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\title{Robustness Algorithms for the Airport Gate Assignment Problem}
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\begin{document}

             \author[1]{Amadeo Asco  Signes}

             \affil[1]{  University of Nottingham}

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\date{\small \em Received: 16 December 2019 Accepted: 2 January 2020 Published: 15 January 2020}

\maketitle


\begin{abstract}
        


Assigning commercial flights to available airport gates can have a major impact on the efficiency of flight schedules as well as on the level of passenger satisfaction with the service. These assignments also depend on the service requirements of flights and the capacity of stand facilities. Unexpected changes also called perturbations, like those due to air traffic delays, severe weather conditions, or equipment failures, may disrupt the initial assignments and increase the difficulty of maintaining smooth operations, which will detrimentally affect costumer satisfaction. The provision of solutions which reduce the potential detrimental effect of perturbations in the stands already assigned on the day of operation is desirable and some approaches are presented here, and compare between them to help identify their performance and trends.

\end{abstract}


\keywords{robustness, airport gate assignment problem, scheduling, heuristics, evolutionary algorithms.}

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\textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}}
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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
ircraft depart from an airport and arrive at their destination airport, from which the aircraft may again depart to yet another airport, and this may be repeated many times a day for each aircraft. During the time between arrival and departure, while the aircraft is still at the airport, it needs to have a space allocated at a stand on the airport airside, where some operations may need to be performed before it is ready to continue its cycle of departure and arrival. The stands next to the airport gates are scarce and expensive resources which must be used efficiently and be assigned to aircraft effectively. The gate assigned to an aircraft arrival may not be the same as that assigned to the same aircraft for departure, and the intermediate parking operation if any is required, between arrival and departure assignments may also be at a different stand, Figure  {\ref 1}.\par
A Fig.  {\ref 1}: Simplified overall process of an aircraft in an airport will detrimentally affect customer satisfaction. The Airport Gate Assignment Problem (AGAP) relates to the assignment of stands to flights already scheduled. The flights have a scheduled arrival and departure time, between which the aircraft is located on either one or several stands sequentially. The movement of an aircraft between stands normally requires the use of tugs (towing trucks) which add extra cost to the aircraft's operations, which airlines would prefer to avoid whenever possible. When an aircraft is assigned to different stands in the same arrival/departure period, to disembark passengers from one stand and embark them on another, then the aircraft must be moved from its assigned stand to the stand assigned to it next until it is located on the stand assigned for its departure. This operation may be executed either by using the aircraft's engines or by tugs, Figure \hyperref[fig_0]{2}.\par
With the increase in passenger traffic volumes and number of flights, the complexity of this task and the number of factors to be considered have increased significantly, and efficient gate utilization has received considerable attention in passed years, e.g. \hyperref[b3]{Ascó (2019)}, \hyperref[b11]{Kim and Feron (2013)}, \hyperref[b18]{Seker and Noyan (2012)}, \hyperref[b10]{Jaehn (2010)}, \hyperref[b12]{Li(2009)}, \hyperref[b9]{Hu and Di Paolo (2007)}. AGenetic Algorithm (GA) was used by  and \hyperref[b8]{Ghazouani et al (2015)}.\par
A flight is said to be in conflict if the departure time of the flight is greater than the arrival time of the next flight at a gate. Reassigning conflicting flights caused by flight delays is one of the major daily tasks for the airport scheduling team. When a delayed flight affects the flight schedule of the subsequent flight then there are two ways it may be corrected: either to reassign the conflicting flight or else reassign the subsequent flight to the conflicting one. A situation may arise when reassigning conflicting flights or the subsequent flight to another gate, where the reassigned flight is interfering with the subsequent flight at the new gate. Thus some reassignments may, therefore, have a downstream effect on the overall schedule, producing more conflicting flights requiring further reassignments, and potentially increasing the difficulty of the problem at a later stage.\par
The model used for the AGAP is based on that proposed in \hyperref[b6]{Dorndorf (2002)}, which considers the problem as a resource constrained project scheduling problem, originally presented in \hyperref[b7]{Dorndorf et al (2000)}, used in Ascó (2013) Chapter 7 and 8 with the Steady State Evolutionary Algorithm (SSEA), and Ascó (2019). The robustness approaches presented and studied for the ABSSAP by \hyperref[b1]{Ascó (2016)} are adapted to the AGAP and are next presented and studied with an adapted implementation of the SSEA for the AGAP. 
\section[{II.}]{II.} 
\section[{Robustness Approaches}]{Robustness Approaches}\par
The approaches considered here take account of the potential disruptions on the day of operation, and are: Total Reduction in Service Time (TRS), Area of Reduction in Service (ARS), Sub-Area of Reduction in Service (SARS), Unsupervised Estimated Stochastic Reduction in Service (UESRS), Reduction in the Number of Conflicts (RNC) and Probability of Conflict Based on the Gap (PCBG), which are described below and more details can be found in \hyperref[b1]{Ascó (2016)}, and Ascó et al (2013). \hyperref[b15]{Mangoubi and Mathaisel (1985)} proposed the use of 'buffer times' between two flights which are consecutively assigned to the same gate in order to obtain robust assignments, defining the reduction in service as that part of buffer time which overlaps with the previous assignment to the same gate. Given the detrimental effects that the reduction in service time has The provision of solutions which reduce the potential detrimental effect of perturbations in the resources already assigned on the day of operation is desirable and was previously studied for the Airport Baggage Sorting Station Assignment Problem (ABSSAP) by \hyperref[b1]{Ascó (2016}\hyperref[b0]{Ascó ( , 2013))}, and it was studied for the AGAP by van Schaijk and Visser (2017), Ascó (2013), \hyperref[b19]{Yan and Tang (2007)}. But this is achieved at the expense of the optimality, although this is far from easy, as the perturbations are not known in advance. It would also be advantageous if the disrupted assignments have no knock on effect, or if any, only a minor one. on the robustness of the assignment as against reallife delays, it is advisable to minimise the total reduction in service time, thus maximising buffer times. 
