optimization of continuous berth scheduling by taking into

Research ArticleOptimization of Continuous Berth Scheduling by Taking intoAccount Double-Line Ship Mooring

Cheng Luo Hongying Fei Dana Sailike Tingyi Xu and Fuzhi Huang

School of Management Shanghai University Shanghai 200444 China

Correspondence should be addressed to Hongying Fei feihyshueducn

Received 2 July 2020 Revised 20 August 2020 Accepted 31 August 2020 Published 18 September 2020

Academic Editor Tingsong Wang

Copyright copy 2020 Cheng Luo et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

ldquoDouble-Line Ship Mooringrdquo (DLSM) mode has been applied as an initiative operation mode for solving berth allocationproblems (BAP) in certain giant container terminals in China In this study a continuous berth scheduling problem with theDLSM model is illustrated and solved with exact and heuristic methods with an objective to minimize the total operation costincluding both the additional transportation cost for vessels not located at their minimum-cost berthing position and the penaltiesfor vessels not being able to leave as planned First of all this problem is formulated as a mixed-integer programming model andsolved by the CPLEX solver for small-size instances Afterwards a particle swarm optimization (PSO) algorithm is developed toobtain good quality solutions within reasonable execution time for large-scale problems Experimental results show that DLSMmode can not only greatly reduce the total operation cost but also significantly improve the efficiency of berth scheduling incomparison with the widely used single-line ship mooring (SLSM) mode (e comparison made between the results obtained bythe proposed PSO algorithm and that obtained by the CPLEX solver for both small-size and large-scale instances are also quiteencouraging To sum up this study can not only validate the effectiveness of DLSMmode for heavy-loaded ports but also provide apowerful decision support tool for the port operators to make good quality berth schedules with the DLSM mode

1 Introduction

With the deepening of economic globalization the worldcontainer transportation volume has increased dramaticallyin recent years [1] As one of the most important compo-nents of container transportation network container ter-minals play an important role in world economyConsidering that the efficiency of berth allocation problemhas great impact on the output of container terminals a lotof studies have been dedicated to the berth schedulingproblems

In general berth schedules are determined by specifyingberthing time and position for the coming vessels by takinginto account various constraints such as the berth capacityannounced arrival time and departure time of containerships and certain specific berthing requirements In order toavoid collisions between vessels the single-line shipmooring(SLSM) mode which specifies that ldquono more than one vesselcan be allocated to the same berth position at the same timerdquo

is normally applied in container terminals over the worldand this rule is regarded as a default in most of the studies onberth scheduling [2]

As one of the most important economic role in the worldeconomy China is continuously developing the economicinnovation resulting in huge container throughput of theinternational hubs in China Yangshan deep-water port islocated in Shanghai a mega city in China As the largest sea-island artificial deep-water port Yangshan deep-water portis an important part of Shanghai International ShippingCenter and its annual throughput has increased constantlysince it was built in the year of 2005

Having been one of the worldrsquos busiest container ter-minals Yangshan deep-water port has applied a so-calledldquoDouble-Line Ship Mooringrdquo (DLSM) mode to build berthschedules since the year of 2019 Different from the widelyused SLSMmode DLSMmode allows two container ships tobe moored simultaneously at the same berth location en-abling more container ships to be moored at their ideal berth

HindawiScientific ProgrammingVolume 2020 Article ID 8863994 11 pageshttpsdoiorg10115520208863994

and cranes to operate two container ships at the same time areasonable way to improve the efficiency of berth operationDespite that the real efficiency of the port with DLSMmodedepends on how sophisticated the operators are according tothe investigation made at Yangshan port because berthscheduling with DLSM is much more complex than with theSLSM system In consequence it is essential to develop aneffective decision support system to help berth operatorsimprove the quality and efficiency of daily schedules with theDLSM mode

(is study aims at minimizing the additional operationcosts for the vessels not placed at their ideal berthing po-sition and penalties for the ships not being able to leavebefore their preplanned departure time for a continuousberth allocation problems (BAP) model with the DLSMmode(emain contributions of this study are as follows (1)as the first study on berth scheduling with the DLSM modeit validates the contribution of DLSM model in comparisonwith the widely used SLSM mode (2) a mixed integerprogramming model is constructed for the continuous BAPwith DLSM mode enabling a benchmark for the futurestudies on such topic (3) a PSO algorithm is developed toobtain good quality DLSM berthing schedules within rea-sonable execution time offering the berth operators at busyports a powerful decision support tool to improve their workefficiency

(e reminder of this paper is structured as followsSection 2 is dedicated to the literature review of related work(e problem description is given in Section 3 In Section 4the targeted problem is formulated as a mixed integerprogramming model which can be solved by CPLEX solverand then the details of the proposed PSO algorithm aregiven in Section 5 Section 6 is dedicated to discussions aboutthe experimental results and this paper ends up withconclusions and perspectives

2 Related Work

According to the literature berth allocation problem (BAP)is regarded as the most important issue faced by themanagement of container terminals As the quality of berthschedule has a great influence on the improvement of portoperation efficiency a lot of researchers have been studyingBAPs and numerous results were published [2] Since Imaiet al [3] firstly addressed a static berth allocation problem(SBAP) and further developed a dynamic BAP model(DBAP for public berth system [4]) numerous studies havebeen published on BAP aiming to optimize the operationefficiencies by taking into account the various constraintsaffecting BAP [2] such as tidal influence [5] vessel servicepriority [6] time-varying water depth [7] and channel flowcontrol [8] As to our knowledge no study on BAPs with theDLSM mode is observed in the literature thus this studyaims to fill the research gaps in this field by taking intoaccount the DLSM mode when constructing berth sched-ules Furthermore this study is specifically dedicated tocontinuous BAP which copes with the real situation ofYangshan deep-water port that we have investigated and canalso be applied to many other huge container hubs

With regard to the methodologies applied to BAPs it canbe observed in the literature that the BAPs are normallyformulated as Mixed Integer Programming (MIP) modelswhich can be solved with commercial programming solversfor small-size problems [2ndash11] Since BAPs are NP-hard lotsof researchers have also taken much effort on developingnear-optimal solutions with efficient heuristics and meta-heuristics such as genetic algorithm and its variants [12ndash15]subgradient optimization method [15] simulated annealing[9] evolutionary algorithms [16 17] particle swarm opti-mization [10] and island-based meta-heuristic algorithm[18]

Particle swarm optimization (PSO) is an evolutionarycomputation technique firstly developed by Eberhart andKennedy [19] Since the particles used in PSO algorithm areonly updated by internal velocity and fewer parametersshould be tuned the principles of PSO can be easily un-derstood and widely adapted to various specific applicationsAccording to the literatures PSO algorithms have beenproven quite efficient in dealing with berth allocation [20]yard allocation [21] quay crane scheduling [10 22] andmany other planning and scheduling problems [23 24] inconsequence a PSO algorithm is also proposed in this studyto develop efficient berth schedules by taking into accountDLSM

