came the everyday reality in the United States and Westernand Central Europe, as well as in some parts of the PacificRim. According to a Federal Aviation Adrnini tration (FAA)report [I] in 1991 23 major U.S. airports experienced morethan 20000 h of annual aircraft flight delays each. The averageairline operating co t of I-h delay is $1600, which impl ies anaverage annual loss of $740 million for the 23 airports. Theprojected growth of the traffic demand will make the situationwor e in the near future if no action are undertaken forcapacity improvements. For example, by 2002 the number ofairports with more than 20000 h of annual delays i projectedto increa e from 23 to 33 if the capacity i kept on the currentlevel. The total annual airline los es for these airports (intoday' co t of delay) would be more than $1 billion.
Europe faces similar if not more acute problems. In 1990,due to airport and airspace congestion, 23.8% of international
Manuscript received January 20. 1995; revised November I, 1996. Recom-mended by Associate Editor. D. W. Repperger. This work was sponsored bythe Federal Aviation Administration and was conducted in the scope of theAdvanced Traffic Management System project at the John A. Volpe ationalTransportation Systems Center. Cambridge, MA.
The author is with the Volpe National Transportation Systems Center,Cambridge, MA 02142 USA.
Publisher Item Identifier S 1063-6536(97)06206-4.
departures within Europe were delayed by more than 15 min[2]. The ituation in Europe i especially complicated sinceits airspace tructure is distributed over a dozen independentcountrie .
It is clear that the phenomenon of growing traffic demandshould be met by a concomitant improvement in airport ca-pacity. The FAA conducts extensive analysis and coordinatesseveral projects to attack the problem.
Possible measures for increasing airport capacity are dis-cussed in [I] and [3]. The long-term programs include con-struction of new airports and expansion of runway systemsat existing airports. The short-term programs consider newoperational method in traffic flow management and capacityutilization a potentially effective measures for improving theexisting capacity resources. Recent analysis showed [4] thatoptimization of the present airport system by the operationaland technological mea ure might result in increasing currenttraffic flow by up to 50%.
This paper considers operational measures for increasingtraffic flow at airport . The work reported in the paper hasbeen conducted in the cope of the Advanced Traffic Man-agement Sy tem (ATMS), the FAA research and developmentprogram that explore, prototype, and evaluates new conceptsin air traffic management automation. The ATMS productsare implemented in the operational real-time Enhanced TrafficManagement System (ETMS), an automated system whichupports the trategic management of air traffic in the United
States. The ETMS ha been installed and used in all FAAARTCC's (Air Route Traffic Control Centers) and TRACON'(Terminal Radar Approach Control Facilities).
Congestion problems occur at an airport whenever trafficdemand exceeds the available capacity. Currently the ETMSMonitor/Alert functionality identifie conge ted periods bycomparing traffic demand and capacity for each IS-min in-terval. Traffic managers strategically control the traffic andresolve the congestion problems by delaying some flights witha ground delay program so that the flow at the airport systemmeets but does not exceed the available capacity.
In this paper, we consider a strategic traffic flow manage-ment (TFM) problem at airports on a IS-min aggregation leveloperating with the predicted traffic demand, traffic flow, andcapacity per 15 min for several hours in advance; flight-by-flight con ideration are beyond the scope of this paper.
1063-6536/97$10.00 © 1997 lEEE
GILBO: OPTIMIZING AIRPORT CAPACITY UTILiZATIO
In [5], a new operational approach to the optimization oftraffic flow at airports wa proposed. The key element ofthe approach is consideration of airport arrival and departurecapacities as interdependent variables whose value depend onarrival/departure ratio in the total airport operation . In contrastto the conventional representation of airport capacity by twoeparate constants (one for arrival capacity and the other for
departure capacity) the airport capacity is represented in [5]by an arrival-departure capacity curve, which determines a setof paired value "arrival capacity-departure capacity" in theentire range of arrival-departure ratios.
The method, presented in [5], is based on the joint consid-eration of the arrival and departure processe at the airport andon the optimal time-dependent allocation of arrival and depar-ture capacitie during an assigned time period. The allocationreflects the dynamics of arrival and departure demand andweather. In other word, the optimization procedure mutuallymatche available capacity and traffic demand. The method,however, was applied only to runway capacity. It did notcon ider the restricted capacity in the near-terminal air pace,in particular, the capacities of arrival and departure fixes.
This paper presents a new optimization model which con-ider the airport (runway ) and arrival and departure fix
capacities jointly a a single system resource. The incomingflow passe through the arrival fixes before landing, and theoutgoing flow passes through the departure fixes after leavingthe runways. The model takes into account the interactionbetween runway capacity and capacity of fixes to optimizethe traffic flow through the airport system.
In general, the total capacity of fixes is greater than theairport runways' capacity. Therefore, one might think that inca e of congestion, the runway capacity, not the capacity offixe, limits the maximum throughput at the airport system.This i true when the traffic demand is distributed more orIe evenly over the fixes. However, extensive analysis of realtraffic at major airports showed that traffic demand, especiallyarrival traffic, i not always evenly distributed over fixes [6].There are time period when some fixes are overloaded whileothers have very small demand. For example, at ChicagoO'Hare International Airport, the demand over arrival fixesis often imbalanced because the traffic comes in waves duringthe day, first westbound and then eastbound, due to the timedifference between the east and west coasts. It may happen thatduring these periods the fixes, not runways, create a bottleneckat the airport system and limit the total traffic.
During periods of conge tion it is very important to properlycoordinate and fully utilize runways and fixes.
The optimization model presented in this paper can be u edby traffic managers and controllers as an automated supporttool for suggesting optimal strategic deci ion on flow manage-ment at airport during period of congestion. In particular, fora given time period, runway configuration, weather forecast,and predicted arrival and departure demand for runways andfixes (input data), one can determine an optimal strategy formanaging arrival/departure traffic at an airport (output), i.e.,how many flights can be accepted (arrivals) and relea ed
491
Arrival Fixes Departure Fixes
Flow via Fix l Queue: AFI•. 1111f-i.e
vr-:DF. ; Flowvia Fuz
, ', ', :DF) :
Flow via rIX2
Flow via FLx3
Flew via Fix n,u..Flow via Fix flcU
Fig. I. The arrival-departure scheme of an airport and its ~
(departures) during congested periods at the airport, how todistribute the arrival and departure flow over the fixes at each15-min interval, and how many flights are to be delayed andfor how long.
To estimate the efficiency of optimal solutions providedby the model, extensive numerical calculations have beenperformed at the Volpe National Transportation Systems Cen-ter [7]. In this paper, we reproduce a fragment of thesecalculations a illustrative examples. In particular, the effectsare illustrated in the example calculated for a congested 3-hperiod at the Chicago O'Hare International airport (ORO).