\section[{a) Total Reduction in Service}]{a) Total Reduction in Service}\par
The arrival and departure flights correspond to the arrival and departure activities respectively. The time between the scheduled arrival time at the stand and the time at which the flight is scheduled to leave is called the base service duration. A predefined period of time, called buffer time, the value of which depends on the flight, is preappended to the flight base starting service time, so that such buffer time may be reduced to allow other assignments to be placed before this flight, but the base service duration must not be affected, Figure  {\ref 3}. The use of buffer service time implies a preference for a greater predetermined service time for each flight, and this buffer time may be obtained from historical information. A reduction in the buffer time for the arrival and departure of aircraft j has been named ?? ?? ?? and ?? ?? ?? respectively, and the sum of these constitutes the reduction in service cost, so this objective can be Fig.  {\ref 3}: Overview of the flight times If the remote parking activity is assigned to the same stand as the departure activity, then the reduction in service for the departure flight is zero. This is a consequence of both activities referring to the same aircraft. 
\section[{b) Distribute Idle Time}]{b) Distribute Idle Time}\par
Different approaches aim to spread the time between the end of one assignment and the start of the next one at the same gate. This time is normally called 'idle time'. This may be achieved by using a penalty function, such as the arctangent function. \hyperref[b5]{Bolat (1999)} proposed the distribution of 'idle time' uniformly amongst gates.\par
A disadvantage of this approach and many others, i.e. UESRS, PCBG and RNC, is that they normally treat all assignments equally, whereas disturbance of a schedule is more likely to have disruptive consequences for the assignment at the time of day when service flight density is greater, which is taken into account by the next two approaches. 
\section[{c) Area Reduction in Service}]{c) Area Reduction in Service}\par
The effect of service reduction is not the same throughout the day, but depends on the time of day. It is more likely that disruptions will occur during periods when the flight density is higher than when fewer flights require servicing, i.e. delay during high flight density is more likely to propagate given that less resources will be available to absorb any reassignment without repercussions on other flights. At the same time, it is these cases where it is most difficult to keep a sufficiently large gap between assignments to the same Airport Gate (gate).\par
Lower Maximum Assignment Point with Parking (LMAPp) is the number of resources required to service a certain number of activities (full assignment is achievable for any other higher number of gates for the same number of flights) when the service starting time (s j ) coincides with the target starting service time (t j ), which it is an adaptation for the AGAP of the Lower Maximum Assignment Point (LMAP) initially introduced in Ascó et al (2013) for the ABSSAP.\par
Upper Maximum Assignment Point with Parking (UMAP p ) is the number of resources required to service those activities when the service starting time (s j ) coincides with the base starting service time (? j ), which it is an adaptation for the AGAP of the Upper Maximum Assignment Point (UMAP) initially introduced in Ascó et al (2013) for the ABSSAP. When considering the extra constrain of parking it may potentially be required extra resources to service the same number of flights in comparison to when there is not parking to consider.\par
The effect of assigning two activities to the same stand, namely too close to each other, has a more detrimental effect in circumstances where there are fewer gates able to accommodate flights. An indication of gate availability on time is provided by the activity density distribution, where no reduction in the target service start time is allowed, see Figure \hyperref[fig_1]{4}. To take account of this, use is made of the area between the reduced service time of the flight density distribution function divided by the Average Assignment Point (AAP) as presented in \hyperref[b1]{Ascó (2016}\hyperref[b0]{Ascó ( , 2013))}.\par
where is one if flight j is assigned to activity i for activity x (arrival, parking, or departure), or zero if flight j is not assigned to stand i during activity x. where no buffer time is used and secondly when the full buffer time is used (buffer time may not be reduced). The cost is that area between both distributions divided by a factor, Figure \hyperref[fig_2]{5}. The factor was estimated to be equal to the UMAP less the LMAP, which cannot be used if both have the same value, that is to say, LMAP equal to UMAP. To avoid this predicament the unfactored subarea is taken, i.e. using the sub-area itself. This will be the approach considered here, as known as This approach is based on reducing the number of conflicts on the day of operation. Given that the real perturbed conditions will not be available until the dayAAP = te ts f u (t) dt t e ? t s\par
(1)A = te ts (f u (t) ? f l (t)) dt (2) 