3 Problem Description

31 Assumptions In general the berth operators are incharge of arranging each vessel arriving at the port to asuitable berth position according to the availability of thewharf resources and respecting the preplanned arrival anddeparture of the vessel According to the practices observedat the targeted container terminal a set of assumptions aredefined as follows

(1) Each vessel arrives at the port on the preplannedarrival time

(2) If a vessel cannot leave the port before the pre-planned departure time the port has to incur apenalty

(3) (e coordinate corresponding to the leftmost end ofthe vessel is used to represent its berthing positionusing the leftmost boundary of the wharf as thecoordinate reference point

(4) Each vessel has a predefined minimum-cost berthingposition which is determined according to the goodsthat will be loadedunloaded at the port and if avessel cannot be berthed at its ideal berthing posi-tion additional operation cost will occur

(5) DLSM mode is applied ie at most two vessels canbe simultaneously moored at the same berth position

(6) When two vessels are moored in double-line thelength of the inner-side ship cannot be longer thanthe vessel moored outside

(7) When two vessels are moored in double-line theinner-side one must be berthed earlier and leave laterthan the outside one

2 Scientific Programming

32 Notations For a better understanding two types ofnotations are applied in this study (1) Latin letters are usedto denote parameters (2) Greek letters are applied to rep-resent decision variables

321 Parameters

V set of the vessels waiting to be allocated at the berthL length of the wharf at the portT planning horizon of this studyM sufficiently large positive numberi j k indices of vessels i j k isinVc1i unit distance cost for transporting containers fromto vessel i i isin Vc2i penalty cost per unit of time caused by the latedeparture of vessel i from the port i isin Vpi predefined minimum-cost berthing position ofvessel i i isin Vearr

i preplanned arrival time of vessel ihi total handling time required by the port to finish thenecessary unloadingunloading operations of vessel ii isin Ve

de p

i preplanned departure time of vessel i i isin Vli length of vessel i i isin V In this study the necessarygap that must be reserved to guarantee the safety is alsointegrated in this value for each vessel

322 Decision variables

μi integer variable representing the actual berthingposition of vessel i i isin Vθi Integer variable representing the actual berthingtime of vessel i ie the moment that this vessel ismoored at its berthing position where it can be op-erated by the cranes i isin Vσij binary variable representing the relative berthingposition of two adjacent vessels which equals 1 if vesseli is moored on the left to vessel j (i j isin V) and 0otherwiseπij binary variable indicating the sequential relation-ship between the berthing time of two vessels whichequals 1 if vessel i is berthed before vessel j (i j isin V)and 0 otherwiseεij binary variable representing the relative line posi-tion of two vessels that are moored in double-linewhich equals to 1 if vessel i is berthed on the inner-sideto vessel j (i j isin V) and 0 otherwise

33TimelineCorresponding to theBerthSchedulingofaVesselAs shown in Figure 1 the timeline of the berth scheduling ofvessel i starts from its arrival time at the port denoted as earr

i and ends up at the moment when all the necessaryunloadingloading operations are completed and the vesselis ready to leave the port for the next destination

It is worth noting that (1) a gap between the preplannedarrival time earr

i and actual berthing time θimay be observedin the condition that no berth location is available on thearrival of the vessel which has to be waiting at the anchorageuntil in-wharf permission is delivered (2) although setupoperations (such as berthing mooring ropes and removingtwist locks) are necessary before and after the cranes op-erating the vessels the setup time of vessel i is integrated intoits overall operation time hi in this study to simplify theexpression because it is observed in the targeted port that thesetup time is generally not vessel-dependent

34 Berth Scheduling with DLSM Mode When the DLSMmode is applied at the port and two vessels will be scheduledto be moored in double-line at the same berthing position itis necessary to determine the relative berthing line positionsbetween these two vessels (rough the interview with theberth operators at the targeted port a general rule is appliedto ensure the berthing safety Vessel i can be moored in theinner-side line to vessel j(i j isin V) as long as the followingconditions can be satisfied (1) the length of vessel i is largerthan that of vessel j (2) vessel i arrives no later than vessel j atthe port (3) the actual departure time of vessel i is not earlierthan that of vessel j (4) the berthing position of the inner-side vessel should not be larger that of the outside mooredship while the coordinate of the rightmost end of the formeralong the wharf must be at least as large as that of the latter

Figure 2 is the three-dimensional schematic diagram of aDLSM example with three vessels where vessels i and j aremoored in double-line

For a better understanding of this example the corre-sponding side-wharf section and time-wharf one are detailedin Figures 3 and 4 respectively It can be observed that vesseli is placed on the inner-side line to vessel j since the fol-lowing conditions are satisfied (1) li ge lj (as shown in Fig-ure 3) (2) θi le θj (as shown in Figure 4) (3) θi + hi ge θj + hj

(as shown in Figure 4) (4) pi lepj pi + li ge pj + lj (asshown in Figure 4) in a berth schedule a vessel can either beplaced at its minimum-cost position (eg vessels j and k) ornonoptimal berthing position (eg vessel i)

As for vessel k since it is moored on the right to the pairof double-lined ships the coordinates of its berthing posi-tion must be larger than those of the rightmost of the inner-side moored ship to avoid overlaps ie μk ge μi + li

4 Mathematical Formulation

41 Construction of the Objective Function As mentioned inSection 1 the objective function of the targeted BAPproblem consists of two parts

(1) Minimization of the additional operation cost forvessels not located at its minimum-cost berthingpositionConsidering that the ports normally predefine theminimum-cost berthing position for each vesselaccording the cargos that will be unloadedloaded tomaximize the operation efficiency it is obvious thatthe larger is the deviation between the berthing

Scientific Programming 3

position and the predefined minimum-cost positionof a vessel the more is operation cost at the wharf inthis study this part of the objective function isformulated as 1113936iisinVc1i|μi minus pi|

(2) Minimization of the penalty cost for vessels notleaving before their preplanned departure timeLet e

de pi denote the preplanned departure time of

vessel i As is known when the actual departure timeof vessel i is later than e

de pi an additional cost will be

incurred due to the influence of such delay on therest of the voyage and if the delay in the shiprsquosdeparture is caused by the inefficient operation of theport the port would have to pay certain fine for suchdelay (erefore this study aims also at minimizingthe total penalties related to the delays of the vesselswithin the planning period and this part of theobjective function can be formulated as 1113936iisinVc2i(θi + hi minus e

de pi )+ where x+ max 0 x

To sum up the objective function can be formulatedas follows

【BAP】min1113944iisinV

c1i μi minus pi

11138681113868111386811138681113868111386811138681113868 + c2i θi + hi minus e

de pi1113872 1113873

+1113882 1113883 (1)

42 Mixed-Integer Programming Model Considering thatformula (1) is nonlinear it should be linearized to meet therequirements of linear programming solver that will beapplied in this study to optimally solve small-size instances

