On Interfacing Multiobjective Optimisation Models ... - Semantic Scholar

London, UK, May 29-31, 2012

ATACCS’2012 | RESEARCH PAPERS

On Interfacing Multiobjective Optimisation Models – the Case of the Airport Gate Assignment Problem Ignacy Kaliszewski

Janusz Miroforidis

Warsaw School of Information Technology ul. Newelska 6 01-447 Warszawa, Poland +48 22 38 10 392 [email protected]

Warsaw School of Information Technology ul. Newelska 6 01-447 Warszawa, Poland Treeffect Co Gdów 1028, 32-420 Gdów, Poland [email protected]

ABSTRACT

is also vital for ATM community in general (see [1] for a state-of-the-art for AGAP). The usual approach to deal with multiple criteria is to optimize the problem with respect to a selected criterion and observe the values of other criteria (cf. [7] where for that purpose a hierarchy of criteria has been proposed).

We present a method to capture decision maker’s preferences in multiobjective problems and we discuss its use as a base for a decision maker – multiobjective optimization model interface. We illustrate the idea on a small but illustrative numerical example of the airport gate assignment problem.

However, there exists a methodology which allows to address multiple criteria decision making problems and the underlying multiobjective optimization problem directly ([2,3,4,6], where other pertaining references are also given). Moreover, this methodology provides for an easy and intuitive capture of decision maker’s preferences and allows in turn determining of solutions which correspond to those preferences best. In other words, the methodology provides for an easy (to understand, command and implement) and intuitive interfacing the multiobjective optimisation models to the decision maker.

Categories and Subject Descriptors

J.2 [Physical Sciences and Engineering]: Aerospace. General Terms

Performance, Reliability, Experimentation. Keywords

Multiobjective problems, decision maker interface, preference capture, airport gate assignment problem INTRODUCTION

In principle, all decision problems within the scope of Air Traffic Management are multiple criteria decision making problems, i.e. each problem involves at least two criteria. Here we focus on a representative problem for the field, namely on the airport gate assignment problem (AGAP). This problem has attracted a lot of research and is well represented in the literature (cf. e.g. [1,7,8]. In relation to AGAP a number of criteria, such as total passenger walking distance, total passenger waiting time, number of apron (not-to-gate) assignments, deviation of the actual schedule from the flight time-table, number of aircraft relocations (towing) ([1]), is always listed, but seldom this problem is considered with respect to more than one criterion in time. Decision making in the presence of multiple criteria has been raised as a challenging issue for the AGAP researchers and practitioners communities, but it

The outline of the paper is as follows. In the next section we present briefly a model of AGAP adequate for small airports. In the subsequent section we present a generic multiple criteria decision making methodology. Lastly, we illustrate the methodology on an elementary but illustrative instance of the presented AGAP model. Some remarks on possible directions of future research conclude. AGAP FOR A SMALL AIRPORT

The problem under consideration is to assign incoming flights to airport gates. If at a given time there is no gate to serve a flight that flight (the corresponding plane) can be directed to wait for a gate or it can be assigned to the airport apron. Both waiting time and the number of flights assigned to the apron are best if zero but in the case of the airport overload they are in an obvious conflict.

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We assume that the airport under consideration:

Copyright 2012 IRIT PRESS, ISBN: 978-2-917490-20-4

93

1.

is small, so gate assignment has no significantly impact on passenger walking distance,

2.

all gates can accommodate any incoming flight,

3.

there are no constraints on neighbouring gate operations.



The model

where is a binary variable equal to 0 if flight assigned to a gate and equal to 1 if otherwise.

A flight is characterised by arrival time and ground time . Arrival times and ground times are discrete with interval . If flight is assigned to gate , at time then this gate is not available for another assignment for period .

There are

There are

∑

, for

There are

Objective functions (4) and (6) together with constraints (1), (2), (3) and (5) constitute a bicriteria model for AGAP at a small airport. Values of objective functions at efficient gate assignment represent rational compromises between waiting time for gate and apron operations.

(1)

constraints of type (1).

Let be a binary variable which is equal to 1 if gate serving a flight at time , and equal to 0 if otherwise. variables

is

The model can accommodate also other objective functions because the multiple criteria decision making methodology we present in the next section can deal with any number of criteria.

.

If flight is assigned to gate at time (i.e. ) then this gate serves that flight for consecutive time intervals ( for ) . This condition is equivalent to for There are

∑

.