\section[{Global Journal of Computer Science and Technology}]{Global Journal of Computer Science and Technology}\par
Volume XX Issue I Version IYear 2020 ( ) C\par
Aj is the density distribution area for the time period from the target service time of flight j (t j ) to the end of service time for the previous flight assigned to the same gate, for example in Figure \hyperref[fig_1]{4}, AAP j is here defined as the mean number of flights over the target start time for flight j, t 18 , and the end of service time for the previous flight assigned to the same gate as flight j, e 9 , Equation  {\ref 1}.A 18 = e9 t18 f u (t) dt.\par
known as Base Sub-Area Reduction in Service (BSARS), Equation  {\ref 2}.\par
the schedule is implemented, these perturbed conditions are simulated by examining a set of perturbed base cases, S, which may be obtained in different ways, such as randomly, e.g. from historical data or calculated using known distribution(s) from information available at the time of generating the assignments.\par
The above version considers all the conflicts to be of the same importance, but it is preferable to have conflicts which do not have repercussions later, that is, can be reassigned to another gate without affecting any of the assignments already in existence. To account for this situation a new variable c js is defined which takes the value 1 if the reassignment of conflicting flight j in a perturbed schedule 's' affects other assignments already in existence, or zero otherwise. The objective is presented as Equation4where the constant, ?, 0 ? ? ? 1, denotes the importance of the conflicting flight repercussions on other assignments; ? = 0 corresponds to the case where no account is taken of any repercussion on other assignments, which corresponds in turn to Equation 3, and ? = 1 corresponds to the cases in which both the conflicting flights and their repercussions on other assignments are considered to be of the same importance. An ? > 1 refers to the cases where more importance is given to the repercussions of a conflict on assignments other than the conflict itself. Calculation of the conflicts is time consuming, and even more so if the effect of the conflict repercussions is also calculated, which is further aggravated by the need to use a large number of schedules in the perturbed set S in order to achieve a good representation of all the potential situations. The execution time is one of the disadvantages of using this approach. 
\section[{f) Probability of Conflicts Based on the Gap}]{f) Probability of Conflicts Based on the Gap}\par
The previous approach would normally require a large number of perturbed data sets, which makes its application very slow. Given that we are still interested in reducing the number of conflicts, but without the heavy cost in speed imposed by the RNC approach, then the probability of having a conflict in a given 'idle time' is used for each flight. This can be easily obtained if the delay distribution is known. In the case of a normal folded distribution (no negative numbers and with zero mean) and independent delays for flights, the probability of a conflict for different standard deviations is shown in Figure  {\ref 6}. This could also be extended to other distributions and to non-zero means.\par
The probability of a conflict for two consecutive flights assigned to the same gate based on the intervening gap modeled with Folded Normal distributions of a zero mean and standard deviation ?.f 2 = ?1 |S| * s S m j=1\par
Fig.  {\ref 6}:\par
Considering a set of perturbed schedules S, which simulate the perturbations on the day of operation. A new variable is introduced c js , which for a given solution of assignments has the value 1 if flight j is in conflict with another flight in the perturbed schedule , or zero otherwise. The average number of conflicts in the set of perturbed schedules S is calculated by Equation  {\ref 3}, which is a measure of the solution robustness. 
\section[{s S}]{s S}\par
When reassigning conflicting flights or subsequent flights to other gates, a situation can arise where the reassigned flight interferes with the subsequent flight at the new gate, a socalled secondary conflict. Some reassignment may therefore have a downstream effect on the overall schedule, producing more conflicting flights, in turn requiring further reassignments, thus potentially increasing the difficulty of the problem later on.f 2 = ?1 |S| * ? ? s S m j=1 (c js + ? * c js ) ? ? ?\par
Independent delays are considered initially, where a conflict between two consecutive flights is independent of other flights assigned to the same gate.\par
The assignments to different gates are independent from the point of view of conflicts. The probability of two consecutive flights having overlapping service times (a conflict), corresponds to the sum of the product between the probability of a sufficiently large delay on the part of the previous flight assigned to that gate, and the probability of the next consecutive assignment to the same gate not being sufficiently delayed, as shown in Figure \hyperref[fig_3]{7}, where ? j (t) is the probability density function for flight j and t 0 = ? k ? e j . 