Let α+i μi minus pi when μi minus pi ge 0 αminus

i pi minus μi whenμi minus pi lt 0 and β+

i and βminusi denote the non-negative and

negative value of (θi + hi minus ede pi ) respectively Furthermore

let β+i 0 when θi + hi minus e

depi lt 0 to ensure that the smaller is

β+i the better is the solution Formula (1) can be linearized to

formulas (2) (3) and (4) and then the mixed-integer pro-grammingmodel of the targeted problem can be constructedas follows

【BAP】min 1113944iisinV

c1i α+i + αminus

i( 1113857 + c2iβ+i1113864 1113865 (2)

subject to

μi minus pi α+i minus αminus

i foralli isin V (3)

θi + hi minus ede pi β+

i minus βminusi foralli isin V (4)

1113944jisinV

εij le 1 foralli isin V (5)

μi + li leL foralli isin V (6)

μi + li le μj + M 1 minus σij1113872 1113873 foralli j isin V ine j (7)

θi + hi le θj + M 1 minus πij1113872 1113873 foralli j isin V ine j (8)

θi le θj + M 1 minus εij1113872 1113873 foralli j isin V ine j (9)

θj + hj le θi + hi + M 1 minus εij1113872 1113873 foralli j isin V ine j (10)

μi le μj + M 1 minus εij1113872 1113873 foralli j isin V ine j (11)

μj + lj le μi + li + M 1 minus εij1113872 1113873 foralli j isin V ine j (12)

σij + σji + πij + πji + εij + εji ge 1 foralli j isin V ine j (13)

σij + εij le 1 foralli j isin V ine j (14)

πij + εij le 1 foralli j isin V ine j (15)

θi ge earri foralli isin V (16)

μi θi α+i αminus

i β+i βminus

i ge 0 foralli isin V (17)

σij πij εijε 0 1 foralli j isin V ine j (18)

eiarr

Waiting (possible) HandlingTime

Ready forleaving

Arrive at berth

Arrive atanchorage

Vessel i

hi

θi

1 2

Figure 1 Timeline corresponding to the berth scheduling of vessel i

Wharf

Side

Vessel j

Vessel iVessel k

Time

Figure 2 Time-side-wharf schematic diagram of a DLSM example

Outside

WharfInside

Side

pi pj (μj) pk (μk)

Vessel j

Vessel i Vessel klk

lj

liμi

Figure 3 Side-wharf section of the DLSM example


Objective function (2) and constraints (3) and (4) in-dicate that the objective of this study is to minimize bothtotal additional operation cost corresponding to vessels notberthing at their minimum-cost berthing position vesselsand the penalty cost incurred when vessels cannot leavebefore their preplanned departure time

Constraint (5) ensures that no more than two vessels canbe berthed at the same position simultaneously Constraint(6) indicates that the rightmost end of each vessel must belimited by the length of the wharf Constraint (7) ensures theposition relationship of two adjacent vessels along the wharfConstraint (8) indicates the sequential relationship betweenthe berthing time of two vessels that will be berthed at thesame position but not in double-line Constraints (9)ndash(12)indicate the conditions to be respected when two vessels areberthed in double-line at the berth as detailed in Section 34Constraints (13) ensure that at least one relationship be-tween two vessels waiting to be berthed within the planningperiod as shown in Figure 4 holds

Constraints (14) and (15) ensure the SLSM mode andDLSM mode cannot be applied to the same pair of vesselssimultaneously ie if vessel i is berthed in double-line withvessel j it can neither be berthed to the left of vessel j nor beberthed before the arrival or after the departure of vessel jConstraint (16) ensures that vessels can only be berthed aftertheir arrival Constraints (17) and(18) define the range ofdecision variables

5 PSO Algorithm for BAP with DLSM Mode

51 Introduction to PSO In PSO algorithms the particleswarm concept originated as a simulation of a simplifiedsocial system by introducing a number of simple enti-tiesmdashthe particlesmdashin the search space where each particlerepresents a solution approach corresponding to a givenposition and velocity which can be used to evaluate theobjective function at its current location

(e movement of each particle is guided by their po-sition according to their own best position and a swarmrsquosbest position which represents the quality of searching andthe velocity decides the direction in which the particle would

move in the next generation (ese particles search foroptimal solutions through updating generations Formulas(19) and (20) represent how velocity and position update inthe classical PSO algorithm respectively

vki d v

kminus1i d + c1r1 pbesti d minus x

kminus1i d1113872 1113873 + c2r2 gbestd minus x

kminus1i d1113872 1113873 (19)

xki d x

kminus1i d + v

kminus1i d (20)

where vki d and vkminus1

i d represent the current and previous flightvelocity of particle i on dimension d in iteration k re-spectively xk

i d and xkminus1i d represent the current and previous

position of particle i on dimension d in iteration k ωkminus 1 isthe inertial weight coefficient which can adjusts the searchrange of solution space c1 and c2 are acceleration weightswhich adjust the learning maximum step length r1 and r2are two random functions with a value range of [0 1] whosefunction is to increase the randomness of the search pbesti d

denotes the best position of particle i on dimension d up toiteration k while gbestd denotes the best position of thewhole swarm on dimension d until iteration k

Considering that the classical PSO algorithm mentionedabove may lead the particles to grow unlimitedly whichinfluences the particlesrsquo convergence to the optimal solutionShi and Eberhart [25] improved the updatingmechanism byintroducing an inertia weight coefficient which can bedynamically adjusted to balance the quality of solution andconvergence velocity of the algorithm as shown in thefollowing formula

vki d ωkminus 1

vkminus1i d + c1r1 pbesti d minus x

kminus1i d1113872 1113873

+ c2r2 gbestd minus xkminus1i d1113872 1113873

(21)

where ωkminus 1 [(cmax minus ckminus1)cmax]lowast (ωmax minus ωmin) + ωminwhich is the inertia weight coefficient ωmax and ωmin denotethe maximum and minimum values of the inertia weightcoefficient respectively and cmax represents the maximumnumber of iterations

(e PSO algorithm proposed in this study is based on theupdating mechanism proposed by Shi and Eberhart (1998)

Wharf

Time

ekdep

eidep

ejdep

ekarr

ejarr

eiarr

pi pj pk

li

lj

lk

Vessel i Vessel k

Vessel jhi

hk

(μj θj)

(μk θk)

(μi θi)

Figure 4 Time-wharf section of the DLSM example


52 Encoding Assuming that n vessels are waiting to bescheduled within the planning period n random numbersdenoted as τi i isin 1 n are randomly generated in therange of 0 and 10 where each random number correspondsto the vessel with the same index

Sort those generated random numbers in descendingorder and then allocate the corresponding vessels to berthpositions one after another ie the greater the randomnumber τi is the earlier vessel i is allocated to a berthposition Ties are broken by selecting the vessel with smallestindex

For a better understanding here illustrated in Figure 5 isthe encoding process with an example of 5 vessels where asolution with the corresponding berthing order of the vesselsas 5-3-1-4-2 is obtained

