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Research ArticleSimulation-Based Optimization for Storage Allocation Problemof Outbound Containers in Automated Container Terminals
Ning Zhao1 Mengjue Xia1 Chao Mi2 Zhicheng Bian3 and Jian Jin34
1Logistics Engineering College Shanghai Maritime University 1550 Haigang Avenue Shanghai 201306 China2Container Supply Chain Technology Engineering Research Center Shanghai Maritime University 1550 Haigang AvenueShanghai 201306 China3Logistics Research Center Shanghai Maritime University 1550 Haigang Avenue Shanghai 201306 China4Taicang Port SIPG ZHENGHE Container Terminal Co Ltd 8 North Circular Road Taicang Port Area Jiangsu 215438 China
Correspondence should be addressed to Chao Mi chaomishmtueducn
Received 11 June 2015 Revised 25 August 2015 Accepted 2 September 2015
Academic Editor Alessandro Gasparetto
Copyright copy 2015 Ning Zhao et al This 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
Storage allocation of outbound containers is a key factor of the performance of container handling system in automated containerterminals Improper storage plans of outbound containers make QC waiting inevitable hence the vessel handling time will belengthened A simulation-based optimization method is proposed in this paper for the storage allocation problem of outboundcontainers in automated container terminals (SAPOBA) A simulation model is built up by Timed-Colored-Petri-Net (TCPN)used to evaluate the QC waiting time of storage plans Two optimization approaches based on Particle SwarmOptimization (PSO)and Genetic Algorithm (GA) are proposed to form the complete simulation-based optimization method Effectiveness of thismethod is verified by experiment as the comparison of the two optimization approaches
1 Introduction
Automated container terminals (ACT) are a category ofcontainer terminals with unmanned equipment which detecttheir surroundings on their own and execute the workingactions in the right moment [1 2] In such terminal con-tainers are loaded onto or unloaded from vessels at quaysideby Quay Cranes (QC) with double trolley stacked into orretrieved out of blocks in storage yard by Automated StackingCranes (ASC) and transported between quayside and storageyard by Automatic Guided Vehicles (AGV) The QC ASCand AGV all operated with no manpower make up thecontainer handling system of ACT with a typical layout asshown in Figure 1 In this figure QC are represented bytwo crossing rectangles ASC are represented by rectangleswith two edges popping out and AGV are represented bysmall rectangles Large rectangles are for blocks strips ofstorage spaces laid perpendicular to the shoreline Two ASCare bound to one block one for the container stacking andretrieving at the seaside of the block and the other for thoseat the land side Grey rectangles at both ends of the blocks areIO points in which containers could be stored temporarily
Outbound containers are containers brought from inlandarea and are to be loaded onto vessels Prior to vesselarrival information of the outbound containers is sent to theterminal in EDI including container numbers and stowagepositions on the vessel In the following days these containersare brought in by container trucks one after another in arandom order On the arrival of every outbound containera block is allocated to it and the landside ASC of that block isto stack the container into the block Within vessel handlingperiod outbound containers are retrieved from block to theseaside IO point by seaside ASC picked up and transportedto quayside byAGV and loaded onto the vessel byQC placedin their stowage positions according to EDI information
Average QC efficiency which means the average numberof containers handled per hour per QC is a main measure-ment for the performance of container handling system ofACT while the upper bound of thismeasurement depends onQC capacity Higher average QC efficiency leads to shortercontainer handling time hence vessels could depart earlierand finish their voyages in fewer days The efficiency of asingle QC reaches the maximum when it concentrates on
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 548762 14 pageshttpdxdoiorg1011552015548762
2 Mathematical Problems in Engineering
Shoreline
QC
AGV
Block
IO points
ASC
Figure 1 A schematic diagram of an ACT
container handling with no time wasted in other actionssuch as moving itself or waiting for an overdue AGV Conse-quently to optimize the performance of container handlingsystem of ACT a key factor is to prevent QC from waitinghence the average QC efficiency could be maximized
Unreasonable distribution of outbound containersamong blocks is a fundamental reason for QC waiting Forevery outbound container stacked in some block the QCand ASC to handle it could be predetermined the QC isdetermined since the stowage position of the container onthe vessel is already known while there is only one ASC ina block retrieving containers to the seaside IO point Inhandling containers a QC works continuously on conditionthat there is always some loaded AGV at its foot everytime the trolley moves back empty to the quayside Owingto the fact that capacity of ASC is commonly lower thanthat of QC the condition for continuous QC working iskept by receiving containers from multiple ASC Howeverthis condition could be failed in case that outboundcontainers of one QC are all stacked in the same blockand QC waiting is unavoidable Still time of QC waitingcould be even longer in case that the ASC of that block isretrieving outbound containers for another QC at the sametime
