Modeling and Optimization of the Capacity Allocation Problem ... - MIS

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Modeling and Optimization of the Capacity Allocation Problem with Constraints Abdellah IDRISSI

Abstract— The Constraint Satisfaction Problem (CSP) is proven more and more promising to model and to solve a great number of real problems. A lot of approaches using constraint reasoning have proposed to solve search problems. In this paper, we propose a modeling of Capacity Allocation Problem of an airport (CAP) and of its fixes (F CAP ) in form of a Constraint Satisfaction Problem. This modeling gives a CSP problem called F CSP AC. Then, in order to solve the F CSP AC, we propose an Optimization model called COF CAP . We present some experimental results through an example followed by its optimized resolution. Finally, we show how to control the solutions of an F CSP AC to assist the airport managers to face unforeseen events and present some future work. Index Terms— Artificial Intelligence, Constraint Optimization, Airport Capacity Allocation Problem, Modeling, Solving.

I. I NTRODUCTION APACITY allocation problems are the core of many realworld scheduling and planning problems. The first steps to solve a capacity allocation problem consist in formulating and modeling it. Modeling is one of the central themes of Artificial Intelligence (AI). A good model generally facilitates the problem-solving. However, modeling is difficult and no general guidelines exist on which factors must be taken into account in a modeling process. Therefore, developing a good model largely remains an art, depending on experiences and tastes. Nevertheless, for a given problem, different modeling techniques can be studied. Among these techniques one can find the constraints satisfaction problems (CSP) [22]. Formally, a CSP can be defined by the triplet (X, D, C), where X = {X1 , ..., Xn }, is the set of n variables; D = {D1 , ..., Dn }, is the set of n domains of values; Di is the domain of values of the variable Xi and C = {C1 , ..., Ce }, is the set of e constraints of the problem, specifying compatible values or excluding incompatible values between variables. To solve a CSP consists in assigning values to variables in order to satisfy all the constraints. Various real problems can be represented in form of a CSP. In this paper, we model the Capacity Allocation Problem of an airport and its fixes (F CAP ) in a CSP and apply CSP techniques to solve it. The CAP of an airport is a key problem in regulating aerial traffic. This problem will become more and more important because aerial traffic will greatly increase in the next years.

C

LaRIA, FRE 2733, Universit´e de Picardie. 33 rue St Leu, 80039 Amiens, France. Emails: [email protected] , [email protected] The second author is partially supported by National 973 Program of China under Grant No. 2005CB321900

and

Chu Min LI

Thus, it is necessary to solve this problem, especially in case of congestion. The congestion occurs at an airport and its fixes when the request of the traffic exceeds the available capacity. The airport managers control the traffic, and solve these problems of congestion by delaying some flights, so that the flow on the airport is regulated, and does not exceed the available capacity. His objective is to minimize the total delay of the planes. The prime objective of the present study is to model the capacity allocation problem of an airport and its fixes in form of a CSP. For this purpose, it is enough to identify the variables, the domains of values which these variables can take, and the constraints of the problem. The second objective is to solve this problem by using the resolution algorithms designed specifically for the CSP, in order to assist the managers of an airport to regulate the arrival and departure demands, and to use efficiently the available capacities of a terminal. These CSP resolution algorithms include the Branch and Bound algorithm and its alternatives, as presented in [19], [20], [21], [24] and [26]. The third objective is to study the complexity of this capacity allocation problem and to show that it is an NPcomplete problem. We will treat the first two objectives in this paper, the third being the subject of the next studies. This paper is organized as follows. Section II defines the capacity allocation problem. In section III we present a modeling of this problem in form of a CSP which we called F CSP AC, and propose an optimization model to solve it in section IV. We give an algorithm in section V and expose in section VI an application example of an F CSP AC. We show in section VII how to assist the managers to regulate traffic and unforeseen events. We conclude in section VIII and discuss some perspectives of this work. II. C APACITY A LLOCATION P ROBLEM A. Presentation An operational scheme of a single airport system that reflects the arrival and departure processes at the airport is shown in figure 1. The system comprises arrival ways called arrival fixes ‘AF’, departure ways called departure fixes ‘DF’, and a runway system. There are two separate sets of arrival and departure fixes located in the near-terminal airspace area (5070 km off the airport), so that the arrival fixes serve only arrival flow, and the departure fixes serve only departure flow. The runway system on the ground serves both arrival and departure flows. The arrival flights are assigned to specific arrival fixes, and, before landing, they should go through the fixes. After leaving

Fig. 1.

