research article a hybrid metaheuristic for multiple...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 742653 8 pageshttpdxdoiorg1011552013742653

Research ArticleA Hybrid Metaheuristic for Multiple Runways Aircraft LandingProblem Based on Bat Algorithm

Jian Xie1 Yongquan Zhou12 and Hongqing Zheng1

1 College of Information Science and Engineering Guangxi University for Nationalities Nanning Guangxi 530006 China2Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis Nanning Guangxi 530006 China

Correspondence should be addressed to Yongquan Zhou yongquanzhou126com

Received 17 May 2013 Accepted 11 July 2013

Academic Editor Xin-She Yang

Copyright copy 2013 Jian Xie et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aircraft landing problem (ALP) is an NP-hard problem the aim of ALP is to minimize the total cost of landing deviation frompredefined target time under the condition of safe landing In this paper the multiple runways case of the static ALP is consideredand a hybrid metaheuristic based on bat algorithm is presented to solve it Moreover four types of landing time assignmentstrategies are applied to allocate the scheduling time and a constructed initialization is used to speed up the convergence rateThe computational results show that the proposed algorithm can obtain the high-quality and comparable solutions for instancesup to 500 aircrafts and also it is capable of finding the optimal solutions for many instances in a short time

1 Introduction

Airport runway scheduling optimization is an ongoingchallenge for air traffic controllers The increasing demandis a challenge which leads to many airports and routesare congested Although investment in infrastructure mayincrease capacity at airports it is an effective solution toimprove planning and scheduling on current infrastructureThe aircraft landing Problem (ALP) is a kind of typical NP-hard problem in airport runway scheduling optimization [1]The ALP consists of determining an optimal schedule oflanding aircrafts on runways and assigning the landing timeof each arriving aircraftThe objective is tominimize the totalcost of landing deviation from predefined target time underthe condition of safe landing A predefined time window andseparation time requirements with other aircraft must meet

The first-come first-served (FCFS) is one of the mostcommon solving methods for ALP especially in the sin-gle runway situation A detailed review of published workaddressing the ALP can be found in [1ndash3] Generallyspeaking for ALP those methods can be broadly classifiedinto three categories exact methods (eg dynamic pro-gramming [4] branch-and-bound [1] and branch-and-price[5]) queueing theory [6] and heuristic or metaheuristic

Metaheuristic includes genetic algorithms [7] ant colonyoptimization [8] simulated annealing [9] scatter search andbionomic algorithms [2] cellular automata optimization [10]and other hybrid metaheuristics [9 11] Briskorn and Stolletzresent integer programmingmodels an aircraft landing prob-lems with aircraft classes [12]

Recently more and more metaheuristics inspired bynature or social phenomenon are proposed and these algo-rithms are increasingly applied to different fields The batalgorithm (BA) is one of the most popular algorithms whichis inspired by the intelligent echolocation behavior of micro-bats when they are foraging [13] Many researchers appliedBA to solve various optimization problems For exampleGandomi et al focus on solving constrained optimizationtasks [14] Yang and Gandomi apply BA to solve manyglobal engineering optimizations [15] Mishra at al use BAto update the weights of a functional link artificial neuralnetwork classifier a model proposed for classification [16]Meanwhile some researchers have improved BA and appliedit to various optimization problems Xie et al proposed abat algorithm based on differential operator and the Levyflights trajectory (DLBA) to solve function optimization andnonlinear equations [17] Wang et al proposed a new batalgorithm with mutation (BAM) to solve the uninhabited

2 Journal of Applied Mathematics

combat air vehicle (UCAV) path planning problem [18] Inthis paper the multiple runways case of the static ALP isconsidered a hybrid metaheuristic based on bat algorithm(HBA for short) is presented to solve it

The rest of this paper is organized as follows Section 2presents the mathematical model of ALP and original batalgorithm Hybrid metaheuristic based on bat algorithm isproposed in detail to solve multiple runways aircraft landingproblem in Section 3 The experimental results of the HBAand comparisons with other previous algorithms are shownin Section 4 In the last section we conclude this paper andpoint out some future work

2 Problem Definitions and Bat Algorithm

21 The Mathematical Formulation of ALP The ALP aimsat finding the best arrangement of sequences runway andcorresponding landing time for a given set of landing aircraftto minimize total cost by following separation requirementsTo illustrate the mathematical formulation some notationsand decision variables are defined as follows

Notations

119899 = the number of planes119898 = the number of runways119878119894119895= the separation time (ge0) between plane 119894 landing

and plane 119895 landing (where the planes 119894 and 119895 land onthe same runways) 119894 = 119895 isin 1 2 119899119904119894119895= the separation time (0 le 119904

119894119895le 119878119894119895) between plane

119894 landing and plane 119895 landing (where the planes 119894 and119895 land on the different runways) 119894 = 119895 isin 1 2 119899119879119894= the target landing time (target time) of plane 119894

119894 isin 1 2 119899119864119894= the earliest landing time of plane 119894 119894 isin

1 2 119899119871119894= the latest landing time of plane 119894 119894 isin 1 2 119899

119888119894= the cost of late landing of plane 119894 119894 isin 1 2 119899

larr997888119888119894= the cost of early landing of plane 119894 119894 isin

1 2 119899

Decision variables

119905119894= the scheduled landing time of plane 119894 119894 isin 1 2 119899

119910119894119895=

1 if plane 119894 lands beforeplane 119895

0 otherwise119894 = 119895 isin 1 2 119899

120574119894119903=

1 if plane 119894 landson runway 119903

0 otherwise119894 isin 1 119899 119903 isin 1 119898

120575119894119895=

1 if planes 119894 and 119895

land on the same runway0 otherwise

119894 = 119895 isin 1 2 119899

120591119894= the delay of landing plane 119894

(landing after the target time) thus 120591

119894= max (0 119905

119894minus 119879119894)

larr997888120591119894= the earliness of landing plane 119894

(landing before the target time) thus larr997888

120591119894= max (0 119879

119894minus 119905119894)

(1)

The mathematical formulation of a mixed-integer pro-gramming for this problem is as follows [1 2 9]

min Z =

119899

sum

119894=1

( 120591119894119888119894+larr997888120591119894

larr997888119888119894) (2)

st 119864119894le 119905119894le 119879119894 119894 = 1 2 119899 (3)

120574119894119895+ 120574119895119894= 1 forall119894 119895 isin 1 2 119899 119894 = 119895 (4)

119905119895ge 119905119894+ 119878119894119895120575119894119895+ 119904119894119895(1 minus 120575

119894119895) minus 119872120575

119894119895

119872 = (119871119894+max (119878

119894119895 119904119894119895) minus 119864119895) forall119894 = 119895 isin 1 2 119899

(5)119898

sum

119903=1

120574119894119903= 1 forall119894 isin 1 2 119899 (6)

120575119894119895ge 120574119894119903+120574119895119903minus1 119894 = 119895 isin 1 2 119899 119903 isin 1 2 119898

(7)

119905119894minus 119879119894= 120591119894minuslarr997888120591119894 119894 = 1 2 119899

0 le 120591119894le 119879119894minus 119864119894 120591119894ge 119879119894minus 119905119894 119894 = 1 2 119899

0 lelarr997888120591119894le 119871119894minus 119879119894larr997888120591119894ge 119905119894minus 119879 119894 = 1 2 119899

(8)

119910119894119895 120574119894119903 120575119894119895isin0 1 119894 =119895 isin 1 2 119899 119903isin1 2 119898

119905119894 120591119894larr997888120591119894ge 0 119894 = 1 2 119899

(9)

Objective function (2) minimizes the total cost of landingdeviation from target time The constraints (3) ensure thatthe scheduled landing time of each aircraft lies within itstime window constraints (4) consider the landing ordereither aircraft 119894 or 119895 must land first separation constraintsmust be ensured by constraints (5) where the role of 119872 isto ensure that the equation is redundant if 119895 lands before119894 the constraints (6) ensure that each aircraft should landon only one runway when aircrafts 119894 and 119895 are assigned toland on the same runway constraints (7) ensure that therunways assigned to aircrafts 119894 and 119895 are identical additionalconstraints (8) are introduced in order to link 120591