53Decoding (e decoding process which is applied in thisstudy to construct the berth schedule corresponding to agiven solution obtained by the proposed PSO algorithmconsists of three steps as follows

(i) Step 1 initialization of the berthing schedule(e initial berth schedule can be generated byarranging each vessel one after another in the orderdefined by the solution to its minimum-cost berthingposition It is worth mentioning that althoughplacing vessels to their pre-defined minimum-costberthing positions can avoid additional operationcosts it is hardly possible for berth operators toarrange all the vessels to their minimum-costberthing positions without overlapping any of themat a busy port In consequence there is a good chancethat the berth schedule obtained at this step is in-feasible due to the overlaps and therefore actionshave to be taken to detect and resolve possibleoverlaps

(ii) Step 2 detection of overlapsConsidering that the overlap between two vesselstakes place if and only if both berthing periods andspaces of these two vessels are partly overlapped theoverlap between two vessels j and k (j k isin V) can bedetected by verifying constraints (22)ndash(25) in Fig-ure 6 an example of three vessels with overlap de-tected between two vessels j and k is shown

μk lt μj + lj k j isin V (22)

θk lt θj + hj k j isin V (23)

μj lt μk + lk k j isin V (24)

θj lt θk + hk k j isin V (25)

(iii) Step 3 overlaps resolvingOnce overlaps are detected the current berthschedule is not yet feasible and thus actions must betaken to remove those overlaps (e procedure

resolving overlaps between two vessels j andk (j k isin V) is as follows

Step 31 removing overlap detected between twovesselsIn this study the overlap detected between twovessels is eliminated by fixing one vessel andmoving the other one towards all possible direc-tions until no overlap is observed between themHere shown in Figure 7 is an example with twooverlapped vessels which are represented withsolid line rectangles Let vessel j be fixed and vesselk can be moved towards four possible directions toeliminate the overlap (i) left (in condition that theleftmost end of vessel k does not exceed the left endof the wharf) (ii) right (in condition that therightmost end of vessel k does not exceed the rightend of the wharf) (iii) up (to delay its berthingtime) and (iv) outside (in condition that theDLSM constraints are satisfied) (e possiblepositions of vessel k after performing thesemovements are mentioned with dashed linerectangles and the rectangle corresponding to theoutside movement is shaded on this time-wharfsectionUpon further analysis of the four movementsmentioned above it can be observed that onlymovement (iii) can result in a feasible solutionbecause (1) movement (i) is not available becausethere is not enough space on the left (dashedrectangle exceeds the left boundary of the wharf)(2) movement (ii) introduces an overlap betweenvessel k and vessel i (3) movement (iv) is notavailable as well because the constraints related tothe DLSM mode as described in Section 4 cannotbe satisfied Nevertheless the feasibility of thesolution obtained by (ii) can be improved bytaking into account the relationship between thevessel being moved ie vessel k and the nearbyvessels that may be overlapped by the newly placedvessel k eg vessel i in the example shown inFigure 7Step 32 improving the feasibility of berthingschedule by taking into account the nearby vesselshaving overlaps with certain moved vessel

Step 1 Generate a random number for each vessel

Step 2 sort τi in descending order

Vessel 1 2 3 4 5

02 01 04 02 08τi

5 3 1 4 2

08 04 02 02 01

Vessel

τi

Figure 5 Schema of the encoding process


Since it is possible to introduce new overlapsbetween the vessel being moved and some of thenearby ships the relationship of all vessels thatmay have overlaps with the newly placed vesselmust be considered to avoid introducing newoverlapsFor a better understanding let us continue the il-lustration with the example mentioned in step 31Since moving vessel k towards right may introducean overlap between the vessels k and i the move-ments of vessel k around vessel i are also consideredto generate possible feasible berthing schedules Asshown in Figure 8 three new berthing schedules canobtained by finding the optimal position of placingvessel k adjacent to vessel i in condition that it doesnot overlap with any other vessels It is worth notingthat the berthing schedule corresponding to theoptimal position above vessel i is not shown inFigure 8 because that berthing schedule can bedominated by at least the one with vessel k on lowerleft ie the berthing schedule corresponding tomovement (i) shown in Figure 8Step 33 accepting the best feasible berthingscheduleCompare all of the possible feasible berthingschedules generated by the adjustments described

in steps 31 and 32 and accept the best one ie thefeasible berthing schedule with the smallest ob-jective value as the one that corresponds to thegiven solution obtained by the proposed PSOalgorithm

54 General Procedure of the Proposed PSO Algorithm(e general procedure of the proposed PSO algorithm is asfollows

(i) Step 1 set up the parameters of the PSO algorithmsuch as the number of particles and the value ofinertia weight coefficient

(ii) Step 2 initialize the position and velocity in al-lowable ranges for each particle and set iterationk 1

(iii) Step 3 calculate the fitness value which is equal tothe objective value of the proposed model for eachparticle

(iv) Step 4 set the local-best value and global-best valuefor each particle where the former equals theparticlersquos current position and the latter the po-sition of the best particle

(v) Step 5 update the velocity and the position foreach particle

(vi) Step 6 update the fitness value for each particle

Time

ljlk

hj Vessel j

Vessel k

li

Vessel i hi

hk(μj θj)

(μk θk)

(μi θi)

Wharfpj pk pi

eiarrejarr

ekarr

Figure 6 An example of three vessels obtained at step 1 with overlap detected

Vessel kVessel jVessel k

Vessel k

Vessel i

Vessel k

Time

Wharf

1 2

34

Figure 7 Illustration of possible movements made to remove theoverlap between two vessels in time-wharf section

Time

Vessel j

Vessel kVessel i

Vessel k

Vessel k

Vessel k

Wharf

2

3

1

Figure 8 Possible movements of vessel k around the nearby vesseli


(vii) Step 7 compare the current fitness value of eachparticle with the local-best one If the currentfitness value of a particle is better update the local-best position of this particle otherwise it remainsunchanged

(viii) Step 8 find out the particle with the best fitnessfunction from the current swam If the current bestfitness value is better than that of the recordedglobal-best one replace the global-best positionwith the position of the current best particleotherwise the global-best one remains unchanged

(xi) Step 9 if the number of iteration k attains thepredefined threshold the PSO algorithm termi-nates and reports the recorded global-best particleas the final solution otherwise set k k+ 1 andreturn to step 3

6 Experimental Results

61 Experimental Settings In this study instances of dif-ferent scales are randomly generated with the method in-troduced by Park and Kim [15] (e length of wharf is set as1200 meters (e planning horizon T is set as 120 time unitswhere the time unit is one hour

(e cost coefficients c1i and c2i are set as 2 and 10 re-spectively as proposed by Meisel and Bierwirth [26] Inorder to ensure that most of the vessels can leave the portbefore their preplanned departure time the value of thepreplanned departure time of a vessel is determined byadding 10 to 20 times of the corresponding operation timeto its preplanned arrival time ie ede p

i earri + hi lowast q (i isin V)

and q is a decimal randomly generated between 10 and 20as proposed by Park and Kim [15] (e generation of theother parameters is detailed in Table 1