Example of an airport system

runways, the arriving flights follow the taxiway to the gates at the terminal. The departing flights are also assigned to the specific fixes. They are directed to the runways, and go through the departure fixes after leaving runways. The arrival queues are formed before the fixes (see figure 1). This means that the flights which are already in the fixes, must be accepted at the runways. If there is an arrival queue, a certain number of flights should be delayed in the air. The departure queue is formed before the runway system, and flights can be delayed either at their gates or on the taxiway. The capacity of arrival and departure fixes is the maximum number of flights that can cross a fix in a 15 minutes interval (or other interval). These capacities determine the operational constraints in the near-terminal airspace. The operational limits on the ground (runways) are characterized by arrival capacity and departure capacity. These capacities are generally variable and interdependent. The traffic demands for an airport and for the fixes of this airport are given by the predicted number of arriving and departing flights per each 15 minutes time interval of a considered period (see table I). Definition 1. Airport capacity is defined respectively as the maximum number of operations (arrivals and departures) that can be performed within a fixed time interval (e.g., 15 minutes or other interval) at a given airport under given conditions such as runway configuration and weather conditions. Definition 2. Fix capacity is defined as the maximum number of flights that can cross a fix in a 15 minutes interval (or other interval) under given conditions such as weather conditions, etc. These capacities determine the operational

TABLE I E XAMPLE OF PREDICTED NUMBER OF ARRIVING AND DEPARTING FLIGHTS PER EACH 15 MINUTES TIME INTERVAL OF A 3 HOURS PERIOD .

constraints in the near-terminal airspace. Definition 3. Airport capacity allocation problem consists in determining a balance, between arrivals and departures, to minimize an adapted objective function for a given time period. The capacity of the airports and the capacity of its fixes become more and more limited compared to the demands and present real problems for the aerial transportation system. The optimal allocation of the existing capacity of an airport and of its fixes during the congestion periods is very important to maximize the departures and to minimize the delay. B. Related Work Aerial traffic is continuously increasing and its regulation becomes more and more difficult, given the limited capacities both in ground (airports) and air (aerial sectors). Various solutions can be used to attack this difficulty: building new airports, physically extending existing runway systems, applying new technologies to increase airport capacity, and optimizing utilization of existing airport capacity [4]. See [13] for some possible measures for increasing airport capacity. On the other hand, aerial traffic regulation should be optimized and automated, especially when treating unforseen

events. In [15], we proposed a modeling and a resolution of the aerial conflicts problems by techniques of constraints network (CSP). This modeling and automatic resolution can be possibly introduced into project (FREER) [7] developed by Eurocontrol following the concept “Free Flight” [8]. At an airport, the air traffic managers generally know the range of arrival/departure capacity tradeoff available for each runway configuration and use the tradeoff for strategic planning of arrival and departure traffic. They control and modify the operational arrival and departure limits during the periods of congestion to better serve the traffic demand and to minimize the delay. However, currently they proceed intuitively because of the lack of automated decision support to determine the best strategies for allocation of airport arrival and departure capacities. An experienced specialist can find the best solutions but, in more complex situations, this task can become impossible. The theoretical aspects of the problem are addressed in [9], [10]. In [16], we proposed a modeling of the airport capacity allocation problems by techniques of constraints network (CSP). We associated an optimization model to it. But, we had not taken into account the capacities of different arrival and departure fixes. In order to optimize and automate the aerial traffic regulation, we propose in this paper a modeling of capacity allocation problem of an airport and its fixes using constraint network (CSP). We also propose an optimization procedure able to give the best allocation of airport (and/or fix) capacity between arrivals and departures under given operational conditions at the fixes to satisfy the predicted traffic demand for a time period and to minimize the delay. We specify this model in terms of a Constraint Satisfaction Problem (CSP) in the next section. III. F CSP AC: A CSP MODEL FOR THE A LLOCATION OF C APACITY AT THE FIXES A. Constraint Satisfaction Problems A CSP is defined by the triplet (X, D, C), with X = {X1 , ..., Xn }, is a set of n variables; D = {D1 , ..., Dn }, is a set of n domains. Di is the domain of values associated with variable Xi and C = {C1 , ..., Ce }, is a set of e constraints of the problem. Solving a CSP consists in assigning values to variables in order to satisfy all the constraints. In general, the CSPs are solved by traditional methods combining a mechanism of search with Backtrack (BT) and a mechanism of reinforcement of consistencies with each node of search. We propose to solve the capacity allocation problem of the airport and of its fixes using CSP resolution algorithms, including the Branch and Bound and its alternatives, as presented in [19], [20], [21], [24] and [26]. B. Problem Formulation Figure 2 shows a simplified scheme of an airport constituted with only one arrival fix AF, one departure fix DF and one terminal T serving both arrival and departure demands during a time interval ‘i’ (‘i’ is a period of 15 minutes). The arrival

Fig. 2.