119894andlarr997888

120591119894to the

decision variable 119905119894 constraints (9) ensure that the decision

variables 119910119894119895 120574119894119903 and 120575

119894119895are binary and the decision variables

119905119894 120591119894 andlarr997888

120591119894are nonnegative respectively

Journal of Applied Mathematics 3

22 Bat Algorithm The basic bat algorithm (BA) is a meta-heuristic proposed by Yang in 2010 [13] Under several idealrules the BA has the following steps

Step 1 Initialize the bat population andother parameters andthese initial individuals are evaluated

Step 2 Each bat individual randomly selects a certain fre-quency 119891

119894of sonic pulse and the position of bat individual is

updated according to their selected frequency The formulasare as follows

119891119894= 119891min + (119891max minus 119891min) 120573

V119905119894= V119905minus1119894

+ (119909119905

119894minus 119909lowast) 119891119894

119909119905

119894= 119909119905minus1

119894+ V119905119894

(10)

where 119909119905

119894and V119905

119894represent the positions and velocities of

individual 119894 in a D-dimensional search space at generation119905 120573 isin [0 1] is a random vector drawn from a uniformdistribution and 119909

lowastis the current global best location

(solution) which is located after comparing all the solutionsamong all the 119899 bats Meanwhile these new individuals areevaluated

Step 3 If a random number is greater than its pulse emissionrate 119877 then a new position is generated around the currentglobal best position for each individual which is the equalto local search The updating formula of local search adopts119909 = 119909

lowast+ 120576 times 119871

119905 where 120576 isin [minus1 1] is a random number and

119871119905= ⟨119871119905

119894⟩ is the average loudness of all the bats at current

generation 119905

Step 4 If the local search is effective and its loudness 119871 isgreater than a random number then the new position isaccepted and its pulse emission rate 119877 and loudness 119871 areupdated where pulse emission rate is increased and loudnessis decreased 119871

119894and 119877

119894are updated by 119871

119905+1

119894= 120572 times 119871

119905

119894 119877119905+1

119894=

1198770

119894times [1 minus exp(minus120574 times 119905)] where 120572 120574 are constants

Step 5 If the termination criterion is met then the algorithmstops otherwise repeat algorithm (go to Step 2)

In general the bat algorithm has three procedures posi-tion updating local search and decreasing the probability oflocal search For the details of BA refer to [13]

3 Hybrid Metaheuristic Based on BatAlgorithm for ALP

Basic bat algorithm is a continuous optimization algorithmwhich is successfully applied to solve real optimizationproblem [15 17] However the standard continuous encodingscheme of BA cannot be used to solveALP directlyThereforein order to solve aircraft landing problem effectively HBA isproposed

31 Solution Representation in HBA In order to apply BAto ALP the first step is to devise a suitable representation

1 2 3 4 5 6 7 8 9 10

1 3 2 2 3 2 1 2 31

Aircraft

Runway

Figure 1 The representation of the candidate solutions

of the candidate solutions for this particular problem Eachindividual is a sequence 119878 (119878 = (119904

1 1199042 119904

119899)) with integer

number 119904119894(119904119894

isin 1 119898) where the integer numberrepresents the runway and the length of this sequence 119878 is thenumber of aircrafts For example if we have three runwaysand ten aircrafts the coded individual with integer is 1rarr3rarr 2rarr 2rarr 3rarr 2rarr 1rarr 2rarr 1rarr 3 Figure 1 shows that theaircrafts 1 7 and 9 land on runway number 1 the aircrafts3 4 6 and 8 land on runway number 2 the aircrafts 2 5 and10 land on runway number 3

32 Landing Time Assignment The assigned landing time(ALT) of each aircraft is the very important step in theALP the purpose is to reduce the total cost of penaltycaused by all aircrafts In this paper the landing time isassigned based on the target landing time 119879

119894assigned to

each aircraft There are four types of assignment strategiescarried out in the proposed algorithm forward assignmentstrategy (FAS) backward assignment strategy (BAS) randomforward assignment strategy (RFAS) and random backwardassignment strategy (RBAS) For each kind of assignmentstrategy firstly all aircrafts are found out on each runwayand the target landing time 119879 of these aircrafts is sorted inascending order namely 119879

1198941

le 1198791198942

le sdot sdot sdot le 119879119894119901minus1

le 119879119894119901

where119901 is the number of aircrafts on this runway

(i) FAS for each two sorted aircrafts 119894119896and 119894119896+1

on arunway the ALT is 119905

119894119896

and 119905119894119896+1

separately If 119879119894119896+1

lt

119905119894119896

+ 119878119894119896119894119896+1

then 119905119894119896+1

= 119905119894119896

+ 119878119894119896119894119896+1

otherwise 119905119894119896+1

=

119879119894119896+1

(ii) BAS for each two sorted aircrafts 119894119896and 119894119896+1

on arunway the ALT islarr997888119905

119894119896

and larr997888

119905119894119896+1

separately If 119879119894119896

gt

larr997888

119905119894119896+1

minus 119878119894119896119894119896+1

then larr997888

119905119894119896

=

larr997888

119905119894119896+1

minus 119878119894119896119894119896+1

otherwiselarr997888

119905119894119896

= 119879119894119896

(iii) RFAS for each two sorted aircrafts 119894119896and 119894119896+1

ona runway the ALT is

119905 119894119896

and

119905 119894119896+1

separately

119905 119894119896

isrounded down to the nearest integer betweenlarr997888

119905119894119896

and119905119894119896

If

119905 119894119896+1

lt

119905 119894119896

+ 119878119894119896119894119896+1

then

119905 119894119896+1

=

119905 119894119896

+ 119878119894119896119894119896+1

(iv) BFAS for each two sorted aircrafts 119894119896and 119894119896+1

ona runway the ALT is 119905

119894119896

and 119905119894119896+1

separately 119905119894119896


119905119894119896

and119905119894119896

If 119905119894119896

lt 119905119894119896+1

minus 119878119894119896119894119896+1

then 119905119894119896

= 119905119894119896+1

minus 119878119894119896119894119896+1

The assigned landing time is effective just while the allconstraints (3)ndash(9) are satisfied and the four kinds of totalcost are compared the best total cost is used as the objectivefunction value


33 Initialization Construction The initialization of a popu-lation is usually performed by randomly selecting a runwayfrom the available runways for each aircraft However thetarget landing time of aircraft is ordered and the timewindowmust ensure considering the safety So a preprocessing isperformed to improve the performance of HBA in theinitialization construction

In initialization 119901119904 individuals are generated where 119901119904 ispopulation size For each individual all target landing time119879119894(119894 = 1 2 119899) is sorted in an ascending order namely

1198791198941

le 1198791198942


le 119879119894119899

The first aircraft 1198941lands on a

runway randomly for each two sorted aircrafts 119894119896and 119894119896+1

if119879119894119896+1

lt 1198791198941

+ 119878119894119896119894119896+1

and then aircraft 119894119896+1

is allocated to otherrunway namely aircrafts 119894

119896and 119894119896+1

on different runwaysOtherwise the aircraft 119894

119896+1is allocated to the runway that

aircraft 119894119896lands on The initialization repeats until the 119901119904

individuals are generatedAccording to the experiments performed this initializa-

tion construction can develop an initial schedule of land-ing aircrafts which has a very good quality However toderive near-optimal solutions this initial schedule should beimproved using the bat algorithm provided in Section 34

34 Hybrid Bat Algorithm In original bat algorithm frame-work the idea is that firstly the bat individual randomlyselects a certain frequency of sonic pulse and the position ofbat individual is updated according to its selected frequencysecondly if a random number is greater than its pulseemission rate 119877 then a new position is generated aroundthe current global best position for each individual whichis equal to local search at last if the local search is effectiveand its loudness 119871 is greater than a random number thenthe new position is accepted and its pulse emission rate119877 and loudness 119871 are updated where pulse emission rateis increased and loudness is decreased In general the batalgorithm has three procedures position updating localsearch and decreasing the probability of local search