This paper presents a research on the storage allocationproblem for outbound containers in ACT (SAPOBA) whichoutputs a storage plan allocating storage spaces to outboundcontainers before they arrive at the terminal It is intendedthat outbound containers are stacked following the storageplan resulting in a container distribution that leads tominimalQCwaiting hence the performance of the containerhandling system could be optimized The rest of the paperis organized as follows Section 2 presents a review of therelated works and an introduction to the methodology usedin this paper A Timed-Colored-Petri-Net (TCPN) model isproposed in Section 3 for simulation of container handlingsystem in ACT and two optimization approaches ParticleSwarm Optimization (PSO) and Genetic Algorithm (GA)are proposed in Section 4 Experiments are conducted inSection 5 to verify the effectiveness of the simulation-based
optimization method and to compare the two optimizationapproaches A conclusion is driven in the last section
2 Related Works and Methodology
21 Related Works SAPOBA has received hardly any atten-tion for the past few years In the field of ACT existingresearch is mainly on problems of inbound containers whichpass through container terminals in a reversed directionSome works are on block selection problem in which everyinbound container is assigned to a block to when they areunloaded from vessel [3 4] Some others are on containerstacking problem in which every inbound container isassigned to a stacking position in a block [5ndash8] Differentfrom outbound containers that are already stacked in someblock in container handling inbound containers could beplaced in either block in the storage yard hence the candidateASC to handle an inbound container is not uniqueThereforethese research results are not applicable to SAPOBA
As a counterpart problem of SAPOBA in another kind ofcontainer terminals inwhich cranes and vehicles are operatedby human labor storage allocation problem of outbound con-tainers inmanual container terminals (SAPOBM) is a hotspotin recent years [9] The idea of SAPOBM is quite similar tothat of SAPOBA With storage space properly assigned tooutbound containers before they arrive at the terminal cycletime of storage yard operations could be reduced Howbeitin manual container terminals yard cranes are not boundto blocks and multiple yard cranes are allowed to work inthe same block hence the SAPOBM is a bit different fromSAPOBA Moreover a general defect of current researchon this problem is that no clear description could be givenof the impact of outbound containersrsquo storage plan on theperformance of container handling system Existing researchpropose mathematical models for their problem in whichobjective functions are constructed for optimal storage planThe aim of these functions could be minimizing total traveldistance of vehicles [10ndash19] maximizing the balance level ofworkload among blocks [11 14ndash17 20ndash27] or maximizing theutilization of storage space [13 15 19 24 26] Since none of
Mathematical Problems in Engineering 3
C
A B
P1T1 P2
P3
T2 P4
Figure 2 A simple Petri Net example
thesemodels is tominimize the QCwaiting time it could notbe guaranteed that the optimal storage plans of these modelswill lead to best performances of container handling systemsin the terminals
It is not an easy work to describe the numerical relation-ship between QC waiting time and storage plan of outboundcontainers using mathematical models while discrete eventsimulation (simulation hereinafter) is a suitable tool toachieve this purpose In addition to the fact that QC waitingis a result of many factors related to QC AGV and ASCactivities AGV are dispatched in real-time and the AGVto transport each container is not determined before vesselhandling starts For the two reasons above evaluation ofstorage plans by mathematical modeling in SAPOBA is quitea difficult task In contrast simulation is a process-basedmethod which could go through the working process of thesystem to bemodeledUsing thismethodQCAGV andASCin the container handling system are treated as individualobjects and the working process could be described as statechanges in events In this way QCwaiting could be expressedby time lags of some events and AGV dispatching couldbe executed following dispatching rules implemented in thesimulation model
In consideration of the facts mentioned above this paperis to solve the SAPOBA using simulation-based optimizationmethod This method is hybrid of simulation and optimiza-tion in which simulation is used to evaluate candidates gen-erated during the optimization procedurewhile optimizationis used to search for a best solution for the problem Theonly difference between simulation-based optimization andpure optimization is the evaluation method of candidates Insimulation-based optimization candidates are evaluated onthe basis of simulation results while in pure optimizationthese evaluations are done by math expressions Just recentlysimulation-based optimizationmethod has been successfullyused to optimize ambulance fleet allocation and base stationlocations [28] to determine the best design and controlfactors of a paint shop production line in an automotivecompany [29] and to decide the best vehicle schedules forhousekeeping in a container transshipment terminal [30] Tothe extent of our knowledge this method has not been usedin ACT