Simplified Scheme of an airport

demands are at the point ‘A’. The departure demands are at the point ‘D’. The point ‘T’ represents the Terminal. The problem consists in satisfying the arrival and departure demands, within the limit of the arrival fix capacity (Cafi ), the limit of the departure fix capacity (Cdfi ) and the limit of the total capacity of the terminal (CTi ) during the time interval ‘i’. In the sequel, we consider that there are ‘j’ arrival fixes AFjdfi with j = 1, ..., naf and ‘k’ departure fixes DFkdfi with k = 1, ..., ndf . Each fix ‘j’ or ‘k’ has its own capacity j k noted Caf or Cdf during the same interval of time ‘i’. i i It should be noted that the sum of the fixes capacities Cai + Cdi of an airport is in general lowerPor equal than naf j the capacity of the terminal CTi where Cai = j=1 Caf and i Pndf k Cdi = k=1 Cdfi . Cai + Cdi ≤ CTi

(1)

CTi is the sum of Pvi and of Poci . CTi = Pvi + Poci

(2)

Pvi is the number of empty places at the terminal and Poci is the number of occupied places at the terminal. If Xjaf ei represents the number of arrival flights accepted at fix j (j = 1, ..., naf ) and Ykdf ei the number of departure flights effectively leaving through fix k (k = 1, ..., ndf ) during time interval ‘i’, then the number of empty places for the time interval ‘i+1’ becomes: naf ndf X X Ykdf ei − Xjaf ei (3) Pvi+1 = Pvi + k=1

j=1

In the same way, the number of occupied places for the time interval ‘i+1’ becomes: naf ndf X X Ykdf ei Poci+1 = Poci + Xjaf ei − (4) j=1

k=1

For the sake of simplification, let us admit in the sequel that CTi is a constant independent of time. We will note it CT such as: CT = Pvi + Poci = Pvi+1 + Poci+1

(5)

We propose hereafter a modeling of this problem in form of a constraint satisfaction problem (CSP). C. Problem Modeling We propose to model this problem of the arrivals/departures at an airport and at its fixes (figure 2) in form of a CSP F CSP AC = (Xi , Di , Ci ) during a time interval ‘i’, where: • Xi = {Xai , Xaei , Ydi , Ydei , Cai , Cdi , Qai , Qdi , X1af , i ..., Xnafi , X1af ei , ..., Xnaf ei , Y1dfi , ..., Yndfi , Y1df ei , ..., Yndf ei , C1afi , ..., Cnafi , C1dfi , ..., Cndfi , Q1afi , ..., Qnafi , Q1dfi , ..., Qndfi , Xtai , Ytdi , Pvi , Poci } is the set of variables such as : – Xai is the number of arrival demands at the point ‘A’ (figure 2). – Xaei is the number of effective arrivals at ‘A’. – Ydi is the number of departure demands at the point ‘D’ (figure 2). – Ydei is the number of effective departures at ‘D’. – Cai is the arrival capacity at the point ‘A’. – Cdi is the departure capacity at the point ‘D’. Cai and Cdi are dictated by the airport managers. – Qai is the number of arrivals delayed at ‘A’. – Qdi is the number of departures delayed at ‘D’. – Xjafi is the number of arrival demands at the fix ‘j’ (figure 1) with j = 1, ..., naf . – Xjaf ei is the number of effective arrivals at the fix ‘j’ with j = 1, ..., naf . – Ykdfi is the number of departure demands at the fix ‘k’ (figure 1) with k = 1, ..., ndf . – Ykdf ei is the number of effective departures at the fix ‘k’ with k = 1, ..., ndf . – Qjafi is the number of arrivals delayed at the fix ‘j’ with j = 1, ..., naf . – Qkdfi is the number of departures delayed at the fix ‘k’ with k = 1, ..., ndf . – Cjafi is the arrival capacity at the fix ‘j’ with j = 1, ..., naf . – Ckdfi is the departure capacity at the fix ‘k’ with k = 1, ..., ndf . – Xtai is the total number of arrival demands for the time interval ‘i’. Note that for each value of ‘i’, we add the delay Qai−1 recorded in the previous interval ‘i-1’ to the predicted arrival demands Xai . Then Xai becomes Xai + Qai−1 which we noted Xtai such as Qa0 = 0. – Ytdi is the total number of departure demands for the time interval ‘i’. For each value of ‘i’, we add the delay Qdi−1 recorded in the previous interval ‘i1’ to the predicted departure demands Ydi . Then Ydi becomes Ydi + Qdi−1 which we noted Ytdi such as Qd0 = 0. – Pvi and Poci are respectively the set of empty and occupied places of the terminal at the point ‘T’ of figure 2 during the same time interval ‘i’. • Di = {DXa , DXae , DYd , DYde , DCa , DCd , DQa i i i i i i i , DQdi , DX1af , ..., DXnaf , DX1af e , ..., DXnaf e , i i i i DY1df , ..., DYndf , DY1df e , ..., DYndf e , DC1af , ..., i