\section[{g) Unsupervised Estimated Stochastic Reduction in Service}]{g) Unsupervised Estimated Stochastic Reduction in Service}\par
Lim and Wang (2005) proposed a stochastic programming model for the AGAP with a robustness cost of conflicts, which is estimated by a function, v(j, k). Flights are ordered by their base starting service time, so the gap between two flights j and k, l(j, k), assigned Fig.  {\ref 8}: Overlap between two flights j and k assigned to the same gate The PCBG is another stochastic approach which uses the predicted delay in activities probability distribution functions to anticipate the delay and take account of it in the assignments. In the case of a normal folded distribution (no negative numbers and with zero mean) and independent delays for flights, the probability of a conflict for different standard deviations.E(p(j, k)) = N i=1 y ij * y ik * v(j, k) ? v min (j, k) v max (j, k) ? v min (j, k) (6) f 2 = ? M ?1 j=1 M k=j+1 E(p(j, k (\textbf{7}\par
to the same gate, where j < k and j, k? [1 . . . M ], is the difference between flight k's target service time and the prior assigned flight j's end service time, Equation \hyperref[formula_7]{5},l(j, k) = t k ? e j (t k = ? k ? B k ) (\textbf{5})\par
where 0, as shown in Figure  {\ref 8}. ) is used to estimate the mean conflict probability between flights j and k assigned to the same gate, which is a function of the gap l(j, k), where larger gaps between assignments to the same gates result in lesser probability of real flight conflicts.\par
) is normalised in Equation  {\ref 6}.\par
The definition of v(j, k) comes from the application domain, in the absence of historical data; some unsupervised estimation functions were introduced in Lim and Wang (2005). Figure \hyperref[fig_5]{9} shows the penalty (y-axis) incurred for different unsupervised estimation functions as a function of the gap (x-axis). Negative values refer to reductions in service time between two assignments to the same baggage sorting l(j, k) = ?r for l(j, k) < v(j, k v(j, k 1. Linear estimation: 2. Exponential estimation: 
\section[{Inverse estimation}]{Inverse estimation}\par
The value of the constant 'b' changes the penalization as shown in Figure \hyperref[fig_5]{9}, so a higher 'b' increases the penalisation and a lower 'b' decreases it. An appropriate value should be selected to properly weight the influence of the potential conflicts. Lim and Wang (2005) used b = 15 minutes, which proved to provide rather poor results when compared with the exponential estimation function, which may partly be caused by the fixed cost when l(j, k) < 0 (dark red dash line, Figure \hyperref[fig_5]{9}), whereas in the exponential estimation function (purple dash line, Figure \hyperref[fig_5]{9}) this is not the case. The value used for 'b' may be too great, and a lower value would make this estimation function provides values closer to those provided by the exponential estimation function which provided fitter solutions in the results presented by \hyperref[b13]{Lim and Wang (2005)}. Consequently, a value b = 6 was seen in the The inverse estimation function as considered by Lim and Wang (2005) treats all gaps smaller than the buffer time equally, which does not represent a real case since smaller gaps between flights are more likely to result in conflicts than larger ones on the day of operation. Given this, and that the exponential estimation function performs best and treats all gaps differently, it is proposed that all of the gaps be treated differently, as shown by the modified version which is herein named 'Offset inverse', Equation  {\ref 11}, which is shown in Figure \hyperref[fig_5]{9} for b = 15 (green line). 
\section[{Sublinear estimation:}]{Sublinear estimation:}\par
This estimation also suffers from the same problem as the Inverse estimation, and may be improved by offsetting its value so that the maximum penalization corresponds to l max and the minimum to l min , Equation 13, which is shown in Figure \hyperref[fig_5]{9} for ? = 0. station (BSS), which are heavily penalised as they may require reassignment should delays occur, whereas positive gaps are penalised less. Wider gaps between two assignments reduce the need to reassign delayed flights, given that the delay has to be larger than the gap in order to affect the following assignment to the same BSS. Similarly, to start the service earlier may not require the flight to be reassigned because the duration of earliness has to be lower than the gap in order to affect the previous assignment. Both earliness and delay probabilities decrease as the gap increases. Sufficiently large gaps may also be used on the day of operation by disrupted flights which need to be reassigned, such that the detrimental effect of disruptions on that day is reduced.v(j, k) = ?l(j, k) (8) v(j, k) = e ?? * l(j,k) (9) v(j, k) = b l(j,k)+b if l(j, k) > 0 1 otherwise (10) v(j, k) = b l(j, k) ? min\{l(j, k)\} + b (11) v(j, k) = cos ? * l(j,k) lmax if l(j, k) > 0 1 otherwise (12) v(j, k) = cos ? * (l(j, k) ? l min + ?) l max ? l min + ? (\textbf{13}\par
The unsupervised estimation functions introduced in Lim and Wang (2005) are presented below and are shown in Figure \hyperref[fig_5]{9}: To penalise all of the flights similarly it is necessary to adapt the previous estimation formulas for v j,k as follow:\par
1. Inverse: 
\section[{Sub-linear:}]{Sub-linear:}\par
In addition in the case of the two consecutive flights j and k the max corresponds to the v(j, k) for minv(j, k) = b l(j, k) + B k + b (14) v(j, k) = cos ? * (l(j, k) + B k + ?) l max + D k + ? (\textbf{15})\par
Preferences' objective, shown in Section 3.1. Also, a table was generated showing the preferences of each handler, which is used in the 'Maximise Handler Preferences' objective, shown in Section 3.2. 
\section[{i. Airlines Gate Preferences}]{i. Airlines Gate Preferences}\par
The following is the data collected from 1 st September 2011 to 31 st August 2012, which it was provided by Heathrow Airport Limited (HAL), and a assigned to each gate, to each airline for the period from 6 th September 2010 to 12 th September 2010 for London Heathrow airport Terminal 4. 
\section[{ii. Handlers Gate Preferences}]{ii. Handlers Gate Preferences}\par
Figure \hyperref[fig_9]{12} shows the overall number of flights assigned to each gate, to each handler for the period from 6 th September 2009 to 12 th September 2009 for London Heathrow airport Terminal 4. 
\section[{IV.}]{IV.} 
\section[{Robustness Results}]{Robustness Results} 
\section[{III.}]{III.} 
\section[{Data}]{Data}\par
A week of records of flight assignments to stands was provided by London Heathrow airport for terminal four, composed of schedules from the 6 th to the 12 th September 2010 (H4T1009dd). Some details are shown in Table \hyperref[tab_1]{2}, which were generated from the data supplied. Using the data summarised in Table \hyperref[tab_1]{2}, tables were generated showing the preferences of each airline, and these were used in the 'Maximise Airline In this section, some experiments are conducted using the robustness approaches summarised in Section 2 for the respective weights shown in Table \hyperref[tab_2]{3} using the SSEA previously presented in Ascó (2013), \hyperref[b2]{Ascó (2018)}. 