(e numerical experiments are programmed in C(VS2017) on a PC with 23GHz Intel Core i5 CPU and 4GBRAM and CPLEX 125 is applied as the programming solverfor small-size instances Both programming solver and theproposed PSO algorithm are set to terminate within 3 hours(10800 s)

62 Comparison between Different Mooring Modes First ofall experiments are conducted to compare between twodifferent mooring modes ie DLSM mode and SLSM modeby considering both objective values and execution time forsmall-size instances ie the instances with up to 25 vessels

As shown in Table 2 it can be observed that the optimalsolutions for both modes can be obtained by CPLEX solverwithin 5 seconds for the instances with no more than 15vessels As for SLSM mode the execution time used to solveinstances with the SLSM mode by the CPLEX solver (asshown in column ldquoCPU1rdquo) increases dramatically when thenumber of vessels is beyond 20 When the DLSM mode isapplied solution for instances with up to 20 vessels can beobtained within 10 seconds and the instances with 25 vesselscan still be obtained within 30 minutes (as shown in columnldquoCPU2rdquo) As shown in column ldquoDiff_CPUrdquo the differentrate of the execution time (Diff_CPU1 (CPU2minusCPU1)

CPU1 lowast 100) varies from minus7916 to minus9887 for theinstances with 20 and 25 vessels and it is reasonable toconclude that the application DLSM mode can greatlyimprove the work efficiency of port operators

With regard to the objective values it can be observedthat the DLSM mode obviously dominates the SLSM modebecause the objective values of solutions with the DLSMmode (shown in column ldquoOBJ2rdquo) are at least as good asthose with the SLSM mode (shown in column ldquoOBJ1rdquo)According to the difference rates shown in column ldquoDif-f_Obj1rdquo (Diff_Obj1 (OBJ2 minusOBJ1)OBJ1 lowast 100) op-eration costs can be reduced in average of 2035 and themaximum reduction rate reaches 3714

To sum up it can be concluded that DLSM mode canhelp the port operators in not only improving their workefficiency but also reducing overall operation costs

It should also be mentioned that the optimal solutionscannot be obtained by CPLEX solver within 3 hours for theinstances with more than 25 vessels for neither of these twomodes (erefore we can conclude that CPLEX solver isonly effective for solving small-scale problems regardless ofwhether DLSM is applied and thus it is necessary to developefficient heuristics to obtain good quality solution withinreasonable execution time for large-scale instances so as tocope with the real requirements of the huge terminal con-tainers such as Yangshan port

63 Comparison between Different Methodologies As men-tioned before the CPLEX solver is just capable of solvingBAP models for small-scale instances with both SLSM andDLSMmodes though much more vessels must be scheduledduring even 120 hours (us in this study a PSO algorithmhas been proposed to obtain good quality solutions withinreasonable execution time for large-scale instances

As shown in Table 3 when comparing the solutionsobtained by CPLEX solver and the proposed PSO algorithmfor instances with DLSM modes we can observe that bothCPLEX solver and the proposed PSO algorithm can get thefinal solution very quickly for the instances within 20vessels As for instances with more vessels the CPLEXsolver becomes more and more inefficient and cannotobtain optimal solutions within three hours for instanceswith beyond 30 vessels though PSO can still get finalsolution within several minutes

With regard to objective values the proposed PSO canobtain optimal solutions for the instances with 8 vessels andmost of the instances with 10 vessels and even one instancewith 20 vessels near-optimal solutions can be obtained forthe rest of the instances with 10 vessels and most of the caseswith 15 vessels and even most of the cases with 25 vessels

Table 1 Parameters used in the experiments

Parameter Distribution type Rangeearr

i Uniform distribution U(1 96)

hi Uniform distribution U(10 24)

pi Uniform distribution U(1 1200)

li Uniform distribution U(150 350)


with quite small difference rate which can be illustratedin column ldquoDiff_Obj2rdquo (Diff_Obj2 (OBJ3 minusOBJ2)OBJ2 lowast 100) It hints that the proposed PSO algorithmis also possible to get solutions of good quality for large-scale instances though further studies should be made totest the condition of such performance

Since it is observed in Table 3 that the gap between so-lutions obtained by the CPLEX solver and the PSO algorithmwith the DLSM mode is relatively significant for some of theinstances a further comparison is made between the resultsobtained by PSO with DLSMmode and the optimal solutionsobtained by the CPLEX solver with the SLSM mode

As shown in Table 4 solutions obtained by the PSOalgorithm with DLSM mode are better than the optimalsolutions obtained by the CPLEX solver with the SLSMmode and the former can save up to 3581 of the cost(Diff_Obj3 (OBJ3minusOBJ1)OBJ1 lowast 100) among all theinstances tested in this study

Considering that hundreds of vessels should be operatedevery day at huge container terminals the proposed PSOwillbe much more practical than CPLEX for supporting thedecision-making of the port operators to not only improvetheir working efficiency but also reduce operation costsrelated to berth scheduling operations

Table 2 Comparison between DLSM and SLSM modes for small-scale instances

InstancesSLSM DLSM

Diff_Obj1 () Diff_CPU1 ()OBJ1 CPU1 (s) OBJ2 CPU2 (s)

8-1 300 02 252 02 minus1600 0008-2 324 03 304 02 minus617 minus33338-3 488 03 488 04 000 333310-1 430 03 430 05 000 666710-2 656 04 656 03 000 minus250010-3 754 06 474 05 minus3714 minus166715-1 2642 44 2020 23 minus2354 minus477315-2 804 22 544 23 minus3234 45515-3 940 05 940 11 000 1200020-1 2816 7332 1842 83 minus3459 minus988720-2 3574 2052 2604 77 minus2714 minus962520-3 1932 1164 1232 36 minus3623 minus969125-1 4532 3630 3384 6687 minus2533 minus815825-2 6886 72452 4616 15101 minus3297 minus791625-3 4440 72801 2936 14210 minus3387 minus8048

Average minus2035 minus2876

Table 3 Comparison between the performance of CPLEX and PSOalgorithm for solving problems with the DLSM mode

InstancesCPLEX PSO

Diff_Obj2()OBJ2 CPU2

(s) OBJ3 CPU3(s)

8-1 252 02 252 82 0008-2 304 02 304 44 0008-3 488 04 488 94 00010-1 430 05 430 101 00010-2 656 03 656 80 00010-3 474 05 484 162 21115-1 2020 23 2052 407 15815-2 544 23 574 413 55115-3 940 11 1066 236 134020-1 1842 83 2102 737 141220-2 2604 77 2974 739 142120-3 1232 36 1232 736 00025-1 3384 6687 3904 1033 153725-2 4616 15101 4854 1228 51625-3 2936 14210 3022 1113 29330-1 Cannot gt3 h 7566 3263 mdash30-2 obtain gt3 h 6820 2511 mdash

