Research ArticleMigrating Birds Optimization for the Seaside Problems atMaritime Container Terminals

Eduardo Lalla-Ruiz Christopher Expoacutesito-Izquierdo Jesica de ArmasBeleacuten Meliaacuten-Batista and J Marcos Moreno-Vega

Department of Computer and Systems Engineering University of La Laguna 38271 La Laguna Spain

Correspondence should be addressed to Eduardo Lalla-Ruiz elallaulles

Received 13 March 2015 Revised 25 May 2015 Accepted 27 May 2015

Academic Editor Wei Fang

Copyright copy 2015 Eduardo Lalla-Ruiz et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Sea freight transportation involvesmoving huge amounts of freights amongmaritime locationswidely spaced bymeans of containervesselsThe time required to serve container vessels is themost relevant indicator when assessing the competitiveness of a maritimecontainer terminal In this paper two main logistic problems stemming from the transshipment of containers in the seaside of amaritime container terminal are addressed namely the Berth Allocation Problem aimed at allocating and scheduling incomingvessels into berthing positions along the quay and the Quay Crane Scheduling Problem whose objective is to schedule the loadingand unloading tasks associated with a container vessel For solving them two Migrating Birds Optimization (MBO) approachesare proposed The MBO is a recently proposed nature-inspired algorithm based on the V-formation flight of migrating birds Inthis algorithm a set of solutions of the problem at hand called birds cooperate among themselves during the search process bysharing information within a V-line formation The computational experiments performed over well-known problem instancesreported in the literature show that the performance of our proposed MBO approaches is highly competitive and presents a betterperformance in terms of running time than the best approximate approach proposed in the literature

1 Introduction

Maritime container terminals and container vessels are themain components involved in sea freight transportationwhere huge amounts of freights are moved among widelyspaced locations Since the international sea freight tradehas undergone a relevant growth over the last few decades(United Nations Conference on Trade And Developmenthttpwwwunctadorg) maritime container terminals haveto better use and schedule their resources in order toefficiently face the operational and technical requirements ofshipping companies In this regard as indicated by Nicolettiet al [1] and Exposito-Izquierdo et al [2] the time requiredto serve container vessels since their arrival until theirdeparture is the most representative indicator used by theshipping companies when assessing the competitiveness of agivenmaritime container terminalMoreover the faster thesetasks are made the earlier the containers are available forwithdrawal by the corresponding companies and therefore

there will be a better management of the whole terminal (Yeo[3])

Several logistic problems are relevant in the productiv-ity of the maritime container terminal One of the mostoutstanding problems within them is the Berth AllocationProblem (BAP) whose purpose is to allocate and schedulethose vessels arriving to port into berthing positions alongthe quay in order to optimize some objective function Thisproblemhas been extensively studied in the literature over thelast years and a multitude of variations has been proposed(Biertwirth and Meisel [4]) One of these variants is theDynamic Berth Allocation Problem (DBAP) proposed byCordeau et al [5] It considers berth and vessel time windowsas well as heterogeneous vessel service times dependingon the assigned berth The other outstanding optimizationproblem in maritime container terminal is the Quay CraneScheduling Problem (QCSP) which is aimed at determiningthe work schedules of the quay cranes allocated to a givencontainer vessel It is worth mentioning that solving the

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 781907 12 pageshttpdxdoiorg1011552015781907

2 Journal of Applied Mathematics

DBAP and the QCSP allows the terminal managers to knowhow to perform the management of the incoming containervessels during a certain planning horizonThis means know-ing the berthing position and berthing time of the vessels andhow the quay cranes are handled during the transshipmentoperations In this regard their efficient solution preventstraffic bottlenecks and enhances the competitiveness of thewhole infrastructure

Logistic operations such as those involved in the DBAPand QCSP require fast and effective solution approaches dueto inherent requirements of the context where they appearIn this backdrop the usage of metaheuristics to find high-quality feasible solutions is advisable For this reason in thiswork we apply and assess a recent nature-inspired meta-heuristic based on the 119881-formation of the migrating birdscalled Migrating Birds Optimization (MBO) proposed byDuman et al [6] This metaheuristic is a population-basedalgorithm where the individuals called birds cooperateamong themselves during the search process by sharinginformation about the explored search space The way theyshare the information is by considering a 119881-formation thatestablishes the relation among birds

The main goal of this work is to propose and evaluatethe use of the MBO technique for solving the main seasideproblems at maritime container terminals With this goal inmind we have selected two of the most relevant problems inthe related literature DBAP and QCSP The computationalresults as well as the comparison with the best approximatealgorithms reported in the literature point out a competitiveperformance ofMBO in terms of objective function value andrunning time This latter feature makes MBO suitable andcompetitive to be included in port decision support systemsIt is worth highlighting the significance of the running timein these systems due to the fact that the aforementionedproblems may have to be solved frequently (i) because of itsdirect link with each other and with problems from otherparts of the container terminal and (ii) to include possiblechanges related to terminal resources and (iii) to assist portmanagers during the negotiations with shipping companies

The remainder of this paper is structured as followsSection 2 introduces the DBAP and QCSP The MBO is pre-sented in Section 3 Afterwards Section 4 describes the appli-cation of MBO to the seaside problems under analysis inthis paper DBAP and QCSP Section 5 discusses the com-putational experiments carried out to assess the suitabilityof MBO Finally Section 6 presents the main conclusionsextracted from the work and suggests several directions forfurther research

2 Seaside Operations

Seaside operations are those related to the transshipment ofcontainers between the container vessels and the maritimecontainer terminal In this context three main problems canbe identified

(i) Berthing of theVesselsEach incoming container vesselhas to be assigned to a position along the quay of thecontainer terminal according to its particular charac-teristics (ie length draft arrival time etc)

(ii) Allocation of Quay CranesA subset of the quay cranesin the container terminal must be allocated to eachberthed vessel in order to perform its loading andunloading operations

(iii) Scheduling of the Quay Cranes The quay cranes allo-cated to a given container vessel have be scheduledfor performing its transshipment operations in such away that the stay is the shortest as possible

As indicated by Bierwirth and Meisel [4] the allocationof quay cranes known as Quay Crane Allocation Problem(QCAP) is tightly related to the BAP due to the fact that thehandling times of the vessels depend on the number of quaycranes assigned to them Consequently the QCAP is usuallyjointly considered with the BAP or QCSP Therefore in thefollowing these two relevant logistic problems namely themanagement of berths and the schedule of quay cranes at aterminal when serving container vessels are addressed

21 Dynamic Berth Allocation Problem The Dynamic BerthAllocation Problem (DBAP) is an NP-hard problem(Cordeau et al [5]) that seeks to identify the berthing positionand berthing time of the container vessels arriving to portover a well-defined time horizon

In the DBAP we are given a set of incoming containervessels 119881 and a set of berths 119861 Each vessel 119894 isin 119881 must beassigned to a berth 119896 isin 119861 Each vessel has a known timewindow [119905V

119894 119905V1015840119894] Similarly each berth has its own time win-

dow [119905119887119896 1199051198871015840

119896] For each vessel 119894 isin 119881 its service time 119904119896

119894

depends on the berth 119896 isin 119861 where it is assigned to That isthe service time of a given vessel may differ from one berth toanother Furthermore each 119894 isin 119881 has a given service prioritydenoted as119901

119894 according to its contractual agreement with the

terminal It should be noted that the higher this value thehigher the priority of the vessel

In a more detailed way the assumptions in the DBAP canbe enumerated as follows

(a) Each berth 119896 isin 119861 can only handle one vessel at a time(b) The service time of each vessel 119894 isin 119881 is determined by

the assigned berth 119896 isin 119861(c) Each vessel 119894 isin 119881 can be served only after its arrival

time 119905V119894

(d) Each vessel 119894 isin 119881 has to be served until its departuretime 119905V1015840

119894

(e) Each vessel 119894 isin 119881 can only be berthed at berth 119896 isin 119861

after 119896 becomes available at time step 119905119887119896

(f) Each vessel 119894 isin 119881 can only be berthed at berth 119896 isin 119861

until 119896 becomes unavailable at time step 1199051198871015840

119896

In order to present the decision variables let us define agraph 119866119896 = (119881

119896 119860119896)forall119896 isin 119861 where 119881119896 = 119881 cup 119900(119896) 119889(119896)

contains a vertex for each vessel as well as the vertices 119900(119896)and 119889(119896) which are the origin and destination nodes for anyroute in the graph The set of arcs is defined as 119860119896 sube 119881

119896times

119881119896 where each one represents the handling time of the vessel

Considering this graph the decision variables defined in theDBAP are as follows

Journal of Applied Mathematics 3

(i) 119909119896119894119895isin 0 1 forall119896 isin 119861 forall(119894 119895) isin 119860

119896 119894 = 119895 set to 1 if vessel119895 is scheduled after vessel 119894 at berth 119896 and 0 otherwise

(ii) 119879119896119894 forall119896 isin 119861 forallV isin 119881 the berthing time of vessel 119894 at

berth 119896 that is the time when the vessel berths(iii) 119879119896

119900(119896) forall119896 isin 119861 starting operation time of berth 119896 that

is the time when the first vessel berths at the berth(iv) 119879119896

119889(119896) forall119896 isin 119861 ending operation time of berth 119896 that

is the time when the last vessel departs at the berth

The objective function (1) aims to minimize the total(weighted) service time of all the vessels defined as the timeelapsed between their arrival to the port and the completionof their handling It should be noted that when vessel 119894 isin 119881

is not assigned to berth 119896 isin 119861 the corresponding term inthe objective function is zero because sum

119895isin119881cup119889(119896)119909119896

119894119895= 0 and

119879119896

119894= 119905119894

minsum

119894isin119881

sum

119896isin119861

119901119894[

[

119879119896

119894minus 119905119894+ 119904119896

119894sum

119895isin119881cup119889(119896)

119909119896

119894119895]

]

(1)

A comprehensive description of the DBAP is provided byCordeau et al [5] Imai et al [9] and Lalla-Ruiz et al [10]

For the sake of clarity we provide an example of a solutionof the DBAP in Figure 1 In this figure an assignment plan isdepicted for 6 container vessels and 3 berths The rectanglesrepresent the vessels Within each rectangle the servicepriority service time and the time windows associated witheach vessel are provided The time windows of the berthsare delimited by the scratched areas For instance berth 1is opened from time step 0 until time step 13 In the figurevessel 6 has to wait for berthing in their respective assignedberths In this regard since its priority is 2 its waiting timewill have less impact on the objective function value thandelaying for example vessel 5

The mathematical formulation of this problem providedin [11] allows solving those instances within reasonablecomputational time However as indicated in [10] this math-ematical model implemented in CPLEX reaches a memoryfault status for problem instances where other characteristicsare taken into account Therefore approximate approachesare required in the following we describe the most recentones de Oliveira et al [12] proposed a clustering search withsimulated annealing and the authors evaluate their approachusing only the large-size problem instances proposed in [5]Their approach allows us to reach high-quality solutions inshort computational times Later Ting et al [13] developeda Particle Swarm Optimization (PSO) and solve the small-and large-size instances proposed in [5] Their approachreports the same quality solutions as [12] in terms of objectivefunction value nevertheless it requires less computationaltime Finally Lalla-Ruiz and Voszlig [14] propose a matheuristicbased on POPMUSIC (Partial Optimization MetaheuristicUnder Special Intensification Conditions)The authors testedtheir approach over the largest instances proposed in [5]exhibiting a high robustness in terms of the average objectivevalues reported by their approach

Tim

e

123456789

10111213141516

Berthing unavailable due to berth TW

QuayBerth 1 Berth 2 Berth 3

Vessel 3

TW = [7 14]

p3 = 1

s1

3= 5

Vessel 1TW = [1 8]

p1 = 4

s1

1= 4

Vessel 6

TW = [2 11]

p6 = 2

s3

6= 6

Vessel 5

TW = [3 12]

p5 = 5

s3

5= 5

Vessel 2TW = [10 15]

p2 = 1

s2

2= 4

Vessel 4

TW = [2 13]

p4 = 1

s2

4= 7

Figure 1 Solution example for |119881| = 6 vessels and |119861| = 3 berths

22 QuayCrane Scheduling Problem TheQuayCrane Sched-uling Problem (QCSP) is stated as determining the finishingtimes of the tasks performed by the available quay cranesallocated to a container vessel berthed at the terminal In thisenvironment a task represents the loadingunloading of agroup of containers ontofrom a given deck or hold of thecontainer vessel at hand Alternative definitions of tasks areproposed by Meisel and Bierwirth [15]

Input data for the QCSP consist of a set of tasks Ω =

1 2 |Ω| (loading or unloading operations associatedwith a container group) and a set of quay cranes 119876 =

1 2 |119876| with similar technical characteristics Each 119905 isin

Ω is located in a certain position along the container vessel119897119905 and has a positive handling time 119901

119905

The objective of the QCSP is to minimize the service timeof the container vessel at handThat is itsmakespan (Kim andPark [16])

min 119888119879 (2)

where 119888119894is the finishing time of the task 119905 isin Ω and 119879 is a

dummy task that represents the end of the serviceThe QCSP has a set of particular constraints which

differentiates it from other well-known scheduling problemsfound in the scientific literature

(a) Each quay crane performs a task without any inter-ruption This means that once a quay crane startsto (un)load the containers related to a given taskthis goes on until all the containers included into therelevant group are (un)loaded

(b) Each quay crane 119902 isin 119876 is only available after itsearliest ready time 119903119902 ge 0

(c) Each quay crane 119902 isin 119876 is initially located on a knownposition 1198971199020

4 Journal of Applied Mathematics

Makespan

1

2

3 4 5

67

8

88

76

66

56

10 8 10 15 67 5 10 0123456789

0 1 2 3 4

1 2 3 41 1 3 4

5 6 7 8 9

0 10 20 30 40 50Time

Bays

1 (10) 2 (8)

3 (10)4 (15)

5 (7) 6 (6) 7 (5)

8 (10)

Task tPosition ltProcessing time pt

QC1 QC2

Figure 2 Example of a QCSP instance and schedule with 8 tasks and 2 quay cranes

(d) Each quay crane 119902 isin 119876 can travel between twoadjacent positions of the container vessel with a traveltime gt 0

(e) Within each position of the container vessel the rel-evant tasks are sorted according to their precedencerelationships For instance unloading tasks must beperformed before loading tasks

(f) The quay cranes are rail mounted This means theycan only move from left to right along the containervessel and vice versa

(g) The quay cranes cannot work in the same position ofthe vessel simultaneously and they cannot cross eachother

(h) The quay cranes have to keep a safety distance 120575 gt

0 between them in order to prevent collisions Thisgives rise to that certain tasks cannot be performedsimultaneously

Figure 2 illustrates an example of an instance for theQCSP with 8 tasks and 2 quay cranes The quay cranes areinitially located in the positions 11989710 = 2 and 119897

20 = 5 of the

container vessel The right figure depicts a schedule for theinstance at hand where the quay cranes are available from thebeginning of the service time 119903119902 = 0 forall119902 isin 119876 Additionallythe quay cranes have to keep a safety distance 120575 = 1 andthey can move between two consecutive positions of thevessel with = 1 In this example tasks 1 2 3 5 6 and7 are performed by quay crane 1 whereas tasks 4 and 8 areperformed by quay crane 2 As can be seen the makespan ofthis schedule is 52 time units

The QCSP is already known to be NP-hard (Sammarraet al [17]) A mathematical formulation for the QCSP isproposed by Legato et al [18] Moreover the QCSP hasbeen suitably dealt in the literature through several papersIt was introduced in the early work by Daganzo [19] andlater approximately addressed by Kim and Park [16] andSammarra et al [17]The computational results of theseworksbrought to light the necessity of developing high efficientoptimization techniques to tackle the QCSP by means ofreasonable computational times In this regard an interestingapproach to solve theQCSPwas put forward byBierwirth and

Meisel [7] In their proposal the authors suggest exploringthe search space of unidirectional schedules A given scheduleis termed unidirectional if all the quay cranes move withsimilar direction of movement along their service time Thisapproach was afterwards deeply exploited by Legato et al [18]and Exposito-Izquierdo et al [8] Lastly the interested readeris referred to the work by Meisel and Bierwirth [15] to obtainan exhaustive review or the related literature Unlike mostof the previous proposals found in the related literature theMBO proposed in this work (see Section 3) allows us to reacha high diversification level of the search space during explo-ration whereas it properly exploits the promising regions inorder to find a large number of local optima solutions Thissuitable balance between diversification and intensification ismainly due to its population-based structure which avoidsstagnation in low-quality regions of the search space

3 Migrating Birds Optimization

The Migrating Birds Optimization (MBO) algorithm wasinitially proposed by Duman et al [6] In that work theauthors propose a nature-inspired metaheuristic based onthe 119881-formation flight of migrating birds This algorithmconsists of a set of individuals where each is associatedto a solution and termed as birds in MBO Moreover theindividuals are aligned in a 119881-formation Figure 3 shows anillustrative scheme of the 119881-formation in which the firstindividual corresponds to the leader bird in the flock and it isrepresented by the doubled circle at the top The remainingcircles represent the rest of the flock The arrows in thefigure represent how the information is shared among theindividuals