22 Methodology Container handling systems in ACT arefull of concurrency and asynchrony Concurrency is a con-cept meaning that multiple independent processes in thissystem may go on simultaneously In the container handlingsystem consisting of multiple QC ASC and AGV any two
objects of them could operate concurrently as long as they arenot handling or transporting the same container Asynchronymeans that processes of different objects upon the sameactivity may occur in different times For example whentransferring a container from anASC to anAGV theASC justmove to the IOpoint andput the container down there whiletheAGV just pick the container up once the container and theAGV itself are both at the IO point In this example the ASCand AGV are for the same container (activity) however theydo not have to be at the IO point at the same time
Flowchart is a traditional formalism to describe theprocesses of a simulation model in concept yet it is notsuitable for simulation modeling of the container handlingsystem in ACT A flowchart describes a process by dividingit into subprocesses and connecting them with certain orderWhen a subprocess ends the next subprocess according toorder starts immediately Thus flowchart could only be usedto describe sequential motions of one object as processesbut not to express the concurrency and asynchrony amongthese processes andmake up a model for the whole containerhandling system
Known for the applicability to express asynchrony andconcurrency Petri Net is a modeling formalism for dis-tributed parallel systems [31] Basic Petri Nets consist ofplaces transitions and directed arcs signified by circlesrectangles and arrows respectively An arc may run froma place to a transaction or in a reversed direction form atransaction to a place In the former case the place is calledthe input place of the transaction while in the latter case theplace is called the output place A number of tokens signifiedby dots may be located in some places and move to someother places following regulations of the Petri Net Supposethat there is a transaction and some input and output placesconnected to it In case that there are enough tokens locatedin the input places of a transaction this transaction firesconsuming a required number of tokens in the input placesand creating new tokens in the output places When used tobuild up a simulation model places in a Petri Net indicatestates of an object while transactions indicate events of thesystem
Petri Net is a formalism which is able to express theconcurrency and asynchrony of a system in nature Figure 2is a simple example of Petri Net consisting of five places (119875
1
1198752 1198753 1198754 and 119875
5) two transactions (119879
1and 119879
2) and five arcs
Three token A B and C are now located in 1198751 1198752 and 119875
3
respectively In this figure places 1198751and 119875
2and transaction
1198791are in the same process while places 119875
3and 119875
4are in two
processes each TokensA andB are independent of each other
4 Mathematical Problems in Engineering
QC
B
IO pointA
Figure 3 Illustration on grids and AGV routing
in the process between 1198751and 119875
2 hence the processes of the
two tokens could go on concurrently Tokens B and C are intwo asynchronous processes since it is not required that theseprocesses should start or end in the same time Howeverthe following process of them (119875
4) will not start till the two
processes both end and the two tokens are in the input placesof transaction 119879
2
Colored Petri Net (CPN) [30] and Timed Petri Net(TPN) [32] are two extensions of Petri Net In a CPNtokens have properties and data values attached to themcalled token color after which distinction between tokens ismade possible [33ndash35] In a TPN transactions are no longerfinished instantaneously On firing of a transaction requiredtokens in the input places are consumed immediately whiletokens in the output places are generated some time later[36 37] In simulating the container handling system in ACTtoken colors could be used to distinguish different QC ASCAGV and containers while transaction times could be usedto indicate the time length of equipment motions
The possible contributions of this paper are listed asfollows
(1) A simulation-based optimizationmethod is proposedto solve the SAPOBA which outputs a storage plan ofoutbound containers that leads to a best performanceof the container handling system in ACT
(2) TCPN is used to build up the simulation modelwhich establishes directly the relationship betweenstorage plans of outbound containers and QCwaitingtime of container handling system
(3) Two optimization methods PSO and GA are com-pared when used in a simulation-based optimizationfor optimal solution
3 TCPN Model for Simulation
31 Model Assumptions Assumptions of the model are listedbelow