i

i

i

i

•

DCnaf , DC1df , ..., DCndf , DQ1af , ..., DQnaf , i i i i i DQ1df , ..., DQndf , DXtd , DYtd , DPvi , DPoci } is the i i i i set of variable domains. We consider that all the domains of these variables are equal and that Di represents a common domain to all these variables. In other words, Di = {0, 1, 2, ..., CT }. Note that the values of arrival and departure capacities are interdependent and are dictated by the airport managers in form of couples (Cai , Cdi ). They can recommend to use for example, at a given period, the set {(18, 29), (24, 24), (26, 19), (28, 15), (17, 30), (20, 27)}. In this example, the first couple means that if Cai = 18 then Cdi = 29 and vice versa. The meaning of other couples is similar. Ci is the set of constraints of the problem. We can formulate them as follows : 0 ≤ Pvi ; Poci

≤ CT

(6)

0 ≤ Pvi + Poci ≤ CT

(7)

0 ≤ Pvi+1 + Poci+1 ≤ CT

(8)

0≤

naf X

Xjaf ei ≤ Pvi

(9)

Ykdf ei ≤ Poci

(10)

j=1

0≤

ndf X k=1

0 ≤ Xjaf ei ≤ Xjafi , ∀j ∈ {1, ..., naf }

(11)

0 ≤ Ykdf ei ≤ Ykdfi , ∀k ∈ {1, ..., ndf }

(12)

0≤

naf X

Xjaf ei +

j=1

ndf X

Ykdf ei ≤ CT

(13)

k=1

Pvi+1 = Pvi +

ndf X

Ykdf ei −

Poci+1 = Poci +

Xjaf ei −

j=1

Xai =

Xjaf ei

(14)

j=1

k=1 naf X

naf X

ndf X

Ykdf ei

(15)

k=1 naf X

Xjafi

(16)

Xjaf ei

(17)

Ykdfi

(18)

Ykdf ei

(19)

j=1

Xaei =

naf X j=1

Ydi =

ndf X k=1

Ydei =

ndf X k=1

C ai =

naf X

Cjafi

(20)

Ckdfi

(21)

j=1

Cd i =

ndf X k=1

Xaei ≤ Cai

(22)

Ydei ≤ Cdi

(23)

Xaei + Ydei ≤ Xai + Ydi

(24)

Xjafi ≤ Cjafi

(25)

Ykdfi ≤ Ckdfi

(26)

Qai =

naf X

Qjafi

(27)

Qkdfi

(28)

j=1

Qdi =

ndf X k=1

Xtai =

naf X

Xtjaf

i

j=1

Xtjaf = Xjafi + Qjafi−1 , ∀j ∈ {1, ..., naf } i

Ytdi =

ndf X k=1

Ytkdf

i

Ytkdf = Ykdfi + Qkdfi−1 , ∀k ∈ {1, ..., ndf } i

Qjafi = Xjafi − Xjaf ei

(29) (30) (31)

D. Problem Solving Given values of (Xai , Ydi ) (arrival and departure demands for a time interval ‘i’) and a number of possible values of (Cai , Cdi ), the problem consists in assigning a value to each variable by respecting the constraints described above. The problem FCSPAC is a problem of decision. It is also an optimization problem. Now our constraint network (CSP) is well identified. In particular the variables, the domains of values of these variables and the constraints of the problem are explicitly defined. It remains to apply one of CSP reasoning procedure (Branch and Bound, Local Search, Backtracking, ...) to solve the capacity allocation problem. We think in particular to use the Branch and Bound algorithm and its alternatives, as presented in [19], [20], [21] [24] and [26]. Let us note that in the case of congestion during Pnaaftime interval ‘i’ the sum of numbers of effective arrivals j=1 (Xjaf ei ) Pndf and of effective departures (Y ) is lower than the ei k=1 kdfP naf (X ) and of sum of numbers of arrival demands jafi j=1 Pndf departure demands k=1 (Ykdfi ). In other words, only a part of the demands, arrivals or departures, is been served (see equation (24)). When solving a CSP, a number of questions should be asked. For example : • Is there any solution for this problem ? In other words, is the problem satisfiable ? • How can we find a solution ? • How can we find all solutions ? • ...