In MBO the leader individual attempts to improve itselfby generating a number of neighbour solutions Then thefollowing individual in the 119881-formation evaluates a numberof its own neighbours and a number of the best discardedneighbour solutions received from the previous individualIn case one of those solutions leads to an improvementwith respect to the solution associated with the individualthen it is replaced by that solution yielding the maximumimprovement Once all the individuals have been consideredthe process is repeated for itermax iterations Once those

Journal of Applied Mathematics 5

(1) Generate 119899birds initial solutions in a random manner and place them on a hypothetical 119881 formation arbitrarily(2) 119892 = 0

(3) while 119892 lt 119870 do(4) for (119897 = 0 119897 lt itermax 119897++) do(5) Try to improve the leading solution by generating and evaluating 120582 neighbours of it(6) 119892 = 119892 + 120582

(7) for all (solutions 119904 in the flock (except leader)) do(8) Try to improve the leading solution by generating and evaluating 120582 minus 120575

neighbours of it and the 120575 unused best neighbours from the solution in the front(9) 119892 = 119892 + (120582 minus 120575)

(10) end for(11) end for(12) Move the leader solution to the end and forward one of the solutions following it to the leader position(13) end while(14) Return best solution in the flock

Algorithm 1 Migrating Birds Optimization algorithm (Duman et al [6])

Leader

Figure 3 Example of the119881-formation of theMBO for 7 individuals(birds)

iterations have been reached the leader individual is movedto the end of one of the lines of the 119881-formation and one ofits direct follower individuals becomes the new leader of theflock For this new formation the process restarts for anotheritermax iterations The complete MBO process is carried outuntil a given a stopping criterion is met

The initial position of the individuals along the 119881-formation depends on the generation order That is thefirst individual generated will be the leader individual andtherefore the leader bird of the flock the second and thirdindividuals generated will be its direct followers and so forthAfter generating the initial population the individuals areorganized into a 119881-formation as shown in Figure 3

The input parameters of the MBO algorithm defined bythe user are the following

119899birds number of individuals termed as birds119870 maximum number of neighbour solutions gener-ated by the individualsitermax number of iterations before changing theleader individuals120582 number of random neighbours generated by eachindividual120575 number of best discarded solutions to share amongindividuals

Algorithm 1 depicts the pseudocode of MBO as reportedby Duman et al [6] The first step is to generate 119899birdsindividuals (line 1) The number of generated solutions bythe population 119892 is set to 0 (line 2) Once the population isgenerated theMBO search process starts (lines 3ndash13) Duringthe search process firstly the leader individual generates 120582random neighbour solutions by means of a neighbourhoodstructure In case the best solution generated leads to animprovement in terms of objective function value the solu-tion associated to the leader is replaced by that neighboursolution (line 5) Secondly each direct follower individualgenerates a 120582 minus 120575 neighbour solutions selected at random(lines 7ndash10) by means of a neighbourhood structure Alsoeach individual receives the best 120575 discarded solutions fromthe individual in front of it If one of the solutions gener-ated or received leads to an improvement of the solutionassociated to the individual then the improved solutionreplaces it (line 8) This 119881-formation is maintained until aprefixed number of iterations itermax is reached Once thatthe leader individual becomes the last solution and one ofits direct follower individuals becomes the new leader (line12) Right after the search process is restarted for otheritermax iterations The MBO search process is executed untila number of neighbour solutions 119870 have been generatedthrough the search process (line 3) For a more detaileddescription of the MBO algorithm the reader is referred tothe work by Duman et al [6]

4 Migrating Birds Optimization forthe Seaside Problems at MaritimeContainer Terminals

In the following we apply the Migrating Birds Optimization(MBO) introduced in Section 3 to the DBAP and the QCSPIn both cases we also evaluate the use of improvementprocedures applied to the best solution provided by MBOThe rationale behind this is to (i) assess the capability ofMBO for pointing out promising regions of the search spacethat can be exploited by using a improvement procedure and

6 Journal of Applied Mathematics

(ii) measure the contribution of an improvement methodbased to the quality of the solutions provided by MBO

41 Migrating Birds Optimization for the DynamicBerth Allocation Problem

411 Solution Representation In the context of the DBAPthe MBO implementation considers a solution 119878DBAP as asequence composed of the vessel identifiers where eachberth is delimited by a 0 The service order of each vesselis determined by its position in the sequence The solutionstructure for the example of Figure 1 for 3 berths and 6 vesselsis as follows 119878DBAP = 1 3 0 4 2 0 5 6

412 Neighbourhood Structures The neighbourhood struc-tures considered in this approach are generated by using thefollowing movements

(a) Reinsertion Movement A vessel 119894 is removed froma berth 119896 and reinserted into another berth 119896

1015840

(forall119896 1198961015840 isin 119861 119894 = 1198941015840) at any of the possible positions

For example in 119878DBAP = 1 3 0 4 2 0 5 6 if vessel1 is removed from its berth then all these possiblereinsertion movements can be performed namely3 0 1 4 2 0 5 6 3 0 4 1 2 0 5 6 3 0 4 2 1 05 6 3 0 4 2 0 1 5 6 3 0 4 2 0 5 1 6 and 3 04 2 0 5 6 1

(b) Interchange Movement It consists of exchanging avessel 119894 assigned to berth 119896 with a vessel 1198941015840 assignedto berth 119896

1015840 (forall119894 1198941015840 isin 119881 119894 = 1198941015840 forall119896 1198961015840 isin 119861 119896 =

1198961015840) For example for the previous solution 119878DBAP =

1 3 0 4 2 0 5 6 if we select vessel 1 the possibleinterchange movements that can be obtained arethe following 4 3 0 1 2 0 5 6 2 3 0 4 1 0 5 65 3 0 4 2 0 1 6 and 6 3 0 4 2 0 5 1

The generation of random neighbour solutions by the indi-viduals is based on the reinsertion move On the other handboth movements are used in the improvement method

413 Improvement Method As discussed in the relevantsection we also analyse the capability of MBO for pointingout promising regions in the solution search space In doingso we applied an improvement method proposed by Lalla-Ruiz et al [10] over the best solution provided by MBO Thismethod consist of the following steps given a solution its bestneighbour solution is obtained by means of the reinsertionmoment Over that best neighbor solution we generate itsneighborhood by means of the interchange movement andreturn the best neighbor solution This process is performeduntil no improvement in terms of the objective function valueis achieved

414 Initial Population For generating the initial popula-tion we use a random greedy method (R-G) proposed byCordeau et al [5] that is given a random vessel permutationthe vessels are assigned one at a time to the best possible berthfollowing that sequence order according to their impact overthe objective function value The use of this method instead

of other proposed initialization procedures reported in theliterature such as First-Come First-Served Greedy (FCFS-G)or generating the solution completely at random is based onthe fact that on the one hand FCFS-G is a deterministicapproach which use will affect the convergence of thealgorithm On the other hand R-G provides better qualitysolutions than generating the initial solutions completely atrandom this is due to the fact that R-G allocates the vesselswithin the random sequence according to the impact over theobjective function value of the solution being constructed

415 Stopping Criterion The stopping criterion for the over-all MBO search process is met when a certain number of 119870generated solutions by the population are reached Moreoveras pointed out in the relevant section for large-size instanceswe included an additional stopping criterion based on amaximum number of iterations 120574 without improvement ofthe best solution obtained

42 Migrating Birds Optimization for the Quay CraneScheduling Problem

421 Solution Representation The solutions of the QCSP arerepresented as sequences composed of the available tasks thatisΩ A given sequence includes zeros in order to delimit thesubsets of tasks performed by the quay cranes This way theleftmost quay crane performs those tasks from the beginningof the sequence up to the first zero the second quay craneperforms those tasks from the first zero up to the secondzero and so forth For instance a solution for the examplepresented in Figure 2 with 2 quay cranes and 8 tasks couldbe as follows 119878QCSP = (1 2 3 5 6 7 0 4 8) In this casethe leftmost quay crane performs the tasks (1 2 3 5 6 7)whereas the other quay crane performs the tasks (4 8)

422 Neighbourhood Structures The neighbourhood struc-tures used by the MBO are based on the following exploringmovements

(a) Reinsertion Movement A task 119905 isin Ω currentlyassigned to a quay crane 119902 isin 119876 is reassigned in such away that 119905 is performed by another quay crane 1199021015840 isin 119876

(where 1199021015840 = 119902)(b) Interchange Movement Given pair tasks 1199051 1199052 isin Ω

assigned to two different quay cranes 119902 isin 119876 and 1199021015840isin

119876 (where 1199021015840 = 119902) the movement exchanges the tasksThis way 1199052 is eventually assigned to 119902 whereas 1199051 isassigned to 1199022

The generation of random neighbour solutions by theindividuals is based on the interchange movement

423 Local Search A local search process based on the bestimprovement strategy is proposed in order to find localoptima solutions during the search This way given a certainfeasible solution of the QCSP at each step the set of neigh-bour solutions found by means of reinsertion movement isgenerated The best neighbour solution replaces the currentsolution until a local optimum is achieved

Journal of Applied Mathematics 7

MBO w IMMBO wo IM

MBO w IMMBO wo IM

02

03

04

Runn

ing

time (

s)

700

800

900

1000

1100

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times7minus03

25times7minus05

25times7minus08

25times7minus03

25times7minus05

25times7minus08

Figure 4 Performance of MBO with and without an improvement procedure when solving the DBAP small-size instances

424 Initial Population The solutions included into theinitial population have been generated at randomThismeansthat each task is assigned to one of the available quay cranesrandomly It is worth mentioning that the tasks are selectedto be assigned from the leftmost up to the rightmost withinthe container vessel

425 Stopping Criterion The stopping criterion for theoverall MBO search process is met when a certain numberof119870 neighbour solutions have been already generated by theindividuals

5 Computational Experiments

This section is devoted to assessing the performance ofthe Migrating Birds Optimization (MBO) introduced in theprevious sectionAll the reported computational experimentshave been conducted on a computer equipped with an IntelDual Core 316GHz and 4GB of RAM

51 Computational Experiments for the DBAP The probleminstances used for evaluating the performance of our MBOapproach are those provided by Cordeau et al [5] Accordingto the authors their instances were generated by takinginto account a statistical analysis of the traffic and berthallocation data at the maritime container terminal of GioiaTauro (Italy) The instances are grouped into sets of 10instances whose sizes range from 25 vessels and 5 berthsup to 60 vessels and 13 berths Moreover in order to fit thespace for this work for the small- and medium-size probleminstances we have selected the 3 hardest solvable instancesof each set with regard to the time required to provide

the optimal solution by the implementation of the mathe-matical formulation (Buhrkal et al [11]) in CPLEX (httpwww-01ibmcomsoftwarecommerceoptimizationcplex-optimizer) to provide the optimal solution For the largeinstance set we selected a representative set of instancesBy taking into account the experiments carried out in thiswork we identified the following parameter values for MBO119899birds = 31 120575 = 3 120582 = 20 itermax = 3 and 119870 = |119873|

3 forthe small- and medium-size instances For the large-sizeinstances we set 119870 = |119873|

25 and an additional stoppingcriterion of a maximum number of 120574 = 10 consecutiveiterations without improvement of the best solutionobtained

511 Improvement Method As previously indicated in Sec-tion 4 an improvement phase (based on that proposed byLalla-Ruiz et al [10]) is applied over the best solution pro-vided by MBO Figures 4 5 and 6 show the computationalperformance in terms of objective function value and compu-tational time of MBO with (MBO wIM) and without (MBOwo IM) improvement method for the small- medium- andlarge-size instances proposed by Cordeau et al [5] As canbe checked the performance exhibited by MBO is similarregardless of size of the instance Furthermore the use of aimprovement procedure leads to an enhancement of the best-known solution through a small increase of the computa-tional time This indicates that the use of the improvementto this method enhances the convergence to the best solutionwithin our complete approach proposed in this work

At the light of this analysis in the following results wereport the computational results provided by the joint use ofMBO with the improvement method

8 Journal of Applied Mathematics

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1200

1400

1600

06

07

08

09

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

Figure 5 Performance of MBO with and without an improvement procedure when solving the DBAP medium-size instances

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1100

1200

1300

1400

2

25

3

35

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

Figure 6 Performance of MBO with and without an improvement procedure when solving the DBAP large-size instances

512 Comparison with the Best Literature Approach Tables1 and 2 present a comparison among the best publishedapproaches for the DBAP namely the mathematical modelpresented by Buhrkal et al [11] the best approximateapproach for this problem consisting of a Particle SwarmOptimization algorithm (PSO) proposed by Ting et al [13]and our MBO The first column shows the characteristics ofthe instances to solve that is the number of vessels (|119881|)and berths (|119861|) and the instance identifier (id) For each

instance the best objective value (Obj) relative error (Gap()) with regard to the optimal value provided by CPLEXand the computational time measured in seconds (119905 (s)) arepresented Furthermore with the aim of assessing the timeimprovement reported byMBO in comparison with PSO thepercentage of time improvement (119905impr) is also reported

As shown in Table 1 MBO reports high-quality solutionsin shorter computational times than the other approachesIt reaches the optimal value for the majority of the problem

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

DBAP and the QCSP allows the terminal managers to knowhow to perform the management of the incoming containervessels during a certain planning horizonThis means know-ing the berthing position and berthing time of the vessels andhow the quay cranes are handled during the transshipmentoperations In this regard their efficient solution preventstraffic bottlenecks and enhances the competitiveness of thewhole infrastructure

Logistic operations such as those involved in the DBAPand QCSP require fast and effective solution approaches dueto inherent requirements of the context where they appearIn this backdrop the usage of metaheuristics to find high-quality feasible solutions is advisable For this reason in thiswork we apply and assess a recent nature-inspired meta-heuristic based on the 119881-formation of the migrating birdscalled Migrating Birds Optimization (MBO) proposed byDuman et al [6] This metaheuristic is a population-basedalgorithm where the individuals called birds cooperateamong themselves during the search process by sharinginformation about the explored search space The way theyshare the information is by considering a 119881-formation thatestablishes the relation among birds

The main goal of this work is to propose and evaluatethe use of the MBO technique for solving the main seasideproblems at maritime container terminals With this goal inmind we have selected two of the most relevant problems inthe related literature DBAP and QCSP The computationalresults as well as the comparison with the best approximatealgorithms reported in the literature point out a competitiveperformance ofMBO in terms of objective function value andrunning time This latter feature makes MBO suitable andcompetitive to be included in port decision support systemsIt is worth highlighting the significance of the running timein these systems due to the fact that the aforementionedproblems may have to be solved frequently (i) because of itsdirect link with each other and with problems from otherparts of the container terminal and (ii) to include possiblechanges related to terminal resources and (iii) to assist portmanagers during the negotiations with shipping companies

The remainder of this paper is structured as followsSection 2 introduces the DBAP and QCSP The MBO is pre-sented in Section 3 Afterwards Section 4 describes the appli-cation of MBO to the seaside problems under analysis inthis paper DBAP and QCSP Section 5 discusses the com-putational experiments carried out to assess the suitabilityof MBO Finally Section 6 presents the main conclusionsextracted from the work and suggests several directions forfurther research

2 Seaside Operations

Seaside operations are those related to the transshipment ofcontainers between the container vessels and the maritimecontainer terminal In this context three main problems canbe identified

(i) Berthing of theVesselsEach incoming container vesselhas to be assigned to a position along the quay of thecontainer terminal according to its particular charac-teristics (ie length draft arrival time etc)

(ii) Allocation of Quay CranesA subset of the quay cranesin the container terminal must be allocated to eachberthed vessel in order to perform its loading andunloading operations

(iii) Scheduling of the Quay Cranes The quay cranes allo-cated to a given container vessel have be scheduledfor performing its transshipment operations in such away that the stay is the shortest as possible

As indicated by Bierwirth and Meisel [4] the allocationof quay cranes known as Quay Crane Allocation Problem(QCAP) is tightly related to the BAP due to the fact that thehandling times of the vessels depend on the number of quaycranes assigned to them Consequently the QCAP is usuallyjointly considered with the BAP or QCSP Therefore in thefollowing these two relevant logistic problems namely themanagement of berths and the schedule of quay cranes at aterminal when serving container vessels are addressed

21 Dynamic Berth Allocation Problem The Dynamic BerthAllocation Problem (DBAP) is an NP-hard problem(Cordeau et al [5]) that seeks to identify the berthing positionand berthing time of the container vessels arriving to portover a well-defined time horizon

In the DBAP we are given a set of incoming containervessels 119881 and a set of berths 119861 Each vessel 119894 isin 119881 must beassigned to a berth 119896 isin 119861 Each vessel has a known timewindow [119905V

119894 119905V1015840119894] Similarly each berth has its own time win-

dow [119905119887119896 1199051198871015840

119896] For each vessel 119894 isin 119881 its service time 119904119896

119894

depends on the berth 119896 isin 119861 where it is assigned to That isthe service time of a given vessel may differ from one berth toanother Furthermore each 119894 isin 119881 has a given service prioritydenoted as119901

119894 according to its contractual agreement with the

terminal It should be noted that the higher this value thehigher the priority of the vessel