(1) Only outbound containers are considered in thismodel
(2) Only 40 ft container are considered in this model20 ft containers could be converted into 40 ft ones inpairs
(3) The capacity of IO points as buffers between ASCand AGV is fixed and is the same No such bufferexists between QC and AGV
(4) Moving range of AGV could be divided into two-dimensional grids of equal square zones Positions ofAGV could be represented by coordinate of the gridwhich they are in as positions of QC and IO points
(5) Capacity of each grid is large enough to hold all theAGV at the same time and traffic congestions arenot considered in this model All AGV move in aconstant speed and they always spend the same timein crossing a current grid
(6) AGV routes are decided step by step following rulesWhen travelling to a destination grid at first an AGVtries to run into a next grid nearer the destination gridin a direction parallel to the shoreline In case that nosuch grid exists the AGV runs into a next grid nearerto the destination grid in a direction perpendicular tothe shoreline
Assumptions (4) to (6) could be illustrated in Figure 3which shows a two-dimensional grid of 6 times 4 square zonesAGV could run into a neighboring grid crossing a dash linewhile full lines are unbridgeable There are 6 IO points and2 QC in these grids Two AGV A and B are traveling to theirdestinations Routes of the two AGV are shown as directedfold lines intersecting at the same grid Following the routesthe two AGV will run into the same grid simultaneouslyhowever their travelling times will never be lengthened
(7) Outbound containers are allowed to be retrieved inbatches Containers in the same batch get the sameearliest time for retrieval before which they are notallowed to be retrieved
(8) Travelling speeds of ASC and AGV are treated asconstants in this model while QC are assumed to stayin a fixed grid during handling process
(9) An AGV is always bound to some ASC In case thatan AGV gets no task at the moment it runs back tothe ASC it is bound to and waits there for a next task
(10) Storage spaces are allocated in slots (spaces in thesame bay and same row of a block) and outbound
Mathematical Problems in Engineering 5
containers are divided into stacks (a group of contain-ers that could be placed in one slot) On this basisstorage plans are treated as one-to-one matching ofslots and stacks
32 Constraints of Storage Plans Storage plan is an allocationof storage spaces to containers a one-to-one matching planin nature Definition and constraints of these plans could beexpressed by math formulas Notations used in the formulasare listed below
119868 total number of stacks indexed by 119894 or 1198941015840 1 le 119894 1198941015840 le119868119869 total number of blocks in the storage yard indexedby 119895 1 le 119895 le 119869119870 total number of bays in a block indexed by 119896 1 le119896 le 119870 The smaller the bay number is the closer thisbay is to the shoreline119862119895119896 number of free slots in the 119896th bay of block 119895
119909119894119895119896 if stack 119894 is allocated to the slot in the 119896th bay of
block 119895 then 119909119894119895119896= 1 otherwise 119909
119894119895119896= 0
A storage plan could be described by a set of variables119909119894119895119896
119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896
isin 1 2 119870
(1)
The constraints are shown in the equations below
sum
119894
119909119894119895119896le 119862119895119896 forall119895 119896 (2)
Constraint (3) means that the total number of stacksallocated to the same bay in the same block should neverexceeds the capacity of that bay
sum
119895119896
119909119894119895119896= 1 forall119894 (3)
Equation (4) means that each stack could be allocatedonly once Outbound containers are to be stacked on theirarrivals in locations as determined in the storage plan
33 Overall Model Notations used in the simulation modelare listed below
119876 total number of QC119860 total number of ASC119862 total number of containers119881 total number of AGV119861 capacity of an IO point119866 total number of zones in themoving range of AGV119875 total number of places indexed by 119901 In this model119875 = 20119879 total number of transactions indexed by 119905 In thismodel 119879 = 13
119867 total number of containers to be handled119889119901 a token in place 119901
119889119901119905 a token that is fired from place 119901 to transaction 119905
1198891015840
119905119901 a token that is fired out of transaction 119905 into place
119901CS color set of tokens an 8-tuple of integers as⟨119888119894119889 119887119894119889 119886119894119889 119902119894119889 119886119894119898 119901119900119904 119886119911 119902119911⟩cid attribute of tokens indicating the container num-berbid attribute of tokens indicating the batch numberaid attribute of tokens indicating the ASC numberqid attribute of tokens indicating the QC numberaim attribute of tokens indicating the destinationzone number of an AGVpos attribute of tokens indicating the current zonenumber of an AGVaz attribute of tokens indicating the zone number ofIO point of an ASC to which an AGV is bound AnAGV moves back to the zone az when it is freeqz attribute of tokens indicating the zone number ofQC joint point119871(1199111 1199112) travel distance between two zones written