(32)

For our F CSP AC, we are interested in the following questions: + Qjafi−1 , ∀j ∈ {1, ..., naf } (33) • Is it possible to serve all the requests without any delay:

Qkdfi = Ykdfi − Ykdf ei + Qkdfi−1 , ∀k ∈ {1, ..., ndf } (34)

N X

Qjafi−1 , for j = 1, ..., naf (equation (33)) is the number of arrival planes delayed at the fix j for j = 1, ..., naf during the time interval ‘i-1’ and Qkdfi−1 , for k = 1, ..., ndf (equation (34)) is the number of departure planes delayed at the fix k for k = 1, ..., ndf during the time interval ‘i-1’. These two numbers should be added respectively to the arrival and departure demands during the time interval ‘i’. So the total number of arrival demands during ‘i’ is the sum of demands effectively expressed Xjaf ei , for j = 1, ..., naf during ‘i’ increased by the number of planes Qjafi−1 , for j = 1, ..., naf which were waiting in the queue due to the fact that they were not been served during ‘i-1’. Similarly, the total departure demand during ‘i’ is the sum of demands really expressed Ykdf ei for k = 1, ..., ndf during ‘i’ increased by the number of planes Qkdfi−1 at the fix k for k = 1, ..., ndf which were waiting in the queue due to the fact that they were not been served during the time interval ‘i-1’. The planes which were waiting in the queue during the time interval ‘i-1’ are served in priority during the time interval ‘i’.

i=1

[

naf X j=1

Qjafi +

ndf X

Qkdfi ] = 0 ?

k=1

Where Qjafi and Qkdfi are respectively the number of arrival planes delayed at the fix j for j = 1, ..., naf and departure planes delayed at the fix k for k = 1, ..., ndf , during the time interval ‘i’ and N is the number of intervals constituting the given time period. •

Is it possible to serve all the requests with the given delay ‘1’ : naf ndf N X X X Qkdfi ] = 1 ? [ Qjafi +

•

...

•

Is it possible to serve all the requests with the given delay ‘r’: naf ndf N X X X [ Qjafi + Qkdfi ] = r ?

i=1

i=1

j=1

j=1

k=1

k=1

•

Is it possible to serve all the requests without exceeding a certain given delay ‘rmax ’: N X

[

i=1

naf X

Qjafi +

j=1

ndf X

Ntar

Qkdfi ] ≤ rmax ?

i=1

[

naf X j=1

naf ndf N X X X Qkdfi ] = [ Qjafi + i=1 j=1

k=1

Our objective is to minimize the sum of the numbers of PN arrival planes delayed ( i=1 [Qjafi ]) ∀j ∈ {1, ..., naf } and to minimize the sum of the numbers of departure planes delayed PN ( i=1 [Qkdfi ]) ∀k ∈ {1, ..., ndf } during a given time period (15*N minutes). In other words, we seek to minimize: N X

is estimated as (Ntar ∗ 15) minutes. Indeed, over one period constituted by N intervals of time one will have:

Qjafi +

ndf X

Qkdfi ]

(35)

Qjafi = Xjafi − Xjaf ei + Qjafi−1 , ∀j ∈ {1, ..., nafi } and Qkdfi = Ykdfi − Ykdf ei + Qkdfi−1 , ∀k ∈ {1, ..., ndfi } are respectively the number of planes arriving and leaving which are delayed during a time interval ‘i=15 min’. The function of penalty becomes:

k=1

ϕF =

subject to the constraints (6)-(34). We recall that:

N X i=1

[

naf X

Qjafi +

j=1

ndf X

Qkdfi ]

(38)

k=1

To optimize the function of penalty thus means to minimize the function ϕF where :

Qjafi = Xjafi − Xjaf ei + Qjafi−1 , ∀j ∈ {1, ..., naf }

naf ndf N X X X ϕF = [ (Xjafi − Xjaf ei ) + (Ykdfi − Ykdf ei )] (39)

and Qkdfi = Ykdfi − Ykdf ei + Qkdfi−1 , ∀k ∈ {1, ..., ndf }

i=1

We propose to place this problem within a general framework of an optimization under time limit constraint. Because we estimate that in the event of breakdown or other, it is well necessary to provide a solution even if it is not of very good quality. This issue will be the subject of our next studies. For the moment we propose below an optimization method under constraints without taking in consideration the time limit constraint. IV. O PTIMIZATION M ODEL When one assigns a capacity to a request consisting of several arrival or departure flight demands, then if this request is totally satisfied, one passes to the next one. If this request is not totally satisfied then we consider that the difference between the request and the available capacity is a penalty p such as (D − C = p), where D is the number of demands in the request, C is the capacity allowed to this request. The problem is to reduce the sum of these penalties of all the fixes. For that, we propose an optimization model which we called COF CAP . A COF CAP is an optimization problem under a set of constraints. It consists to reduce to the minimum a function of penalty ϕ under a set of constraints C. This function of penalty ϕ is the sum of the penalties of the unsatisfied constraints. Let us note p(c) the penalty of an unsatisfied constraint. Then the problem COF CAP can be characterized by: minimize [ϕ] where X ϕ= p(c|c ∈ C and c unsatisf ied)

where

k=1

(36) (37)