This paper has been organized as follows. Section II de-scribes a general scheme of arrival-departure system of asingle airport. A mathematical optimization model is presentedin Section III. Section IV contain numerical examples.
II. ARRIVAL-DEPARTURE SYSTEM OF A SINGLE AIRPORT
A simplified operational scheme of a single airport systemthat reflects the arrival and departure proce se at the airportand its fixes is shown in Fig. I.
The system comprises naf arrival fixes AF, ndf departurefixes OF, and a runway system. There are two separate sets ofarrival and departure fixes located in the near-terminal airspacearea (50-70 km off the airport) so that the arrival fixes serveonly arrival flow, and the departure fixes serve only departureflow. The runway system on the ground serves both ani valand departure flows.
The arrival flights are assigned to specific arrival fixes,and, before landing, they should pass the fixes. After leavingrunways, the arriving flights follow the taxiways to the gate atthe terminal. The departure flights, after leaving the gate, areheaded for the runways, and, after leaving runways, go throughthe departure fixes. The departing flights are also assigned tothe specific fixes.
The arrival queues are formed before the fixe (ee Fig. 1).1This means that the flights which passed through the fixes, Imust be accepted at the runways. If there is an arrival queue,a certain amount of flights should be delayed. Some of them \are to be delayed in the air and some of them on the ground atthe departure airport . The departure queue is formed before
492
v dep. cap.2
15
10
5
arr. cap.0 5 10 15 u
(a)
v dep. cap.20-
15
10
5
an. cap.0 5 10 15 u
(b)
Fig. 2. Airport arrival-departure capacity curves.
Time period of intere t, con isting ofdi crete-time intervals of length 6.
(e.g., 6. = 15 rnin); T = N 6..A set of time intervals.
IEEE TRA SACfIO S ON CO TROL SYSTEMS TECH OLOGY, VOL. 5. 0.5, SEPTEMBER 1997
the runway system, and flights can be delayed either at theirgate or on the taxiway.
The arrival and departure fixe have constant capacities(service rates), which show the maximum number of flightsthat can cross a fix in a IS-min interval (or other interval).The e capacitie determine the operational constraint in thenear-terminal air pace.
The operational limits on the ground (runways) are char-acterized by arrival capacity and departure capacity. Thesecapacitie are generally variable and interdependent.
There are a number of major airports with runway configu-rations that practice the tradeoff between arrival and departurecapacitie . For the e configuration the arrival capacity u anddeparture capacity v are interdependent and can be representedby a functional relationship v = ¢(u). Generally, the functionis a piecewise linear convex one. Graphical repre entation ofthe function on the "arrival capacity-departure capacity" planeis called the airport capacity curve [5], [8]-[10]. Fig. 2(a)illustrate a IS-min capacity curve with the tradeoff area. Therepre entation of airport runway capacity through the capacitycurves i a key factor in the optimization model.
For a runway configuration, which is not able to performthe tradeoff the capacity curve degenerates into a rectangle[Fig. 2(b»). There is no tradeoff area, and the runway configu-ration ha constant arrival and departure capacities regardlessof the arrival-departure ratio. In Fig. 2(b), the arrival anddeparture capacities are equal to 15 and 17 flight per 15 min,respectively.
The traffic demand for the airport and fixe are given bythe predicted number of arriving and departing flights per eachl S-min interval of the time period of interest.
An optimization model for managing arrival and departuretraffic at a single airport system i now pre ented.
m. MATHEMATICAL MODEL OF A SINGLE AlRPORT SYSTEM
A. Notation
T
1= {1 2 ... N}" ,<P= P{¢(1)(u),¢(2)(u), ... , ¢(J\J) (u)}
naJndfJ = {1, .. · ,nailK = {1, ... ,ndf}
Ft
A set of M airport capacitycurve that represent the opera-tional limits for all runwayconfigurations under various weatherconditions.An arrival-departure capacity curvethat determines the airport operationallimits at the ith time interval; ¢i(U) E<P
umber of arrival fixe.umber of departure fixe .
A set of arrival fixes.A et of departure fixes.Capacity of the jth arrival fix cor-responding to the ith interval at theairport, i E I j E J.Capacity of the kth departure fix cor-responding to the ith interval at theairport, i E I, k E K.Arrival demand through the jth fixfor the ith time interval at the airport,i E 1,j E J.Departure demand through the kth fixfor the ith time interval at the airport,iEI,kEK.Queue at the jth arrival fix for thebeginning of the ith time interval atthe airport, i E I, j E J.Total airport arrival queue at the be-ginning of the ith time interval, i =1,2,···,N+l.A fraction of the departure queue atthe airport at the beginning of theith time interval, cau ed by the kthdeparture fix, i E 1,k E K.Total airport departure queue at thebeginning of the ith time interval,i = 1,2,···, N + l.Airport (runways) arrival capacity atthe ith time interval, i E I.Airport (runways) departure capacityat the ith time interval, i E I.
GILBO: OPTIMIZI G AIRPORT CAPACITY UTILIZATIO
Flow through the jth arrival fix forthe ith time interval at airport, i EI,j E J.Flow through the kth departure fixfor the ith time interval at airport.iEI,kEK.
B. Assumptions and Simplifications
In this paper, a deterministic single airport model icon id-ered. It i as umed that the following input data are given:
• the time period T for which the traffic managementproblem i to be olved;
• the airport capacity curves for each time interval of theperiod in accordance with a predicted chedule of runwayconfiguration and weather forecast;
• the number of arrival and departure fixe and their ca-pacitie ;
• predicted arrival and departure demand for the airport andthe arrival and departure fixes at each time interval.
There are several a umption and simplifications connectedwith the arrival and departure fixe.
• All the flights assigned to the arrival fixes land at the amede tination airport and there are no other flights followingthrough the arrival fixes to other airports.
• All the flights a igned to the departure fixes are origi-nated from the ame airport and there are no other flightscro ing the departure fixes which are originated fromother airports.
• A flight, which is as igned to a specific arrival or depar-ture fix, mu t fly through the fix and cannot be reassignedto another fix.
• All demand and flows through the fixes are related topecific time interval at the airport.
The latter make it easy to match the demand and capacitiesof the fixe to the demand and capacities of the airport for eachtime interval and hence to keep the demands and flow throughthe fixes and the runway consistent.
For example, if a{ is the arrival demand at the fix j for thetime interval i at the airport then the total demand ai at theairport for the time interval j i equal to the sum of demandat all fixe
no!
a, = 'L.a{i=l
where naJ i a number of arrival fixes.Similarly, if df i the departure demand through the fix j
for the time interval i at the airport then the total demand d,at the airport for the time interval i is equal to the um ofdemand at all fixes
nd/
d.= "dkt ~ t
k'=1
where ndJ i the number of departure fixe .Similar implification ha been also applied to the traffic
flows wI and zf through the fixe .