30-3 theoptimal gt3 h 8248 2114 mdash

35-1 solution 7898 2722 mdash35-2 12006 2731 mdash35-3 12760 2795 mdash40-1 15800 3541 mdash40-2 15640 5652 mdash40-3 20558 4742 mdash45-1 26150 6852 mdash45-2 35374 7446 mdash45-3 31726 6442 mdash

Table 4 Comparison between optimal solutions with SLSM modeand solutions obtained by PSO with the DLSM mode

InstancesCPLEX-SLSM PSO-DLSM

Diff_Obj3 ()OBJ1 CPU1 (s) OBJ3 CPU3 (s)

8-1 300 02 252 82 minus16008-2 324 03 304 44 minus6178-3 488 03 488 94 00010-1 430 03 430 101 00010-2 656 04 656 80 00010-3 754 06 484 162 minus358115-1 2642 44 2052 407 minus223315-2 804 22 574 413 minus286115-3 940 05 1066 236 134020-1 2816 7332 2102 737 minus253620-2 3574 2052 2974 739 minus167920-3 1932 1164 1232 736 minus362325-1 4532 3630 3904 1033 minus138625-2 6886 72452 4854 1228 minus295125-3 4440 72801 3022 1113 minus3194Average minus1661


7 Conclusions and Perspectives

(e study aims at minimizing the total operation cost of thecontinuous berth scheduling problem by taking into accountthe Double-Line Shipping Mooring (DLSM) mode whereboth the additional operation cost for vessels not moored attheir minimum-cost berthing position and penalty costrelated to vessels not being able to leave before its pre-planned departure time are considered

(e problem is firstly formulated as a mixed integerprogramming model and solved by the CPLEX solver forsmall-scale instances As for larger size instances that cannotbe optimally solved by CPLEX solver a PSO algorithm isproposed to obtain good quality solutions within reasonableexecution time

Numerical experiments are conducted to compare not onlythe efficiency between the traditional Single-Line ShippingMooring (SLSM) mode and the innovative DLSM mode butalso the performances between CPLEX solver and the pro-posed PSO algorithm It can be concluded with the experi-mental results that (1) DLSM mode outperforms the SLSMmode in reducing not only total operation cost but also exe-cution time (2) (e proposed PSO algorithm can generateoptimal or near-optimal solution for small-scale instances (3)(e proposed PSO algorithm is much more efficient than theCPLEX solver for large-scale instances which copes with therequirements of berthing management in Yangshan Deep-Water Port one of the busiest container terminals in the world

To sum up as the first research dedicated to BAP withDLSM mode this study can help not only in validating theadvantages of DLSM mode but also offering an efficientdecision support tool to berth operators in busy ports toimprove their working efficiency

Motivated by the results obtained in this study it isinteresting to keep improving the efficiency of the proposedalgorithm and to apply such method in the targeted port

Data Availability

All the experimental data can be generated with the rulesdescribed in the paper

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] Q Meng S Wang H Andersson and K (un ldquoContain-ership routing and scheduling in liner shipping overview andfuture research directionsrdquo Transportation Science vol 48no 2 pp 265ndash280 2014

[2] D Kizilay and D T Eliiyi ldquoA comprehensive review of quaycrane scheduling yard operations and integrations thereof incontainer terminalsrdquo Flexible Services and ManufacturingJournal 2020

[3] A Imai K I Nagaiwa and C W Tat ldquoEfficient planning ofberth allocation for container terminals in Asiardquo Journal ofAdvanced Transportation vol 31 no 1 pp 75ndash94 1997

[4] A Imai E Nishimura and S Papadimitriou ldquo(e dynamicberth allocation problem for a container portrdquo TransportationResearch Part B Methodological vol 35 no 4 pp 401ndash4172001

[5] V H Barros T S Costa A C M Oliveira andL A N Lorena ldquoModel and heuristic for berth allocation intidal bulk ports with stock level constraintsrdquo Computers ampIndustrial Engineering vol 60 no 4 pp 606ndash613 2011

[6] L Dai and L Tang ldquoBerth allocation with service priority forcontainer terminal of hub portrdquo in Proceedings of the 2008 4thInternational Conference on Wireless Communications Net-working and Mobile Computing pp 1ndash4 Logs Engineering ampManagement Dalian China October 2008

[7] T Qin Y Du and M Sha ldquoEvaluating the solution per-formance of IP and CP for berth allocation with time-varyingwater depthrdquo Transportation Research Part E Logistics andTransportation Review vol 87 pp 167ndash185 2016

[8] L Zhen Z Liang D Zhuge L H Lee and E P Chew ldquoDailyberth planning in a tidal port with channel flow controlrdquoTransportation Research Part B Methodological vol 106pp 193ndash217 2017

[9] K H Kim and K C Moon ldquoBerth scheduling by simulatedannealingrdquo Transportation Research Part B Methodologicalvol 37 no 6 pp 541ndash560 2003

[10] B C Jos M Harimanikandan C Rajendran and H ZieglerldquoMinimum cost berth allocation problem in maritime lo-gistics new mixed integer programming modelsrdquo IndianAcademy of SciencesSadhana vol 44 p 149 2019

[11] L Zhen H Hu W Wang X Shi and C Ma ldquoCranesscheduling in frame bridges based automated container ter-minalsrdquo Transportation Research Part C Emerging Technol-ogies vol 97 pp 369ndash384 2018

[12] E Lalla-Ruiz J L Gonzalez-Velarde B Melian-Batista andJ MMoreno-Vega ldquoBiased random key genetic algorithm forthe tactical berth allocation problemrdquo Applied Soft Com-puting vol 22 pp 60ndash76 2014

[13] E Nishimura A Imai and S Papadimitriou ldquoBerth alloca-tion planning in the public berth system by genetic algo-rithmsrdquo European Journal of Operational Research vol 131no 2 pp 282ndash292 2001

[14] S R Seyedalizadeh Ganji A Babazadeh and N ArabshahildquoAnalysis of the continuous berth allocation problem incontainer ports using a genetic algorithmrdquo Journal of MarineScience and Technology vol 15 no 4 pp 408ndash416 2010

[15] Y-M Park and K H Kim ldquoA scheduling method for berthand quay cranesrdquo OR Spectrum vol 25 no 1 pp 1ndash23 2003

[16] M A Dulebenets ldquoApplication of evolutionary computationfor berth scheduling at marine container terminals parametertuning versus parameter controlrdquo IEEE Transactions on In-telligent Transportation Systems vol 19 no 1 pp 25ndash37 2018

[17] M A Dulebenets ldquoAn adaptive island evolutionary algorithmfor the berth scheduling problemrdquo Memetic Computingvol 12 no 1 pp 51ndash72 2020

[18] M Kavoosi M A Dulebenets O Abioye et al ldquoBerthscheduling at marine container terminals a universal island-based metaheuristic approachrdquo Maritime Business Reviewvol 5 no 1 pp 30ndash66 2020

[19] R C Eberhart and J Kennedy ldquoA new optimizer usingparticle swarm theoryrdquo in Proceeding of the 6th InternationalSymposium on Micromachine and Human Science pp 39ndash43Nagoya Japan October 1995