In addition, the penalty can be the total time of delay. It can be also the total number of planes delayed (Ntar ) over all the period considered. The total time of delay for this last

j=1

k=1

V. A LGORITHM The resolution and the optimization of an F CSP AC problem proceed in several stages: • Initially they consist of the distribution of the arrival demands of an airport on its arrival fixes and of the distribution of the departure demands on its departure fixes. • In the second time they consist to select the pertinent couple of arrival and departure capacity (value) (Cai , Cdi ) among the various couples of capacities (dictated by airport managers) to affect at the same time to arrival and departure demands (variables) of the different fixes. As mentioned above, this choice is selected according to the heuristic which allows to respect all the constraints and to minimize the number of delayed planes. • In the third time they consist of the distribution of this pertinent couple of arrival and departure capacity (value) (Cai , Cdi ) found above respectively on the capacities of arrival fixes C1afi , ..., Cnafi and on the capacities of departure fixes C1dfi , ..., Cndfi . • Finally they consist of the assignment of these capacities respectively to the variables Xjaei for j=1, ..., nafi and Ykdei for k=1, ..., ndfi so that all the constraints (6)-(34) PN Pnaf Pdaf are satisfied and that i=1 [ j=1i Qjafi + k=1i Qkdfi ] is minimized. • The delays are the sum of differences between the predicted demands for arrival and the arrival planes effectively served added to the sum of differences between the predicted demands for departure and the planes effectively leaved. In other words, the total number of planes delayed is ϕF where: ϕF =

naf N X X i=1 j=1

(Xtjaf − Xtjaf e ) + i

i

ndf X

(Ytkdf − Ytkdf e )

k=1

i

i

We implemented this approach and compared it to the other methods suggested in the literature (in particular [10] and [11]). The results obtained for the various instances of the problem show that our approach gives solutions at least as good as those found by other techniques. We present below an example of F CSP AC problem and one of its optimal solutions satisfying constraints (6)-(34) and minimizing N X i=1

[

naf X j=1

Qjafi +

ndf X

Qkdfi ]

k=1

VI. E XAMPLE OF AN F CSP AC PROBLEM FOLLOWED BY ITS RESOLUTION AND ITS OPTIMIZATION

A. Presentation of an F CSP AC problem Given a time period composed of a number of time intervals, the predicted arrival and departure demands in each of these intervals, and a set of compatible repartitions of airport capacity between arrival and departure flights, our problem is to determine the real repartition of the demands among the fixes, and the real repartition of arrival and departure capacities among fixes, according to the actual demands of each fix. We illustrate the resolution of this problem by an example in this section. We consider that there are 4 arrival fixes noted af1, af2, af3 and af4 and 4 departure fixes noted df1, df2, df3 and df4 at the airport (figure 1). In this example, the period is 3 hours (12h00-15h00) and the number of intervals N forming this period is equal to 12. The predicted arrival and departure demands are those given in table I and the available capacities are values (domain) given as an example in subsection C of section III (D = {(18, 29), (24, 24), (26, 19), (28, 15), (17, 30), (20, 27)}). In table II, column 0 shows the time intervals of 15 minutes. For each interval, columns 1 and 2 respectively show the number of predicted arrival demands Xai and departure demands Ydi of the airport. These predicted demands will be distributed among the different fixes. The arrival predicted demands of the airport will be distributed among the arrival demands of its fixes X1afi , X2afi , X3afi and X4afi as indicated respectively in columns 3, 4, 5 and 6 for arrival fixes af1, af2, af3 and af4 (see table II). In the same way, the departure predicted demands of the airport will be distributed among the departure demands of its fixes Y1dfi , Y2dfi , Y3dfi and Y4dfi as indicated respectively in columns 7, 8, 9 and 10 for departure fixes df1, df2, df3 and df4 (table II). In table III, columns 11, 12, 13 and 14 present respectively arrival flows served (or effective) X1af ei , X2af ei , X3af ei and X4af ei at the fixes af1, af2, af3 and af4. Columns 15, 16, 17 and 18 of the same table III present respectively departure flows Y1df ei , Y2df ei , Y3df ei and Y4df ei at the fixes df1, df2, df3 and df4. In table IV, columns 21, 22, 23 and 24 present respectively the arrival demands which were not served (queues or delays) Q1afi , Q2afi , Q3afi and Q4afi at the arrival fixes af1, af2,