In this paper the frequency 119891 is a runway 119891 isin 1 119898The position updating is different from continuous batalgorithmTheposition updating is used to assign the landingsequence which is performed as follows

(i) select a frequency 119891 randomly namely select arunway randomly

(ii) select an aircraft randomly on selected runway thenassign this aircraft to other runway

There is an example used to illustrate the procedure inFigure 2 If the frequency 119891 = 2 and the second aircraft isselected then this aircraft is assigned to runway number 3

For the local search part this procedure is controlled bypulse emission rate 119877 The 119877 is equivalent to the probabilityof performing local search and the 119877 is updated by

119877 (119905) = (1 + exp(minus

5

119905maxtimes (119905 minus

119905max2

)))

minus1

(11)

where 119905 denotes the 119905th generation 119905max is the maximalgeneration The rate 119877 is similar to sigmoid function The

1 3 2 3 2 1 2 31

1 3 2 3 2 1 2 31

Before

After

2

3

Figure 2 An individual with ten aircrafts before and after positionupdating

1 3 2 3 2 1 2 1

1 3 2 3 2 1 2 1

Before

After

2 3

3 2

Figure 3 An individual with ten aircrafts before and after swap localsearch

purpose of the local search is to enhance the solutiongenerated and the operation is performed on the currentglobal best individual in bat algorithm In this paper twotypes of local search are presented the swap and loop sub-sequence inserting (LSI)

The purpose of swap is mildly mutating the currentglobal best individual so that an improved solution can befound out around the current optimal solution The swap isillustrated in Figure 3 where the third and tenth componentsare randomly chosen to be exchanged note that the twoselected aircrafts land on different runwaysThus the value ofthe third component is changed from 2 to 3 while the valueof the tenth component is switched from 3 to 2

The LSI is a variant of inserting operation the purpose isto mutate the current global best individual in a large extentso that the diversity of population can be ensured It caneffectively avoid prematurity and greatly improve efficiencyof global search The LSI is illustrated in Figure 4 a startpoint is randomly chosen (ninth component) and a randomlength of subsequence is determined (5) so the sub-sequencecan be determined (1rarr 3rarr 1rarr 3) A random insert point ischosen in remainder sub-sequence (third fifth components)then the selected sub-sequence is inserted into remaindersub-sequence before insert point

The local search systematically explores different neigh-borhood structuresThe swap and LSI are preformed accord-ing to the pulse emission rate 119877 In other words if arandom number is greater than the 119877 the swap is performedotherwise the LSI is performed

The loudness 119871119894of bat individual 119894 determines the

accepted probability a solution generated by local searchin original bat algorithm Meanwhile it also dominatesthe updating of pulse emission rate 119877 and loudness 119871 incontinuous bat algorithm However in this paper a runwaybalance (RB) operation is performed according to the valueof loudness 119871 for each individual

The intention of RB operation is to balance the load ofeach runway Firstly the aircrafts are counted on each run-way an aircraft selected randomly on runwaywithmaximumaircrafts is assigned to a runway with minimum aircraftsIf the amount of aircrafts on runway (maximum aircraftsor minimum aircrafts) is equal then a runway is selected


1 3 2 2 1 2 31

2 2 1 1 3 2 21

Before

After

2

33 3

3

Figure 4 An individual with ten aircrafts before and after LSI localsearch

randomly This process is stopped until the difference ofaircrafts amount on each runway is not more than one

The loudness 119871119894is updated by (12)

119871119905+1

119894= 120572 times 119871

119905

119894 (12)

where 120572 is a constant and initial value of 1198711119894

isin (1 2) If arandom number is less than its loudness 119871

119905

119894 then the RB

operation is performed otherwise each aircraft is assignedto a random runway

The iterative process is repeated until the terminationcriterion is met Algorithm 1 shows the pseudo-code of HBAbased on the framework of bat algorithm for ALP The lines1ndash3 the bat population and other parameters are initialized(the initialization of population uses a method describedin Section 33) and these initial individuals are evaluated(Section 32 detailedly described this procedure) In lines 5ndash9 the individuals are updated by selecting frequency (thispart corresponds to the position updating of BA this methodis described in Section 34) and these new individualsare evaluated Lines 10ndash15 show the local search and thesesolutions generated by local search are evaluated In lines 16ndash24 the runway balance operation is carried out Lines 25-26are the judgment of termination criterion and the output ofresults

4 Simulation Results and Comparisons

Thesimulation experiment is extensively investigated by a lar-ge number of benchmark instances these well-studied pro-blems are taken from the web OR-Library (last update June2012 httppeoplebrunelacuksimmastjjbjebinfohtml) areference site which contains detailed information regardinga large number of benchmark instances In this paper thewhole 13 instances from OR-Library are selected theseinstances have been widely used as benchmarks to certify theperformance of algorithms by many researchers [2 9]

All computational experiments are conductedwithMAT-LAB 2012a on a 30GHz Athlon PC with 20GB memoryThere are two kinds of termination criterion for differentinstances For instances where the value of optimal (exact)solution (119881opt) is known the algorithm is repeated untilthe objective function value is equal to the 119881opt if theobjective function value is greater than the 119881opt when themaximum generation 119905max(= 200) is met then the algorithmalso is terminated For instances where the value of 119881opt isnot known the termination criterion is set as maximumgeneration 119905max = 1000

41 Sensitivity Analysis Since performance is affected bythe settings of the parameter values used in metaheuristicsa sensitivity analysis is conducted to examine the effectof different parameter values on the proposed HBA Twoparameters that is population size 119901119904 and parameter 120572 areused to investigate the performance with respect to differentvalues which are based on the average results obtained frominstances involving from 10 to 50 aircrafts and each instancerun ten times The principal figures of merit for comparisonof different parameter values are the average percentagegap 119866avg () associated with the best solution found andthe average computational time CTavg in seconds for exactsolution is reachedThe percentage gap119866() is measurementcriteria referenced [2] Figures 5 and 6 show the statisticalresult where 119866avg is shown on the left 119910-axis and CTavg isshown on the right 119910-axis and different parameter values (119901119904and 120572) are listed in on the horizontal 119909-axis

The performance of the HBA in terms of differentpopulation numbers is shown in Figure 5 It shows that betterresults can be obtained with a larger population However itdoes not improve significantly when the value of 119901119904 is equalto or greater than 10 On the other hand the computationaltime increases steadily when the population rises

Figure 6 illustrates the performance effect of the HBAwith a decreasing 120572 value It indicates that the averagesolution quality deteriorates when the value of 120572 decreasesgradually In some cases increasing the value above 09 mayactually worsen the average objective value This is probablybecause the search is extremely random and RB operation iscarried out rarely

42 Comparisons of Results Based on the results of thesensitivity analysis the parameter values are set in theproposed HBA (ie 119901119904= 10 120572= 09) for comparison withother algorithms for solving the ALP with multiple runwaysthat were proposed and tested to be valid in previous studiesThe computational results obtained are shown in Tables 1 and2 In Tables 1 and 2 for each problem the instance (Ins)the number of aircrafts (119899) the number of runways (119898)the value of optimal solution (119881opt) the value of the best-known solution (119881best) if the optimal solution is not knownthe objective function value (119874) obtained and the percentagegap (119866) associated with the best solution found over the 15replications and the average execution time (119879) in seconds for15 replications Note that in order to compare the results the119904119894119895is set zero in accordance with the previous literature where

119866 is calculated on the basis of 119881opt or 119881best 119866 = 100 lowast (119874 minus

119881opt)119881opt One complication is that for some problem where119881opt or119881best is zero the percentage gap is defined as zero if andonly if the best solution found as zero is also zero otherwiseit is undefined (nd)