In a more detailed way the assumptions in the DBAP canbe enumerated as follows

(a) Each berth 119896 isin 119861 can only handle one vessel at a time(b) The service time of each vessel 119894 isin 119881 is determined by

the assigned berth 119896 isin 119861(c) Each vessel 119894 isin 119881 can be served only after its arrival

time 119905V119894

(d) Each vessel 119894 isin 119881 has to be served until its departuretime 119905V1015840

119894

(e) Each vessel 119894 isin 119881 can only be berthed at berth 119896 isin 119861

after 119896 becomes available at time step 119905119887119896

(f) Each vessel 119894 isin 119881 can only be berthed at berth 119896 isin 119861

until 119896 becomes unavailable at time step 1199051198871015840

119896

In order to present the decision variables let us define agraph 119866119896 = (119881

119896 119860119896)forall119896 isin 119861 where 119881119896 = 119881 cup 119900(119896) 119889(119896)

contains a vertex for each vessel as well as the vertices 119900(119896)and 119889(119896) which are the origin and destination nodes for anyroute in the graph The set of arcs is defined as 119860119896 sube 119881

119896times

119881119896 where each one represents the handling time of the vessel

Considering this graph the decision variables defined in theDBAP are as follows

Journal of Applied Mathematics 3

(i) 119909119896119894119895isin 0 1 forall119896 isin 119861 forall(119894 119895) isin 119860

119896 119894 = 119895 set to 1 if vessel119895 is scheduled after vessel 119894 at berth 119896 and 0 otherwise

(ii) 119879119896119894 forall119896 isin 119861 forallV isin 119881 the berthing time of vessel 119894 at

berth 119896 that is the time when the vessel berths(iii) 119879119896

119900(119896) forall119896 isin 119861 starting operation time of berth 119896 that

is the time when the first vessel berths at the berth(iv) 119879119896

119889(119896) forall119896 isin 119861 ending operation time of berth 119896 that

is the time when the last vessel departs at the berth

The objective function (1) aims to minimize the total(weighted) service time of all the vessels defined as the timeelapsed between their arrival to the port and the completionof their handling It should be noted that when vessel 119894 isin 119881

is not assigned to berth 119896 isin 119861 the corresponding term inthe objective function is zero because sum

119895isin119881cup119889(119896)119909119896

119894119895= 0 and

119879119896

119894= 119905119894

minsum

119894isin119881

sum

119896isin119861

119901119894[

[

119879119896

119894minus 119905119894+ 119904119896

119894sum

119895isin119881cup119889(119896)

119909119896

119894119895]

]

(1)

A comprehensive description of the DBAP is provided byCordeau et al [5] Imai et al [9] and Lalla-Ruiz et al [10]

For the sake of clarity we provide an example of a solutionof the DBAP in Figure 1 In this figure an assignment plan isdepicted for 6 container vessels and 3 berths The rectanglesrepresent the vessels Within each rectangle the servicepriority service time and the time windows associated witheach vessel are provided The time windows of the berthsare delimited by the scratched areas For instance berth 1is opened from time step 0 until time step 13 In the figurevessel 6 has to wait for berthing in their respective assignedberths In this regard since its priority is 2 its waiting timewill have less impact on the objective function value thandelaying for example vessel 5

The mathematical formulation of this problem providedin [11] allows solving those instances within reasonablecomputational time However as indicated in [10] this math-ematical model implemented in CPLEX reaches a memoryfault status for problem instances where other characteristicsare taken into account Therefore approximate approachesare required in the following we describe the most recentones de Oliveira et al [12] proposed a clustering search withsimulated annealing and the authors evaluate their approachusing only the large-size problem instances proposed in [5]Their approach allows us to reach high-quality solutions inshort computational times Later Ting et al [13] developeda Particle Swarm Optimization (PSO) and solve the small-and large-size instances proposed in [5] Their approachreports the same quality solutions as [12] in terms of objectivefunction value nevertheless it requires less computationaltime Finally Lalla-Ruiz and Voszlig [14] propose a matheuristicbased on POPMUSIC (Partial Optimization MetaheuristicUnder Special Intensification Conditions)The authors testedtheir approach over the largest instances proposed in [5]exhibiting a high robustness in terms of the average objectivevalues reported by their approach

Tim

e

123456789

10111213141516

Berthing unavailable due to berth TW

QuayBerth 1 Berth 2 Berth 3

Vessel 3

TW = [7 14]

p3 = 1

s1

3= 5

Vessel 1TW = [1 8]

p1 = 4

s1

1= 4

Vessel 6

TW = [2 11]

p6 = 2

s3

6= 6

Vessel 5

TW = [3 12]

p5 = 5

s3

5= 5

Vessel 2TW = [10 15]

p2 = 1

s2

2= 4

Vessel 4

TW = [2 13]

p4 = 1

s2

4= 7

Figure 1 Solution example for |119881| = 6 vessels and |119861| = 3 berths

22 QuayCrane Scheduling Problem TheQuayCrane Sched-uling Problem (QCSP) is stated as determining the finishingtimes of the tasks performed by the available quay cranesallocated to a container vessel berthed at the terminal In thisenvironment a task represents the loadingunloading of agroup of containers ontofrom a given deck or hold of thecontainer vessel at hand Alternative definitions of tasks areproposed by Meisel and Bierwirth [15]

Input data for the QCSP consist of a set of tasks Ω =

1 2 |Ω| (loading or unloading operations associatedwith a container group) and a set of quay cranes 119876 =

1 2 |119876| with similar technical characteristics Each 119905 isin

Ω is located in a certain position along the container vessel119897119905 and has a positive handling time 119901

119905

The objective of the QCSP is to minimize the service timeof the container vessel at handThat is itsmakespan (Kim andPark [16])

min 119888119879 (2)

where 119888119894is the finishing time of the task 119905 isin Ω and 119879 is a

dummy task that represents the end of the serviceThe QCSP has a set of particular constraints which

differentiates it from other well-known scheduling problemsfound in the scientific literature

(a) Each quay crane performs a task without any inter-ruption This means that once a quay crane startsto (un)load the containers related to a given taskthis goes on until all the containers included into therelevant group are (un)loaded

(b) Each quay crane 119902 isin 119876 is only available after itsearliest ready time 119903119902 ge 0

(c) Each quay crane 119902 isin 119876 is initially located on a knownposition 1198971199020

4 Journal of Applied Mathematics

Makespan

1

2

3 4 5

67

8

88

76

66

56

10 8 10 15 67 5 10 0123456789

0 1 2 3 4

1 2 3 41 1 3 4

5 6 7 8 9

0 10 20 30 40 50Time

Bays

1 (10) 2 (8)

3 (10)4 (15)

5 (7) 6 (6) 7 (5)

8 (10)

Task tPosition ltProcessing time pt

QC1 QC2

Figure 2 Example of a QCSP instance and schedule with 8 tasks and 2 quay cranes

(d) Each quay crane 119902 isin 119876 can travel between twoadjacent positions of the container vessel with a traveltime gt 0

(e) Within each position of the container vessel the rel-evant tasks are sorted according to their precedencerelationships For instance unloading tasks must beperformed before loading tasks

(f) The quay cranes are rail mounted This means theycan only move from left to right along the containervessel and vice versa

(g) The quay cranes cannot work in the same position ofthe vessel simultaneously and they cannot cross eachother

(h) The quay cranes have to keep a safety distance 120575 gt

0 between them in order to prevent collisions Thisgives rise to that certain tasks cannot be performedsimultaneously

Figure 2 illustrates an example of an instance for theQCSP with 8 tasks and 2 quay cranes The quay cranes areinitially located in the positions 11989710 = 2 and 119897

20 = 5 of the

container vessel The right figure depicts a schedule for theinstance at hand where the quay cranes are available from thebeginning of the service time 119903119902 = 0 forall119902 isin 119876 Additionallythe quay cranes have to keep a safety distance 120575 = 1 andthey can move between two consecutive positions of thevessel with = 1 In this example tasks 1 2 3 5 6 and7 are performed by quay crane 1 whereas tasks 4 and 8 areperformed by quay crane 2 As can be seen the makespan ofthis schedule is 52 time units

The QCSP is already known to be NP-hard (Sammarraet al [17]) A mathematical formulation for the QCSP isproposed by Legato et al [18] Moreover the QCSP hasbeen suitably dealt in the literature through several papersIt was introduced in the early work by Daganzo [19] andlater approximately addressed by Kim and Park [16] andSammarra et al [17]The computational results of theseworksbrought to light the necessity of developing high efficientoptimization techniques to tackle the QCSP by means ofreasonable computational times In this regard an interestingapproach to solve theQCSPwas put forward byBierwirth and

Meisel [7] In their proposal the authors suggest exploringthe search space of unidirectional schedules A given scheduleis termed unidirectional if all the quay cranes move withsimilar direction of movement along their service time Thisapproach was afterwards deeply exploited by Legato et al [18]and Exposito-Izquierdo et al [8] Lastly the interested readeris referred to the work by Meisel and Bierwirth [15] to obtainan exhaustive review or the related literature Unlike mostof the previous proposals found in the related literature theMBO proposed in this work (see Section 3) allows us to reacha high diversification level of the search space during explo-ration whereas it properly exploits the promising regions inorder to find a large number of local optima solutions Thissuitable balance between diversification and intensification ismainly due to its population-based structure which avoidsstagnation in low-quality regions of the search space

3 Migrating Birds Optimization

The Migrating Birds Optimization (MBO) algorithm wasinitially proposed by Duman et al [6] In that work theauthors propose a nature-inspired metaheuristic based onthe 119881-formation flight of migrating birds This algorithmconsists of a set of individuals where each is associatedto a solution and termed as birds in MBO Moreover theindividuals are aligned in a 119881-formation Figure 3 shows anillustrative scheme of the 119881-formation in which the firstindividual corresponds to the leader bird in the flock and it isrepresented by the doubled circle at the top The remainingcircles represent the rest of the flock The arrows in thefigure represent how the information is shared among theindividuals

In MBO the leader individual attempts to improve itselfby generating a number of neighbour solutions Then thefollowing individual in the 119881-formation evaluates a numberof its own neighbours and a number of the best discardedneighbour solutions received from the previous individualIn case one of those solutions leads to an improvementwith respect to the solution associated with the individualthen it is replaced by that solution yielding the maximumimprovement Once all the individuals have been consideredthe process is repeated for itermax iterations Once those

Journal of Applied Mathematics 5

(1) Generate 119899birds initial solutions in a random manner and place them on a hypothetical 119881 formation arbitrarily(2) 119892 = 0

(3) while 119892 lt 119870 do(4) for (119897 = 0 119897 lt itermax 119897++) do(5) Try to improve the leading solution by generating and evaluating 120582 neighbours of it(6) 119892 = 119892 + 120582

(7) for all (solutions 119904 in the flock (except leader)) do(8) Try to improve the leading solution by generating and evaluating 120582 minus 120575

neighbours of it and the 120575 unused best neighbours from the solution in the front(9) 119892 = 119892 + (120582 minus 120575)

(10) end for(11) end for(12) Move the leader solution to the end and forward one of the solutions following it to the leader position(13) end while(14) Return best solution in the flock

Algorithm 1 Migrating Birds Optimization algorithm (Duman et al [6])

Leader

Figure 3 Example of the119881-formation of theMBO for 7 individuals(birds)

iterations have been reached the leader individual is movedto the end of one of the lines of the 119881-formation and one ofits direct follower individuals becomes the new leader of theflock For this new formation the process restarts for anotheritermax iterations The complete MBO process is carried outuntil a given a stopping criterion is met

The initial position of the individuals along the 119881-formation depends on the generation order That is thefirst individual generated will be the leader individual andtherefore the leader bird of the flock the second and thirdindividuals generated will be its direct followers and so forthAfter generating the initial population the individuals areorganized into a 119881-formation as shown in Figure 3

The input parameters of the MBO algorithm defined bythe user are the following

119899birds number of individuals termed as birds119870 maximum number of neighbour solutions gener-ated by the individualsitermax number of iterations before changing theleader individuals120582 number of random neighbours generated by eachindividual120575 number of best discarded solutions to share amongindividuals

Algorithm 1 depicts the pseudocode of MBO as reportedby Duman et al [6] The first step is to generate 119899birdsindividuals (line 1) The number of generated solutions bythe population 119892 is set to 0 (line 2) Once the population isgenerated theMBO search process starts (lines 3ndash13) Duringthe search process firstly the leader individual generates 120582random neighbour solutions by means of a neighbourhoodstructure In case the best solution generated leads to animprovement in terms of objective function value the solu-tion associated to the leader is replaced by that neighboursolution (line 5) Secondly each direct follower individualgenerates a 120582 minus 120575 neighbour solutions selected at random(lines 7ndash10) by means of a neighbourhood structure Alsoeach individual receives the best 120575 discarded solutions fromthe individual in front of it If one of the solutions gener-ated or received leads to an improvement of the solutionassociated to the individual then the improved solutionreplaces it (line 8) This 119881-formation is maintained until aprefixed number of iterations itermax is reached Once thatthe leader individual becomes the last solution and one ofits direct follower individuals becomes the new leader (line12) Right after the search process is restarted for otheritermax iterations The MBO search process is executed untila number of neighbour solutions 119870 have been generatedthrough the search process (line 3) For a more detaileddescription of the MBO algorithm the reader is referred tothe work by Duman et al [6]

4 Migrating Birds Optimization forthe Seaside Problems at MaritimeContainer Terminals

In the following we apply the Migrating Birds Optimization(MBO) introduced in Section 3 to the DBAP and the QCSPIn both cases we also evaluate the use of improvementprocedures applied to the best solution provided by MBOThe rationale behind this is to (i) assess the capability ofMBO for pointing out promising regions of the search spacethat can be exploited by using a improvement procedure and

6 Journal of Applied Mathematics

(ii) measure the contribution of an improvement methodbased to the quality of the solutions provided by MBO

41 Migrating Birds Optimization for the DynamicBerth Allocation Problem

411 Solution Representation In the context of the DBAPthe MBO implementation considers a solution 119878DBAP as asequence composed of the vessel identifiers where eachberth is delimited by a 0 The service order of each vesselis determined by its position in the sequence The solutionstructure for the example of Figure 1 for 3 berths and 6 vesselsis as follows 119878DBAP = 1 3 0 4 2 0 5 6

412 Neighbourhood Structures The neighbourhood struc-tures considered in this approach are generated by using thefollowing movements

(a) Reinsertion Movement A vessel 119894 is removed froma berth 119896 and reinserted into another berth 119896

1015840

(forall119896 1198961015840 isin 119861 119894 = 1198941015840) at any of the possible positions

For example in 119878DBAP = 1 3 0 4 2 0 5 6 if vessel1 is removed from its berth then all these possiblereinsertion movements can be performed namely3 0 1 4 2 0 5 6 3 0 4 1 2 0 5 6 3 0 4 2 1 05 6 3 0 4 2 0 1 5 6 3 0 4 2 0 5 1 6 and 3 04 2 0 5 6 1

(b) Interchange Movement It consists of exchanging avessel 119894 assigned to berth 119896 with a vessel 1198941015840 assignedto berth 119896

1015840 (forall119894 1198941015840 isin 119881 119894 = 1198941015840 forall119896 1198961015840 isin 119861 119896 =

1198961015840) For example for the previous solution 119878DBAP =

1 3 0 4 2 0 5 6 if we select vessel 1 the possibleinterchange movements that can be obtained arethe following 4 3 0 1 2 0 5 6 2 3 0 4 1 0 5 65 3 0 4 2 0 1 6 and 6 3 0 4 2 0 5 1

The generation of random neighbour solutions by the indi-viduals is based on the reinsertion move On the other handboth movements are used in the improvement method

413 Improvement Method As discussed in the relevantsection we also analyse the capability of MBO for pointingout promising regions in the solution search space In doingso we applied an improvement method proposed by Lalla-Ruiz et al [10] over the best solution provided by MBO Thismethod consist of the following steps given a solution its bestneighbour solution is obtained by means of the reinsertionmoment Over that best neighbor solution we generate itsneighborhood by means of the interchange movement andreturn the best neighbor solution This process is performeduntil no improvement in terms of the objective function valueis achieved

414 Initial Population For generating the initial popula-tion we use a random greedy method (R-G) proposed byCordeau et al [5] that is given a random vessel permutationthe vessels are assigned one at a time to the best possible berthfollowing that sequence order according to their impact overthe objective function value The use of this method instead

of other proposed initialization procedures reported in theliterature such as First-Come First-Served Greedy (FCFS-G)or generating the solution completely at random is based onthe fact that on the one hand FCFS-G is a deterministicapproach which use will affect the convergence of thealgorithm On the other hand R-G provides better qualitysolutions than generating the initial solutions completely atrandom this is due to the fact that R-G allocates the vesselswithin the random sequence according to the impact over theobjective function value of the solution being constructed

415 Stopping Criterion The stopping criterion for the over-all MBO search process is met when a certain number of 119870generated solutions by the population are reached Moreoveras pointed out in the relevant section for large-size instanceswe included an additional stopping criterion based on amaximum number of iterations 120574 without improvement ofthe best solution obtained