\section[{\{v(j, k)\}}]{\{v(j, k)\}}\par
The gap definition used takes account of the buffer time, as the target service duration is the base service duration (T k ) plus the buffer time (B k ) for the flight. This makes the estimation functions dependent on the buffer time of each flight, as shown in Figure10for two buffer times of 30 min and 15 min each. When the buffer time is the same irrespective of the flights, \hyperref[b13]{Lim and Wang (2005)}, the cost is the same irrespective of the flight, depending only on the separation between consecutive flight assignments, but this is not the situation when the buffer time depends on the flight, namely long, medium or short distance flights, which are the cases studied here.B k = B? k? [1 . . .M], as considered in\{l(i, k)\} = ?B k .\par
summary of the data is shown on Table \hyperref[tab_0]{1}.\par
Figure \hyperref[fig_7]{11} shows the overall number of flights  
\section[{Global Journal of Computer Science and Technology}]{Global Journal of Computer Science and Technology}\par
Volume XX Issue I Version I   These weights are smaller than the weights used in the ABSSAP because they have been normalised, Equation  {\ref 16}. The results are summarised in tables which only show the robustness approaches which, at least in one instance of the disruptions for a given standard deviation, provide statistically significantly less collisions than other approaches evaluated, and cannot be said to be statistically worse than any of the approaches considered. The tables show for each standard deviation the number of times an approach cannot be said to be statistically significantly worse than any of the other approaches. The last column provides the sum of each result for each of the standard deviations. The case where all instances in a given standard deviation cannot be said to be statistically worse than any other are shown in bold text, and in underlined text for those cases which provide the highest number of all the approaches considered.\par
The results for the different robustness approaches, when applied to the data sets in Table \hyperref[tab_1]{2} (data sets from HAL for London Heathrow airport apparent statistical difference between them for short disruptions (10 min standard deviation). For longer disruptions it is the UESRS with exponential unsupervised estimation function with ? = 0.03, weight of 0.3125, with and without TRS approach which performs best for each of the similar disruptions considered. These results correspond to data sets where there is a sufficient number of gates for assignment to all of the activities (N < UMAP p ). No general gain is shown by combining the base approach with TRS. Nevertheless, there seems to be no detriment in combining with TRS either. The approaches ARS and BSARS do not perform well in any instance when either used alone or combined with TRS, which has also been observed when the rate of activities per gate increases presented in \hyperref[b13]{Lim and Wang (2005)}, namely, when the number of gates is greater than the UMAP, the exponential unsupervised estimation function performs better, but only when compared with the other unsupervised estimation functions.\par
Table  {\ref 6} shows the summary results for the new data sets with an extra 37 groups for the same number These data sets are equivalent to a reduction in the number of gates available per group, representing more activities for the same number of resources. The UESRS approaches alone or in combination with TRS still perform well for low disruptions (particularly with the exponential estimation function with ? = 0 05), and is subsequently performed better for longer disruptions. The ARS and BSARS also achieved solutions with statistically significantly less collision when they were used together with the TRS than when used alone, but not when compared to UESRS and PCBG(x). Terminal 4) are summarised in Table \hyperref[tab_3]{4}. There is no (Table  {\ref 6}). These results also corroborate those of gates (a summary of data sets is shown in Table \hyperref[tab_4]{5}). The empirical results show, when comparing the results of Tables \hyperref[tab_3]{3 and 4}, that combining the approaches with TRS helps to reduce the number of collisions where there is a lower number of gates per activity. These results suggest that when fewer resources (gates) are available the increase in the influence of the buffer time is advantageous, given that there is more chance of future disruptions as there is less 'idle time' available for the overall problem. It is therefore anticipated that combining both UESRS and such as ARS and BSARS, should also further improve the results. The ARS and BSARS are tailored to take account of the influence of the flights distribution over time, so increasing the penalty in periods where there is a higher demand for gates, which the experiments indicate improved results. 
\section[{Global Journal of Computer Science and Technology}]{Global Journal of Computer Science and Technology}\par
Volume XX Issue I Version I The summary of the statistical significance of the different robustness approaches is shown in Tables \hyperref[tab_6]{7 and 8}, where the PCBG uses the same standard deviation as the normal distribution which was used to generate the perturbed schedules. The LMAP, UMAP, LMAP p and UMAP p from Table \hyperref[tab_1]{2}  To speed up execution of the PCBG robustness approach instead of using the density function for the distribution (folded normal distribution), a pre-generated table of the accumulative probabilities was used for up to four times the standard deviation.\par
It should be noted that given that the PCBG used considers standard deviations equal to those used to build the perturbed data sets it may be considered biased and be expected to perform better. However the results obtained for data sets with a sufficient number of gates to assign all the activities shows that the UESRS performs better for different unsupervised functions than PCBG for all the disruption standard deviations considered.\par
V. 