Table 1 presents results for a set of instances involvingfrom 10 to 50 aircrafts SS is scatter search the bionomicalgorithm is marked as BA1 [2] and IACA is improvedant colony algorithm [8] heuristic is an effective heuristicalgorithm in [1] FCFS is first-comefirst-served and the resultis reference [2] According to Table 1 the percentage gap119866 of SS and BA1 is better than HBA however the average


(1) Initialize the ps 119905 = 1 bat population and other parameters(2) Construct initial bat population (33 Initialization construction)(3) Assign landing time and evaluate each individual (32 Landing time assignment)(4) repeat(5) for 119894 = 1ps do(6) Determine frequency 119891(7) Update each bat individual(8) end(9) Assign landing time and evaluate each individual (32 Landing time assignment)(10) if rand gt 119877(119905) then(11) Carry out swap local search operation(12) else(13) Carry out LSI local search operation(14) end(15) Assign landing time and evaluate each individual (32 Landing time assignment)(16) Compute loudness of each individual by (12)(17) for 119894 = 1ps do(18) if rand lt 119871

119894then

(19) Carry out RB operation Runway balance operation(20) else(21) Assign each aircraft to a random runway(22) end(23) end(24) Assign landing time and evaluate each individual (32 Landing time assignment)(25) until 119905 = 119905max(26) Output result and plot

Algorithm 1 The pseudo-code of HBA for ALP

0 5 10 15 20 25 30 355

505

51

515

52

525

Population size

Aver

age p

erce

ntag

e gap

()

02

04

06

08

1

12

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg

Figure 5 The effect of the value of population size 119901119904 with respectto relative computational time and average percentage gap

execution time 119879 of SS and BA1 is longer than the 119879 ofHBA and the average execution time of HBA only expends043 seconds by contrast the SS and BA1 expend 64 and77 seconds respectively The comparison of results betweenHBA and heuristic shows the solutions obtained by HBA aresuperior to the solutions obtained by heuristic Meanwhilethe average execution time is approximate between HBA and

1 09 08 07 06 05 04 035

52

54

Aver

age p

erce

ntag

e gap

()

04

05

06

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg

Parameter 120572

Figure 6 The effect of the value of 120572 with respect to relativecomputational time and average percentage gap

heuristic Comparing with IACA and FCFS it is evidentlyshown that the HBA is effective

Table 2 presents results for a set of instances involvingfrom 100 to 500 aircrafts The termination criterion is set asmaximum generation 119905max = 1000 The IACA and heuristicdid not select these instances for testing From Table 2 evenfor the larger instances theHBA found that optimal solutions


Table 1 The comparisons of computational results for first group instances

Ins 119899 119898 119881optHBA SS BA1 IACA Heuristic FCFS

119874 119866 119879 119866 119879 119866 119879 119866 119879 119866 119879 119866

Airland1 10 2 90 90 0 008 0 24 0 45 0 01 0 01 03 0 0 0 011 0 39 0 34 0 00 0 01 0

Airland2 15 2 210 210 0 009 0 45 0 49 0 06 0 01 9523 0 0 0 010 0 46 0 43 0 04 0 01 0

Airland3 20 2 60 60 0 009 0 48 0 58 0 113 0 01 116673 0 0 0 010 0 62 0 63 0 93 0 02 nd

Airland4 202 640 640 0 055 0 52 0 55 0 79 0 01 03 130 130 0 014 0 46 0 57 0 64 0 01 04 0 0 0 014 0 56 0 52 0 58 0 02 0

Airland5 202 650 890 3692 144 0 50 308 61 1231 24 6462 01 16923 170 170 0 016 0 54 0 43 0 70 4118 01 5884 0 0 0 021 0 56 0 68 0 32 0 02 nd

Airland6 30 2 554 636 1480 161 0 70 361 101 5108 38 5921 01 59213 0 0 0 030 0 54 0 87 0 122 0 02 0

Airland7 44 2 0 0 0 009 0 118 0 124 0 552 0 02 0

Airland8 50 2 135 180 3333 201 0 121 0 196 2222 1681 8889 02 425933 0 0 0 016 0 139 0 181 nd 1082 0 06 nd

Average 500 043 0 64 039 77 535 236 1493 02 4530

Table 2 The comparisons of computational results for second group instances

Ins 119899 119898 119881bestHBA SS BA1 FCFS

119874 119866 119879 119866 119879 119866 119879 119866

Airland9 1002 45292 49949 1028 160 567 243 5473 487 172603 7575 7703 169 165 0 390 8746 466 342814 0 0 0 166 0 336 nd 439 nd

Airland10 150

2 128873 140745 921 186 787 608 2595 845 103473 22079 22413 151 210 888 668 19588 803 552164 3422 3422 0 206 1674 647 29240 788 3473495 0 0 0 231 0 607 nd 762 nd

Airland11 200

2 154084 167395 864 232 919 959 3854 1287 129803 28082 28064 minus006 261 2159 1021 29009 1203 764254 5453 5453 0 274 277 993 47447 1168 3947885 0 0 0 272 0 956 nd 1158 nd

Airland12 250

2 196139 248226 2656 283 1880 1266 5018 1835 137423 29004 24379 minus1595 315 1748 1454 19801 1710 903974 349 244 minus3009 333 27163 1445 1321691 1688 70752445 0 0 0 346 0 1386 nd 1662 nd

Airland13 500

2 550196 518406 minus578 580 372 3836 3747 5379 56913 110851 75515 minus3188 607 198 4560 18269 5158 462604 18846 9003 minus5223 637 2298 4413 118681 4977 2027945 735 0 minus100 659 0 4421 2230844 4887 5262871

Average minus937 340 2154 1590 257600 1932 909710

in several cases are prominent We can clearly find that thepercentage deviation 119866 from the best-known solutions isnegative which indicates the solutions found by HBA arebetter than the best-known solutionsThe average percentagegap 119866 of HBA is minus937 for this group of instances howeverthe average percentage gap119866 of SS BA1 and FCFS is positive

and is much greater than the 119866 of HBA On the other handthe average execution time119879 expended byHBA ismuch lesserthan the average execution time 119879 of SS and BA1 expended

Figure 7 illustrates the performance of the HBA theSS the BA1 the IACA and the heuristic with respect tothe computational time HBA and heuristic perform almost


0 5 10 15 20 25 30 350

100

200

300

400

500

600

Instance no

Aver

age e

xecu

tion

time (

s)

HBASSBA1

IACAHeuristic

Figure 7 The computational time of each test case

the same on instances up to 50 aircrafts For the largerinstances up to 500 aircrafts the computational time of HBAis shortened observably which also demonstrates that theHBA has a faster convergence rate

5 Conclusions

In this paper we considered the multiple runways aircraftlanding problem with the objective of minimizing the totaldeviation of landing time from the target time In orderto solve the larger instances involving up to 500 aircraftand multiple runways a hybrid metaheuristic based on batalgorithm (HBA for short) has been implementedThe HBAincludes a problem-dependent initialization constructionand several local search operations are integrated into theframework of bat algorithmThe computational results of theHBA show that the proposed algorithm is very effective andcompetitive and can obtain solutions with high quality forinstances up to 500 aircrafts in a short time Moreover thelanding time assignment of each aircraft is a key for solvingALP several excellent assignment strategies need to bepresented in our further work meanwhile the aircraft take-off problem (ATP) in airport runway scheduling problem alsois our future work

Acknowledgment

This work is supported by National Science Foundation ofChina under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 the Funded by Open ResearchFund Program of Key Lab of Intelligent Perception andImage Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100 and the Innovation Project ofGuangxiGraduate Education underGrant no YCSZ2012063

References

[1] J E Beasley M Krishnamoorthy Y M Sharaiha and DAbramson ldquoScheduling aircraft landingsmdashthe static caserdquoTransportation Science vol 34 no 2 pp 180ndash197 2000

[2] H Pinol and J E Beasley ldquoScatter search and bionomicalgorithms for the aircraft landing problemrdquo European Journalof Operational Research vol 171 no 2 pp 439ndash462 2006

[3] J A Bennell M Mesgarpour and C N Potts ldquoAirport runwayschedulingrdquo Annals of Operations Research vol 204 pp 249ndash270 2013