[20] C-J Ting K-C Wu and H Chou ldquoParticle swarm opti-mization algorithm for the berth allocation problemrdquo ExpertSystems with Application vol 41 no 4 pp 1543ndash1550 2014


[21] L Zhen ldquoModeling of yard congestion and optimization ofyard template in container portsrdquo Transportation ResearchPart B Methodological vol 90 pp 83ndash104 2016

[22] P Guo W Cheng and Y Wang ldquoA modified generalizedextremal optimization algorithm for the quay crane sched-uling problem with interference constraintsrdquo EngineeringOptimization vol 46 pp 1411ndash1429 2014

[23] H-P Hsu and C-N Wang ldquoResources planning for con-tainer terminal in a maritime supply chain using multipleparticle swarms optimization (MPSO)rdquo Mathematics vol 8no 5 p 764 2020

[24] M Zhong Y Yang Y Zhou and O Postolache ldquoAdaptiveautotuning mathematical approaches for integrated optimi-zation of automated container terminalrdquo MathematicalProblems in Engineering vol 2019 Article ID 764167014 pages 2019

[25] Y Shi and R Eberhart ldquoA modified particle swarm opti-mizerrdquo in Proceedings of the IEEE world congress on Com-putational Intelligence pp 69ndash73 Anchorage AK USA 1998

[26] F Meisel and C Bierwirth ldquoHeuristics for the integration ofcrane productivity in the berth allocation problemrdquo Trans-portation Research Part E Logistics and Transportation Re-view vol 45 no 1 pp 196ndash209 2009


and cranes to operate two container ships at the same time areasonable way to improve the efficiency of berth operationDespite that the real efficiency of the port with DLSMmodedepends on how sophisticated the operators are according tothe investigation made at Yangshan port because berthscheduling with DLSM is much more complex than with theSLSM system In consequence it is essential to develop aneffective decision support system to help berth operatorsimprove the quality and efficiency of daily schedules with theDLSM mode

(is study aims at minimizing the additional operationcosts for the vessels not placed at their ideal berthing po-sition and penalties for the ships not being able to leavebefore their preplanned departure time for a continuousberth allocation problems (BAP) model with the DLSMmode(emain contributions of this study are as follows (1)as the first study on berth scheduling with the DLSM modeit validates the contribution of DLSM model in comparisonwith the widely used SLSM mode (2) a mixed integerprogramming model is constructed for the continuous BAPwith DLSM mode enabling a benchmark for the futurestudies on such topic (3) a PSO algorithm is developed toobtain good quality DLSM berthing schedules within rea-sonable execution time offering the berth operators at busyports a powerful decision support tool to improve their workefficiency

(e reminder of this paper is structured as followsSection 2 is dedicated to the literature review of related work(e problem description is given in Section 3 In Section 4the targeted problem is formulated as a mixed integerprogramming model which can be solved by CPLEX solverand then the details of the proposed PSO algorithm aregiven in Section 5 Section 6 is dedicated to discussions aboutthe experimental results and this paper ends up withconclusions and perspectives

2 Related Work

According to the literature berth allocation problem (BAP)is regarded as the most important issue faced by themanagement of container terminals As the quality of berthschedule has a great influence on the improvement of portoperation efficiency a lot of researchers have been studyingBAPs and numerous results were published [2] Since Imaiet al [3] firstly addressed a static berth allocation problem(SBAP) and further developed a dynamic BAP model(DBAP for public berth system [4]) numerous studies havebeen published on BAP aiming to optimize the operationefficiencies by taking into account the various constraintsaffecting BAP [2] such as tidal influence [5] vessel servicepriority [6] time-varying water depth [7] and channel flowcontrol [8] As to our knowledge no study on BAPs with theDLSM mode is observed in the literature thus this studyaims to fill the research gaps in this field by taking intoaccount the DLSM mode when constructing berth sched-ules Furthermore this study is specifically dedicated tocontinuous BAP which copes with the real situation ofYangshan deep-water port that we have investigated and canalso be applied to many other huge container hubs

With regard to the methodologies applied to BAPs it canbe observed in the literature that the BAPs are normallyformulated as Mixed Integer Programming (MIP) modelswhich can be solved with commercial programming solversfor small-size problems [2ndash11] Since BAPs are NP-hard lotsof researchers have also taken much effort on developingnear-optimal solutions with efficient heuristics and meta-heuristics such as genetic algorithm and its variants [12ndash15]subgradient optimization method [15] simulated annealing[9] evolutionary algorithms [16 17] particle swarm opti-mization [10] and island-based meta-heuristic algorithm[18]

Particle swarm optimization (PSO) is an evolutionarycomputation technique firstly developed by Eberhart andKennedy [19] Since the particles used in PSO algorithm areonly updated by internal velocity and fewer parametersshould be tuned the principles of PSO can be easily un-derstood and widely adapted to various specific applicationsAccording to the literatures PSO algorithms have beenproven quite efficient in dealing with berth allocation [20]yard allocation [21] quay crane scheduling [10 22] andmany other planning and scheduling problems [23 24] inconsequence a PSO algorithm is also proposed in this studyto develop efficient berth schedules by taking into accountDLSM

3 Problem Description

31 Assumptions In general the berth operators are incharge of arranging each vessel arriving at the port to asuitable berth position according to the availability of thewharf resources and respecting the preplanned arrival anddeparture of the vessel According to the practices observedat the targeted container terminal a set of assumptions aredefined as follows

(1) Each vessel arrives at the port on the preplannedarrival time

(2) If a vessel cannot leave the port before the pre-planned departure time the port has to incur apenalty

(3) (e coordinate corresponding to the leftmost end ofthe vessel is used to represent its berthing positionusing the leftmost boundary of the wharf as thecoordinate reference point

(4) Each vessel has a predefined minimum-cost berthingposition which is determined according to the goodsthat will be loadedunloaded at the port and if avessel cannot be berthed at its ideal berthing posi-tion additional operation cost will occur

(5) DLSM mode is applied ie at most two vessels canbe simultaneously moored at the same berth position

(6) When two vessels are moored in double-line thelength of the inner-side ship cannot be longer thanthe vessel moored outside

(7) When two vessels are moored in double-line theinner-side one must be berthed earlier and leave laterthan the outside one



321 Parameters



de p


























c1i μi minus pi

11138681113868111386811138681113868111386811138681113868 + c2i θi + hi minus e

de pi1113872 1113873

+1113882 1113883 (1)











c1i α+i + αminus

i( 1113857 + c2iβ+i1113864 1113865 (2)

subject to





1113944jisinV














i β+i βminus



eiarr


Ready forleaving

Arrive at berth

Arrive atanchorage

Vessel i

hi

θi

1 2


Wharf

Side

Vessel j

Vessel iVessel k

Time


Outside

WharfInside

Side


Vessel j

Vessel i Vessel klk

lj

liμi










vki d v



kminus1i d1113872 1113873 (19)

xki d x

kminus1i d + v

kminus1i d (20)







vki d ωkminus 1


kminus1i d1113872 1113873


(21)