493
C. Dynamics of Arrival-Departure Processesat the Airport System
The following equation and inequalities determine thedynamics of arrival and departure processes at the airportsy tem.
I) Flow balance at the arrival fixes
Xi - Xi + i ii+l - i ai - Wi' i E i, j E J (I)
with the given initial conditions xi. X~+1 i anout tanding queue at the end of time period T, i.e.,number of flight a igned to arrival fix j that aredelayed beyond the period T.According to the e equations, the number of flight ina queue at the jth fix at the beginning of the (i + l)thinterval is equal to the difference between the demandat the ith interval (which includes the 'inherited" queuefrom the previous lots and the original demand for theslot) and the number of aircraft left the fix during theith interval.
2) The nonnegativity conditions for the queue (I)
wI ~ xl + ai, i E I. j E J. (2)
3) At each time interval, the total arriving flow (from allarrival fixe) can not exceed the runway arrival capacity
n./
'L.w{ ~ Uj.
i=li E I. (3)
4) Flow balance for departure fixes
Yk yk dk k;+1 = i + i - Zi I i E I. k E K (4)
with the given initial conditions r: y~+l is an out-standing queue at the end of time period T, i.e. numberof flights a igned to departure fix k that are delayedbeyond the period T.
5) The nonnegativity of the queues (4)
k < y.k· dkZ, = 1 + r : i E I, k E K. (5)
6) At each time interval, the total departing flow (throughall departure fixes) cannot exceed the runway departurecapacity
nd/
" k < A...( .)Z:; Zi = 'fit U1. ,
k=l7) At each time interval, the flows through the fixes cannot
exceed the fix capacitie
i E I. (6)
wI ~F~i'k <r:Zi = oc.
i E I,i E I,
jEJk E K.
(7)(8)
8) Constraint for runway arrival capacities at each timeinterval
iEI (9)where U, i the upper bound for the arrival capacity atthe ith interval.
494 IEEE TRANSACTIONS ON CO TROL SYSTEMS TECH OLOGY, VOL. 5. 0.5. SEPTEMBER 1997
9) The total airport arrival and departure queues at the at the airport over a period Tbeginning of the (i + l)th interval are obtained by [ 1ummation of queues at arrival and departure fixe , mi~,~~1ze A L•.=1 Xi+1 + B L'.__1Yi+1 (14)
re pectively,n"f
Xi+1 = L xl+1'j=1ndf
Yi+l = L 1-i:l,k=l
10) The non negativity and integrality conditions
iEI
i E I.
U· wj z~ are nonnegative and integer1., t' 1.
iEI,jE.J,kEK. (12)
D. Optimization Model
Fir t of all we formulate an optimization criterion. One ofthe conventional measure of quality of air traffic managementis the total aircraft flight delay time, which is calculated asa urn of delay time of all flights con idered. The amountof delay ub tantiaJly depends on how well the availablecapacity i utilized to meet the traffic demand, e peciallyduring the conge ted periods. Therefore a meaningful criterionof optimality could be the minimization of total aircraft flightdelay time. In case of discrete time, timing accuracy of eachflight i within the range of the time di creteness. In particular,with IS-min discreteness, the delay time can only be expressedthrough the number of IS-min blocks.
In turn, the total number of IS-min block in the total aircraftflight delay time can be expre ed through the queues at theend of each IS-min interval of the time period T. A simpleanaly i of propagation of queues at the end of each is-mininterval over a period T hows that, if all the flights have beenassigned within the con idered time period T, i.e., there is noout tanding flights left unserved by the end of the period, thenthe total number of l S-min blocks in the total aircraft flightdelay time i equal to the sum of queues at the end of eachIS-min interval over a period of time T (we will call it thecumulative queue). Hence, the total aircraft flight delay timei equal to the cumulative queue multiplied by IS min. In thisca e, minimization of total delay is equivalent to minimizationof the cumulative queue.
The queues at the end of each IS-min interval are ea ilycalculated as the difference between demand and capacity (thequeue i equal to zero if demand i Ie or equal to capacity).A queue how the number of flight that cannot be erved ata time interval and should be delayed to some later interval .
According to 2.1 notation, cumulative arrival and departurequeues at the airport over a period of time T are, respectively,
N N
LXi+l and LYi+l. (13)i=l i=l
The queues Xi+l and Yi+l can be expressed throughdemands and capacities by using (1), (4), (10), and (11).
A an optimality criteria, we will con ider the minimum ofa linear function of cumulative arrival and departure queue
(10) where A and B are nonnegative weight coefficient; u,w.and z denote the et of deci ion variables, the airport arrivalcapacitie {ud, and flow {w{} and {zf} through the arri valand departure fixe, re pectively.
If at the end of time period T there are no arrival and depar-ture queue (XN+l = 0 and YN+l = 0) then (14) minimizesalso a weighted sum of total arrival and departure aircraftflight delay. Generally, there can be outstanding queues atthe end of period T, and (14) includes both intermediate andoutstanding queue.
The coefficients A and B in the objective function (14)can have various meanings. For example, A and B candenote an average cost of a unit of time of delay for arrivaland departures, respectively. In this case, (14) minimizes anaverage cost of total arrival and departure delays for the etof Rights considered.
Another application of coefficients A and B i to u e thema control parameter of the model. By varying their valuesit is po ible to vary relative impact of arrival and departurequeues or delays in the objective function (14), which in turncan affect the optimal strategies of managing traffic Rowandallocation of arrival and departure delay at the airport. It iconvenient to normalize the coefficient by dividing (14) by(A + B). Then instead of (14), we can write
(I I)
N
minimize L[QXi+1 + (1 - Q)Yi+ll (IS)u,w,z i=l
where Q = Aj(A + B), 0 ::; Q ::; l.The normalization made it po sible to reduce number of
parameter from two (A and B) to one (0).Coefficient Q varies from zero to one. While increasing the
weight Q for cumulative arrival queue in (IS), the weight(1 - Q) for cumulative departure queue decrease and viceversa, so that varying Q we can increase or decrea e an impactof arrival or departure component in the objective function.Therefore, it is pos ible to interpret the coefficient Q a atradeoff parameter between arrival and departures. It can bealso a ociated with the priority rate for arrivals. In extremeca e of Q = 1 or 0 = 0, we give a full priority to arrivalor departure, respectively optimizing only arrival or onlydeparture operation. In ca e of Q = 0.5, we a sume equalpriority for arrivals and departures (or give no priority to anyof the two operations), and minimize the sum of cumulativearrival and departure queues or the urn of total aircraft Rightdelays for all arrival and departure flights at the airport overa period T. Thus the coefficient Q may be used as a policyparameter that reflects the operational priori tie at the airport.