[4] A R Brentnall and R C H Cheng ldquoSome effects of air-craft arrival sequence algorithmsrdquo Journal of the OperationalResearch Society vol 60 no 7 pp 962ndash972 2009

[5] M Wen Algorithms of Scheduling Aircraft Landing ProblemTechnical University of Denmark DTU Lyngby Denmark2005

[6] N Bauerle O Engelhardt-Funke and M Kolonko ldquoOn thewaiting time of arriving aircrafts and the capacity of airportswith one or two runwaysrdquo European Journal of OperationalResearch vol 177 no 2 pp 1180ndash1196 2006

[7] X-B Hu and E Di Paolo ldquoAn efficient genetic algorithmwith uniform crossover for air traffic controlrdquo Computers andOperations Research vol 36 no 1 pp 245ndash259 2009

[8] G Bencheikh J Boukachour and A E H Alaoui ldquoImprovedant colony algorithm to solve the aircraft landing problemrdquoInternational Journal of Computer Theory and Engineering vol3 no 2 pp 224ndash233 2011

[9] A Salehipour M Modarres and N L Moslemi ldquoAn efficienthybrid meta-heuristic for aircraft landing problemrdquo Computersamp Operations Research vol 40 no 1 pp 207ndash213 2013

[10] S-P Yu X-B Cao and J Zhang ldquoA real-time schedule methodfor aircraft landing scheduling problem based on cellularautomationrdquo Applied Soft Computing Journal vol 11 no 4 pp3485ndash3493 2011

[11] G Hancerliogullari G Rabadi A H Al-Salem and M Khar-beche ldquoGreedy algorithms and metaheuristics for a multiplerunway combined arrival-departure aircraft sequencing prob-lemrdquo Journal of Air Transport Management vol 32 pp 39ndash482013

[12] D Briskorn and R Stolletz ldquoAircraft landing problems withaircraft classesrdquo Journal of Scheduling pp 1ndash15 2013

[13] X S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization (NICSOrsquo10) pp 65ndash74 Springer Berlin Germany 2010

[14] A H Gandomi X S Yang A H Alavi and S TalataharildquoBat algorithm for constrained optimization tasksrdquo NeuralComputing and Applications vol 22 no 6 pp 1239ndash1255 2013

[15] X S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[16] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012

[17] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator andLevy-flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013

[18] G Wang L Guo H Duan L Liu and H Wang ldquoA bat algo-rithm with mutation for UCAV path planningrdquo The ScientificWorld Journal vol 2012 Article ID 418946 15 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


combat air vehicle (UCAV) path planning problem [18] Inthis paper the multiple runways case of the static ALP isconsidered a hybrid metaheuristic based on bat algorithm(HBA for short) is presented to solve it

The rest of this paper is organized as follows Section 2presents the mathematical model of ALP and original batalgorithm Hybrid metaheuristic based on bat algorithm isproposed in detail to solve multiple runways aircraft landingproblem in Section 3 The experimental results of the HBAand comparisons with other previous algorithms are shownin Section 4 In the last section we conclude this paper andpoint out some future work

2 Problem Definitions and Bat Algorithm

21 The Mathematical Formulation of ALP The ALP aimsat finding the best arrangement of sequences runway andcorresponding landing time for a given set of landing aircraftto minimize total cost by following separation requirementsTo illustrate the mathematical formulation some notationsand decision variables are defined as follows

Notations

119899 = the number of planes119898 = the number of runways119878119894119895= the separation time (ge0) between plane 119894 landing

and plane 119895 landing (where the planes 119894 and 119895 land onthe same runways) 119894 = 119895 isin 1 2 119899119904119894119895= the separation time (0 le 119904

119894119895le 119878119894119895) between plane

119894 landing and plane 119895 landing (where the planes 119894 and119895 land on the different runways) 119894 = 119895 isin 1 2 119899119879119894= the target landing time (target time) of plane 119894

119894 isin 1 2 119899119864119894= the earliest landing time of plane 119894 119894 isin

1 2 119899119871119894= the latest landing time of plane 119894 119894 isin 1 2 119899

119888119894= the cost of late landing of plane 119894 119894 isin 1 2 119899

larr997888119888119894= the cost of early landing of plane 119894 119894 isin

1 2 119899

Decision variables

119905119894= the scheduled landing time of plane 119894 119894 isin 1 2 119899

119910119894119895=

1 if plane 119894 lands beforeplane 119895

0 otherwise119894 = 119895 isin 1 2 119899

120574119894119903=

1 if plane 119894 landson runway 119903

0 otherwise119894 isin 1 119899 119903 isin 1 119898

120575119894119895=

1 if planes 119894 and 119895

land on the same runway0 otherwise

119894 = 119895 isin 1 2 119899

120591119894= the delay of landing plane 119894

(landing after the target time) thus 120591

119894= max (0 119905

119894minus 119879119894)

larr997888120591119894= the earliness of landing plane 119894

(landing before the target time) thus larr997888

120591119894= max (0 119879

119894minus 119905119894)

(1)

The mathematical formulation of a mixed-integer pro-gramming for this problem is as follows [1 2 9]

min Z =

119899

sum

119894=1

( 120591119894119888119894+larr997888120591119894

larr997888119888119894) (2)

st 119864119894le 119905119894le 119879119894 119894 = 1 2 119899 (3)

120574119894119895+ 120574119895119894= 1 forall119894 119895 isin 1 2 119899 119894 = 119895 (4)

119905119895ge 119905119894+ 119878119894119895120575119894119895+ 119904119894119895(1 minus 120575

119894119895) minus 119872120575

119894119895

119872 = (119871119894+max (119878

119894119895 119904119894119895) minus 119864119895) forall119894 = 119895 isin 1 2 119899

(5)119898

sum

119903=1

120574119894119903= 1 forall119894 isin 1 2 119899 (6)

120575119894119895ge 120574119894119903+120574119895119903minus1 119894 = 119895 isin 1 2 119899 119903 isin 1 2 119898

(7)

119905119894minus 119879119894= 120591119894minuslarr997888120591119894 119894 = 1 2 119899

0 le 120591119894le 119879119894minus 119864119894 120591119894ge 119879119894minus 119905119894 119894 = 1 2 119899

0 lelarr997888120591119894le 119871119894minus 119879119894larr997888120591119894ge 119905119894minus 119879 119894 = 1 2 119899

(8)

119910119894119895 120574119894119903 120575119894119895isin0 1 119894 =119895 isin 1 2 119899 119903isin1 2 119898

119905119894 120591119894larr997888120591119894ge 0 119894 = 1 2 119899

(9)

Objective function (2) minimizes the total cost of landingdeviation from target time The constraints (3) ensure thatthe scheduled landing time of each aircraft lies within itstime window constraints (4) consider the landing ordereither aircraft 119894 or 119895 must land first separation constraintsmust be ensured by constraints (5) where the role of 119872 isto ensure that the equation is redundant if 119895 lands before119894 the constraints (6) ensure that each aircraft should landon only one runway when aircrafts 119894 and 119895 are assigned toland on the same runway constraints (7) ensure that therunways assigned to aircrafts 119894 and 119895 are identical additionalconstraints (8) are introduced in order to link 120591