Time widows

The model presented is all-integer and can be made linear by linearization of the objective function . The problems to be solved are of considerable size even for modest values of and . Although we have no influence on the magnitude of and , we can decrease the magnitude of significantly employing the concept of time windows.

(2)


No more that one flight can be assigned to a gate at a time, so ∑ There are

Observe that in the model the apron is a buffer which absorbs all flights which cannot wait sufficiently long for being assigned to a gate. It is reasonable to assume, as we do to define the objective function , that there is an upper bound on waiting time. Hence, with that assumption in place, in the model any flight is assigned to the apron at latest at its arrival time plus Suppose that time horizon of AGAP is divided into time windows of equal size such that window width is greater than or equal to , where is the end of that time window into which falls. Then, in the model a flight whose arrival time is in time window q will never compete for a gate assignment with a flight whose arrival 1. By this observation the model time is in time window for AGAP can be solved separately in a time window for all flights with arrival times in that window. Gate assignments in that time window which overlap with the

. (3)


If flight is assigned to a gate at its arrival time then there is no waiting time. Otherwise, the waiting time for flight equals ∑

∑

∑

and the maximal waiting time over all flights is ∑

∑ ∑

where

,

(4)

is an upper bound for admissible waiting time.

If flight is not assigned to a gate then it is assigned to the apron, so ∑

∑

, for

(6)

as the objective function to be minimised, at optimality with respect to (i.e. when variables and are optimal with respect to ) or at efficiency (i.e. when variables and are efficient with respect to and ), the number of variables taking value 1 will be minimal but not less then the value dictated by constraints (5).

is the

.

constraints of type (5). ∑

A flight can be assigned at most once to at most one gate, so ∑

and

With

Let be a binary variable which is equal to 1 if flight is assigned to gate at time , and equal to 0 if otherwise. Such an assignment cannot be made before flight arrives, i.e. if , hence for variables are undefined. There are ∑ variables , where number of time intervals in the time horizon .

variables

is

1

(5)

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To illustrate this, suppose that in time window with its end at a flight has arrival time at . Assume . Then the size of time windows should be at least 4 . With the size of time windows 3 that flight will not compete for a gate assignment in time window .



next time window can be represented in time window by fixing the corresponding variables to 1 but this requires that AGAP have to be solved sequentially in time windows, starting from the first time window.

capture of decision maker’s preferences. Observe that element ̂, where ̂ , represents maximal values of objective functions which can be attained if they are maximised separately.

Solving AGAP in time windows does not guarantee optimality with respect to the whole time horizon but makes the whole problem much more manageable from the computational point of view. We shall return to this issue in the concluding section.

To assists the decision maker in the search for the most preferred variant one can employ the optimisation problem (8). Here we assume the minimum of the decision maker rationality, namely we assume that the decision maker prefers an efficient variant to a non-efficient one.

THE MULTIOBJECTIVE METHODOLOGY

Suppose that an element such that ̂ does not exists which is rather a standard with conflicting criteria (if otherwise, is the most preferred variant). Then, the decision maker knows that whatever efficient variant he (or she) selects he has to compromise on values of objective functions with respect to values ̂ , . He can define his acceptable compromises on values ̂ , , and search for an efficient variant which corresponds to this compromise in three ways:

Let denote a (decision) variant (solution), variants, a set of feasible variants, multiobjective optimisation problem is:

a space of . Then the

(7) , where

, , , , , are objective functions (criteria); denotes the operator of deriving all efficient (as defined below) variants in . Variant ̅ of is efficient, if implies ̅ .

̅

where

),

, and

(8)

is such that

.

On the first glance, the objective function in (8) ( ( ) ) seems to be difficult to handle. However, observe that optimisation problem (8) is equivalent to

providing a reference point

3.

providing weights

.

.

(9)

Way 3. An experienced decision maker can define a vector of concessions in terms of weights , in optimisation problem (8). Vector concessions and vector of weights are related by formula (10).

,

The optimisation problem (8) if solved with ,

.

(10)

has the following property;

Besides the potential ability to derive each efficient variant, optimisation problem (8) provides for an easy and intuitive 2

,

̂

, )

2.

Way 2. A reference point ( ̂ , (it is irrelevant whether there exists an element such that or not) specifies explicitly a compromise between values of objective functions with respect to values ̂ , , which the decision maker regards as agreeable. A reference point specifies indirectly a vector of concessions:

By the “only if” part of this result no efficient variant is excluded from being derived by solving an instance of optimisation problem (8). In contrast to that, maximisation of a weighted sum of objective functions over does not possess, in general (and especially in the case of problems with discrete variables), this property.