TABLE II A RRIVAL AND DEPARTURE DEMANDS FORMULATED AT AN AIRPORT ( COLUMNS 1 AND 2). T HESE ARRIVAL AND DEPARTURE DEMANDS ( COLUMNS 1 AND 2) OF THE AIRPORT ARE RESPECTIVELY DISTRIBUTED ON ITS 4 ARRIVAL FIXES ( COLUMNS 3,...,6) AND ON ITS 4 DEPARTURE FIXES ( COLUMNS 7,...,10).

af3 and af4. Columns 25, 26, 27 and 28 of the same table IV present respectively the departure demands which were not served Q1dfi , Q2dfi , Q3dfi and Q4dfi at the departure fixes df1, df2, df3 and df4. The arrival/departure demands which were not served are queues which must be taken into account during the next time interval. B. Example of the F CSP AC problem resolution The problem consists in distributing the values of column 1 on columns 3, 4, 5 and 6 of arrival fixes af1, af2, af3 and af4 (figure 1) and in distributing the values of column 2 on columns 7, 8, 9 and 10 of departure fixes df1, df2, df3 and df4. This distribution is arbitrary and is more or less equitable. Then, it acts of the distribution of the arrival capacities of the terminal on the capacities of arrival fixes which are in columns 11, 12, 13 and 14, as well as the distribution of the departure capacities of the terminal on the capacities of departure fixes which are in columns 15, 16, 17 and 18. But before that, let us notice, for each line, the choice of the capacity (value) among the various capacities to affect to

TABLE III

TABLE IV

A RRIVAL AND DEPARTURE CAPACITIES DICTATED BY THE MANAGERS OF AN AIRPORT ( MENTIONED HERE IN COLUMNS 19 AND 20). F OR EACH LINE , ONE COUPLE OF CAPACITY FORMED RESPECTIVELY BY ARRIVAL

R EAL DELAYS RECORDED ON THE DIFFERENT FIXES OF THE AIRPORT.

AND DEPARTURE CAPACITY IS SELECTED AND DISTRIBUTED

4 ARRIVAL FIXES ( COLUMNS 11,...,14) AND ON ITS 4 DEPARTURE FIXES ( COLUMNS 15,...,18).

RESPECTIVELY ON ITS

arrival and departure demands (variables) of the different fixes. This choice is selected according to the heuristic which allows to respect all the constraints and to minimize the number of delayed planes. The resolution is done in an incremental way and follows the law of line per line. In columns 21, 22, 23 and 24 of table IV we recover at each time interval i the number of the planes which had required to arrive at the one of the arrival fixes af1, af2, af3 and af4 but which has not been served. These demands which has not been served will respectively constitute arrival queues for these same fixes for the next time interval i+1. In the same way, in columns 25, 26, 27 and 28 of table IV we recover at each time interval i the number of the planes which had required to leave at one of the departure fixes df1, df2, df3 and df4 but which has not been served. These demands which has not been served will respectively constitute departure queues for these same fixes for the next time interval i+1. The objective of this study is to reduce at the minimum the

sum of the numbers of delayed planes which are in columns 21, 22, 23 and 24 for arrival fixes and in columns 25, 26, 27 and 28 for departure fixes. In a general way, we seek to reduce the total sum of all these planes delayed. C. Example of the F CSP AC problem optimization For the example of the problem presented above, the function of penalty ϕF which is the sum of the columns 21, 22, 23 and 24 for arrival fixes and of the columns 25, 26, 27 and 28 for the departure fixes (table IV) will be written: ϕF =

12 X 4 4 X X [ Qjafi + Qkdfi ] i=1 j=1

(40)

k=1

That is to say : ϕF =

naf ndf 12 X X X (Ykdfi − Ykdf ei )] (41) [ (Xjafi − Xjaf ei ) + i=1 j=1

k=1

number of the arrival planes delayed is: PThe 12 Pnaf =4 [ j=1 (Xjafi − Xjaf ei )] = 37 + 35 + 30 + 41 = 143 i=1 (columns 21, 22, 23 and 24), and The number of the departure planes delayed is:

P12 Pndf =4 i=1 [ k=1 (Ykdfi − Ykdf ei )] = 18 + 21 + 17 + 21 = 77 (columns 25, 26, 27 and 28). The total number of planes delayed is 143 + 77 = 220. We find the same results just like if we had considered the existence of only one arrival fix and only one departure fix and one terminal (figure 2) as we proposed it in [17].