119894andlarr997888

120591119894to the

decision variable 119905119894 constraints (9) ensure that the decision

variables 119910119894119895 120574119894119903 and 120575

119894119895are binary and the decision variables

119905119894 120591119894 andlarr997888

120591119894are nonnegative respectively







119891119894= 119891min + (119891max minus 119891min) 120573

V119905119894= V119905minus1119894

+ (119909119905

119894minus 119909lowast) 119891119894

119909119905

119894= 119909119905minus1

119894+ V119905119894

(10)

where 119909119905

119894and V119905








119871119905= ⟨119871119905


generation 119905


119894and 119877


119905+1

119894= 120572 times 119871

119905

119894 119877119905+1

119894=

1198770







1 2 3 4 5 6 7 8 9 10

1 3 2 2 3 2 1 2 31

Aircraft

Runway



1 1199042 119904


number 119904119894(119904119894



119894assigned to


1198941

le 1198791198942


le 119879119894119901




119894119896

and 119905119894119896+1

separately If 119879119894119896+1

lt

119905119894119896

+ 119878119894119896119894119896+1

then 119905119894119896+1

= 119905119894119896

+ 119878119894119896119894119896+1

otherwise 119905119894119896+1

=

119879119894119896+1



119894119896

and larr997888

119905119894119896+1

separately If 119879119894119896

gt

larr997888

119905119894119896+1

minus 119878119894119896119894119896+1

then larr997888

119905119894119896

=

larr997888

119905119894119896+1

minus 119878119894119896119894119896+1

otherwiselarr997888

119905119894119896

= 119879119894119896



119905 119894119896

and

119905 119894119896+1

separately

119905 119894119896


119905119894119896

and119905119894119896

If

119905 119894119896+1

lt

119905 119894119896

+ 119878119894119896119894119896+1

then

119905 119894119896+1

=

119905 119894119896

+ 119878119894119896119894119896+1



119894119896

and 119905119894119896+1

separately 119905119894119896


119905119894119896

and119905119894119896

If 119905119894119896

lt 119905119894119896+1

minus 119878119894119896119894119896+1

then 119905119894119896

= 119905119894119896+1

minus 119878119894119896119894119896+1





1198791198941

le 1198791198942


le 119879119894119899



if119879119894119896+1

lt 1198791198941

+ 119878119894119896119894119896+1



119896and 119894119896+1












119877 (119905) = (1 + exp(minus

5


119905max2

)))

minus1

(11)


1 3 2 3 2 1 2 31

1 3 2 3 2 1 2 31

Before

After

2

3


1 3 2 3 2 1 2 1

1 3 2 3 2 1 2 1

Before

After

2 3

3 2










1 3 2 2 1 2 31

2 2 1 1 3 2 21

Before

After

2

33 3

3




119871119905+1

119894= 120572 times 119871

119905

119894 (12)



119905

119894 then the RB















119894then



0 5 10 15 20 25 30 355

505

51

515

52

525

Population size

Aver

age p

erce

ntag

e gap

()

02

04

06

08

1

12

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg



1 09 08 07 06 05 04 035

52

54

Aver

age p

erce

ntag

e gap

()

04

05

06

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg

Parameter 120572







119874 119866 119879 119866 119879 119866 119879 119866 119879 119866 119879 119866

Airland1 10 2 90 90 0 008 0 24 0 45 0 01 0 01 03 0 0 0 011 0 39 0 34 0 00 0 01 0

Airland2 15 2 210 210 0 009 0 45 0 49 0 06 0 01 9523 0 0 0 010 0 46 0 43 0 04 0 01 0

Airland3 20 2 60 60 0 009 0 48 0 58 0 113 0 01 116673 0 0 0 010 0 62 0 63 0 93 0 02 nd

Airland4 202 640 640 0 055 0 52 0 55 0 79 0 01 03 130 130 0 014 0 46 0 57 0 64 0 01 04 0 0 0 014 0 56 0 52 0 58 0 02 0

Airland5 202 650 890 3692 144 0 50 308 61 1231 24 6462 01 16923 170 170 0 016 0 54 0 43 0 70 4118 01 5884 0 0 0 021 0 56 0 68 0 32 0 02 nd

Airland6 30 2 554 636 1480 161 0 70 361 101 5108 38 5921 01 59213 0 0 0 030 0 54 0 87 0 122 0 02 0

Airland7 44 2 0 0 0 009 0 118 0 124 0 552 0 02 0

Airland8 50 2 135 180 3333 201 0 121 0 196 2222 1681 8889 02 425933 0 0 0 016 0 139 0 181 nd 1082 0 06 nd

Average 500 043 0 64 039 77 535 236 1493 02 4530



119874 119866 119879 119866 119879 119866 119879 119866

Airland9 1002 45292 49949 1028 160 567 243 5473 487 172603 7575 7703 169 165 0 390 8746 466 342814 0 0 0 166 0 336 nd 439 nd

Airland10 150

2 128873 140745 921 186 787 608 2595 845 103473 22079 22413 151 210 888 668 19588 803 552164 3422 3422 0 206 1674 647 29240 788 3473495 0 0 0 231 0 607 nd 762 nd

Airland11 200

2 154084 167395 864 232 919 959 3854 1287 129803 28082 28064 minus006 261 2159 1021 29009 1203 764254 5453 5453 0 274 277 993 47447 1168 3947885 0 0 0 272 0 956 nd 1158 nd

Airland12 250

2 196139 248226 2656 283 1880 1266 5018 1835 137423 29004 24379 minus1595 315 1748 1454 19801 1710 903974 349 244 minus3009 333 27163 1445 1321691 1688 70752445 0 0 0 346 0 1386 nd 1662 nd

Airland13 500


Average minus937 340 2154 1590 257600 1932 909710





0 5 10 15 20 25 30 350

100

200

300

400

500

600

Instance no

Aver

age e

xecu

tion

time (

s)

HBASSBA1

IACAHeuristic



5 Conclusions


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119891119894= 119891min + (119891max minus 119891min) 120573

V119905119894= V119905minus1119894

+ (119909119905

119894minus 119909lowast) 119891119894

119909119905

119894= 119909119905minus1

119894+ V119905119894

(10)

where 119909119905

119894and V119905








119871119905= ⟨119871119905


generation 119905


119894and 119877


119905+1

119894= 120572 times 119871

119905

119894 119877119905+1

119894=

1198770







1 2 3 4 5 6 7 8 9 10

1 3 2 2 3 2 1 2 31

Aircraft

Runway



1 1199042 119904


number 119904119894(119904119894



119894assigned to


1198941

le 1198791198942


le 119879119894119901




119894119896

and 119905119894119896+1

separately If 119879119894119896+1

lt

119905119894119896

+ 119878119894119896119894119896+1

then 119905119894119896+1

= 119905119894119896

+ 119878119894119896119894119896+1

otherwise 119905119894119896+1

=

119879119894119896+1



119894119896

and larr997888

119905119894119896+1

separately If 119879119894119896

gt

larr997888

119905119894119896+1

minus 119878119894119896119894119896+1

then larr997888

119905119894119896

=

larr997888

119905119894119896+1

minus 119878119894119896119894119896+1

otherwiselarr997888

119905119894119896

= 119879119894119896



119905 119894119896

and

119905 119894119896+1

separately

119905 119894119896


119905119894119896

and119905119894119896

If

119905 119894119896+1

lt

119905 119894119896

+ 119878119894119896119894119896+1

then

119905 119894119896+1

=

119905 119894119896

+ 119878119894119896119894119896+1



119894119896

and 119905119894119896+1

separately 119905119894119896


119905119894119896

and119905119894119896

If 119905119894119896

lt 119905119894119896+1

minus 119878119894119896119894119896+1

then 119905119894119896

= 119905119894119896+1

minus 119878119894119896119894119896+1





1198791198941

le 1198791198942


le 119879119894119899



if119879119894119896+1

lt 1198791198941

+ 119878119894119896119894119896+1



119896and 119894119896+1












119877 (119905) = (1 + exp(minus

5


119905max2

)))

minus1

(11)


1 3 2 3 2 1 2 31

1 3 2 3 2 1 2 31

Before

After

2

3


1 3 2 3 2 1 2 1

1 3 2 3 2 1 2 1

Before

After

2 3

3 2










1 3 2 2 1 2 31

2 2 1 1 3 2 21

Before

After

2

33 3

3




119871119905+1

119894= 120572 times 119871

119905

119894 (12)



119905

119894 then the RB















119894then



0 5 10 15 20 25 30 355

505

51

515

52

525

Population size

Aver

age p

erce

ntag

e gap

()

02

04

06

08

1

12

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg



1 09 08 07 06 05 04 035

52

54

Aver

age p

erce

ntag

e gap

()