\section[{Conclusions}]{Conclusions}\par
Different algorithms and their parameters were studied to find characteristics which could be used to identify the algorithm and parameters most appropriate to the AGAP. Both the model and algorithms are derived by modifying those presented in previous work, and are based on the specific characteristics of the problem.\par
Table 6: Summary of statistical significance of AGAP robustness (significance level = 0.05) using perturbed schedules generated from normal distributions of 10, 20 and 30 min standard deviations (x), SSEA1 for new data sets N4T1009dd with 37 extra groups each.\par
The time an aircraft expends parked at a gate has a considerable effect on the operations which take place up stream in the overall airport operation, especially when some of the resources required, such as gates, are limited. Delays in starting the departure sequencing may have important effects on the departure itself, which in turn may also require other aircraft to extend the time during which they are held at the gates. This could well affect other flights arriving which have had the same gates assigned to them. It would be therefore advisable to account for the effect of potential disturbances in the assignment plan and so some approaches were considered. The number of conflicts in perturbed schedules were used as a means of comparing the performance of different approaches. Different conditions were considered from when there are too few gates to assign to all flights, represented by LMAPp, when there are sufficient gates to service all the flight but at the expenses of a reduction on buffer time, represented by UMAPp, and when there are sufficient gates to assign all flight to the available gates without having to reduce the buffer time. It was concluded that the empirical results indicate that the PCBG did not provide such good results as the UESRS regarding those conflicts where there are plentiful gates to which to assign activities. PCBG performance improved as the number of gates available to service the activities is reduced. Furthermore, it was noted that the close relationship between the PCBG approach and the perturbed base schedule used to calculate the conflicts, and which provides some bias in favour of the PCBG, may be reduced or removed if the buffer times, considered for the other robustness approaches, are modified accordingly. The combination of UESRS and PCBG with TRS provides good solutions, and there is still some potential for combining UESRS and PCBG with other approaches, such as ARS and BSARS, which take account of other problem characteristics which both UESRS and PCBG do not, so potentially further improving the robustness of the solutions reached. \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Fig. 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Fig. 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{7}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Fig. 7 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4})}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{9}\includegraphics[]{image-7.png}
\caption{\label{fig_5}Fig. 9 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{10}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Fig. 10 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{11}\includegraphics[]{image-9.png}
\caption{\label{fig_7}Fig. 11 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-10.png}
\caption{\label{fig_8}Global}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{12}\includegraphics[]{image-11.png}
\caption{\label{fig_9}Fig. 12 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-12.png}
\caption{\label{fig_10}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-13.png}
\caption{\label{figure13}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.435016835016835\textwidth}P{0.011447811447811446\textwidth}P{0.03720538720538721\textwidth}P{0.051515151515151514\textwidth}P{0.05437710437710438\textwidth}P{0.09730639730639731\textwidth}P{0.0601010101010101\textwidth}P{0.10303030303030303\textwidth}}
ID\tabcellsep Date\tabcellsep LMAP\tabcellsep UMAP\tabcellsep LMAPp\tabcellsep \multicolumn{2}{l}{UMAPp No. Activities}\tabcellsep No. Parking Activities\\