There is another application of the coefficient Q. It is wellknown (see, e.g., [10]) that in the real world, the maximumarrival capacity is u ually les than the maximum departurecapacity and thus the airport capacity curve are asymmetric.
GILBO: OPTIMIZING AIRPORT CAPACITY TILIZATIO
If the difference between maximum arrival and departurecapacities is significant, then even for equal priority for arrivalsand departure [0: = 0.5 in (15)] the allocation of airportoperations for arrival and departures can be more favorableto departure . The effect of a ymmetry can be com pen atedby increasing parameter 0: above 0.5.
In (15), coefficient 0: icon tant for all time intervals over aperiod T. In a more general ca e, the coefficient 0: can be time-dependent, i.e., 0: = O:i.i = 1,2 "', . It may be connectedwith changing operational policie at the airport for some timeegment of a period T, and as igning various arrival priority
rate at various time interval may reflect the change. Thepossibility to vary the parameter 0: makes the model morereali tic and more flexible in providing alternative olution.In thi ca e, criterion (15) tran form to
N
minimize L[o:;Xi+l + (1 - O:i)Yi+l] (16)u,w,z
i=1
where 0 < O:i < 1.The criterion (16) can be further modified as follows:
N
minimize L 'YdO:iXi+l + (1 - O:t)Yi+d (17)u1w,z i=1
with additional parameter 'Yi. i = 1,2"", N.The parameter 'Yi can be introduced to reflect relative
importance of or difference in values of various time intervals.For example. it can be connected with the reliability inpredicting the traffic and/or airport capacity. Generally, formore distant time interval, that are farther into the future,the reliability of the forecast decrea es. Therefore for thoseintervals the mailer values of 'Yi can be a igned.
Criteria (14)-(16) are the special cases of (17) and can beea ily obtained from (J 7) by the corresponding a signment ofcoefficient O:i and 'Yi.
For all ver ion of optimality criteria, the optimization isachieved by controlling arrival and departure flows throughthe fixes and runways at each time interval through the properallocation of arrival and departure resources.
The decision variables comprise:airport arrival capacities Ui (i = 1,2.··· .N);
• ( * naI) flow wI through arrival fixe (i =1,2,···. 'j = 1,2,···.naI);
• ( * ndI) flow zf through departure fixe (i1.2,···.Njk = 1,2"" ndI)'
There are N * (na.I + ndI + 1) decision variables altogether.Now we can formulate the following optimization problem:
determine the optimal values of airport arrival capacities andthe flows through the arrival and departure fixes which satisfythe optimality criterion (17) (or any other from (J4)-(l6)),subject to (1) through (12).
After the optimal values of the airport arrival capacities u.,have been determined the corre ponding departure capacitiesVi are determined through the airport capacity curve
Vi = trunc <Pi(Ui). i = 1.2.··· .
495
There are variou methods to obtain the optimal solutions.All numerical result pre ented in thi paper were derivedusing the integer linear program techniques.
The deci ion variable are pre ent in the optimization cri-teria (14)-(17) implicitly. Keeping in mind that the criteria(14 )-( 16) are the the pecial ca e of (17), let us transform theoptimization problem (17), subject to (I )-(12), to another formwith the deci ion variables represented explicitly in both theoptimization criteria and the constraints. The transformationi very u eful methodologically because it helps establishthe equivalence between the minimization of queues andmaximization of flows. The duality relation can be also u efulfor computational purposes.
Using the recurrent relationships (I) and (4), the queue atthe arrival and departure fixe can be expressed through thedecision variables and through the original demand and initialcondition as follow :
i i
Xi - Xi + '\' i '\' ii+l - 1 Lap - LWp'
p=1 p=1i E I, j E J (19)
Yk yk '\' dk '\' ki+l = 1 + L p - L zp'p=1 p=1
i E I. k E K. (20)
Then, instead of inequalities (2) and (5) the following non-negativity condition for the queues can be obtained directlyfrom (19) and (20):
Lwt ~xt + La~, i E I, j E J (21)p=1 p=l
·i
L k <v: Ldk i E I, k E K. (22)zp = 1 + p :p=1 p=1
After a series of transformations in the criterion (17) using(10), (I I), (19) and (20), and taking into account the expres-sion (3), (6)-(9), (12), (19) and (20), the optimization problem
formulated a follow:
subject to
i i
Lwt ~X{ + Lat,p=l p=1
i E I, j E J (24)
i E I (25)
( 18)
'\' k < yk '\' dkLZp = 1 + L p'p=1 p=1ndl'\' k <,/..( .)LZ'i ='1-', U"
k=1w{ ~ F{i'
k <FkZi = tu-
(28)
(29)
i E I. k E K (26)
iEI (27)
jEJk EK
i E I.i E 1.
496 IEEE TRA SACTIONS ON CO TROL SYSTEMS TECHNOLOGY. VOL. 5. NO.5. SEPTEMBER 1997
i E 1
tti, wj, zf are nonnegative and integer
i E I.j E J k E K
h x! yk' r! Fk U· .were l' 1, Ai' oc. i are given nonnegative con tantsand cPi(tt) are given nonnegative functions, i E 1,.7 E J, k EK.
The optimization problem (23)-(31) is equivalent to(1)-( 12), (17). It mean that the problem (I)-(12), (17) tominimize a weighted um of arrival and departure queues atthe airport is equivalent to the problem (23)-(31) to maximizethe weighted sum of arrival and departure flows at the airport.
In the ca e of con tant weight coefficients /i and ai (i.e./i = "t, ai = a) for the entire period of time considered, theoptimization criteria (23) is transformed to
Criterion (32) corresponds to criterion (15) which minimizea weighted um of cumulative arrival and departure queues (ora weighted um of total arrival and departure delays) over aperiod T.
IV. UMERICAL EXAMPLES
The presented optimization model has been developed inthe cope of the FAA Advanced Traffic Management System(ATMS). To assess it potential benefit, exten ive numericalexperiments have been performed for several major U.S.airports u ing the real data [7].