(

providing a vector of concessions ,

Way 1. Components of a vector of concessions ( ) specify concessions the decision maker accepts to make with respect to ̂ . Components of vector can be defined in absolute values (“the decision maker is willing to make a concession of units on the value ̂ ”) or in relative values (“the decision maker is willing to make a concession of per cent on the value ̂ ”).

It is a well established result ([2,4,6]) that variant ̅ is efficient 2 if and only if it solves the optimisation problem (

1.

Actually, variant ̅ is weakly efficient but for the sake of conciseness we do not make this distinction here, for a formal treatment of this issue cf. [2,4,6].

95

-

it finds an efficient variant half line ̂ a variant exists,

such that is on , whenever such

-

otherwise, it finds an efficient variant

such that

London, UK, May 29-31, 2012 (̂ is on half line

) ̂

ATACCS’2012 | RESEARCH PAPERS ̂ .

In the adapted problem we have ̂ , so we can take, quite arbitrarily, (remember the necessity of replacement of ̂ by , as explained in the previous section).

̃ , where ̃

To avoid dividing by zero in formula (10), in formula (9) and in all pertaining considerations we are to replace ̂ by , but since in the definition of ( ̂ ,) can be taken arbitrarily small, so the difference between ̂ and can be made insignificant.

The clearly best combination4: zero waiting time and zero apron assignments is not possible in our example problem, so the decision maker has to compromise, i.e. to accept assignments which are worse than that combination with respect to at least one objective function.

SOLVING AGAP

Consider the following example. In a time horizon of 2 hours there are 5 flights scheduled as in Table 1. Discretisation interval is minutes. All ground times are equal to minutes. The upper bound on waiting time is 30 minutes.

As presented in the previous section, the decision maker can define his favourable compromises in three ways. Here we show how he can act along each of these ways. 1. Suppose that the decision maker is willing to make concessions on the (impossible) best combination and he defines such concessions by (the vector of) favourable concessions: (1 apron assignment, 10 minutes waiting time). Hence . By formula (10) . With these weights the objective function of optimisation problem (8) has the smallest value for assignment number 3 (apron assignments: 2, waiting time: 20), so this assignment (variant) is the solution of the optimisation problem (see Fig. 1). Suppose now that the decision maker is willing to make concessions on the (impossible) best combination but this time he defines such concessions by (the vector of) favourable concessions: (1 apron assignment, 3 minutes waiting time). Hence . By formula (10) . With these weights the objective function of optimisation problem (8) has the smallest value for assignment number 4 (apron assignments: 3, waiting time: 0), so now this assignment (variant) is the solution of the optimisation problem.

Table 1 TIME WINDOW i FLIGHT ARRIVAL TIME 1 0:05 2 0:15 3 0:30 4 0:40 5 0:45

We have generated by inspection all gate assignments (variants) for which the waiting time is less or equal 30 minutes There are 4 such assignments, as in Table 2. Table 2

1 2 3 4

FLIGHT-GATE ASSIGNMENTS GATE 1 GATE 2 APRON WAITING TIME 1,3 2,4 5 30 1,4 2,5 3 25 1,4 2 3,5 20 1 2 3,4,5 0

2. Suppose that the decision maker specifies explicitly a compromise between apron assignments and waiting time he would like to achieve or at least to mimic as closely as possible: 1 apron assignment, 10 minutes waiting time. Observe that in the example considered there is no such assignment, nevertheless reference point (signs have to be reverted for the adapted problem) captures the decision maker’s preferences at this point as described in the previous section. By formula (9) and by formula (10) . With these weights the objective function of optimisation problem (8) has the smallest value for assignment number 3 (apron assignments: 2, waiting time: 20), so this assignment (variant) is the solution of the optimisation problem. Suppose now that the decision maker specifies explicitly another compromise between apron assignments and waiting time he would like to achieve or at least to mimic as closely as possible: 0 apron assignment, 20 minutes waiting time. Hence, and by formula (9) and by formula (10) . With these weights the objective

We illustrate on this example the multiobjective optimisation methodology to capture decision maker’s preferences presented in the previous section. The methodology has been exposed for problems where all objective functions are maximised whereas in the example objective functions: the number of apron assignments and the waiting time are to be minimised. We can either adapt the presentation of the methodology to the example or vice versa. As there are many possible combinations of “min” or “max” type criteria we prefer to have one methodology presentation, as above, and to adapt in each case the problem at hand3. So, to have in our example both objective functions in the “max” form, we simply change signs of values of objective functions. For example, if we maximise in the adapted problem apron assignments then -1 apron assignments is better than -3 apron assignments, which is a purely technical convention.