42 Migrating Birds Optimization for the Quay CraneScheduling Problem

421 Solution Representation The solutions of the QCSP arerepresented as sequences composed of the available tasks thatisΩ A given sequence includes zeros in order to delimit thesubsets of tasks performed by the quay cranes This way theleftmost quay crane performs those tasks from the beginningof the sequence up to the first zero the second quay craneperforms those tasks from the first zero up to the secondzero and so forth For instance a solution for the examplepresented in Figure 2 with 2 quay cranes and 8 tasks couldbe as follows 119878QCSP = (1 2 3 5 6 7 0 4 8) In this casethe leftmost quay crane performs the tasks (1 2 3 5 6 7)whereas the other quay crane performs the tasks (4 8)

422 Neighbourhood Structures The neighbourhood struc-tures used by the MBO are based on the following exploringmovements

(a) Reinsertion Movement A task 119905 isin Ω currentlyassigned to a quay crane 119902 isin 119876 is reassigned in such away that 119905 is performed by another quay crane 1199021015840 isin 119876

(where 1199021015840 = 119902)(b) Interchange Movement Given pair tasks 1199051 1199052 isin Ω

assigned to two different quay cranes 119902 isin 119876 and 1199021015840isin

119876 (where 1199021015840 = 119902) the movement exchanges the tasksThis way 1199052 is eventually assigned to 119902 whereas 1199051 isassigned to 1199022

The generation of random neighbour solutions by theindividuals is based on the interchange movement

423 Local Search A local search process based on the bestimprovement strategy is proposed in order to find localoptima solutions during the search This way given a certainfeasible solution of the QCSP at each step the set of neigh-bour solutions found by means of reinsertion movement isgenerated The best neighbour solution replaces the currentsolution until a local optimum is achieved

Journal of Applied Mathematics 7

MBO w IMMBO wo IM

MBO w IMMBO wo IM

02

03

04

Runn

ing

time (

s)

700

800

900

1000

1100

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times7minus03

25times7minus05

25times7minus08

25times7minus03

25times7minus05

25times7minus08

Figure 4 Performance of MBO with and without an improvement procedure when solving the DBAP small-size instances

424 Initial Population The solutions included into theinitial population have been generated at randomThismeansthat each task is assigned to one of the available quay cranesrandomly It is worth mentioning that the tasks are selectedto be assigned from the leftmost up to the rightmost withinthe container vessel

425 Stopping Criterion The stopping criterion for theoverall MBO search process is met when a certain numberof119870 neighbour solutions have been already generated by theindividuals

5 Computational Experiments

This section is devoted to assessing the performance ofthe Migrating Birds Optimization (MBO) introduced in theprevious sectionAll the reported computational experimentshave been conducted on a computer equipped with an IntelDual Core 316GHz and 4GB of RAM

51 Computational Experiments for the DBAP The probleminstances used for evaluating the performance of our MBOapproach are those provided by Cordeau et al [5] Accordingto the authors their instances were generated by takinginto account a statistical analysis of the traffic and berthallocation data at the maritime container terminal of GioiaTauro (Italy) The instances are grouped into sets of 10instances whose sizes range from 25 vessels and 5 berthsup to 60 vessels and 13 berths Moreover in order to fit thespace for this work for the small- and medium-size probleminstances we have selected the 3 hardest solvable instancesof each set with regard to the time required to provide

the optimal solution by the implementation of the mathe-matical formulation (Buhrkal et al [11]) in CPLEX (httpwww-01ibmcomsoftwarecommerceoptimizationcplex-optimizer) to provide the optimal solution For the largeinstance set we selected a representative set of instancesBy taking into account the experiments carried out in thiswork we identified the following parameter values for MBO119899birds = 31 120575 = 3 120582 = 20 itermax = 3 and 119870 = |119873|

3 forthe small- and medium-size instances For the large-sizeinstances we set 119870 = |119873|

25 and an additional stoppingcriterion of a maximum number of 120574 = 10 consecutiveiterations without improvement of the best solutionobtained

511 Improvement Method As previously indicated in Sec-tion 4 an improvement phase (based on that proposed byLalla-Ruiz et al [10]) is applied over the best solution pro-vided by MBO Figures 4 5 and 6 show the computationalperformance in terms of objective function value and compu-tational time of MBO with (MBO wIM) and without (MBOwo IM) improvement method for the small- medium- andlarge-size instances proposed by Cordeau et al [5] As canbe checked the performance exhibited by MBO is similarregardless of size of the instance Furthermore the use of aimprovement procedure leads to an enhancement of the best-known solution through a small increase of the computa-tional time This indicates that the use of the improvementto this method enhances the convergence to the best solutionwithin our complete approach proposed in this work

At the light of this analysis in the following results wereport the computational results provided by the joint use ofMBO with the improvement method

8 Journal of Applied Mathematics

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1200

1400

1600

06

07

08

09

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

Figure 5 Performance of MBO with and without an improvement procedure when solving the DBAP medium-size instances

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1100

1200

1300

1400

2

25

3

35

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

Figure 6 Performance of MBO with and without an improvement procedure when solving the DBAP large-size instances

512 Comparison with the Best Literature Approach Tables1 and 2 present a comparison among the best publishedapproaches for the DBAP namely the mathematical modelpresented by Buhrkal et al [11] the best approximateapproach for this problem consisting of a Particle SwarmOptimization algorithm (PSO) proposed by Ting et al [13]and our MBO The first column shows the characteristics ofthe instances to solve that is the number of vessels (|119881|)and berths (|119861|) and the instance identifier (id) For each

instance the best objective value (Obj) relative error (Gap()) with regard to the optimal value provided by CPLEXand the computational time measured in seconds (119905 (s)) arepresented Furthermore with the aim of assessing the timeimprovement reported byMBO in comparison with PSO thepercentage of time improvement (119905impr) is also reported

As shown in Table 1 MBO reports high-quality solutionsin shorter computational times than the other approachesIt reaches the optimal value for the majority of the problem

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

(i) 119909119896119894119895isin 0 1 forall119896 isin 119861 forall(119894 119895) isin 119860

119896 119894 = 119895 set to 1 if vessel119895 is scheduled after vessel 119894 at berth 119896 and 0 otherwise

(ii) 119879119896119894 forall119896 isin 119861 forallV isin 119881 the berthing time of vessel 119894 at

berth 119896 that is the time when the vessel berths(iii) 119879119896

119900(119896) forall119896 isin 119861 starting operation time of berth 119896 that

is the time when the first vessel berths at the berth(iv) 119879119896

119889(119896) forall119896 isin 119861 ending operation time of berth 119896 that

is the time when the last vessel departs at the berth

The objective function (1) aims to minimize the total(weighted) service time of all the vessels defined as the timeelapsed between their arrival to the port and the completionof their handling It should be noted that when vessel 119894 isin 119881

is not assigned to berth 119896 isin 119861 the corresponding term inthe objective function is zero because sum

119895isin119881cup119889(119896)119909119896

119894119895= 0 and

119879119896

119894= 119905119894

minsum

119894isin119881

sum

119896isin119861

119901119894[

[

119879119896

119894minus 119905119894+ 119904119896

119894sum

119895isin119881cup119889(119896)

119909119896

119894119895]

]

(1)

A comprehensive description of the DBAP is provided byCordeau et al [5] Imai et al [9] and Lalla-Ruiz et al [10]

For the sake of clarity we provide an example of a solutionof the DBAP in Figure 1 In this figure an assignment plan isdepicted for 6 container vessels and 3 berths The rectanglesrepresent the vessels Within each rectangle the servicepriority service time and the time windows associated witheach vessel are provided The time windows of the berthsare delimited by the scratched areas For instance berth 1is opened from time step 0 until time step 13 In the figurevessel 6 has to wait for berthing in their respective assignedberths In this regard since its priority is 2 its waiting timewill have less impact on the objective function value thandelaying for example vessel 5

The mathematical formulation of this problem providedin [11] allows solving those instances within reasonablecomputational time However as indicated in [10] this math-ematical model implemented in CPLEX reaches a memoryfault status for problem instances where other characteristicsare taken into account Therefore approximate approachesare required in the following we describe the most recentones de Oliveira et al [12] proposed a clustering search withsimulated annealing and the authors evaluate their approachusing only the large-size problem instances proposed in [5]Their approach allows us to reach high-quality solutions inshort computational times Later Ting et al [13] developeda Particle Swarm Optimization (PSO) and solve the small-and large-size instances proposed in [5] Their approachreports the same quality solutions as [12] in terms of objectivefunction value nevertheless it requires less computationaltime Finally Lalla-Ruiz and Voszlig [14] propose a matheuristicbased on POPMUSIC (Partial Optimization MetaheuristicUnder Special Intensification Conditions)The authors testedtheir approach over the largest instances proposed in [5]exhibiting a high robustness in terms of the average objectivevalues reported by their approach

Tim

e

123456789

10111213141516

Berthing unavailable due to berth TW

QuayBerth 1 Berth 2 Berth 3

Vessel 3

TW = [7 14]

p3 = 1

s1

3= 5

Vessel 1TW = [1 8]

p1 = 4

s1

1= 4

Vessel 6

TW = [2 11]

p6 = 2

s3

6= 6

Vessel 5

TW = [3 12]

p5 = 5

s3

5= 5

Vessel 2TW = [10 15]

p2 = 1

s2

2= 4

Vessel 4

TW = [2 13]

p4 = 1

s2

4= 7

Figure 1 Solution example for |119881| = 6 vessels and |119861| = 3 berths

22 QuayCrane Scheduling Problem TheQuayCrane Sched-uling Problem (QCSP) is stated as determining the finishingtimes of the tasks performed by the available quay cranesallocated to a container vessel berthed at the terminal In thisenvironment a task represents the loadingunloading of agroup of containers ontofrom a given deck or hold of thecontainer vessel at hand Alternative definitions of tasks areproposed by Meisel and Bierwirth [15]

Input data for the QCSP consist of a set of tasks Ω =

1 2 |Ω| (loading or unloading operations associatedwith a container group) and a set of quay cranes 119876 =

1 2 |119876| with similar technical characteristics Each 119905 isin

Ω is located in a certain position along the container vessel119897119905 and has a positive handling time 119901

119905

The objective of the QCSP is to minimize the service timeof the container vessel at handThat is itsmakespan (Kim andPark [16])

min 119888119879 (2)

where 119888119894is the finishing time of the task 119905 isin Ω and 119879 is a

dummy task that represents the end of the serviceThe QCSP has a set of particular constraints which

differentiates it from other well-known scheduling problemsfound in the scientific literature

(a) Each quay crane performs a task without any inter-ruption This means that once a quay crane startsto (un)load the containers related to a given taskthis goes on until all the containers included into therelevant group are (un)loaded

(b) Each quay crane 119902 isin 119876 is only available after itsearliest ready time 119903119902 ge 0

(c) Each quay crane 119902 isin 119876 is initially located on a knownposition 1198971199020

4 Journal of Applied Mathematics

Makespan

1

2

3 4 5

67

8

88

76

66

56

10 8 10 15 67 5 10 0123456789

0 1 2 3 4

1 2 3 41 1 3 4

5 6 7 8 9

0 10 20 30 40 50Time

Bays

1 (10) 2 (8)

3 (10)4 (15)

5 (7) 6 (6) 7 (5)

8 (10)

Task tPosition ltProcessing time pt

QC1 QC2

Figure 2 Example of a QCSP instance and schedule with 8 tasks and 2 quay cranes

(d) Each quay crane 119902 isin 119876 can travel between twoadjacent positions of the container vessel with a traveltime gt 0

(e) Within each position of the container vessel the rel-evant tasks are sorted according to their precedencerelationships For instance unloading tasks must beperformed before loading tasks

(f) The quay cranes are rail mounted This means theycan only move from left to right along the containervessel and vice versa

(g) The quay cranes cannot work in the same position ofthe vessel simultaneously and they cannot cross eachother

(h) The quay cranes have to keep a safety distance 120575 gt

0 between them in order to prevent collisions Thisgives rise to that certain tasks cannot be performedsimultaneously

Figure 2 illustrates an example of an instance for theQCSP with 8 tasks and 2 quay cranes The quay cranes areinitially located in the positions 11989710 = 2 and 119897

20 = 5 of the

container vessel The right figure depicts a schedule for theinstance at hand where the quay cranes are available from thebeginning of the service time 119903119902 = 0 forall119902 isin 119876 Additionallythe quay cranes have to keep a safety distance 120575 = 1 andthey can move between two consecutive positions of thevessel with = 1 In this example tasks 1 2 3 5 6 and7 are performed by quay crane 1 whereas tasks 4 and 8 areperformed by quay crane 2 As can be seen the makespan ofthis schedule is 52 time units

The QCSP is already known to be NP-hard (Sammarraet al [17]) A mathematical formulation for the QCSP isproposed by Legato et al [18] Moreover the QCSP hasbeen suitably dealt in the literature through several papersIt was introduced in the early work by Daganzo [19] andlater approximately addressed by Kim and Park [16] andSammarra et al [17]The computational results of theseworksbrought to light the necessity of developing high efficientoptimization techniques to tackle the QCSP by means ofreasonable computational times In this regard an interestingapproach to solve theQCSPwas put forward byBierwirth and

Meisel [7] In their proposal the authors suggest exploringthe search space of unidirectional schedules A given scheduleis termed unidirectional if all the quay cranes move withsimilar direction of movement along their service time Thisapproach was afterwards deeply exploited by Legato et al [18]and Exposito-Izquierdo et al [8] Lastly the interested readeris referred to the work by Meisel and Bierwirth [15] to obtainan exhaustive review or the related literature Unlike mostof the previous proposals found in the related literature theMBO proposed in this work (see Section 3) allows us to reacha high diversification level of the search space during explo-ration whereas it properly exploits the promising regions inorder to find a large number of local optima solutions Thissuitable balance between diversification and intensification ismainly due to its population-based structure which avoidsstagnation in low-quality regions of the search space

3 Migrating Birds Optimization

The Migrating Birds Optimization (MBO) algorithm wasinitially proposed by Duman et al [6] In that work theauthors propose a nature-inspired metaheuristic based onthe 119881-formation flight of migrating birds This algorithmconsists of a set of individuals where each is associatedto a solution and termed as birds in MBO Moreover theindividuals are aligned in a 119881-formation Figure 3 shows anillustrative scheme of the 119881-formation in which the firstindividual corresponds to the leader bird in the flock and it isrepresented by the doubled circle at the top The remainingcircles represent the rest of the flock The arrows in thefigure represent how the information is shared among theindividuals

In MBO the leader individual attempts to improve itselfby generating a number of neighbour solutions Then thefollowing individual in the 119881-formation evaluates a numberof its own neighbours and a number of the best discardedneighbour solutions received from the previous individualIn case one of those solutions leads to an improvementwith respect to the solution associated with the individualthen it is replaced by that solution yielding the maximumimprovement Once all the individuals have been consideredthe process is repeated for itermax iterations Once those

Journal of Applied Mathematics 5

(1) Generate 119899birds initial solutions in a random manner and place them on a hypothetical 119881 formation arbitrarily(2) 119892 = 0

(3) while 119892 lt 119870 do(4) for (119897 = 0 119897 lt itermax 119897++) do(5) Try to improve the leading solution by generating and evaluating 120582 neighbours of it(6) 119892 = 119892 + 120582

(7) for all (solutions 119904 in the flock (except leader)) do(8) Try to improve the leading solution by generating and evaluating 120582 minus 120575

neighbours of it and the 120575 unused best neighbours from the solution in the front(9) 119892 = 119892 + (120582 minus 120575)

(10) end for(11) end for(12) Move the leader solution to the end and forward one of the solutions following it to the leader position(13) end while(14) Return best solution in the flock

Algorithm 1 Migrating Birds Optimization algorithm (Duman et al [6])

Leader

Figure 3 Example of the119881-formation of theMBO for 7 individuals(birds)

iterations have been reached the leader individual is movedto the end of one of the lines of the 119881-formation and one ofits direct follower individuals becomes the new leader of theflock For this new formation the process restarts for anotheritermax iterations The complete MBO process is carried outuntil a given a stopping criterion is met

The initial position of the individuals along the 119881-formation depends on the generation order That is thefirst individual generated will be the leader individual andtherefore the leader bird of the flock the second and thirdindividuals generated will be its direct followers and so forthAfter generating the initial population the individuals areorganized into a 119881-formation as shown in Figure 3

The input parameters of the MBO algorithm defined bythe user are the following

119899birds number of individuals termed as birds119870 maximum number of neighbour solutions gener-ated by the individualsitermax number of iterations before changing theleader individuals120582 number of random neighbours generated by eachindividual120575 number of best discarded solutions to share amongindividuals

Algorithm 1 depicts the pseudocode of MBO as reportedby Duman et al [6] The first step is to generate 119899birdsindividuals (line 1) The number of generated solutions bythe population 119892 is set to 0 (line 2) Once the population isgenerated theMBO search process starts (lines 3ndash13) Duringthe search process firstly the leader individual generates 120582random neighbour solutions by means of a neighbourhoodstructure In case the best solution generated leads to animprovement in terms of objective function value the solu-tion associated to the leader is replaced by that neighboursolution (line 5) Secondly each direct follower individualgenerates a 120582 minus 120575 neighbour solutions selected at random(lines 7ndash10) by means of a neighbourhood structure Alsoeach individual receives the best 120575 discarded solutions fromthe individual in front of it If one of the solutions gener-ated or received leads to an improvement of the solutionassociated to the individual then the improved solutionreplaces it (line 8) This 119881-formation is maintained until aprefixed number of iterations itermax is reached Once thatthe leader individual becomes the last solution and one ofits direct follower individuals becomes the new leader (line12) Right after the search process is restarted for otheritermax iterations The MBO search process is executed untila number of neighbour solutions 119870 have been generatedthrough the search process (line 3) For a more detaileddescription of the MBO algorithm the reader is referred tothe work by Duman et al [6]