\multicolumn{2}{l}{H4T100906 6 Sept 2010}\tabcellsep 8\tabcellsep 10\tabcellsep 17\tabcellsep 19\tabcellsep 118\tabcellsep 15\\
\multicolumn{2}{l}{H4T100907 7 Sept 2010}\tabcellsep 11\tabcellsep 14\tabcellsep 18\tabcellsep 20\tabcellsep 120\tabcellsep 15\\
\multicolumn{2}{l}{H4T100908 8 Sept 2010}\tabcellsep 7\tabcellsep 10\tabcellsep 16\tabcellsep 18\tabcellsep 119\tabcellsep 16\\
\multicolumn{2}{l}{H4T100909 9 Sept 2010}\tabcellsep 8\tabcellsep 10\tabcellsep 18\tabcellsep 20\tabcellsep 119\tabcellsep 15\\
\multicolumn{2}{l}{H4T100910 10 Sept 2010}\tabcellsep 9\tabcellsep 12\tabcellsep 15\tabcellsep 18\tabcellsep 120\tabcellsep 15\\
\multicolumn{2}{l}{H4T100911 11 Sept 2010}\tabcellsep 9\tabcellsep 10\tabcellsep 16\tabcellsep 16\tabcellsep 110\tabcellsep 11\\
\multicolumn{2}{l}{H4T100912 12 Sept 2010}\tabcellsep 11\tabcellsep 11\tabcellsep 18\tabcellsep 19\tabcellsep 117\tabcellsep 15\end{longtable} \par
 
\caption{\label{tab_1}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.05141776937618148\textwidth}P{0.10604914933837428\textwidth}P{0.12533081285444234\textwidth}P{0.567202268431002\textwidth}}
\tabcellsep \tabcellsep \tabcellsep Parameters\\
Approach\tabcellsep Weight\tabcellsep \tabcellsep Name Values\\
ARS\tabcellsep 0.00025\tabcellsep Buffer Time\tabcellsep 15 min long-haul and 10 min others\\
ATRS\tabcellsep 0.00025\tabcellsep Buffer Time\tabcellsep 15 min long-haul and 10 min others\\
BSARS\tabcellsep 0.00025\tabcellsep Buffer Time\tabcellsep 15 min long-haul and 10 min others\\
PCBG\tabcellsep 0.225 and 0.3125\tabcellsep Std. deviation\tabcellsep 10, 20 and 30 min\\
TRS\tabcellsep 0.00025\tabcellsep Buffer Time\tabcellsep 15 min long-haul and 10 min others\\
\tabcellsep \tabcellsep \tabcellsep Exp 0.03, Exp 0.05, Inverse 6, Inverse 15, Linear, Offset\\
\tabcellsep \tabcellsep Estimator\tabcellsep Inverse 6, Offset Inverse 15, Offset-\\
UESRS\tabcellsep 0.225 and 0.3125\tabcellsep \tabcellsep Sub linear 0, Offset Sub linear 1000 and Sub linear\\
\tabcellsep \tabcellsep Buffer Time\tabcellsep 15 min long-haul and 10 min others\end{longtable} \par
 
\caption{\label{tab_2}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.6518062397372741\textwidth}P{0.08234811165845649\textwidth}P{0.029310344827586206\textwidth}P{0.029310344827586206\textwidth}P{0.05722495894909688\textwidth}}
Approach\tabcellsep \multicolumn{3}{l}{Standard Deviation (x) in min 10 20 30}\tabcellsep Total\\
0.225PCBG(x)+0.00025TRS(2)\tabcellsep 7\tabcellsep 1\tabcellsep 1\tabcellsep 9\\
0.3125PCBG(x)+0.00025TRS(2)\tabcellsep 7\tabcellsep 1\tabcellsep 4\tabcellsep 12\\
0.225PCBG(x)\tabcellsep 7\tabcellsep 0\tabcellsep 2\tabcellsep 9\\
0.3125PCBG(x)\tabcellsep 7\tabcellsep 1\tabcellsep 4\tabcellsep 12\\
0.225UESRS(E0.03)+0.00025TRS(2)\tabcellsep 7\tabcellsep 4\tabcellsep 5\tabcellsep 16\\
0.225UESRS(E0.05)+0.00025TRS(2)\tabcellsep 7\tabcellsep 1\tabcellsep 0\tabcellsep 8\\
0.225UESRS(I4)+0.00025TRS(2)\tabcellsep 7\tabcellsep 1\tabcellsep 0\tabcellsep 8\\
0.225UESRS(I6)+0.00025TRS(2)\tabcellsep 7\tabcellsep 3\tabcellsep 0\tabcellsep 10\\
0.3125UESRS(E0.03)+0.00025TRS(2)\tabcellsep 7\tabcellsep 6\tabcellsep 6\tabcellsep 19\\
0.3125UESRS(E0.05)+0.00025TRS(2)\tabcellsep 7\tabcellsep 2\tabcellsep 2\tabcellsep 11\\
0.3125UESRS(I4)+0.00025TRS(2)\tabcellsep 7\tabcellsep 3\tabcellsep 1\tabcellsep 11\\
0.3125UESRS(I6)+0.00025TRS(2)\tabcellsep 7\tabcellsep 2\tabcellsep 1\tabcellsep 10\\
0.225UESRS(E0.03)\tabcellsep 7\tabcellsep 5\tabcellsep 6\tabcellsep 15\\
0.225UESRS(E0.05)\tabcellsep 7\tabcellsep 3\tabcellsep 1\tabcellsep 11\\
0.225UESRS(I4)\tabcellsep 7\tabcellsep 1\tabcellsep 0\tabcellsep 8\\
0.225UESRS(I6)\tabcellsep 7\tabcellsep 3\tabcellsep 0\tabcellsep 10\\
0.225UESRS(I15)\tabcellsep 7\tabcellsep 3\tabcellsep 3\tabcellsep 12\\
0.3125UESRS(E0.03)\tabcellsep 7\tabcellsep 6\tabcellsep 6\tabcellsep 19\\
0.3125UESRS(I4)\tabcellsep 7\tabcellsep 1\tabcellsep 0\tabcellsep 8\\
0.3125UESRS(I6)\tabcellsep 7\tabcellsep 3\tabcellsep 2\tabcellsep 12\\
0.3125UESRS(I15)\tabcellsep 7\tabcellsep 6\tabcellsep 5\tabcellsep 18\end{longtable} \par
 