04

05

06

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg

Parameter 120572







119874 119866 119879 119866 119879 119866 119879 119866 119879 119866 119879 119866

Airland1 10 2 90 90 0 008 0 24 0 45 0 01 0 01 03 0 0 0 011 0 39 0 34 0 00 0 01 0

Airland2 15 2 210 210 0 009 0 45 0 49 0 06 0 01 9523 0 0 0 010 0 46 0 43 0 04 0 01 0

Airland3 20 2 60 60 0 009 0 48 0 58 0 113 0 01 116673 0 0 0 010 0 62 0 63 0 93 0 02 nd

Airland4 202 640 640 0 055 0 52 0 55 0 79 0 01 03 130 130 0 014 0 46 0 57 0 64 0 01 04 0 0 0 014 0 56 0 52 0 58 0 02 0

Airland5 202 650 890 3692 144 0 50 308 61 1231 24 6462 01 16923 170 170 0 016 0 54 0 43 0 70 4118 01 5884 0 0 0 021 0 56 0 68 0 32 0 02 nd

Airland6 30 2 554 636 1480 161 0 70 361 101 5108 38 5921 01 59213 0 0 0 030 0 54 0 87 0 122 0 02 0

Airland7 44 2 0 0 0 009 0 118 0 124 0 552 0 02 0

Airland8 50 2 135 180 3333 201 0 121 0 196 2222 1681 8889 02 425933 0 0 0 016 0 139 0 181 nd 1082 0 06 nd

Average 500 043 0 64 039 77 535 236 1493 02 4530



119874 119866 119879 119866 119879 119866 119879 119866

Airland9 1002 45292 49949 1028 160 567 243 5473 487 172603 7575 7703 169 165 0 390 8746 466 342814 0 0 0 166 0 336 nd 439 nd

Airland10 150

2 128873 140745 921 186 787 608 2595 845 103473 22079 22413 151 210 888 668 19588 803 552164 3422 3422 0 206 1674 647 29240 788 3473495 0 0 0 231 0 607 nd 762 nd

Airland11 200

2 154084 167395 864 232 919 959 3854 1287 129803 28082 28064 minus006 261 2159 1021 29009 1203 764254 5453 5453 0 274 277 993 47447 1168 3947885 0 0 0 272 0 956 nd 1158 nd

Airland12 250

2 196139 248226 2656 283 1880 1266 5018 1835 137423 29004 24379 minus1595 315 1748 1454 19801 1710 903974 349 244 minus3009 333 27163 1445 1321691 1688 70752445 0 0 0 346 0 1386 nd 1662 nd

Airland13 500


Average minus937 340 2154 1590 257600 1932 909710





0 5 10 15 20 25 30 350

100

200

300

400

500

600

Instance no

Aver

age e

xecu

tion

time (

s)

HBASSBA1

IACAHeuristic



5 Conclusions


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198791198941

le 1198791198942


le 119879119894119899



if119879119894119896+1

lt 1198791198941

+ 119878119894119896119894119896+1



119896and 119894119896+1












119877 (119905) = (1 + exp(minus

5


119905max2

)))

minus1

(11)


1 3 2 3 2 1 2 31

1 3 2 3 2 1 2 31

Before

After

2

3


1 3 2 3 2 1 2 1

1 3 2 3 2 1 2 1

Before

After

2 3

3 2










1 3 2 2 1 2 31

2 2 1 1 3 2 21

Before

After

2

33 3

3




119871119905+1

119894= 120572 times 119871

119905

119894 (12)



119905

119894 then the RB















119894then



0 5 10 15 20 25 30 355

505

51

515

52

525

Population size

Aver

age p

erce

ntag

e gap

()

02

04

06

08

1

12

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg



1 09 08 07 06 05 04 035

52

54

Aver

age p

erce

ntag

e gap

()

04

05

06

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg

Parameter 120572







119874 119866 119879 119866 119879 119866 119879 119866 119879 119866 119879 119866

Airland1 10 2 90 90 0 008 0 24 0 45 0 01 0 01 03 0 0 0 011 0 39 0 34 0 00 0 01 0

Airland2 15 2 210 210 0 009 0 45 0 49 0 06 0 01 9523 0 0 0 010 0 46 0 43 0 04 0 01 0

Airland3 20 2 60 60 0 009 0 48 0 58 0 113 0 01 116673 0 0 0 010 0 62 0 63 0 93 0 02 nd

Airland4 202 640 640 0 055 0 52 0 55 0 79 0 01 03 130 130 0 014 0 46 0 57 0 64 0 01 04 0 0 0 014 0 56 0 52 0 58 0 02 0

Airland5 202 650 890 3692 144 0 50 308 61 1231 24 6462 01 16923 170 170 0 016 0 54 0 43 0 70 4118 01 5884 0 0 0 021 0 56 0 68 0 32 0 02 nd

Airland6 30 2 554 636 1480 161 0 70 361 101 5108 38 5921 01 59213 0 0 0 030 0 54 0 87 0 122 0 02 0

Airland7 44 2 0 0 0 009 0 118 0 124 0 552 0 02 0

Airland8 50 2 135 180 3333 201 0 121 0 196 2222 1681 8889 02 425933 0 0 0 016 0 139 0 181 nd 1082 0 06 nd

Average 500 043 0 64 039 77 535 236 1493 02 4530



119874 119866 119879 119866 119879 119866 119879 119866

Airland9 1002 45292 49949 1028 160 567 243 5473 487 172603 7575 7703 169 165 0 390 8746 466 342814 0 0 0 166 0 336 nd 439 nd

Airland10 150

2 128873 140745 921 186 787 608 2595 845 103473 22079 22413 151 210 888 668 19588 803 552164 3422 3422 0 206 1674 647 29240 788 3473495 0 0 0 231 0 607 nd 762 nd

Airland11 200

2 154084 167395 864 232 919 959 3854 1287 129803 28082 28064 minus006 261 2159 1021 29009 1203 764254 5453 5453 0 274 277 993 47447 1168 3947885 0 0 0 272 0 956 nd 1158 nd

Airland12 250

2 196139 248226 2656 283 1880 1266 5018 1835 137423 29004 24379 minus1595 315 1748 1454 19801 1710 903974 349 244 minus3009 333 27163 1445 1321691 1688 70752445 0 0 0 346 0 1386 nd 1662 nd

Airland13 500


Average minus937 340 2154 1590 257600 1932 909710





0 5 10 15 20 25 30 350

100

200

300

400

500

600

Instance no

Aver

age e

xecu

tion

time (

s)

HBASSBA1

IACAHeuristic



5 Conclusions


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 3 2 2 1 2 31

2 2 1 1 3 2 21

Before

After

2

33 3

3




119871119905+1

119894= 120572 times 119871

119905

119894 (12)



119905

119894 then the RB















119894then



0 5 10 15 20 25 30 355

505

51

515

52

525

Population size

Aver

age p

erce

ntag

e gap

()

02

04

06

08

1

12

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg



1 09 08 07 06 05 04 035

52

54

Aver

age p

erce

ntag

e gap

()

04

05

06

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg

Parameter 120572







119874 119866 119879 119866 119879 119866 119879 119866 119879 119866 119879 119866

Airland1 10 2 90 90 0 008 0 24 0 45 0 01 0 01 03 0 0 0 011 0 39 0 34 0 00 0 01 0

Airland2 15 2 210 210 0 009 0 45 0 49 0 06 0 01 9523 0 0 0 010 0 46 0 43 0 04 0 01 0

Airland3 20 2 60 60 0 009 0 48 0 58 0 113 0 01 116673 0 0 0 010 0 62 0 63 0 93 0 02 nd

Airland4 202 640 640 0 055 0 52 0 55 0 79 0 01 03 130 130 0 014 0 46 0 57 0 64 0 01 04 0 0 0 014 0 56 0 52 0 58 0 02 0

Airland5 202 650 890 3692 144 0 50 308 61 1231 24 6462 01 16923 170 170 0 016 0 54 0 43 0 70 4118 01 5884 0 0 0 021 0 56 0 68 0 32 0 02 nd