4 Migrating Birds Optimization forthe Seaside Problems at MaritimeContainer Terminals

In the following we apply the Migrating Birds Optimization(MBO) introduced in Section 3 to the DBAP and the QCSPIn both cases we also evaluate the use of improvementprocedures applied to the best solution provided by MBOThe rationale behind this is to (i) assess the capability ofMBO for pointing out promising regions of the search spacethat can be exploited by using a improvement procedure and

6 Journal of Applied Mathematics

(ii) measure the contribution of an improvement methodbased to the quality of the solutions provided by MBO

41 Migrating Birds Optimization for the DynamicBerth Allocation Problem

411 Solution Representation In the context of the DBAPthe MBO implementation considers a solution 119878DBAP as asequence composed of the vessel identifiers where eachberth is delimited by a 0 The service order of each vesselis determined by its position in the sequence The solutionstructure for the example of Figure 1 for 3 berths and 6 vesselsis as follows 119878DBAP = 1 3 0 4 2 0 5 6

412 Neighbourhood Structures The neighbourhood struc-tures considered in this approach are generated by using thefollowing movements

(a) Reinsertion Movement A vessel 119894 is removed froma berth 119896 and reinserted into another berth 119896

1015840

(forall119896 1198961015840 isin 119861 119894 = 1198941015840) at any of the possible positions

For example in 119878DBAP = 1 3 0 4 2 0 5 6 if vessel1 is removed from its berth then all these possiblereinsertion movements can be performed namely3 0 1 4 2 0 5 6 3 0 4 1 2 0 5 6 3 0 4 2 1 05 6 3 0 4 2 0 1 5 6 3 0 4 2 0 5 1 6 and 3 04 2 0 5 6 1

(b) Interchange Movement It consists of exchanging avessel 119894 assigned to berth 119896 with a vessel 1198941015840 assignedto berth 119896

1015840 (forall119894 1198941015840 isin 119881 119894 = 1198941015840 forall119896 1198961015840 isin 119861 119896 =

1198961015840) For example for the previous solution 119878DBAP =

1 3 0 4 2 0 5 6 if we select vessel 1 the possibleinterchange movements that can be obtained arethe following 4 3 0 1 2 0 5 6 2 3 0 4 1 0 5 65 3 0 4 2 0 1 6 and 6 3 0 4 2 0 5 1

The generation of random neighbour solutions by the indi-viduals is based on the reinsertion move On the other handboth movements are used in the improvement method

413 Improvement Method As discussed in the relevantsection we also analyse the capability of MBO for pointingout promising regions in the solution search space In doingso we applied an improvement method proposed by Lalla-Ruiz et al [10] over the best solution provided by MBO Thismethod consist of the following steps given a solution its bestneighbour solution is obtained by means of the reinsertionmoment Over that best neighbor solution we generate itsneighborhood by means of the interchange movement andreturn the best neighbor solution This process is performeduntil no improvement in terms of the objective function valueis achieved

414 Initial Population For generating the initial popula-tion we use a random greedy method (R-G) proposed byCordeau et al [5] that is given a random vessel permutationthe vessels are assigned one at a time to the best possible berthfollowing that sequence order according to their impact overthe objective function value The use of this method instead

of other proposed initialization procedures reported in theliterature such as First-Come First-Served Greedy (FCFS-G)or generating the solution completely at random is based onthe fact that on the one hand FCFS-G is a deterministicapproach which use will affect the convergence of thealgorithm On the other hand R-G provides better qualitysolutions than generating the initial solutions completely atrandom this is due to the fact that R-G allocates the vesselswithin the random sequence according to the impact over theobjective function value of the solution being constructed

415 Stopping Criterion The stopping criterion for the over-all MBO search process is met when a certain number of 119870generated solutions by the population are reached Moreoveras pointed out in the relevant section for large-size instanceswe included an additional stopping criterion based on amaximum number of iterations 120574 without improvement ofthe best solution obtained

42 Migrating Birds Optimization for the Quay CraneScheduling Problem

421 Solution Representation The solutions of the QCSP arerepresented as sequences composed of the available tasks thatisΩ A given sequence includes zeros in order to delimit thesubsets of tasks performed by the quay cranes This way theleftmost quay crane performs those tasks from the beginningof the sequence up to the first zero the second quay craneperforms those tasks from the first zero up to the secondzero and so forth For instance a solution for the examplepresented in Figure 2 with 2 quay cranes and 8 tasks couldbe as follows 119878QCSP = (1 2 3 5 6 7 0 4 8) In this casethe leftmost quay crane performs the tasks (1 2 3 5 6 7)whereas the other quay crane performs the tasks (4 8)

422 Neighbourhood Structures The neighbourhood struc-tures used by the MBO are based on the following exploringmovements

(a) Reinsertion Movement A task 119905 isin Ω currentlyassigned to a quay crane 119902 isin 119876 is reassigned in such away that 119905 is performed by another quay crane 1199021015840 isin 119876

(where 1199021015840 = 119902)(b) Interchange Movement Given pair tasks 1199051 1199052 isin Ω

assigned to two different quay cranes 119902 isin 119876 and 1199021015840isin

119876 (where 1199021015840 = 119902) the movement exchanges the tasksThis way 1199052 is eventually assigned to 119902 whereas 1199051 isassigned to 1199022

The generation of random neighbour solutions by theindividuals is based on the interchange movement

423 Local Search A local search process based on the bestimprovement strategy is proposed in order to find localoptima solutions during the search This way given a certainfeasible solution of the QCSP at each step the set of neigh-bour solutions found by means of reinsertion movement isgenerated The best neighbour solution replaces the currentsolution until a local optimum is achieved

Journal of Applied Mathematics 7

MBO w IMMBO wo IM

MBO w IMMBO wo IM

02

03

04

Runn

ing

time (

s)

700

800

900

1000

1100

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times7minus03

25times7minus05

25times7minus08

25times7minus03

25times7minus05

25times7minus08

Figure 4 Performance of MBO with and without an improvement procedure when solving the DBAP small-size instances

424 Initial Population The solutions included into theinitial population have been generated at randomThismeansthat each task is assigned to one of the available quay cranesrandomly It is worth mentioning that the tasks are selectedto be assigned from the leftmost up to the rightmost withinthe container vessel

425 Stopping Criterion The stopping criterion for theoverall MBO search process is met when a certain numberof119870 neighbour solutions have been already generated by theindividuals

5 Computational Experiments

This section is devoted to assessing the performance ofthe Migrating Birds Optimization (MBO) introduced in theprevious sectionAll the reported computational experimentshave been conducted on a computer equipped with an IntelDual Core 316GHz and 4GB of RAM

51 Computational Experiments for the DBAP The probleminstances used for evaluating the performance of our MBOapproach are those provided by Cordeau et al [5] Accordingto the authors their instances were generated by takinginto account a statistical analysis of the traffic and berthallocation data at the maritime container terminal of GioiaTauro (Italy) The instances are grouped into sets of 10instances whose sizes range from 25 vessels and 5 berthsup to 60 vessels and 13 berths Moreover in order to fit thespace for this work for the small- and medium-size probleminstances we have selected the 3 hardest solvable instancesof each set with regard to the time required to provide

the optimal solution by the implementation of the mathe-matical formulation (Buhrkal et al [11]) in CPLEX (httpwww-01ibmcomsoftwarecommerceoptimizationcplex-optimizer) to provide the optimal solution For the largeinstance set we selected a representative set of instancesBy taking into account the experiments carried out in thiswork we identified the following parameter values for MBO119899birds = 31 120575 = 3 120582 = 20 itermax = 3 and 119870 = |119873|

3 forthe small- and medium-size instances For the large-sizeinstances we set 119870 = |119873|

25 and an additional stoppingcriterion of a maximum number of 120574 = 10 consecutiveiterations without improvement of the best solutionobtained

511 Improvement Method As previously indicated in Sec-tion 4 an improvement phase (based on that proposed byLalla-Ruiz et al [10]) is applied over the best solution pro-vided by MBO Figures 4 5 and 6 show the computationalperformance in terms of objective function value and compu-tational time of MBO with (MBO wIM) and without (MBOwo IM) improvement method for the small- medium- andlarge-size instances proposed by Cordeau et al [5] As canbe checked the performance exhibited by MBO is similarregardless of size of the instance Furthermore the use of aimprovement procedure leads to an enhancement of the best-known solution through a small increase of the computa-tional time This indicates that the use of the improvementto this method enhances the convergence to the best solutionwithin our complete approach proposed in this work

At the light of this analysis in the following results wereport the computational results provided by the joint use ofMBO with the improvement method

8 Journal of Applied Mathematics

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1200

1400

1600

06

07

08

09

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

Figure 5 Performance of MBO with and without an improvement procedure when solving the DBAP medium-size instances

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1100

1200

1300

1400

2

25

3

35

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

Figure 6 Performance of MBO with and without an improvement procedure when solving the DBAP large-size instances

512 Comparison with the Best Literature Approach Tables1 and 2 present a comparison among the best publishedapproaches for the DBAP namely the mathematical modelpresented by Buhrkal et al [11] the best approximateapproach for this problem consisting of a Particle SwarmOptimization algorithm (PSO) proposed by Ting et al [13]and our MBO The first column shows the characteristics ofthe instances to solve that is the number of vessels (|119881|)and berths (|119861|) and the instance identifier (id) For each

instance the best objective value (Obj) relative error (Gap()) with regard to the optimal value provided by CPLEXand the computational time measured in seconds (119905 (s)) arepresented Furthermore with the aim of assessing the timeimprovement reported byMBO in comparison with PSO thepercentage of time improvement (119905impr) is also reported

As shown in Table 1 MBO reports high-quality solutionsin shorter computational times than the other approachesIt reaches the optimal value for the majority of the problem

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Makespan

1

2

3 4 5

67

8

88

76

66

56

10 8 10 15 67 5 10 0123456789

0 1 2 3 4

1 2 3 41 1 3 4

5 6 7 8 9

0 10 20 30 40 50Time

Bays

1 (10) 2 (8)

3 (10)4 (15)

5 (7) 6 (6) 7 (5)

8 (10)

Task tPosition ltProcessing time pt

QC1 QC2

Figure 2 Example of a QCSP instance and schedule with 8 tasks and 2 quay cranes

(d) Each quay crane 119902 isin 119876 can travel between twoadjacent positions of the container vessel with a traveltime gt 0

(e) Within each position of the container vessel the rel-evant tasks are sorted according to their precedencerelationships For instance unloading tasks must beperformed before loading tasks

(f) The quay cranes are rail mounted This means theycan only move from left to right along the containervessel and vice versa

(g) The quay cranes cannot work in the same position ofthe vessel simultaneously and they cannot cross eachother

(h) The quay cranes have to keep a safety distance 120575 gt

0 between them in order to prevent collisions Thisgives rise to that certain tasks cannot be performedsimultaneously

Figure 2 illustrates an example of an instance for theQCSP with 8 tasks and 2 quay cranes The quay cranes areinitially located in the positions 11989710 = 2 and 119897

20 = 5 of the

container vessel The right figure depicts a schedule for theinstance at hand where the quay cranes are available from thebeginning of the service time 119903119902 = 0 forall119902 isin 119876 Additionallythe quay cranes have to keep a safety distance 120575 = 1 andthey can move between two consecutive positions of thevessel with = 1 In this example tasks 1 2 3 5 6 and7 are performed by quay crane 1 whereas tasks 4 and 8 areperformed by quay crane 2 As can be seen the makespan ofthis schedule is 52 time units

The QCSP is already known to be NP-hard (Sammarraet al [17]) A mathematical formulation for the QCSP isproposed by Legato et al [18] Moreover the QCSP hasbeen suitably dealt in the literature through several papersIt was introduced in the early work by Daganzo [19] andlater approximately addressed by Kim and Park [16] andSammarra et al [17]The computational results of theseworksbrought to light the necessity of developing high efficientoptimization techniques to tackle the QCSP by means ofreasonable computational times In this regard an interestingapproach to solve theQCSPwas put forward byBierwirth and

Meisel [7] In their proposal the authors suggest exploringthe search space of unidirectional schedules A given scheduleis termed unidirectional if all the quay cranes move withsimilar direction of movement along their service time Thisapproach was afterwards deeply exploited by Legato et al [18]and Exposito-Izquierdo et al [8] Lastly the interested readeris referred to the work by Meisel and Bierwirth [15] to obtainan exhaustive review or the related literature Unlike mostof the previous proposals found in the related literature theMBO proposed in this work (see Section 3) allows us to reacha high diversification level of the search space during explo-ration whereas it properly exploits the promising regions inorder to find a large number of local optima solutions Thissuitable balance between diversification and intensification ismainly due to its population-based structure which avoidsstagnation in low-quality regions of the search space

3 Migrating Birds Optimization

The Migrating Birds Optimization (MBO) algorithm wasinitially proposed by Duman et al [6] In that work theauthors propose a nature-inspired metaheuristic based onthe 119881-formation flight of migrating birds This algorithmconsists of a set of individuals where each is associatedto a solution and termed as birds in MBO Moreover theindividuals are aligned in a 119881-formation Figure 3 shows anillustrative scheme of the 119881-formation in which the firstindividual corresponds to the leader bird in the flock and it isrepresented by the doubled circle at the top The remainingcircles represent the rest of the flock The arrows in thefigure represent how the information is shared among theindividuals

In MBO the leader individual attempts to improve itselfby generating a number of neighbour solutions Then thefollowing individual in the 119881-formation evaluates a numberof its own neighbours and a number of the best discardedneighbour solutions received from the previous individualIn case one of those solutions leads to an improvementwith respect to the solution associated with the individualthen it is replaced by that solution yielding the maximumimprovement Once all the individuals have been consideredthe process is repeated for itermax iterations Once those

Journal of Applied Mathematics 5

(1) Generate 119899birds initial solutions in a random manner and place them on a hypothetical 119881 formation arbitrarily(2) 119892 = 0

(3) while 119892 lt 119870 do(4) for (119897 = 0 119897 lt itermax 119897++) do(5) Try to improve the leading solution by generating and evaluating 120582 neighbours of it(6) 119892 = 119892 + 120582

(7) for all (solutions 119904 in the flock (except leader)) do(8) Try to improve the leading solution by generating and evaluating 120582 minus 120575

neighbours of it and the 120575 unused best neighbours from the solution in the front(9) 119892 = 119892 + (120582 minus 120575)

(10) end for(11) end for(12) Move the leader solution to the end and forward one of the solutions following it to the leader position(13) end while(14) Return best solution in the flock

Algorithm 1 Migrating Birds Optimization algorithm (Duman et al [6])

Leader

Figure 3 Example of the119881-formation of theMBO for 7 individuals(birds)

iterations have been reached the leader individual is movedto the end of one of the lines of the 119881-formation and one ofits direct follower individuals becomes the new leader of theflock For this new formation the process restarts for anotheritermax iterations The complete MBO process is carried outuntil a given a stopping criterion is met

The initial position of the individuals along the 119881-formation depends on the generation order That is thefirst individual generated will be the leader individual andtherefore the leader bird of the flock the second and thirdindividuals generated will be its direct followers and so forthAfter generating the initial population the individuals areorganized into a 119881-formation as shown in Figure 3

The input parameters of the MBO algorithm defined bythe user are the following

119899birds number of individuals termed as birds119870 maximum number of neighbour solutions gener-ated by the individualsitermax number of iterations before changing theleader individuals120582 number of random neighbours generated by eachindividual120575 number of best discarded solutions to share amongindividuals

Algorithm 1 depicts the pseudocode of MBO as reportedby Duman et al [6] The first step is to generate 119899birdsindividuals (line 1) The number of generated solutions bythe population 119892 is set to 0 (line 2) Once the population isgenerated theMBO search process starts (lines 3ndash13) Duringthe search process firstly the leader individual generates 120582random neighbour solutions by means of a neighbourhoodstructure In case the best solution generated leads to animprovement in terms of objective function value the solu-tion associated to the leader is replaced by that neighboursolution (line 5) Secondly each direct follower individualgenerates a 120582 minus 120575 neighbour solutions selected at random(lines 7ndash10) by means of a neighbourhood structure Alsoeach individual receives the best 120575 discarded solutions fromthe individual in front of it If one of the solutions gener-ated or received leads to an improvement of the solutionassociated to the individual then the improved solutionreplaces it (line 8) This 119881-formation is maintained until aprefixed number of iterations itermax is reached Once thatthe leader individual becomes the last solution and one ofits direct follower individuals becomes the new leader (line12) Right after the search process is restarted for otheritermax iterations The MBO search process is executed untila number of neighbour solutions 119870 have been generatedthrough the search process (line 3) For a more detaileddescription of the MBO algorithm the reader is referred tothe work by Duman et al [6]