\caption{\label{tab_3}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.1866554054054054\textwidth}P{0.2412162162162162\textwidth}P{0.05168918918918919\textwidth}P{0.05168918918918919\textwidth}P{0.05456081081081081\textwidth}P{0.05456081081081081\textwidth}P{0.10337837837837838\textwidth}P{0.10625\textwidth}}
ID\tabcellsep Date\tabcellsep LMAP\tabcellsep UMAP\tabcellsep LMAPp\tabcellsep UMAPp\tabcellsep No. Activ-ities\tabcellsep No. Parking Activi-ties\\
N4T100906\tabcellsep 6 Sept 2010\tabcellsep 17\tabcellsep 20\tabcellsep 23\tabcellsep 26\tabcellsep 164\tabcellsep 21\\
N4T100907\tabcellsep 7 Sept 2010\tabcellsep 21\tabcellsep 23\tabcellsep 25\tabcellsep 28\tabcellsep 160\tabcellsep 19\\
N4T100908\tabcellsep 8 Sept 2010\tabcellsep 18\tabcellsep 20\tabcellsep 23\tabcellsep 25\tabcellsep 169\tabcellsep 24\\
N4T100909\tabcellsep 9 Sept 2010\tabcellsep 21\tabcellsep 21\tabcellsep 28\tabcellsep 28\tabcellsep 168\tabcellsep 22\\
N4T100910\tabcellsep 10 Sept 2010\tabcellsep 19\tabcellsep 20\tabcellsep 20\tabcellsep 21\tabcellsep 164\tabcellsep 21\\
N4T100911\tabcellsep 11 Sept 2010\tabcellsep 19\tabcellsep 21\tabcellsep 21\tabcellsep 21\tabcellsep 154\tabcellsep 15\\
N4T100912\tabcellsep 12 Sept 2010\tabcellsep 19\tabcellsep 21\tabcellsep 23\tabcellsep 24\tabcellsep 167\tabcellsep 22\end{longtable} \par
 
\caption{\label{tab_4}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.6626878130217028\textwidth}P{0.08372287145242069\textwidth}P{0.02979966611018364\textwidth}P{0.02979966611018364\textwidth}P{0.043989983305509184\textwidth}}
Approach\tabcellsep \multicolumn{3}{l}{Standard Deviation (x) in min 10 20 30}\tabcellsep Total\\
0.225PCBG(x)+0.00025TRS(2)\tabcellsep 3\tabcellsep 3\tabcellsep 5\tabcellsep 11\\
0.3125PCBG(x)+0.00025TRS(2)\tabcellsep 2\tabcellsep 6\tabcellsep 6\tabcellsep 14\\
0.225PCBG(x)\tabcellsep 2\tabcellsep 4\tabcellsep 6\tabcellsep 12\\
0.3125PCBG(x)\tabcellsep 3\tabcellsep 5\tabcellsep 5\tabcellsep 13\\
0.225UESRS(E0.03)+0.00025TRS(2)\tabcellsep 5\tabcellsep 0\tabcellsep 0\tabcellsep 5\\
0.225UESRS(E0.05)+0.00025TRS(2)\tabcellsep 6\tabcellsep 1\tabcellsep 0\tabcellsep 7\\
0.225UESRS(I4)+0.00025TRS(2)\tabcellsep 3\tabcellsep 0\tabcellsep 0\tabcellsep 3\\
0.225UESRS(I6)+0.00025TRS(2)\tabcellsep 5\tabcellsep 0\tabcellsep 5\tabcellsep 5\\
0.3125UESRS(E0.03)+0.00025TRS(2)\tabcellsep 6\tabcellsep 2\tabcellsep 1\tabcellsep 9\\
0.3125UESRS(E0.05)+0.00025TRS(2)\tabcellsep 6\tabcellsep 4\tabcellsep 2\tabcellsep 12\\
0.3125UESRS(I4)+0.00025TRS(2)\tabcellsep 5\tabcellsep 0\tabcellsep 0\tabcellsep 5\\
0.3125UESRS(I6)+0.00025TRS(2)\tabcellsep 4\tabcellsep 1\tabcellsep 0\tabcellsep 5\\
0.225UESRS(E0.03)\tabcellsep 3\tabcellsep 0\tabcellsep 0\tabcellsep 3\\
0.225UESRS(E0.05)\tabcellsep 4\tabcellsep 1\tabcellsep 0\tabcellsep 5\\
0.225UESRS(I4)\tabcellsep 4\tabcellsep 0\tabcellsep 0\tabcellsep 4\\
0.225UESRS(I6)\tabcellsep 3\tabcellsep 0\tabcellsep 0\tabcellsep 3\\
0.225UESRS(I15)\tabcellsep 4\tabcellsep 1\tabcellsep 0\tabcellsep 5\\
0.3125UESRS(E0.03)\tabcellsep 5\tabcellsep 2\tabcellsep 0\tabcellsep 7\\
0.3125UESRS(I4)\tabcellsep 5\tabcellsep 0\tabcellsep 0\tabcellsep 5\\
0.3125UESRS(I6)\tabcellsep 4\tabcellsep 1\tabcellsep 0\tabcellsep 5\\
0.3125UESRS(I15)\tabcellsep 3\tabcellsep 1\tabcellsep 0\tabcellsep 4\end{longtable} \par
 
\caption{\label{tab_5}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{7} \par 
\begin{longtable}{P{0.1568452380952381\textwidth}P{0.12142857142857141\textwidth}P{0.11636904761904762\textwidth}P{0.11636904761904762\textwidth}P{0.11636904761904762\textwidth}P{0.11636904761904762\textwidth}P{0.10625\textwidth}}
H4T100906\tabcellsep H4T100907\tabcellsep H4T100908\tabcellsep H4T100909\tabcellsep H4T100910\tabcellsep H4T100911\tabcellsep H4T100912\\
(8, 10, 17,\tabcellsep (11, 14, 18,\tabcellsep (7, 10, 16,\tabcellsep (8, 10, 18,\tabcellsep (9, 12, 15,\tabcellsep (9. 10, 16,\tabcellsep (11, 11, 18,\\
19)\tabcellsep 20)\tabcellsep 18)\tabcellsep 20)\tabcellsep 18)\tabcellsep 16)\tabcellsep \\
Approach\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_6}Table 7 :}\end{figure}
 			\footnote{© 2020 Global Journals} 		 		\backmatter   			 
\subsection[{Acknowledgements}]{Acknowledgements}\par
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