In this section, we de cribe several examples calculatedfor the Chicago O'Hare International Airport (ORD), one ofthe busie t airport . Heavy traffic was predicted over the 3-h period on February 12, 1993 from 16:45 to 19:45 localtime. During this period, four arrival fixes and four departurefixe were supposed to be used for the incoming and outgoingflows, respectively. The airport ha ix runways that are u edin different combination or runway configurations. Some ofthe configuration allow the arrival/departure tradeoff withincertain limit and ome of them do not. In thi section, weuppose that during the 3-h period, a runway configuration
with the tradeoff capability will be used.The airport capacity curves for VFR and IFR operational
conditions are shown in Fig. 3. The coordinate of vertice ofthe curves how ome capacity values (the fir t number corre-sponds to the arrival capacity). For example, the coordinatesof vertices of the VFR curve (17, 30), (24, 24), and (28, 15)show that under the maximum departure capacity of 30 flightsper 15 min, the arrival capacity is equal to 17 flight per 15min. Under the maximum arrival capacity of 28 flight per15 min, the departure capacity is 15 flights per 15 min. For a50/50 arrival-departure mix, the airport capacitie for arrivalsand departures are identical and equal to 24 flight per 15 min.According to Fig. 3, the IFR capacitie are approximately 30%les than VFR capacitie .
(30) # of dep/15min
30 +-"'----.:.
- -- -- -----------
(31) 20
20,11
28,15
10
o30 # of arr/15min10 20
Fig. 3. Airport capacity curves for ORO.
Capacities of the fixes are as umed to be the same for arrivaland departure fixes and are equal to ten flights per 15 min foreach fix.
Table I shows the predicted arrival and departure demandat the airport di tributed through the fixes for each 15-mininterval of the 3-h period.
As we can ee from the table, the demand for arrivals anddeparture are di tributed nonuniformly over the 3-h period(see column for the airport demands). The highly congestedintervals are alternated with the relatively quiet ones.
The first 30 min, from 16:45 to 17:15, are extremelyconge ted for both arrival and departures. The arrival anddeparture demands for this half hour are 64 (26 + 38) and68 (36 + 32) flights, respectively, which ubstantially exceedthe airport capacity. For the next half hour, there is still ahigh arrival demand (71 flights) and relatively low departuredemand (24 flight ). The following 45 min are characterizedby low demands (33 arrival and 34 departures). The demandincrease at the next 45 min (85 arrivals and 89 departures).The la t half hour is relatively calm with 25 arrival and 14departure flight in demand.
Below we pre ent some computational results of the op-timization problem (32), subject to (J) through (12), for thedemand data presented in Table I with the following value ofthe parameters: N = 12 (12 intervals of 15-min each in the 3-hperiod), naJ = ndJ = 4 (four arrival and four departure fixes).
The results include the optimal strategies of managing thearrival and departure flow calculated separately for two valueof parameter a (0.5 and 0.7) and for two weather scenario,which were forecasted for the 3-h period. The weather wastaken into account in the optimization model by u ing the VFRand IFR capacity curve from Fig. 3 at the corresponding timeegment . For each trategy the arrival and departure queues
were calculated. To illustrate the propagation of the queues atthe airport and fixes over a 3-h period, the numerical resultsare shown in eparate tables.
A. VFR Weather Conditions
Case 1: Arrival Priority Rate a = 0.5: The optimal solu-tion for thi case is hown in Table II(a). The table containsthe optimal allocation of arrival and departure flows at theairport and the distribution of the flow through the fixe at
GILBO: OPTlMlZI G AIRPORT CAPACITY UTILIZATION 497
TABLE I
ARRIVAL DEMAND DEPARTIJRE DEMAND
TIME FIXES AIRPORT' FIXES AIRPORT1 2 3 4 (TOTAL) 1 2 3 4 (TOTAL)
16:45-17:00 10 11 1 4 26 9 9 9 9 36
17:00-17:15 13 14 5 6 38 8 8 8 8 32
17:15-17:30 15 12 7 8 42 2 2 2 3 9
17:30-17:45 9 15 2 3 29 4 4 4 3 15
17:45-18:00 2 2 2 0 6 2 2 2 1 7
18:00-18:15 1 1 5 6 13 1 3 3 3 10
18:15-18:30 4 0 4 6 14 4 4 4 5 17
18:30-18:45 2 3 7 8 20 9 8 8 8 33
18:45-19:00 5 2 14 19 40 8 8 10 8 34
19:00-19:15 2 2 12 9 25 5 7 5 5 22
19:15-19:30 3 6 2 2 13 3 3 3 4 13
19:30-19:45 2 6 0 4 12 1 0 0 0 1
TOTAL 68 74 61 75 278 56 58 58 57 229
each 15-min interval. The weather conditions are expressed interm of the operational category in the OP. CAT. column.
Optimal value of airport capacities are shown in two right-hand columns. As we can ee from the table, the optimalairport capacities are not constant over the period of timecon idered. They vary to best satisfy the original demand bytrading off the arri val and departure operation at each 15-mininterval.
The queue values at the airport and at the fixes at the endof each 15-min interval are presented in Table Iltb).
Table neb) shows that the original demand has been sati fiedwithin the 3-h time frame: there are neither arrival nor depar-ture queues at the end of the last 15-min interval. Cumulativearrival and departure queues at the end of the 3-h period are143 and 77 flight, re pectively. It al 0 mean that the totalarrival delay and the total departure delay are, respectively,equal to 143 and 77 15-min intervals.
Case 2: Arrival Priority Rate a = 0.7: Let u increase thearrival priority rate from 0.5 to 0.7 to get a new optimalstrategy for managing the flows that is more favorable toarrivals.
For a = 0.7, the optimal value of airport capacities and theflows through the fixes and the airport are presented in TableIlI(a). The corresponding queues are shown in Table Illfb).
Increasing the value of parameter a from 0.5 to 0.7 changedthe allocation of arrival and departure capacities at the airportat each 15-min interval and, a a result, changed the allocationof arrival and departure flows at runways and the distributionof flow through the fixe . The arrival operations have beenimproved at the expen e of departures.
Although, according to Tables Ilta) and lII(a), the cumu-lative arrival capacity increa ed insignificantly (from 281 to286), the arrival queues, and, hence, the total arrival delay,decrea ed significantly [ ee Tables Iltb) and ill(b)]. The total
arrival delay was reduced from 143 to 94 15-min intervals(more than 34%); the maximum arrival queue at the airportwas reduced from 37 to 26. Thi effect was achieved dueto the rational allocation of arrival capacities at each 15-mininterval without dramatic increa e in the total (cumulative)arrival capacity.
At the same time, the cumulative departure capacity isdecreased from 279 to 256, the total departure delay increasedfrom 77 to 185 I5-min intervals, and the maximum departurequeue increa ed from 20 to 32. evertheles, the wholedeparture demand as well as arrival demand is satisfied 0
that there is neither arrival nor departure flights left unservedwithin the 3-h period.
Other strategie of the utilization of runways and fix capaci-ties can be obtained by varying parameter a. This would allowa traffic manager to generate several alternative strategies andchoose the best of them.