4 Migrating Birds Optimization forthe Seaside Problems at MaritimeContainer Terminals

In the following we apply the Migrating Birds Optimization(MBO) introduced in Section 3 to the DBAP and the QCSPIn both cases we also evaluate the use of improvementprocedures applied to the best solution provided by MBOThe rationale behind this is to (i) assess the capability ofMBO for pointing out promising regions of the search spacethat can be exploited by using a improvement procedure and

6 Journal of Applied Mathematics

(ii) measure the contribution of an improvement methodbased to the quality of the solutions provided by MBO

41 Migrating Birds Optimization for the DynamicBerth Allocation Problem

411 Solution Representation In the context of the DBAPthe MBO implementation considers a solution 119878DBAP as asequence composed of the vessel identifiers where eachberth is delimited by a 0 The service order of each vesselis determined by its position in the sequence The solutionstructure for the example of Figure 1 for 3 berths and 6 vesselsis as follows 119878DBAP = 1 3 0 4 2 0 5 6

412 Neighbourhood Structures The neighbourhood struc-tures considered in this approach are generated by using thefollowing movements

(a) Reinsertion Movement A vessel 119894 is removed froma berth 119896 and reinserted into another berth 119896

1015840

(forall119896 1198961015840 isin 119861 119894 = 1198941015840) at any of the possible positions

For example in 119878DBAP = 1 3 0 4 2 0 5 6 if vessel1 is removed from its berth then all these possiblereinsertion movements can be performed namely3 0 1 4 2 0 5 6 3 0 4 1 2 0 5 6 3 0 4 2 1 05 6 3 0 4 2 0 1 5 6 3 0 4 2 0 5 1 6 and 3 04 2 0 5 6 1

(b) Interchange Movement It consists of exchanging avessel 119894 assigned to berth 119896 with a vessel 1198941015840 assignedto berth 119896

1015840 (forall119894 1198941015840 isin 119881 119894 = 1198941015840 forall119896 1198961015840 isin 119861 119896 =

1198961015840) For example for the previous solution 119878DBAP =

1 3 0 4 2 0 5 6 if we select vessel 1 the possibleinterchange movements that can be obtained arethe following 4 3 0 1 2 0 5 6 2 3 0 4 1 0 5 65 3 0 4 2 0 1 6 and 6 3 0 4 2 0 5 1

The generation of random neighbour solutions by the indi-viduals is based on the reinsertion move On the other handboth movements are used in the improvement method

413 Improvement Method As discussed in the relevantsection we also analyse the capability of MBO for pointingout promising regions in the solution search space In doingso we applied an improvement method proposed by Lalla-Ruiz et al [10] over the best solution provided by MBO Thismethod consist of the following steps given a solution its bestneighbour solution is obtained by means of the reinsertionmoment Over that best neighbor solution we generate itsneighborhood by means of the interchange movement andreturn the best neighbor solution This process is performeduntil no improvement in terms of the objective function valueis achieved

414 Initial Population For generating the initial popula-tion we use a random greedy method (R-G) proposed byCordeau et al [5] that is given a random vessel permutationthe vessels are assigned one at a time to the best possible berthfollowing that sequence order according to their impact overthe objective function value The use of this method instead

of other proposed initialization procedures reported in theliterature such as First-Come First-Served Greedy (FCFS-G)or generating the solution completely at random is based onthe fact that on the one hand FCFS-G is a deterministicapproach which use will affect the convergence of thealgorithm On the other hand R-G provides better qualitysolutions than generating the initial solutions completely atrandom this is due to the fact that R-G allocates the vesselswithin the random sequence according to the impact over theobjective function value of the solution being constructed

415 Stopping Criterion The stopping criterion for the over-all MBO search process is met when a certain number of 119870generated solutions by the population are reached Moreoveras pointed out in the relevant section for large-size instanceswe included an additional stopping criterion based on amaximum number of iterations 120574 without improvement ofthe best solution obtained

42 Migrating Birds Optimization for the Quay CraneScheduling Problem

421 Solution Representation The solutions of the QCSP arerepresented as sequences composed of the available tasks thatisΩ A given sequence includes zeros in order to delimit thesubsets of tasks performed by the quay cranes This way theleftmost quay crane performs those tasks from the beginningof the sequence up to the first zero the second quay craneperforms those tasks from the first zero up to the secondzero and so forth For instance a solution for the examplepresented in Figure 2 with 2 quay cranes and 8 tasks couldbe as follows 119878QCSP = (1 2 3 5 6 7 0 4 8) In this casethe leftmost quay crane performs the tasks (1 2 3 5 6 7)whereas the other quay crane performs the tasks (4 8)

422 Neighbourhood Structures The neighbourhood struc-tures used by the MBO are based on the following exploringmovements

(a) Reinsertion Movement A task 119905 isin Ω currentlyassigned to a quay crane 119902 isin 119876 is reassigned in such away that 119905 is performed by another quay crane 1199021015840 isin 119876

(where 1199021015840 = 119902)(b) Interchange Movement Given pair tasks 1199051 1199052 isin Ω

assigned to two different quay cranes 119902 isin 119876 and 1199021015840isin

119876 (where 1199021015840 = 119902) the movement exchanges the tasksThis way 1199052 is eventually assigned to 119902 whereas 1199051 isassigned to 1199022

The generation of random neighbour solutions by theindividuals is based on the interchange movement

423 Local Search A local search process based on the bestimprovement strategy is proposed in order to find localoptima solutions during the search This way given a certainfeasible solution of the QCSP at each step the set of neigh-bour solutions found by means of reinsertion movement isgenerated The best neighbour solution replaces the currentsolution until a local optimum is achieved

Journal of Applied Mathematics 7

MBO w IMMBO wo IM

MBO w IMMBO wo IM

02

03

04

Runn

ing

time (

s)

700

800

900

1000

1100

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times7minus03

25times7minus05

25times7minus08

25times7minus03

25times7minus05

25times7minus08

Figure 4 Performance of MBO with and without an improvement procedure when solving the DBAP small-size instances

424 Initial Population The solutions included into theinitial population have been generated at randomThismeansthat each task is assigned to one of the available quay cranesrandomly It is worth mentioning that the tasks are selectedto be assigned from the leftmost up to the rightmost withinthe container vessel

425 Stopping Criterion The stopping criterion for theoverall MBO search process is met when a certain numberof119870 neighbour solutions have been already generated by theindividuals

5 Computational Experiments

This section is devoted to assessing the performance ofthe Migrating Birds Optimization (MBO) introduced in theprevious sectionAll the reported computational experimentshave been conducted on a computer equipped with an IntelDual Core 316GHz and 4GB of RAM

51 Computational Experiments for the DBAP The probleminstances used for evaluating the performance of our MBOapproach are those provided by Cordeau et al [5] Accordingto the authors their instances were generated by takinginto account a statistical analysis of the traffic and berthallocation data at the maritime container terminal of GioiaTauro (Italy) The instances are grouped into sets of 10instances whose sizes range from 25 vessels and 5 berthsup to 60 vessels and 13 berths Moreover in order to fit thespace for this work for the small- and medium-size probleminstances we have selected the 3 hardest solvable instancesof each set with regard to the time required to provide

the optimal solution by the implementation of the mathe-matical formulation (Buhrkal et al [11]) in CPLEX (httpwww-01ibmcomsoftwarecommerceoptimizationcplex-optimizer) to provide the optimal solution For the largeinstance set we selected a representative set of instancesBy taking into account the experiments carried out in thiswork we identified the following parameter values for MBO119899birds = 31 120575 = 3 120582 = 20 itermax = 3 and 119870 = |119873|

3 forthe small- and medium-size instances For the large-sizeinstances we set 119870 = |119873|

25 and an additional stoppingcriterion of a maximum number of 120574 = 10 consecutiveiterations without improvement of the best solutionobtained

511 Improvement Method As previously indicated in Sec-tion 4 an improvement phase (based on that proposed byLalla-Ruiz et al [10]) is applied over the best solution pro-vided by MBO Figures 4 5 and 6 show the computationalperformance in terms of objective function value and compu-tational time of MBO with (MBO wIM) and without (MBOwo IM) improvement method for the small- medium- andlarge-size instances proposed by Cordeau et al [5] As canbe checked the performance exhibited by MBO is similarregardless of size of the instance Furthermore the use of aimprovement procedure leads to an enhancement of the best-known solution through a small increase of the computa-tional time This indicates that the use of the improvementto this method enhances the convergence to the best solutionwithin our complete approach proposed in this work

At the light of this analysis in the following results wereport the computational results provided by the joint use ofMBO with the improvement method

8 Journal of Applied Mathematics

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1200

1400

1600

06

07

08

09

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

Figure 5 Performance of MBO with and without an improvement procedure when solving the DBAP medium-size instances

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1100

1200

1300

1400

2

25

3

35

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

Figure 6 Performance of MBO with and without an improvement procedure when solving the DBAP large-size instances

512 Comparison with the Best Literature Approach Tables1 and 2 present a comparison among the best publishedapproaches for the DBAP namely the mathematical modelpresented by Buhrkal et al [11] the best approximateapproach for this problem consisting of a Particle SwarmOptimization algorithm (PSO) proposed by Ting et al [13]and our MBO The first column shows the characteristics ofthe instances to solve that is the number of vessels (|119881|)and berths (|119861|) and the instance identifier (id) For each

instance the best objective value (Obj) relative error (Gap()) with regard to the optimal value provided by CPLEXand the computational time measured in seconds (119905 (s)) arepresented Furthermore with the aim of assessing the timeimprovement reported byMBO in comparison with PSO thepercentage of time improvement (119905impr) is also reported

As shown in Table 1 MBO reports high-quality solutionsin shorter computational times than the other approachesIt reaches the optimal value for the majority of the problem

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

(1) Generate 119899birds initial solutions in a random manner and place them on a hypothetical 119881 formation arbitrarily(2) 119892 = 0

(3) while 119892 lt 119870 do(4) for (119897 = 0 119897 lt itermax 119897++) do(5) Try to improve the leading solution by generating and evaluating 120582 neighbours of it(6) 119892 = 119892 + 120582

(7) for all (solutions 119904 in the flock (except leader)) do(8) Try to improve the leading solution by generating and evaluating 120582 minus 120575

neighbours of it and the 120575 unused best neighbours from the solution in the front(9) 119892 = 119892 + (120582 minus 120575)

(10) end for(11) end for(12) Move the leader solution to the end and forward one of the solutions following it to the leader position(13) end while(14) Return best solution in the flock

Algorithm 1 Migrating Birds Optimization algorithm (Duman et al [6])

Leader

Figure 3 Example of the119881-formation of theMBO for 7 individuals(birds)

iterations have been reached the leader individual is movedto the end of one of the lines of the 119881-formation and one ofits direct follower individuals becomes the new leader of theflock For this new formation the process restarts for anotheritermax iterations The complete MBO process is carried outuntil a given a stopping criterion is met

The initial position of the individuals along the 119881-formation depends on the generation order That is thefirst individual generated will be the leader individual andtherefore the leader bird of the flock the second and thirdindividuals generated will be its direct followers and so forthAfter generating the initial population the individuals areorganized into a 119881-formation as shown in Figure 3

The input parameters of the MBO algorithm defined bythe user are the following

119899birds number of individuals termed as birds119870 maximum number of neighbour solutions gener-ated by the individualsitermax number of iterations before changing theleader individuals120582 number of random neighbours generated by eachindividual120575 number of best discarded solutions to share amongindividuals

Algorithm 1 depicts the pseudocode of MBO as reportedby Duman et al [6] The first step is to generate 119899birdsindividuals (line 1) The number of generated solutions bythe population 119892 is set to 0 (line 2) Once the population isgenerated theMBO search process starts (lines 3ndash13) Duringthe search process firstly the leader individual generates 120582random neighbour solutions by means of a neighbourhoodstructure In case the best solution generated leads to animprovement in terms of objective function value the solu-tion associated to the leader is replaced by that neighboursolution (line 5) Secondly each direct follower individualgenerates a 120582 minus 120575 neighbour solutions selected at random(lines 7ndash10) by means of a neighbourhood structure Alsoeach individual receives the best 120575 discarded solutions fromthe individual in front of it If one of the solutions gener-ated or received leads to an improvement of the solutionassociated to the individual then the improved solutionreplaces it (line 8) This 119881-formation is maintained until aprefixed number of iterations itermax is reached Once thatthe leader individual becomes the last solution and one ofits direct follower individuals becomes the new leader (line12) Right after the search process is restarted for otheritermax iterations The MBO search process is executed untila number of neighbour solutions 119870 have been generatedthrough the search process (line 3) For a more detaileddescription of the MBO algorithm the reader is referred tothe work by Duman et al [6]

4 Migrating Birds Optimization forthe Seaside Problems at MaritimeContainer Terminals

In the following we apply the Migrating Birds Optimization(MBO) introduced in Section 3 to the DBAP and the QCSPIn both cases we also evaluate the use of improvementprocedures applied to the best solution provided by MBOThe rationale behind this is to (i) assess the capability ofMBO for pointing out promising regions of the search spacethat can be exploited by using a improvement procedure and

6 Journal of Applied Mathematics

(ii) measure the contribution of an improvement methodbased to the quality of the solutions provided by MBO

41 Migrating Birds Optimization for the DynamicBerth Allocation Problem

411 Solution Representation In the context of the DBAPthe MBO implementation considers a solution 119878DBAP as asequence composed of the vessel identifiers where eachberth is delimited by a 0 The service order of each vesselis determined by its position in the sequence The solutionstructure for the example of Figure 1 for 3 berths and 6 vesselsis as follows 119878DBAP = 1 3 0 4 2 0 5 6

412 Neighbourhood Structures The neighbourhood struc-tures considered in this approach are generated by using thefollowing movements

(a) Reinsertion Movement A vessel 119894 is removed froma berth 119896 and reinserted into another berth 119896

1015840

(forall119896 1198961015840 isin 119861 119894 = 1198941015840) at any of the possible positions

For example in 119878DBAP = 1 3 0 4 2 0 5 6 if vessel1 is removed from its berth then all these possiblereinsertion movements can be performed namely3 0 1 4 2 0 5 6 3 0 4 1 2 0 5 6 3 0 4 2 1 05 6 3 0 4 2 0 1 5 6 3 0 4 2 0 5 1 6 and 3 04 2 0 5 6 1

(b) Interchange Movement It consists of exchanging avessel 119894 assigned to berth 119896 with a vessel 1198941015840 assignedto berth 119896

1015840 (forall119894 1198941015840 isin 119881 119894 = 1198941015840 forall119896 1198961015840 isin 119861 119896 =

1198961015840) For example for the previous solution 119878DBAP =

1 3 0 4 2 0 5 6 if we select vessel 1 the possibleinterchange movements that can be obtained arethe following 4 3 0 1 2 0 5 6 2 3 0 4 1 0 5 65 3 0 4 2 0 1 6 and 6 3 0 4 2 0 5 1

The generation of random neighbour solutions by the indi-viduals is based on the reinsertion move On the other handboth movements are used in the improvement method

413 Improvement Method As discussed in the relevantsection we also analyse the capability of MBO for pointingout promising regions in the solution search space In doingso we applied an improvement method proposed by Lalla-Ruiz et al [10] over the best solution provided by MBO Thismethod consist of the following steps given a solution its bestneighbour solution is obtained by means of the reinsertionmoment Over that best neighbor solution we generate itsneighborhood by means of the interchange movement andreturn the best neighbor solution This process is performeduntil no improvement in terms of the objective function valueis achieved

414 Initial Population For generating the initial popula-tion we use a random greedy method (R-G) proposed byCordeau et al [5] that is given a random vessel permutationthe vessels are assigned one at a time to the best possible berthfollowing that sequence order according to their impact overthe objective function value The use of this method instead

of other proposed initialization procedures reported in theliterature such as First-Come First-Served Greedy (FCFS-G)or generating the solution completely at random is based onthe fact that on the one hand FCFS-G is a deterministicapproach which use will affect the convergence of thealgorithm On the other hand R-G provides better qualitysolutions than generating the initial solutions completely atrandom this is due to the fact that R-G allocates the vesselswithin the random sequence according to the impact over theobjective function value of the solution being constructed

415 Stopping Criterion The stopping criterion for the over-all MBO search process is met when a certain number of 119870generated solutions by the population are reached Moreoveras pointed out in the relevant section for large-size instanceswe included an additional stopping criterion based on amaximum number of iterations 120574 without improvement ofthe best solution obtained

42 Migrating Birds Optimization for the Quay CraneScheduling Problem

421 Solution Representation The solutions of the QCSP arerepresented as sequences composed of the available tasks thatisΩ A given sequence includes zeros in order to delimit thesubsets of tasks performed by the quay cranes This way theleftmost quay crane performs those tasks from the beginningof the sequence up to the first zero the second quay craneperforms those tasks from the first zero up to the secondzero and so forth For instance a solution for the examplepresented in Figure 2 with 2 quay cranes and 8 tasks couldbe as follows 119878QCSP = (1 2 3 5 6 7 0 4 8) In this casethe leftmost quay crane performs the tasks (1 2 3 5 6 7)whereas the other quay crane performs the tasks (4 8)