Wharf

Time

ekdep

eidep

ejdep

ekarr

ejarr

eiarr

pi pj pk

li

lj

lk

Vessel i Vessel k

Vessel jhi

hk

(μj θj)

(μk θk)

(μi θi)


















Vessel 1 2 3 4 5

02 01 04 02 08τi

5 3 1 4 2

08 04 02 02 01

Vessel

τi












Time

ljlk

hj Vessel j

Vessel k

li

Vessel i hi

hk(μj θj)

(μk θk)

(μi θi)

Wharfpj pk pi

eiarrejarr

ekarr



Vessel k

Vessel i

Vessel k

Time

Wharf

1 2

34


Time

Vessel j

Vessel kVessel i

Vessel k

Vessel k

Vessel k

Wharf

2

3

1

































InstancesSLSM DLSM





InstancesCPLEX PSO


(s) OBJ3 CPU3(s)















Data Availability




References






























321 Parameters



de p


























c1i μi minus pi

11138681113868111386811138681113868111386811138681113868 + c2i θi + hi minus e

de pi1113872 1113873

+1113882 1113883 (1)











c1i α+i + αminus

i( 1113857 + c2iβ+i1113864 1113865 (2)

subject to





1113944jisinV














i β+i βminus



eiarr


Ready forleaving

Arrive at berth

Arrive atanchorage

Vessel i

hi

θi

1 2


Wharf

Side

Vessel j

Vessel iVessel k

Time


Outside

WharfInside

Side


Vessel j

Vessel i Vessel klk

lj

liμi










vki d v



kminus1i d1113872 1113873 (19)

xki d x

kminus1i d + v

kminus1i d (20)







vki d ωkminus 1


kminus1i d1113872 1113873


(21)



Wharf

Time

ekdep

eidep

ejdep

ekarr

ejarr

eiarr

pi pj pk

li

lj

lk

Vessel i Vessel k

Vessel jhi

hk

(μj θj)

(μk θk)

(μi θi)


















Vessel 1 2 3 4 5

02 01 04 02 08τi

5 3 1 4 2

08 04 02 02 01

Vessel

τi












Time

ljlk

hj Vessel j

Vessel k

li

Vessel i hi

hk(μj θj)

(μk θk)

(μi θi)

Wharfpj pk pi

eiarrejarr

ekarr



Vessel k

Vessel i

Vessel k

Time

Wharf

1 2

34


Time

Vessel j

Vessel kVessel i

Vessel k

Vessel k

Vessel k

Wharf

2

3

1

































InstancesSLSM DLSM





InstancesCPLEX PSO


(s) OBJ3 CPU3(s)















Data Availability




References






































c1i μi minus pi

11138681113868111386811138681113868111386811138681113868 + c2i θi + hi minus e

de pi1113872 1113873

+1113882 1113883 (1)











c1i α+i + αminus

i( 1113857 + c2iβ+i1113864 1113865 (2)

subject to





1113944jisinV














i β+i βminus



eiarr


Ready forleaving

Arrive at berth

Arrive atanchorage

Vessel i

hi

θi

1 2


Wharf

Side

Vessel j

Vessel iVessel k

Time


Outside

WharfInside

Side


Vessel j

Vessel i Vessel klk

lj

liμi










vki d v



kminus1i d1113872 1113873 (19)

xki d x

kminus1i d + v

kminus1i d (20)







vki d ωkminus 1


kminus1i d1113872 1113873


(21)



Wharf

Time

ekdep

eidep

ejdep

ekarr

ejarr

eiarr

pi pj pk

li

lj

lk

Vessel i Vessel k

Vessel jhi

hk

(μj θj)

(μk θk)

(μi θi)


















Vessel 1 2 3 4 5

02 01 04 02 08τi

5 3 1 4 2

08 04 02 02 01

Vessel

τi












Time

ljlk

hj Vessel j

Vessel k

li

Vessel i hi

hk(μj θj)

(μk θk)

(μi θi)

Wharfpj pk pi

eiarrejarr

ekarr



Vessel k

Vessel i

Vessel k

Time

Wharf

1 2

34


Time

Vessel j

Vessel kVessel i

Vessel k

Vessel k

Vessel k

Wharf

2

3

1

































InstancesSLSM DLSM





InstancesCPLEX PSO


(s) OBJ3 CPU3(s)















Data Availability




References




































vki d v



kminus1i d1113872 1113873 (19)

xki d x

kminus1i d + v

kminus1i d (20)







vki d ωkminus 1


kminus1i d1113872 1113873


(21)



Wharf

Time

ekdep

eidep

ejdep

ekarr

ejarr

eiarr

pi pj pk

li

lj

lk

Vessel i Vessel k

Vessel jhi

hk

(μj θj)

(μk θk)

(μi θi)


















Vessel 1 2 3 4 5

02 01 04 02 08τi

5 3 1 4 2

08 04 02 02 01

Vessel

τi












Time

ljlk

hj Vessel j

Vessel k

li

Vessel i hi

hk(μj θj)

(μk θk)

(μi θi)

Wharfpj pk pi

eiarrejarr

ekarr



Vessel k

Vessel i

Vessel k

Time

Wharf

1 2

34


Time

Vessel j

Vessel kVessel i

Vessel k

Vessel k

Vessel k

Wharf

2

3

1

































InstancesSLSM DLSM





InstancesCPLEX PSO


(s) OBJ3 CPU3(s)















Data Availability




References












































Vessel 1 2 3 4 5

02 01 04 02 08τi

5 3 1 4 2

08 04 02 02 01

Vessel

τi












Time

ljlk

hj Vessel j

Vessel k

li

Vessel i hi

hk(μj θj)

(μk θk)

(μi θi)

Wharfpj pk pi

eiarrejarr

ekarr



Vessel k

Vessel i

Vessel k

Time

Wharf

1 2

34


Time

Vessel j

Vessel kVessel i

Vessel k

Vessel k

Vessel k

Wharf

2

3

1

































InstancesSLSM DLSM





InstancesCPLEX PSO


(s) OBJ3 CPU3(s)















Data Availability




References






































Time

ljlk

hj Vessel j

Vessel k

li

Vessel i hi

hk(μj θj)

(μk θk)

(μi θi)

Wharfpj pk pi

eiarrejarr

ekarr



Vessel k

Vessel i

Vessel k

Time

Wharf

1 2

34


Time

Vessel j

Vessel kVessel i

Vessel k

Vessel k

Vessel k

Wharf

2

3

1

































InstancesSLSM DLSM





InstancesCPLEX PSO


(s) OBJ3 CPU3(s)















Data Availability




References



























































InstancesSLSM DLSM





InstancesCPLEX PSO


(s) OBJ3 CPU3(s)















Data Availability




References



































Data Availability




References





























optimization of continuous berth scheduling by taking into

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