B. Changeable Weather
Consider another weather cenario. Suppose that accordingto the weather forecast the IFR conditions are predicted forthe first hour of the 3-h period, and the VFR conditions forthe remaining 2 h.Case 3: Changeable Weather, Arrival Priority Rate a = 0.5:The optimal values of airport capacities and the flows throughthe fixe and the airport for a = 0.5 are presented in TableIV(a). The corresponding queues are shown in Table IV(b).
Tables IV(a) and IV(b) reflect the effect of reduced airportcapacity during the first hour on the overall optimal strategyof managing traffic through the runways and fixes.
The reduction resulted in a ignificant increase of the arrivaland departure queues at the end of the first hour in compari onwith the VFR conditions. The arrival queue increa ed from
49 IEEE TRA SACTIO S 0 CONTROL SYSTEMS TECH OLOGY. VOL. 5. O. 5, SEPTEMBER 1997
TABLE II(a) OPTIMALSOLUTIONFOR ORO (VFR. = 0.5). (b) QUE ESATORO (VFR, a = 0.5)
OP. ARRIVAL FLOW DEPARTURE FLOW AIRPORT
TIME CAT. FIXES FIXES CAPACITYAIRPORT AIRPORT
1 2 3 4 (TOTAL) 1 2 3 4 (TOTAL) ARR OEP
16:45-17:00 9 10 1 4 24 6 6 6 6 24 24 24
17:00-17:15 VFR 10 10 3 1 24 6 6 6 6 24 24 24
17:15-17:30 8 10 2 4 24 6 6 6 6 24 24 24
17:30--17:45 8 10 3 5 26 5 5 5 4 19 26 19
17:45-18:00 10 10 4 4 28 2 2 2 2 8 28 15
18:00-18:15 5 5 9 9 28 1 3 3 3 10 28 15
18:15-18:30 4 0 4 6 14 4 4 4 5 17 17 30
18:30--18:45 2 3 7 8 20 7 7 7 6 27 20 27
18:45-19:00 3 1 10 10 24 6 5 7 6 24 24 24
19:00-19:15 3 I 10 10 24 6 6 6 6 24 24 24
19:15-19:30 2 5 7 10 24 6 6 6 6 24 24 24
19:30--19:45 4 9 1 4 18 1 2 0 1 4 18 29
TOTAL 68 74 61 75 278 56 58 58 57 229 281 279
(a)
ARRIVAL QUEUES DEPARTURE QUEUES
TIME FIXES AIRPORT FIXES AIRPORT1 2 3 4 (TOTAL) 1 2 3 4 (TOTAL)
16:45-17:00 1 1 0 0 2 3 3 3 3 12
17:00--17:15 4 5 2 5 16 5 5 5 5 20
17:15-17:30 11 7 7 9 34 1 1 1 2 5
17:30--17:45 12 12 6 7 37 0 0 a 1 1
17:45-18:00 4 4 4 3 15 0 a 0 0 0
18:00--18:15 0 0 0 0 0 0 0 0 0 0
18:15-18:30 a 0 0 0 0 0 a 0 0 a18:30--18:45 0 0 0 0 0 2 1 1 2 6
18:45-19:00 2 1 4 9 16 4 4 4 4 16
19:00--19:15 1 2 6 8 17 3 5 3 3 14
19: 15-19:30 2 3 1 0 6 0 2 0 1 3
19:30--19:45 0 0 0 0 0 0 0 0 0 aTOTAL 37 35 39 41 143 18 21 17 21 77
37 to 67 flights, and departure queue increa ed from I to 24flight [ee Tables II(b) and IY(b)].
Significant reduction in the airport capacity during the fir thour affected the total airport operation for the 3-h period. Be-cause of the reduction, total anival and departure queue anddelays increa ed dramatically. Moreover, the arrival demandwa not completely ati fied within the 3-h period, and at theend of the period eight arrival flight left unserved [see TableIY(b)). At the same time the departure demand wa completelysatisfied, and there i no outstanding departure queue at theend of the 3-h period.
(b)
If the outstanding arrival queue of eight flights is notsatisfactory for a traffic manager, it is pos ible to obtain thealternative strategie which are more favorable to arrivals byincreasing parameter a. The quantitative effect of increasingthe arrival priority rate from 0.5 to 0.7 to improve the arrivaloperations i illustrated in Tables Yea) and Y(b).
The compari on of optimal olution for Q = 0.5 andQ = 0.7 from Tables IY(a) and Yea) shows that during thefirst hour under the IFR condition, the optimal arrival capacityincrea ed from 68 to 80 flight /h, and the departure capacitydecreased from 68 to 44 flights/h. As a re ult, by the end of
GILBO: OPTIMIZI G AIRPORT CAPACITY UTILIZATIO 499
TABLE m(a) OPTIMALSOLUTIO FORORD (VFR, Ct = 0.7). (b) QUEUESATORD (YFR, Ct = 0.7)
OP. ARRIVAL FLOW DEPARTIJRE FLOW AIRPORT
TIME CAT. FIXES FIXES CAPACITYAIRPORT AIRPORT
1 2 3 4 (TOTAL) 1 2 3 4 (TOTAL) ARR DEP
16:45-17:00 10 10 1 4 25 5 5 5 6 21 25 21
17:00-17:15 VFR 10 10 4 4 28 4 4 4 3 15 28 15
17:15-17:30 10 10 3 5 28 3 4 4 4 15 28 15
17:30-17:45 10 10 3 5 28 4 4 4 3 15 28 15
17:45-18:00 9 10 6 3 28 4 4 4 3 15 28 15
18:00-18:15 1 5 5 6 17 6 7 7 7 27 17 3018:15-18:30 4 0 4 6 14 4 4 4 6 18 17 30
18:30-18:45 2 3 7 8 20 7 7 7 6 27 20 27
18:45-19:00 5 2 10 10 27 4 4 5 4 17 27 17
19:00-19:15 2 2 10 10 24 6 6 6 6 24 24 24
19:15-19:30 3 6 8 10 27 4 4 4 5 17 27 17
19:30-19:45 2 6 0 4 12 5 5 4 4 18 17 30
TOTAL 68 74 61 75 278 56 58 58 57 229 286 256
(a)
ARRIVAL QUEUES DEPAR1URE QUEUES
TIME FIXES AIRPORT FlXES AIRPORT1 2 3 4 (TOTAL) 1 2 3 4 (TOTAL)
16:45-17:00 0 1 0 0 1 4 4 4 3 15
17:00-17:15 3 5 1 2 11 8 8 8 8 32
17:15-17:30 8 7 5 5 25 7 6 6 7 26
17:30-17:45 7 12 4 3 26 7 6 6 7 26
17:45-18:00 0 4 0 0 4 5 4 4 5 18
18:00-18:15 0 0 0 0 0 0 0 0 1 118:15-18:30 0 0 0 0 0 0 0 0 0 0
18:30-18:45 0 0 0 0 0 2 1 1 2 6
18:45-19:00 0 0 4 9 13 6 5 6 6 23
19:00-19:15 0 0 6 8 14 5 6 5 5 21
19:15-19:30 0 0 0 0 0 4 5 4 4 17
19:30-19:45 0 0 0 0 0 0 0 0 0 0
TOTAL 18 29 20 27 94 48 45 44 48 185
(b)
the first hour the arrival queue decrea ed from 67 to 55 flights,but the departure queue increased from 24 to 48 flights [seeTables IV(b) and V(b)].