422 Neighbourhood Structures The neighbourhood struc-tures used by the MBO are based on the following exploringmovements

(a) Reinsertion Movement A task 119905 isin Ω currentlyassigned to a quay crane 119902 isin 119876 is reassigned in such away that 119905 is performed by another quay crane 1199021015840 isin 119876

(where 1199021015840 = 119902)(b) Interchange Movement Given pair tasks 1199051 1199052 isin Ω

assigned to two different quay cranes 119902 isin 119876 and 1199021015840isin

119876 (where 1199021015840 = 119902) the movement exchanges the tasksThis way 1199052 is eventually assigned to 119902 whereas 1199051 isassigned to 1199022

The generation of random neighbour solutions by theindividuals is based on the interchange movement

423 Local Search A local search process based on the bestimprovement strategy is proposed in order to find localoptima solutions during the search This way given a certainfeasible solution of the QCSP at each step the set of neigh-bour solutions found by means of reinsertion movement isgenerated The best neighbour solution replaces the currentsolution until a local optimum is achieved

Journal of Applied Mathematics 7

MBO w IMMBO wo IM

MBO w IMMBO wo IM

02

03

04

Runn

ing

time (

s)

700

800

900

1000

1100

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times7minus03

25times7minus05

25times7minus08

25times7minus03

25times7minus05

25times7minus08

Figure 4 Performance of MBO with and without an improvement procedure when solving the DBAP small-size instances

424 Initial Population The solutions included into theinitial population have been generated at randomThismeansthat each task is assigned to one of the available quay cranesrandomly It is worth mentioning that the tasks are selectedto be assigned from the leftmost up to the rightmost withinthe container vessel

425 Stopping Criterion The stopping criterion for theoverall MBO search process is met when a certain numberof119870 neighbour solutions have been already generated by theindividuals

5 Computational Experiments

This section is devoted to assessing the performance ofthe Migrating Birds Optimization (MBO) introduced in theprevious sectionAll the reported computational experimentshave been conducted on a computer equipped with an IntelDual Core 316GHz and 4GB of RAM

51 Computational Experiments for the DBAP The probleminstances used for evaluating the performance of our MBOapproach are those provided by Cordeau et al [5] Accordingto the authors their instances were generated by takinginto account a statistical analysis of the traffic and berthallocation data at the maritime container terminal of GioiaTauro (Italy) The instances are grouped into sets of 10instances whose sizes range from 25 vessels and 5 berthsup to 60 vessels and 13 berths Moreover in order to fit thespace for this work for the small- and medium-size probleminstances we have selected the 3 hardest solvable instancesof each set with regard to the time required to provide

the optimal solution by the implementation of the mathe-matical formulation (Buhrkal et al [11]) in CPLEX (httpwww-01ibmcomsoftwarecommerceoptimizationcplex-optimizer) to provide the optimal solution For the largeinstance set we selected a representative set of instancesBy taking into account the experiments carried out in thiswork we identified the following parameter values for MBO119899birds = 31 120575 = 3 120582 = 20 itermax = 3 and 119870 = |119873|

3 forthe small- and medium-size instances For the large-sizeinstances we set 119870 = |119873|

25 and an additional stoppingcriterion of a maximum number of 120574 = 10 consecutiveiterations without improvement of the best solutionobtained

511 Improvement Method As previously indicated in Sec-tion 4 an improvement phase (based on that proposed byLalla-Ruiz et al [10]) is applied over the best solution pro-vided by MBO Figures 4 5 and 6 show the computationalperformance in terms of objective function value and compu-tational time of MBO with (MBO wIM) and without (MBOwo IM) improvement method for the small- medium- andlarge-size instances proposed by Cordeau et al [5] As canbe checked the performance exhibited by MBO is similarregardless of size of the instance Furthermore the use of aimprovement procedure leads to an enhancement of the best-known solution through a small increase of the computa-tional time This indicates that the use of the improvementto this method enhances the convergence to the best solutionwithin our complete approach proposed in this work

At the light of this analysis in the following results wereport the computational results provided by the joint use ofMBO with the improvement method

8 Journal of Applied Mathematics

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1200

1400

1600

06

07

08

09

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

Figure 5 Performance of MBO with and without an improvement procedure when solving the DBAP medium-size instances

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1100

1200

1300

1400

2

25

3

35

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

Figure 6 Performance of MBO with and without an improvement procedure when solving the DBAP large-size instances

512 Comparison with the Best Literature Approach Tables1 and 2 present a comparison among the best publishedapproaches for the DBAP namely the mathematical modelpresented by Buhrkal et al [11] the best approximateapproach for this problem consisting of a Particle SwarmOptimization algorithm (PSO) proposed by Ting et al [13]and our MBO The first column shows the characteristics ofthe instances to solve that is the number of vessels (|119881|)and berths (|119861|) and the instance identifier (id) For each

instance the best objective value (Obj) relative error (Gap()) with regard to the optimal value provided by CPLEXand the computational time measured in seconds (119905 (s)) arepresented Furthermore with the aim of assessing the timeimprovement reported byMBO in comparison with PSO thepercentage of time improvement (119905impr) is also reported

As shown in Table 1 MBO reports high-quality solutionsin shorter computational times than the other approachesIt reaches the optimal value for the majority of the problem

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

(ii) measure the contribution of an improvement methodbased to the quality of the solutions provided by MBO

41 Migrating Birds Optimization for the DynamicBerth Allocation Problem

411 Solution Representation In the context of the DBAPthe MBO implementation considers a solution 119878DBAP as asequence composed of the vessel identifiers where eachberth is delimited by a 0 The service order of each vesselis determined by its position in the sequence The solutionstructure for the example of Figure 1 for 3 berths and 6 vesselsis as follows 119878DBAP = 1 3 0 4 2 0 5 6

412 Neighbourhood Structures The neighbourhood struc-tures considered in this approach are generated by using thefollowing movements

(a) Reinsertion Movement A vessel 119894 is removed froma berth 119896 and reinserted into another berth 119896

1015840

(forall119896 1198961015840 isin 119861 119894 = 1198941015840) at any of the possible positions

For example in 119878DBAP = 1 3 0 4 2 0 5 6 if vessel1 is removed from its berth then all these possiblereinsertion movements can be performed namely3 0 1 4 2 0 5 6 3 0 4 1 2 0 5 6 3 0 4 2 1 05 6 3 0 4 2 0 1 5 6 3 0 4 2 0 5 1 6 and 3 04 2 0 5 6 1

(b) Interchange Movement It consists of exchanging avessel 119894 assigned to berth 119896 with a vessel 1198941015840 assignedto berth 119896

1015840 (forall119894 1198941015840 isin 119881 119894 = 1198941015840 forall119896 1198961015840 isin 119861 119896 =

1198961015840) For example for the previous solution 119878DBAP =

1 3 0 4 2 0 5 6 if we select vessel 1 the possibleinterchange movements that can be obtained arethe following 4 3 0 1 2 0 5 6 2 3 0 4 1 0 5 65 3 0 4 2 0 1 6 and 6 3 0 4 2 0 5 1

The generation of random neighbour solutions by the indi-viduals is based on the reinsertion move On the other handboth movements are used in the improvement method

413 Improvement Method As discussed in the relevantsection we also analyse the capability of MBO for pointingout promising regions in the solution search space In doingso we applied an improvement method proposed by Lalla-Ruiz et al [10] over the best solution provided by MBO Thismethod consist of the following steps given a solution its bestneighbour solution is obtained by means of the reinsertionmoment Over that best neighbor solution we generate itsneighborhood by means of the interchange movement andreturn the best neighbor solution This process is performeduntil no improvement in terms of the objective function valueis achieved

414 Initial Population For generating the initial popula-tion we use a random greedy method (R-G) proposed byCordeau et al [5] that is given a random vessel permutationthe vessels are assigned one at a time to the best possible berthfollowing that sequence order according to their impact overthe objective function value The use of this method instead

of other proposed initialization procedures reported in theliterature such as First-Come First-Served Greedy (FCFS-G)or generating the solution completely at random is based onthe fact that on the one hand FCFS-G is a deterministicapproach which use will affect the convergence of thealgorithm On the other hand R-G provides better qualitysolutions than generating the initial solutions completely atrandom this is due to the fact that R-G allocates the vesselswithin the random sequence according to the impact over theobjective function value of the solution being constructed

415 Stopping Criterion The stopping criterion for the over-all MBO search process is met when a certain number of 119870generated solutions by the population are reached Moreoveras pointed out in the relevant section for large-size instanceswe included an additional stopping criterion based on amaximum number of iterations 120574 without improvement ofthe best solution obtained

42 Migrating Birds Optimization for the Quay CraneScheduling Problem

421 Solution Representation The solutions of the QCSP arerepresented as sequences composed of the available tasks thatisΩ A given sequence includes zeros in order to delimit thesubsets of tasks performed by the quay cranes This way theleftmost quay crane performs those tasks from the beginningof the sequence up to the first zero the second quay craneperforms those tasks from the first zero up to the secondzero and so forth For instance a solution for the examplepresented in Figure 2 with 2 quay cranes and 8 tasks couldbe as follows 119878QCSP = (1 2 3 5 6 7 0 4 8) In this casethe leftmost quay crane performs the tasks (1 2 3 5 6 7)whereas the other quay crane performs the tasks (4 8)

422 Neighbourhood Structures The neighbourhood struc-tures used by the MBO are based on the following exploringmovements

(a) Reinsertion Movement A task 119905 isin Ω currentlyassigned to a quay crane 119902 isin 119876 is reassigned in such away that 119905 is performed by another quay crane 1199021015840 isin 119876

(where 1199021015840 = 119902)(b) Interchange Movement Given pair tasks 1199051 1199052 isin Ω

assigned to two different quay cranes 119902 isin 119876 and 1199021015840isin

119876 (where 1199021015840 = 119902) the movement exchanges the tasksThis way 1199052 is eventually assigned to 119902 whereas 1199051 isassigned to 1199022

The generation of random neighbour solutions by theindividuals is based on the interchange movement

423 Local Search A local search process based on the bestimprovement strategy is proposed in order to find localoptima solutions during the search This way given a certainfeasible solution of the QCSP at each step the set of neigh-bour solutions found by means of reinsertion movement isgenerated The best neighbour solution replaces the currentsolution until a local optimum is achieved

Journal of Applied Mathematics 7

MBO w IMMBO wo IM

MBO w IMMBO wo IM

02

03

04

Runn

ing

time (

s)

700

800

900

1000

1100

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times7minus03

25times7minus05

25times7minus08

25times7minus03

25times7minus05

25times7minus08

Figure 4 Performance of MBO with and without an improvement procedure when solving the DBAP small-size instances

424 Initial Population The solutions included into theinitial population have been generated at randomThismeansthat each task is assigned to one of the available quay cranesrandomly It is worth mentioning that the tasks are selectedto be assigned from the leftmost up to the rightmost withinthe container vessel

425 Stopping Criterion The stopping criterion for theoverall MBO search process is met when a certain numberof119870 neighbour solutions have been already generated by theindividuals

5 Computational Experiments

This section is devoted to assessing the performance ofthe Migrating Birds Optimization (MBO) introduced in theprevious sectionAll the reported computational experimentshave been conducted on a computer equipped with an IntelDual Core 316GHz and 4GB of RAM

51 Computational Experiments for the DBAP The probleminstances used for evaluating the performance of our MBOapproach are those provided by Cordeau et al [5] Accordingto the authors their instances were generated by takinginto account a statistical analysis of the traffic and berthallocation data at the maritime container terminal of GioiaTauro (Italy) The instances are grouped into sets of 10instances whose sizes range from 25 vessels and 5 berthsup to 60 vessels and 13 berths Moreover in order to fit thespace for this work for the small- and medium-size probleminstances we have selected the 3 hardest solvable instancesof each set with regard to the time required to provide

the optimal solution by the implementation of the mathe-matical formulation (Buhrkal et al [11]) in CPLEX (httpwww-01ibmcomsoftwarecommerceoptimizationcplex-optimizer) to provide the optimal solution For the largeinstance set we selected a representative set of instancesBy taking into account the experiments carried out in thiswork we identified the following parameter values for MBO119899birds = 31 120575 = 3 120582 = 20 itermax = 3 and 119870 = |119873|

3 forthe small- and medium-size instances For the large-sizeinstances we set 119870 = |119873|

25 and an additional stoppingcriterion of a maximum number of 120574 = 10 consecutiveiterations without improvement of the best solutionobtained

511 Improvement Method As previously indicated in Sec-tion 4 an improvement phase (based on that proposed byLalla-Ruiz et al [10]) is applied over the best solution pro-vided by MBO Figures 4 5 and 6 show the computationalperformance in terms of objective function value and compu-tational time of MBO with (MBO wIM) and without (MBOwo IM) improvement method for the small- medium- andlarge-size instances proposed by Cordeau et al [5] As canbe checked the performance exhibited by MBO is similarregardless of size of the instance Furthermore the use of aimprovement procedure leads to an enhancement of the best-known solution through a small increase of the computa-tional time This indicates that the use of the improvementto this method enhances the convergence to the best solutionwithin our complete approach proposed in this work

At the light of this analysis in the following results wereport the computational results provided by the joint use ofMBO with the improvement method

8 Journal of Applied Mathematics

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1200

1400

1600

06

07

08

09

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

Figure 5 Performance of MBO with and without an improvement procedure when solving the DBAP medium-size instances

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1100

1200

1300

1400

2

25

3

35

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

Figure 6 Performance of MBO with and without an improvement procedure when solving the DBAP large-size instances

512 Comparison with the Best Literature Approach Tables1 and 2 present a comparison among the best publishedapproaches for the DBAP namely the mathematical modelpresented by Buhrkal et al [11] the best approximateapproach for this problem consisting of a Particle SwarmOptimization algorithm (PSO) proposed by Ting et al [13]and our MBO The first column shows the characteristics ofthe instances to solve that is the number of vessels (|119881|)and berths (|119861|) and the instance identifier (id) For each

instance the best objective value (Obj) relative error (Gap()) with regard to the optimal value provided by CPLEXand the computational time measured in seconds (119905 (s)) arepresented Furthermore with the aim of assessing the timeimprovement reported byMBO in comparison with PSO thepercentage of time improvement (119905impr) is also reported

As shown in Table 1 MBO reports high-quality solutionsin shorter computational times than the other approachesIt reaches the optimal value for the majority of the problem

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

MBO w IMMBO wo IM

MBO w IMMBO wo IM

02

03

04

Runn

ing

time (

s)

700

800

900

1000

1100

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times5minus01

25times5minus05

25times5minus10

25times10minus02

25times10minus03

25times10minus05

25times7minus03

25times7minus05

25times7minus08

25times7minus03

25times7minus05

25times7minus08

Figure 4 Performance of MBO with and without an improvement procedure when solving the DBAP small-size instances

424 Initial Population The solutions included into theinitial population have been generated at randomThismeansthat each task is assigned to one of the available quay cranesrandomly It is worth mentioning that the tasks are selectedto be assigned from the leftmost up to the rightmost withinthe container vessel

425 Stopping Criterion The stopping criterion for theoverall MBO search process is met when a certain numberof119870 neighbour solutions have been already generated by theindividuals

5 Computational Experiments

This section is devoted to assessing the performance ofthe Migrating Birds Optimization (MBO) introduced in theprevious sectionAll the reported computational experimentshave been conducted on a computer equipped with an IntelDual Core 316GHz and 4GB of RAM

51 Computational Experiments for the DBAP The probleminstances used for evaluating the performance of our MBOapproach are those provided by Cordeau et al [5] Accordingto the authors their instances were generated by takinginto account a statistical analysis of the traffic and berthallocation data at the maritime container terminal of GioiaTauro (Italy) The instances are grouped into sets of 10instances whose sizes range from 25 vessels and 5 berthsup to 60 vessels and 13 berths Moreover in order to fit thespace for this work for the small- and medium-size probleminstances we have selected the 3 hardest solvable instancesof each set with regard to the time required to provide

the optimal solution by the implementation of the mathe-matical formulation (Buhrkal et al [11]) in CPLEX (httpwww-01ibmcomsoftwarecommerceoptimizationcplex-optimizer) to provide the optimal solution For the largeinstance set we selected a representative set of instancesBy taking into account the experiments carried out in thiswork we identified the following parameter values for MBO119899birds = 31 120575 = 3 120582 = 20 itermax = 3 and 119870 = |119873|

3 forthe small- and medium-size instances For the large-sizeinstances we set 119870 = |119873|

25 and an additional stoppingcriterion of a maximum number of 120574 = 10 consecutiveiterations without improvement of the best solutionobtained

511 Improvement Method As previously indicated in Sec-tion 4 an improvement phase (based on that proposed byLalla-Ruiz et al [10]) is applied over the best solution pro-vided by MBO Figures 4 5 and 6 show the computationalperformance in terms of objective function value and compu-tational time of MBO with (MBO wIM) and without (MBOwo IM) improvement method for the small- medium- andlarge-size instances proposed by Cordeau et al [5] As canbe checked the performance exhibited by MBO is similarregardless of size of the instance Furthermore the use of aimprovement procedure leads to an enhancement of the best-known solution through a small increase of the computa-tional time This indicates that the use of the improvementto this method enhances the convergence to the best solutionwithin our complete approach proposed in this work

At the light of this analysis in the following results wereport the computational results provided by the joint use ofMBO with the improvement method