The following formula allows us to solve partly this problem of congestion and to ensure a certain balance between arrivals and departures. N X

[αi Poci −γi Ydei +βi Pvi −δi Xaei ] ≤

i=1

VII. C ONTROL AND REGULATION OF CAPACITIES We saw in inequation (7) that 0 ≤ Pvi + Poci ≤ CTi . Let us suppose that one wishes to reduce or increase the capacities of an airport and of its fixes for a some reason. In otherwise, one wishes to control and regulate these capacities. Then in this case it is necessary to introduce some control parameters. The formula above becomes : αi Poci + βi Pvi = λi CTi

(42)

The parameters αi , βi and λi control the capacities. They depend on the wishes of regulation expressed by the airport managers according to certain conditions weather, priorities, authorities, rush hours, etc. They have values between 0 and 1 such that αi + βi = 1. For example, if αi = 0.5, βi = 0.5 and λi = 0.25, we will have 0.5Poci + 0.5Pvi = 0.25CTi . This formula becomes Poci + Pvi = 0.5CTi . The total capacity is not entirely available: meaning that only half of the airport capacity is available. This restriction can occur because of certain conditions like weather, authorities, etc. It is thus necessary to adapt these new capacities for the arrival and departure demands. We are thus brought to introduce these parameters into all the formulas which we put until now. More precisely, we will have the following formulas: αi Xaei ≤ λi Cai ≤ βi Pvi

(43)

αi0 Ydei ≤ λ0i Cdi ≤ βi0 Poci

(44)

Such that Cai and Cdi are the total capacities respectively of arrival and departure fixes (figure 1) during the time interval ‘i’ and αi0 , λ0i and βi0 are parameters of control which can be respectively equal or different of αi , βi and λi . Let us note that Cai ≤ CTai and that Cdi ≤ CTdi where CTai and CTdi are respectively the arrival and departure capacities of the terminal ‘T’ during ‘i’ such as CTai +CTdi = CTi . We point out that CTi is the total capacity of the terminal. Over one period of (15*N) minutes, we will have : N X

[αi Xaei + αi0 Ydei ] ≤

i=1

N X [λi Cai + λ0i Cdi ] i=1

≤

N X [βi Pvi + βi0 Poci ]

(45)

i=1

Concretely if the capacity is rather large so that no congestion takes place, the problem would not arise. But if there are more arrival demands than departure demands during one time period, the problem of the congestion appears. In this case, it is necessary to minimize the number of delayed planes or to maximize departure flows as well as possible.

N X [λi CTdi +µi CTai ] i=1

(46) N represents the number of intervals over one period of (15*N) minutes; Poci is the number of occupied places during the time interval ‘i’; Pvi is the number of empty places during the time interval ‘i’; Ydei is the number of effective departures during ‘i’; Xaei is the number of effective arrivals during ‘i’; CTdi is the departure capacity at the terminal during ‘i’; CTai is the arrival capacity at the terminal during ‘i’; αi , βi , γi , δi , λi and µi are parameters of regulation. The response to this load balancing problem between arrivals and departures consists in minimizing the following function which we call the new objective function. N X min { [αi Poci − γi Ydei ] + [βi Pvi − δi Xaei ]}

(47)

i=1

We will treat in details these concepts of optimization in a forthcoming paper.

VIII. C ONCLUSION In this paper, we presented a modelling of the capacity allocation problem of an airport (CAP) and of its fixes (F CAP ) in form of a constraint satisfaction problem (CSP) called F CSP AC. The variables represent the number of arrival and departure demands, the number of planes effectively arrived and leaved, the number of arrival and departure planes delayed, the number of demands for arrivals at the different fixes, the number of effective arrivals at the different fixes, the number of departure demands at the different fixes, the number of effective departures at the different fixes, the number of arrivals delayed at the different fixes, the number of departures delayed at the different fixes, as well as the number of empty and occupied places at the terminal. All these variables admit a common domain which is Di = {0, 1, 2, ..., CT }. The values which can take the variables Cai and Cdi are dictated by airport managers. The constraints of the problem are the various equations and inequations governing the available capacities at the airport. Thereafter, we proposed an optimization model under constraints for the capacity allocation problem of the fixes. We showed an example of illustration implementing our procedures of modelling and optimization. The experimental results showed that our approach gives results at least as good as those found in the literature using other techniques. We also showed how to control and regulate these capacities in order to assist the managers to deal with unforeseen events.

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