Increa ing the arrival priority rate from 0.5 to 0.7 providedthe optimal capacity allocation which improved the overallarrival operations during the 3-h period. At the end of theperiod the total arrival demand was completely satisfied,the cumulative arrival queue decreased from 386 to 257flight and the total arrival delay decreased from at least386 to 257 15-min intervals. This improvement, however,
was achieved at the expense of the departure operation .Departure demand wa not completely satisfied within the 3-hperiod, and at the end of the period, the outstanding departurequeue increased from zero to seven flights. Additionally, thecumulative departure queue and total departure delay increasedsignificantly.
C. Effect of Fix Constraints on Utilization of Airport Capacity
In this section we illustrate the effect of a finite capacityof near-terminal airspace, in particular, the limited capacity of
500 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO.5, SEPTEMBER 1997
TABLE rv(a) OPTlMALSOLUTIO, FOR ORO (VFR AND [FR, Q = 0.5). (b) QUEUESATORO (VFR A D IFR, Q = 0.5)
OP. ARRIVAL FLOW DEPARTURE FLOW AIRPORT
TIME CAT. FIXES FIXES CAPAC[TYAIRPORT AIRPORT
] 2 3 4 (fOTAL) 1 2 3 4 (TOTAL) ARR DEP
16:45-17:00 7 9 0 1 17 4 4 4 5 17 17 17
17:00--17:15 IFR 7 7 2 1 17 4 4 5 4 17 17 17
17:15-17:30 6 4 3 4 17 5 5 3 4 17 17 1717:30--17:45 5 9 ] 2 17 4 4 5 4 17 17 l7
17:45-18:00 10 10 2 2 2A 6 6 6 6 2A 2A 24VFR
18:00--18:15 7 6 4 10 27 3 5 5 4 17 27 1718:15-18:30 5 3 9 10 27 4 4 4 5 17 27 171 :30--18:45 2 3 9 10 2A 6 6 6 6 24 2A 24
18:45-19:00 4 2 8 10 24 6 6 7 5 2A 2A 2A19:00--19:15 3 2 10 9 24 6 6 6 6 24 2A 2A
19:15-19:30 4 7 7 6 2A 6 6 6 6 2A 24 2A
19:30--19:45 6 10 4 8 28 2 2 1 2 7 28 15
TOTAL 66 72 59 73 270 56 58 58 57 229 270 237
(a)
ARRIVAL QUEUES DEPARTURE QUEUES
TIME FIXES AIRPORT FIXES AIRPORT
1 2 3 4 (TOTAL) 1 2 3 4 (TOTAL)
16:45-l7:oo 3 2 1 3 9 5 5 5 4 19
17:00--17:15 9 9 4 8 30 9 9 8 8 34
17:15-17:30 18 17 8 12 55 6 6 7 7 26
l7:30--17:45 22 23 9 13 67 6 6 6 6 24
17:45-18:00 14 15 9 11 49 2 2 2 1 7
18:00--18:15 8 10 10 7 35 0 0 0 0 018:15-18:30 7 7 5 3 22 0 0 0 0 0
18:30--18:45 7 7 3 1 18 3 2 2 2 9
18:45-19:00 8 7 9 10 34 5 4 5 5 19
19:00--19:15 7 7 11 10 35 4 5 4 4 17
19:15-19:30 6 6 6 6 2A 1 2 1 2 6
19:30--19:45 2 2 2 2 8 0 0 0 0 0
TOTAL 111 112 77 86 386 41 41 40 39 161
(b)
arrival and departure fixes, on the utilization of the runwaycapacity.
The effect is illustrated in the scope of the above exam-ple by comparison of the optimal allocation of arrival anddeparture traffic flows at the airport and delays under VFRcondition in two ca e: I) limited capacity of fixe (tenflight per 15 min for each fix) and 2) unlimited capacity offixes.
Table VI hows the optimal value of total airport trafficflow and queue at each 15-min interval calculated underlimited and unlimited capacitie of fixe for a = 0.7.
In thi table. the values that are different in both ca e areshown by the bold font.
The difference in optimal re ults for the first 15-min intervalcan be ea ily explained, if we calculate the maximum flowthrough the fixe, u ing demand data from Table 1. Maximumarrival flow through the fixe with unlimited and limited (tenflight per 15 min) capacities are equal to 26 and 25 flights,respectively. Both values are within the limits of runwayarrival capacity. However, because of fix con traints, theoriginal demand of 26 arrival flights could not be completelyati fied. Maximum flow through departure fixe i the ame in
GILBO: OPTIMIZl G AIRPORT CAPACITY UTILIZATION 501
TABLE V(a) OPTIMAL SOLUTIO FOR ORD (VFR AND IFR, a = 0.7). (b) QUEUES AT ORD (VFR A 0 IFR, a = 0.7)
OP. ARRIVAL FLOW DEPARTURE FLOW AIRPORT
TIME CAT. FIXES FIXES CAPACITYAIRPORT AIRPORT
1 2 3 4 (TOTAL) 1 2 3 4 (TOTAL) ARR DEP
16:45-17:00 8 10 0 2 20 3 3 3 2 11 20 11
17:00-17:15 IFR 8 9 1 2 20 3 3 2 3 11 20 11
17:15-17:30 8 4 3 5 20 2 2 3 4 11 20 11
17:30-17:45 4 10 3 3 20 3 3 3 2 11 20 11
17:45-18:00 10 10 4 4 28 4 4 4 3 15 28 15VFR
18:00-18:15 7 7 6 6 26 4 5 5 5 19 26 1918:15-18:30 4 5 7 8 24 5 6 6 7 24 24 24
18:30-18:45 4 1 9 10 24 6 6 6 6 24 24 24
18:45-19:00 5 2 7 10 24 6 5 7 6 24 24 24
19:00-19:15 3 2 10 9 24 6 7 6 5 24 24 24
19:15-19:30 2 5 8 9 24 6 6 6 6 24 24 24
19:30-19:45 5 9 3 7 24 6 6 6 6 24 24 24
TOTAL 68 74 61 75 278 54 56 57 55 222 278 223
(a)
ARRIVAL QUEUES DEPARTURE QUEUES
TIME FIXES AIRPORT FIXES AIRPORT1 2 3 4 (TOTAL) 1 2 3 4 (TOTAL)
16:45-17:00 2 1 1 2 6 6 6 6 7 25
17:00-17:15 7 6 5 6 24 11 11 12 12 46
17:15-17:30 14 14 9 9 46 11 11 11 11 44
17:30-17:45 19 19 8 9 55 12 12 12 12 48
17:45-18:00 11 11 6 5 33 10 10 10 10 40
18:00-18:15 5 5 5 5 20 7 8 8 8 31
18:15-18:30 5 0 2 3 10 6 6 6 6 24
18:30-18:45 3 2 0 1 6 9 8 8 8 33
18:45-19:00 3 2 7 10 22 11 11 11 10 43
19:00-19:15 2 2 9 10 23 10 11 10 10 41
19:15-19:30 3 3 3 3 12 7 8 7 8 30
19:30-19:45 0 0 0 0 0 2 2 1 2 7
TOTAL 74 65 55 63 257 102 104 102 104 412
(b)
both cases and equal to 36 flights, which exceed the runwaydeparture capacity. Reduction in arrival flow from 26 to 25flights was compensated for by increa ing departure flow from19 to 21 flights due to the tradeoff between runway arrival anddeparture capacitie . Similar situations affected the optimalsolution for some of the subsequent intervals as hown in theremainder of Table VI.