8 Journal of Applied Mathematics

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1200

1400

1600

06

07

08

09

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

Figure 5 Performance of MBO with and without an improvement procedure when solving the DBAP medium-size instances

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1100

1200

1300

1400

2

25

3

35

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

Figure 6 Performance of MBO with and without an improvement procedure when solving the DBAP large-size instances

512 Comparison with the Best Literature Approach Tables1 and 2 present a comparison among the best publishedapproaches for the DBAP namely the mathematical modelpresented by Buhrkal et al [11] the best approximateapproach for this problem consisting of a Particle SwarmOptimization algorithm (PSO) proposed by Ting et al [13]and our MBO The first column shows the characteristics ofthe instances to solve that is the number of vessels (|119881|)and berths (|119861|) and the instance identifier (id) For each

instance the best objective value (Obj) relative error (Gap()) with regard to the optimal value provided by CPLEXand the computational time measured in seconds (119905 (s)) arepresented Furthermore with the aim of assessing the timeimprovement reported byMBO in comparison with PSO thepercentage of time improvement (119905impr) is also reported

As shown in Table 1 MBO reports high-quality solutionsin shorter computational times than the other approachesIt reaches the optimal value for the majority of the problem

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1200

1400

1600

06

07

08

09

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

35times7minus02

35times7minus06

35times7minus08

35times10minus01

35times10minus05

35times10minus09

Figure 5 Performance of MBO with and without an improvement procedure when solving the DBAP medium-size instances

MBO w IMMBO wo IM

MBO w IMMBO wo IM

1100

1200

1300

1400

2

25

3

35

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

Instance Instance

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

60times13minus01

60times13minus02

60times13minus03

60times13minus04

60times13minus05

60times13minus06

60times13minus07

60times13minus08

60times13minus09

60times13minus10

Figure 6 Performance of MBO with and without an improvement procedure when solving the DBAP large-size instances

512 Comparison with the Best Literature Approach Tables1 and 2 present a comparison among the best publishedapproaches for the DBAP namely the mathematical modelpresented by Buhrkal et al [11] the best approximateapproach for this problem consisting of a Particle SwarmOptimization algorithm (PSO) proposed by Ting et al [13]and our MBO The first column shows the characteristics ofthe instances to solve that is the number of vessels (|119881|)and berths (|119861|) and the instance identifier (id) For each

instance the best objective value (Obj) relative error (Gap()) with regard to the optimal value provided by CPLEXand the computational time measured in seconds (119905 (s)) arepresented Furthermore with the aim of assessing the timeimprovement reported byMBO in comparison with PSO thepercentage of time improvement (119905impr) is also reported

As shown in Table 1 MBO reports high-quality solutionsin shorter computational times than the other approachesIt reaches the optimal value for the majority of the problem

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

Table 1 Computational results for a representative set of small- and medium-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Obj 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

25 51 759 599 759 000 075 759 000 029 minus6133

5 955 697 955 000 086 955 000 027 minus6977

10 1073 638 1073 000 073 1073 000 032 minus5479

25 73 807 428 807 000 097 807 000 024 minus7526

5 725 385 725 000 044 725 000 027 minus3864

8 768 393 768 000 105 768 000 025 minus7619

25 102 727 699 727 000 075 727 000 036 minus5200

3 761 612 761 000 056 761 000 034 minus3929

5 840 677 840 000 045 840 000 029 minus3556

35 72 1192 1593 1192 000 491 1198 050 081 minus8350

6 1686 2916 1686 000 328 1692 036 077 minus7652

8 1318 1752 1318 000 239 1324 046 073 minus6946

35 101 1124 1998 1124 000 158 1124 000 091 minus4241

5 1349 2231 1349 000 153 1350 007 091 minus4052

9 1311 2945 1311 000 281 1313 015 097 minus6548

Average 102633 1238 102633 000 154 102773 010 052 minus5871

Table 2 Computational results for a representative set of large-size instances proposed by Cordeau et al [5]

Instance CPLEX PSO MBO|119881| |119861| id Opt 119905 (s) Obj Gap () 119905 (s) Obj Gap () 119905 (s) 119905impr ()

60 13

1 1409 1792 1409 000 1111 1411 014 342 minus6922

2 1261 1577 1261 000 789 1261 000 352 minus5539

3 1129 1354 1129 000 748 1129 000 363 minus5147

4 1302 1448 1302 000 603 1302 000 381 minus3682

5 1207 1721 1207 000 584 1207 000 313 minus4640

6 1261 1385 1261 000 767 1261 000 346 minus5489

7 1279 1460 1279 000 75 1279 000 305 minus5933

8 1299 1421 1299 000 994 1299 000 330 minus6680

9 1444 1651 1444 000 425 1444 000 348 minus1812

10 1213 1416 1213 000 52 1213 000 340 minus3462

Average 128040 1523 128040 000 729 128060 001 342 minus4931

instances Although MBO is not able to provide the optimalsolutions in five cases (ie 35 times 7 minus 2 35 times 7 minus 6 35 times 7 minus 835 times 10 minus 1 and 35 times 10 minus 9) it presents a very competitiveperformance with a time range enough for improvementIn this regard the maximum gap in those cases is 050Furthermore MBO is able to reduce on average about the58 of the time required by PSO

Moreover when we evaluate larger problem instancesas those reported in Table 2 we can point out the relevanttime improvement reported by MBO over PSO which is onaverage of almost 50 In this regard as shown in this tablethe quality of the solutions is similar to the PSO MBO isable to provide the optimal solutions in the majority of thecases In the unique case where MBO does not provide theoptimal solution it is able to provide a solution with a gapof 007 It should be also pointed out that the time benefit

presented by MBOmakes it suitable as a solution method forbeing applied either individually or included into integratedschemes in which the berth allocation is required and has tobe executed frequently

52 Computational Experiments for the QCSP In order toassess the suitability of MBO when solving the QCSP wehave considered 40 instances (k13ndashk52) of those proposedby Bierwirth and Meisel [7] These instances have differentnumber of tasks (from 10 up to 25) and quay cranes (from2 up to 3) which allow encompassing real-world scenariosIt is worth mentioning that as done in previous works inthis experiment we have established that the quay cranes areavailable from the beginning of the service time (ie 119903119902 = 0forall119902 isin 119876) and have to keep a safety distance 120575 = 1 duringthe service Moreover by preliminary tests the following

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

MBO w LSMBO wo LS

MBO w LSMBO wo LS

700

800

900

1000

34

35

36

37

38

Runn

ing

time (

s)

Obj

ectiv

e fun

ctio

n va

lue

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

k43

k44

k45

k46

k47

k48

k49

k50

k51

k52

Instance

Figure 7 Performance of MBO with and without a local search procedure when solving the QCSP k43ndashk52 problem instances

parameter values have been used in the execution of MBO119899birds = 31 120575 = 1 120582 = 10 itermax = 5 and119870 = |119873|

3

521 Local Search As done in previous works (eg Sam-marra et al [17] Exposito-Izquierdo et al [8]) a local searchprocess is applied to the best solution provided by MBOTherationale behind this is to assess the capacity ofMBO to pointout promising regions in the search space Figure 7 shows thecomputational performance in terms of objective functionvalue and computational time ofMBOwith (MBOwLS) andwithout (MBOwo LS) local search for the instances k43ndashk52(for the other instances k13ndashk42 regardless of the use or notof the local search our algorithm provides the best knownsolution) As shown in the figure the performance exhibitedby the MBO with and without LS is similar independentlyof the instance tackled In this regard the use of a localsearch leads to a very small improvement of some solutionsin some cases (see k51 and k52) requiring only an slightlyincrease of the computational time This fact may indicatethat in some cases the solution provided byMBO is already alocal optimum Finally although LS contributes to improvingthe quality of the solution not using it may not affectsubstantively the quality of the solution Nevertheless dueto small computational time required in the following theMBO computational results reported for the QCSP instancesare the ones obtained with local search

522 Comparison with the Best Literature Approach Table 3shows a comparison between the optimal solutions reportedby Bierwirth and Meisel [7] the Estimation DistributionAlgorithm (EDA) proposed by Exposito-Izquierdo et al [8]and our MBO In each case we report the objective functionvalue of the best found solution and the computational timemeasured in seconds In the case of the MBO we alsoreport the gap in the objective function value compared with

the optimal solution and computational time compared withthose reported by the EDA

As can be checked in Table 3 our MBO has reported(near-)optimal solutions for all the instances under analysisOnly in one instance (k45) the optimal solution was notreached In those cases MBO reports a gap of 036 andan overall gap of only 001 for all the instances consideredThe quality of the solutions reported by MBO indicates thatour approach is highly effective in realistic scenarios Finallywhen carrying out an analysis of the computational timeswe realize that MBO requires short computational timesrequiring atmost 384 secondsThis fact constitutes a relevantimprovement in comparison with the EDA which requiresmore than 16 seconds in some instance (k50) This timeadvantagemust be suitably consideredwhen addressing prac-tical scenarios where the QCSP has to be solved dynamically

6 Conclusions and Further Research

In this paper we have presented a Migrating Birds Opti-mization-based approach for addressing two essential seasideproblems at maritime container terminals the DynamicBerth Allocation Problem (DBAP) and Quay Crane Schedul-ing Problem (QCSP) It is noticeable from the computationalexperiments that the proposed algorithm is able to reporthigh-quality solutions by means of short computationaltimes In this regard the time advantage makes MBOpromising and competitive as solutionmethodwhen tacklingseaside operations either individually or embedded into realdecision-support systemswhere this problemhas to be solvedfrequently Moreover since our approach includes the use ofan improvement method (in the case of DBAP) and a localsearch (in the case of QCSP) over the best solution providedby MBO we have assessed the contribution of them to thequality of the solution provided In this regard their use

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

Table 3 Comparison among the optimal solutions (UDS Bierwirthand Meisel [7]) the Estimation Distribution Algorithm (EDAExposito-Izquierdo et al [8]) and ourMBOwhen solving theQCSP

UDS EDA MBOObj 119905 (s) Obj 119905 (s) Obj 119905 (s) Gap ()

11989613 453 mdash 453 008 453 017 00011989614 546 mdash 546 009 546 018 00011989615 513 mdash 513 008 513 017 00011989616 312 mdash 312 056 312 017 00011989617 453 mdash 453 008 453 017 00011989618 375 mdash 375 007 375 016 00011989619 543 mdash 543 008 543 018 00011989620 399 mdash 399 009 399 019 00011989621 465 mdash 465 007 465 017 00011989622 540 mdash 540 013 540 020 00011989623 576 mdash 576 027 576 040 00011989624 666 mdash 666 039 666 038 00011989625 738 mdash 738 025 738 035 00011989626 639 mdash 639 033 639 037 00011989627 657 mdash 657 029 657 035 00011989628 531 mdash 531 027 531 034 00011989629 807 mdash 807 031 807 036 00011989630 891 mdash 891 022 891 040 00011989631 570 mdash 570 026 570 037 00011989632 591 mdash 591 037 591 038 00011989633 603 mdash 603 912 603 113 00011989634 717 mdash 717 924 717 116 00011989635 684 mdash 684 448 684 109 00011989636 678 mdash 678 762 678 113 00011989637 510 mdash 510 409 510 111 00011989638 618 mdash 618 666 618 109 00011989639 513 mdash 513 654 513 114 00011989640 564 mdash 564 714 564 111 00011989641 588 mdash 588 666 588 113 00011989642 573 mdash 573 630 573 118 00011989643 876 126 876 1260 876 374 00011989644 822 120 822 1140 822 363 00011989645 834 108 834 840 837 352 03611989646 690 114 690 960 690 366 00011989647 792 102 792 1020 792 351 00011989648 639 114 639 840 639 344 00011989649 894 108 894 1320 894 380 00011989650 741 102 741 1680 741 384 00011989651 798 102 798 1200 798 352 00011989652 960 102 960 1320 960 369 000Avg 63398 1098 63398 470 63405 133 001

allows an enhancement in the quality of the solutions in termsof objective function value through a small increase of thecomputational time

Furthermore the inherent dynamism of the seaside oper-ations at maritime container terminals highly impacts on

the performance of the technical equipment and conse-quently on the involved transportation modes Thus havingeffective and fast algorithms to reach high-quality solutionsis aspired by terminal managers In this context at the lightof the computational results presented along this paper wecan claim that usingMBO is suitable and advisable to be usedin practical contexts with the goal of providing an adequateservice to the incoming container vessels

A multitude of lines are open for further research Inthe future we are going to test the performance of MBOin other heterogeneous transportation problems such asVehicle Routing Problem due to its generalist standpointIn this regard we are also going to study how differentinteraction schemes impact on the performance of MBO

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been partially funded by the Spanish Ministryof Economy and Competitiveness (Project TIN2012-32608)The authors thank the funding granted to the University of LaLaguna (ULL) by the Canary Agency of Investigation Inno-vation and Information Society (ACIISI) 85 cofinancedwith the help of European Social Fund Eduardo Lalla-Ruizand Christopher Exposito-Izquierdo would like to thank theCanary Government for the financial support they receivethrough their doctoral grants

References

[1] L Nicoletti A Chiurco C Arango and R Diaz ldquoHybridapproach for container terminals performances evaluationand analysisrdquo International Journal of Simulation and ProcessModelling vol 9 no 1-2 pp 104ndash112 2014

[2] C Exposito-Izquierdo E Lalla-Ruiz B Melian Batista and JM Moreno-Vega ldquoOptimization of container terminal prob-lems an integrated solution approachrdquo in Computer AidedSystemsTheorymdashEUROCAST 2013 R Moreno-Dıaz F Pichlerand A Quesada-Arencibia Eds vol 8111 of Lecture Notes inComputer Science pp 324ndash331 Springer Berlin Germany 2013

[3] H-J Yeo ldquoCompetitiveness of Asian container terminalsrdquo TheAsian Journal of Shipping and Logistics vol 26 no 2 pp 225ndash246 2010

[4] C Bierwirth and F Meisel ldquoA survey of berth allocation andquay crane scheduling problems in container terminalsrdquo Euro-pean Journal of Operational Research vol 202 no 3 pp 615ndash6272010

[5] J-F Cordeau G Laporte P Legato and LMoccia ldquoModels andtabu search heuristics for the berth-allocation problemrdquo Trans-portation Science vol 39 no 4 pp 526ndash538 2005

[6] E Duman M Uysal and A F Alkaya ldquoMigrating birds opti-mization a newmetaheuristic approach and its performance onquadratic assignment problemrdquo Information Sciences vol 217pp 65ndash77 2012

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

[7] C Bierwirth and F Meisel ldquoA fast heuristic for quay cranescheduling with interference constraintsrdquo Journal of Schedulingvol 12 no 4 pp 345ndash360 2009

[8] C Exposito-Izquierdo J L Gonzalez-Velarde B Melian-Batista and J MMoreno-Vega ldquoHybrid estimation of distribu-tion algorithm for the quay crane scheduling problemrdquo AppliedSoft Computing Journal vol 13 no 10 pp 4063ndash4076 2013

[9] A Imai E Nishimura and S Papadimitriou ldquoThe dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash417 2001

[10] E Lalla-Ruiz B Melian-Batista and J Marcos Moreno-VegaldquoArtificial intelligence hybrid heuristic based on tabu search forthe dynamic berth allocation problemrdquo Engineering Applica-tions of Artificial Intelligence vol 25 no 6 pp 1132ndash1141 2012

[11] K Buhrkal S Zuglian S Ropke J Larsen and R Lusby ldquoMod-els for the discrete berth allocation problem a computationalcomparisonrdquo Transportation Research Part E Logistics andTransportation Review vol 47 no 4 pp 461ndash473 2011

[12] R M de Oliveira G R Mauri and L A N Lorena ldquoClusteringsearch for the berth allocation problemrdquo Expert Systems withApplications vol 39 no 5 pp 5499ndash5505 2012

[13] C-J Ting K-C Wu and H Chou ldquoParticle swarm optimiza-tion algorithm for the berth allocation problemrdquo Expert Systemswith Applications vol 41 no 4 pp 1543ndash1550 2014

[14] E Lalla-Ruiz and S Voszlig ldquoPopmusic as a matheuristic for theberth allocation problemrdquo Annals of Mathematics and ArtificialIntelligence pp 1ndash17 2014

[15] F Meisel and C Bierwirth ldquoA unified approach for the evalua-tion of quay crane schedulingmodels and algorithmsrdquoComput-ers amp Operations Research vol 38 no 3 pp 683ndash693 2011

[16] K H Kim and Y-M Park ldquoA crane scheduling method for portcontainer terminalsrdquo European Journal of Operational Researchvol 156 no 3 pp 752ndash768 2004

[17] M Sammarra J-F Cordeau G Laporte and M F Monaco ldquoAtabu search heuristic for the quay crane scheduling problemrdquoJournal of Scheduling vol 10 no 4-5 pp 327ndash336 2007

[18] P Legato R Trunfio and F Meisel ldquoModeling and solvingrich quay crane scheduling problemsrdquo Computers amp OperationsResearch vol 39 no 9 pp 2063ndash2078 2012

[19] C F Daganzo ldquoThe crane scheduling problemrdquo TransportationResearch Part B Methodological vol 23 no 3 pp 159ndash175 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of