Airland6 30 2 554 636 1480 161 0 70 361 101 5108 38 5921 01 59213 0 0 0 030 0 54 0 87 0 122 0 02 0

Airland7 44 2 0 0 0 009 0 118 0 124 0 552 0 02 0

Airland8 50 2 135 180 3333 201 0 121 0 196 2222 1681 8889 02 425933 0 0 0 016 0 139 0 181 nd 1082 0 06 nd

Average 500 043 0 64 039 77 535 236 1493 02 4530



119874 119866 119879 119866 119879 119866 119879 119866

Airland9 1002 45292 49949 1028 160 567 243 5473 487 172603 7575 7703 169 165 0 390 8746 466 342814 0 0 0 166 0 336 nd 439 nd

Airland10 150

2 128873 140745 921 186 787 608 2595 845 103473 22079 22413 151 210 888 668 19588 803 552164 3422 3422 0 206 1674 647 29240 788 3473495 0 0 0 231 0 607 nd 762 nd

Airland11 200

2 154084 167395 864 232 919 959 3854 1287 129803 28082 28064 minus006 261 2159 1021 29009 1203 764254 5453 5453 0 274 277 993 47447 1168 3947885 0 0 0 272 0 956 nd 1158 nd

Airland12 250

2 196139 248226 2656 283 1880 1266 5018 1835 137423 29004 24379 minus1595 315 1748 1454 19801 1710 903974 349 244 minus3009 333 27163 1445 1321691 1688 70752445 0 0 0 346 0 1386 nd 1662 nd

Airland13 500


Average minus937 340 2154 1590 257600 1932 909710





0 5 10 15 20 25 30 350

100

200

300

400

500

600

Instance no

Aver

age e

xecu

tion

time (

s)

HBASSBA1

IACAHeuristic



5 Conclusions


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894then



0 5 10 15 20 25 30 355

505

51

515

52

525

Population size

Aver

age p

erce

ntag

e gap

()

02

04

06

08

1

12

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg



1 09 08 07 06 05 04 035

52

54

Aver

age p

erce

ntag

e gap

()

04

05

06

Aver

age c

ompu

tatio

nal t

ime (

s)

GavgCTavg

Parameter 120572







119874 119866 119879 119866 119879 119866 119879 119866 119879 119866 119879 119866

Airland1 10 2 90 90 0 008 0 24 0 45 0 01 0 01 03 0 0 0 011 0 39 0 34 0 00 0 01 0

Airland2 15 2 210 210 0 009 0 45 0 49 0 06 0 01 9523 0 0 0 010 0 46 0 43 0 04 0 01 0

Airland3 20 2 60 60 0 009 0 48 0 58 0 113 0 01 116673 0 0 0 010 0 62 0 63 0 93 0 02 nd

Airland4 202 640 640 0 055 0 52 0 55 0 79 0 01 03 130 130 0 014 0 46 0 57 0 64 0 01 04 0 0 0 014 0 56 0 52 0 58 0 02 0

Airland5 202 650 890 3692 144 0 50 308 61 1231 24 6462 01 16923 170 170 0 016 0 54 0 43 0 70 4118 01 5884 0 0 0 021 0 56 0 68 0 32 0 02 nd

Airland6 30 2 554 636 1480 161 0 70 361 101 5108 38 5921 01 59213 0 0 0 030 0 54 0 87 0 122 0 02 0

Airland7 44 2 0 0 0 009 0 118 0 124 0 552 0 02 0

Airland8 50 2 135 180 3333 201 0 121 0 196 2222 1681 8889 02 425933 0 0 0 016 0 139 0 181 nd 1082 0 06 nd

Average 500 043 0 64 039 77 535 236 1493 02 4530



119874 119866 119879 119866 119879 119866 119879 119866

Airland9 1002 45292 49949 1028 160 567 243 5473 487 172603 7575 7703 169 165 0 390 8746 466 342814 0 0 0 166 0 336 nd 439 nd

Airland10 150

2 128873 140745 921 186 787 608 2595 845 103473 22079 22413 151 210 888 668 19588 803 552164 3422 3422 0 206 1674 647 29240 788 3473495 0 0 0 231 0 607 nd 762 nd

Airland11 200

2 154084 167395 864 232 919 959 3854 1287 129803 28082 28064 minus006 261 2159 1021 29009 1203 764254 5453 5453 0 274 277 993 47447 1168 3947885 0 0 0 272 0 956 nd 1158 nd

Airland12 250

2 196139 248226 2656 283 1880 1266 5018 1835 137423 29004 24379 minus1595 315 1748 1454 19801 1710 903974 349 244 minus3009 333 27163 1445 1321691 1688 70752445 0 0 0 346 0 1386 nd 1662 nd

Airland13 500


Average minus937 340 2154 1590 257600 1932 909710





0 5 10 15 20 25 30 350

100

200

300

400

500

600

Instance no

Aver

age e

xecu

tion

time (

s)

HBASSBA1

IACAHeuristic



5 Conclusions


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119874 119866 119879 119866 119879 119866 119879 119866 119879 119866 119879 119866

Airland1 10 2 90 90 0 008 0 24 0 45 0 01 0 01 03 0 0 0 011 0 39 0 34 0 00 0 01 0

Airland2 15 2 210 210 0 009 0 45 0 49 0 06 0 01 9523 0 0 0 010 0 46 0 43 0 04 0 01 0

Airland3 20 2 60 60 0 009 0 48 0 58 0 113 0 01 116673 0 0 0 010 0 62 0 63 0 93 0 02 nd

Airland4 202 640 640 0 055 0 52 0 55 0 79 0 01 03 130 130 0 014 0 46 0 57 0 64 0 01 04 0 0 0 014 0 56 0 52 0 58 0 02 0

Airland5 202 650 890 3692 144 0 50 308 61 1231 24 6462 01 16923 170 170 0 016 0 54 0 43 0 70 4118 01 5884 0 0 0 021 0 56 0 68 0 32 0 02 nd

Airland6 30 2 554 636 1480 161 0 70 361 101 5108 38 5921 01 59213 0 0 0 030 0 54 0 87 0 122 0 02 0

Airland7 44 2 0 0 0 009 0 118 0 124 0 552 0 02 0

Airland8 50 2 135 180 3333 201 0 121 0 196 2222 1681 8889 02 425933 0 0 0 016 0 139 0 181 nd 1082 0 06 nd

Average 500 043 0 64 039 77 535 236 1493 02 4530



119874 119866 119879 119866 119879 119866 119879 119866

Airland9 1002 45292 49949 1028 160 567 243 5473 487 172603 7575 7703 169 165 0 390 8746 466 342814 0 0 0 166 0 336 nd 439 nd

Airland10 150

2 128873 140745 921 186 787 608 2595 845 103473 22079 22413 151 210 888 668 19588 803 552164 3422 3422 0 206 1674 647 29240 788 3473495 0 0 0 231 0 607 nd 762 nd

Airland11 200

2 154084 167395 864 232 919 959 3854 1287 129803 28082 28064 minus006 261 2159 1021 29009 1203 764254 5453 5453 0 274 277 993 47447 1168 3947885 0 0 0 272 0 956 nd 1158 nd

Airland12 250

2 196139 248226 2656 283 1880 1266 5018 1835 137423 29004 24379 minus1595 315 1748 1454 19801 1710 903974 349 244 minus3009 333 27163 1445 1321691 1688 70752445 0 0 0 346 0 1386 nd 1662 nd

Airland13 500


Average minus937 340 2154 1590 257600 1932 909710





0 5 10 15 20 25 30 350

100

200

300

400

500

600

Instance no

Aver

age e

xecu

tion

time (

s)

HBASSBA1

IACAHeuristic



5 Conclusions


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 350

100

200

300

400

500

600

Instance no

Aver

age e

xecu

tion

time (

s)

HBASSBA1

IACAHeuristic



5 Conclusions


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a hybrid metaheuristic for multiple...

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