The difference in optimal allocation of airport capacityand its utilization for the limited and unlimited capacity offixes resulted in different quality of managing the arrival and
departure traffic. The quantitative effect is illustrated in TableVll, where the total arrival and departure delays are hown.In case of unlimited capacity of fixes, the total arrival anddeparture delay time are equal to 85 and 203 15-min intervals,respectively. Under the limited capacity of fixes, the optimalolution provides greater total arrival delay of 94 intervals.
At the same time the total departure delay is reduced from203 to 185 intervals. The optimization procedure automaticallyreallocates the airport arrival and departure resources becau eof the fixes con traint .
502 IEEE TRA SACfIONS ON CONTROL YSTEMS TECH OLOGY. VOL. 5. 0.5. SEPTEMBER 1997
TABLE VIOPTIMAL ALLOATIO OF ARRIVAL AND DEPARTURE FLOWS A D QUE ES (0 = 0.7)
UNLIMITED CAPACITY OF FIXES LIMITED CAPACITYOF FIXES
TIME TRAFFIC FLOW QUEUES TRAFFIC FLOW QUEUES
ARR DEP ARR DEP ARR DEP ARR DEP
16:45-17:00 26 19 0 17 25 21 1 15
17:00-17:15 28 15 10 34 28 15 11 32
17:15-17:30 28 15 24 28 28 15 25 26
17:30-17:45 28 15 25 28 28 15 26 26
17:45-18:00 28 15 3 20 28 15 4 18
18:00-18:15 16 30 0 0 17 27 0 118:]5-18:30 14 17 0 0 14 18 0 0
18:30-]8:45 20 27 0 6 20 27 0 6
18:45-19:00 28 15 12 25 27 17 13 23
]9:00-19:] 5 26 19 11 28 24 24 14 21
19:15-19:30 24 24 0 17 27 17 0 17
19:30-19:45 12 18 0 0 12 18 0 0
TOTAL 278 229 85 203 278 229 94 185
For equal arrival and departure priorities (a = 0.5) 1 theoptimal allocation of airport capacity proved to be the amefor the limited and unlimited capacity of fixes. In thi case thecapacity of fixe of ten flights per 15 min was not restrictivefor the utilization of runway capacity. The optimal valuesof arrival and departure flows and the airport capacitie arepre ented in Table ll(a).
The e examples illustrate the abilities of the proposedmodel to determine the optimal strategie for utilization ofthe operational re ources at the airport and near-terminalair pace in accordance with the dynamics of traffic demandand weather. They al 0 illustrate how the e resource interactto provide the optimal traffic flow at airport .
V. Co CLUSIONS
In this paper, a problem has been formulated to opumizethe utilization of airport runways and near-terminal air pacecapacitie to improve the efficiency of managing arrival anddeparture traffic at airports. Runways and arrival and departurefixes were considered as an integrated unit and a single y ternre ource.
It ha been hown that the limited capacity of fixes andimbalance in di tribution of demand over the fixes with someoverloaded and orne underloaded fixe can ignificantly affectthe utilization of airport capacity. eglecting the fix con traintsin the e ca e can re ult in overly optimi tic, nonrealizablescenario of managing traffic at the airport. The optimizationmodel pre ented automatically find the best trategies forutilization of runway and near-terminal air pace resourcesduring congested period . The model allocates the e resourcesbetween arrival and departures so that no available lot are10 t.
TABLE VIITOTAL DELAY TIMES FOR o = 0.•
Unlimited Limitedcapacity capacityof fixes of fixes
Total delayArrival 85 94time
(number ofIS-minute Departure 203 185intervals)
ACKNOWLEDGME T
The author thanks hi colleagues at the Volpe Center fortheir time and attention helping to refine this paper. Specialthanks to L. McCabe, R. Oie en, E. Roberts and R. D. Wright.The author thank the A ociate Editor D. Repperger andthree anonymous reviewers for their valuable and in ightfulcomment and sugge tions, which were useful in the revisionof thi paper.
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GILBO: OPTLMIZfNG AIRPORT CAPACITY UTlLlZATION 503
l7J E. P. Gilbo, "Optimization of airport capacity and arrival-departure Howthrough the runway-fix system," Volpe Center, Cambridge, MA, InterimTech. Rep., Feb. 1994.
[8] G. F. ewell, "Airport capacity and delays," Transponarion Sci., vol.3. no. 3, pp. 201-241, 1979.
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Eugene P. Gilbo received the M.S. degree in me-chanical engineering and the Ph.D. and Doctorof Sciences degrees in electrical engineering andapplied mathematics from SI. Petersburg (formerlyLeningrad) Polytechnic Institute in 1960, 1965, and1975, respectively.
After immigrating to the United States from theSoviet Union in 1988, He was with Unisys Cor-poration, Cambridge, MA, until 1995. In 1995, hejoined the Volpe National Transportation SystemsCenter of the U.S. Department of Transportation,
Cambridge, MA. He is the author or coauthor of more than 80 scienti ficpublications including the monograph with I. B. Chelpanov, Signal ProcessingBased on Order Statistics (Mo cow: Soviet Radio Publishing House, 1975(in Russian)]. His major research interests include system optimization,scheduling, and robu tness in statistics and control. His current re earchconcentrates on the optimization of air traffic flow management trategieswith an emphasi on airport operations.
Dr. Gilbo is a member of the Institute for Operations Research andManagement Science (INFORMS).