3

We prefer so for the sake of the clarity of the methodology presentation. However, in practical application, for the sake of the decision maker comfort, the opposite is clearly preferred.

4

96

In the scope of multiple criteria decision making this combination is called the ideal point.



function of optimisation problem (8) has the smallest value for assignment number 2 (apron assignments: 1, waiting time: 25), so this assignment (variant) is the solution of the optimisation problem. 3. Suppose that the decision maker specifies directly two vectors of weights where from his experience with the problem solved many times in the past he knows that the first vector leads to assignments with a small number of apron assignments whereas the second leads to assignments with low waiting time. Let those vectors be: and . In the first case the objective function of optimisation problem (8) has the smallest value for assignment number 2 (apron assignments: 1, waiting time: 25), so this assignment (variant) is the solution of the optimisation problem. In the second case the objective function of optimisation problem (8) has the smallest value for assignment number 4 (apron assignments: 3, waiting time: 0), so this assignment (variant) is the solution of the optimisation problem.

A-34

It still may happen though that the resulting problem is not tractable in reasonable time by exact optimisation methods. Then one can resort to heuristics. In [5} an Evolutionary Multiobjective Optimisation method which accounts for the preference capture methodology as presented above has been recently proposed. The methodology to capture decision maker’s preferences presented in this paper allows for ad-hoc adapting the strategy for the search in a changing decision making environment. In the example we have shown how decision maker’s preferences can influence the property of assignments derived. Search strategy can change from one time window to the next, as dictated, e.g. by the weather conditions which if favourable may result in a preference to more apron assignments and if otherwise may result in a preference to less apron assignments at the cost of greater waiting times.

5

Assignments -3,5

overall assignment optimality in a block of time windows only during rush hours. There are two ways to cope which such relations. First, AGAP can be solved for a whole block of congested consecutive time windows at once, if the size of the resulting problem is manageable. Second, if the adopted assignment for time window consumes so much gate capacity in the time window that in that window no assignment with acceptable waiting time and number of apron assignments exist, the assignment in time window can be corrected (“worsened”) to allow for a better assignment for these two blocks considered jointly.

0 -2,5

-2

-1,5

-1

τ

-0,5

-5

0

0,5

-10 -15

A3

-20 A2

-25

A1

-30 -35

Waiting time

REFERENCES

1. Dorndorf, U., Drexel A., Nikulin, Y., Pesch E. Flight gate scheduling: State-of-the-art and recent development. Omega 35, 2007, 326-334. 2. Ehrgott, M. Multicriteria Optimization. Springer, 2005. 3. Kaliszewski, I. Out of the mist – towards decisionmaker-friendly Multiple Criteria Decision Making support. European Journal of Operational Research 158, 2004, 93–307. 4. Kaliszewski, I. Soft Computing for Complex Multiple Criteria Decision Making. Springer, 2006. 5. Kaliszewski, I., Miroforidis, J., Podkopaev, D. Interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy. European Journal of Operational Research 216, 2012, 293–307. 6. Miettinen, K.M. Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, 1999. 7. Şeker, M. Stochastic optimization models for the airport gate assignment problem. Transportation Research Part E, 48, 2011, 438-459. 8. Yan, S., Huo Ch-M. Optimization of multiple objective gate assignments. Transportation Research Part A, 35, 2001, 413-432

Figure 1. A graphical interpretation of searching for optimal solution in the example problem with the vector of concessions (1,00 , 10,00) (A # stands for the assignment number #). Here the smallest value of the objective function in optimisation problem (8) is attained for assignment 3, so this assignment is the optimal (and efficient) solution. CONCLUDING REMARKS

Solving sequential AGAP in time windows is attractive from the point of view of robustness of assignments. Arrival times are to be known only within some time before the beginning of the corresponding time window and not before the beginning of the time horizon . Such an approach absorbs flight delays in much less rigid form than when solving AGAP for the whole time horizon at once. As already mentioned, solving sequentially AGAP in time windows does not guarantee optimality with respect to the whole time horizon . On the other hand, in time windows which are far from each other mutual relations between assignments can be expected to be weak. For example, consider 24 hours time horizon starting 0:00 and ending 23:59. At a small (local) airport early morning and late night hours are seldom congested. Mutual relations between assignments may be of significant impact for the 97