as 1199111and 1199112
119872(119901 exp) number of tokens in places 119901 as definedby expressions exp in the bracket If there is noexpression in the brackets this natation means thetotal number of tokens in places 119901119873(1199111 1199112) the next zone of 119911
1when running to zone
1199112
120575 average QC waiting time as output of the model119862119873119902 total number of containers handled by QC 119902
indexed by 119899119905119904119902119899 start time of the 119899th handling of QC 119902 according
to simulation119905119890119902119899 end time of the 119899th handling of QC 119902 according
to simulation
The overall TCPN model is shown in Figure 4 Circlesin this figure are for places in which dashed circles indicateboundaries between different kinds of equipment As forplaces with limited capacity the maximal number of tokensallowed is marked at the upper left of the circle Meanings ofplaces in Figure 1 are listed in Table 1 Squares and rectanglesare for transactions the former fire instantaneously whilethe latter fire with some time Square brackets are for firingconditions of transactions In case that there is a squarebracket over or under a squarerectangle the transactionwill not fire till the conditions are met Arrows are for arcseach with the number of tokens to or from a transactionmarked Marks of arrows could be divided into two parts Forarrows from a place to a transaction the first part of the markindicates number of tokens consumed by the transaction and
6 Mathematical Problems in Engineering
VV
11
Q
Q
A
AA
V
V
A
V
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
P18
1998400d99840024 1998400d32
1998400d41 1998400d99840013 1998400d103
M(1 d1bid =
d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d1810
1998400d114
1998400d1612
1998400d1310
1998400d2010
1998400d129
1998400d1713
1998400d127
1998400d96
1998400d118
1998400d88
1998400d76
1998400d1911
1998400d99840035
1998400d9984001118
1998400d9984001019
1998400d9984001011
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d998400610 1998400d99840068
1998400d99840089
1998400d998400814
1998400d998400620
1998400d998400713
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65
1998400d54
1998400d99840057
1998400d998400412
T5
T4
P11
T10T11 P19
P9
P8
P17
P13
T8
P20
P15
P16
P12
P7
T6
T7
T13
T12
T9 P14
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A[d76aid =
d96aid]
[d88cid =d118 cid]
M(1 d1bid =
d41bid)998400d99840012
Figure 4 TCPN model for a handling system in ACT
the second part is written as ldquo119889119901119905rdquo with 119901 for the origin place
and 119905 for the destination transaction indicating every tokengoing through the arc The two parts are put together to be awhole mark with a back single quotation set between themMarks of arcs in an opposed direction are formed in a similarway while in these marks the tokens are written as ldquo1198891015840
119905119901rdquo for
distinctionMeanings of places in Figure 1 are listed in Table 1 while
the transactions and arcs are to be explained in partialmodels
In the beginning some colored tokens are scatteredinitially in some places in the TCPN model which is calledthe initial marking of the model as is shown in Table 2Tokens located in 119875
1are for containers to be handled Each
container gets a unique container number and the batchnumbers are known as well In addition the ASC and QC tohandle the containers are known as well (aid and qid) as thezonenumber ofQC joint point (qz)The token in119875
2is used to
allow container retrievals in batches and no color is assignedto it Tokens in 119875
10are for ASC hence the ASC number of
each token (aid) is known and unique as the zone number ofIO points (az) Tokens in 119875
9are used to indicate the residual
capacity of IO points hence tokens in this place all get anASC number (aid) and the amount of tokens that have thesame ASC number is fixed Tokens in 119875
18are for the QC
hence each of these tokens gets a unique QC number (qid)Tokens in 119875
11are for the AGV which are always located in
some zone (pos) and bound to some ASC (az) There is onlyone token in 119875
17with no color used to update the positions
of AGV periodically The TCPN model ends with a similarmarking as in the beginning yet no container is left in place1198751When simulation ends the total QC waiting time could
be calculated as sum of time intervals between the time a QC
finishes a current container handling and the time the QCstarts a next container handling As a result the average QCwaiting time as the output of the simulation runs and theoptimization objective of the SAPOBA could be calculatedfollowing (1) as follows
Min 120575 = Min 1119876sum
119902
(
119862119873119902
sum
119899=2
(119905119904119902119899minus 119905119890119902(119899minus1)
)) (4)
34 Partial Models The overall TCPN model could bedivided into three parts with those dashed circles in Figure 4as boundaries corresponding to ASC AGV and QC respec-tively Partial model of ASC is shown in Figure 5 Notice thatthe position of 119875
20is adjusted
There are 5 transactions in this partial model Transition1198791is to allow a new batch of outbound containers to be
retrieved by ASC and transaction 1198792is the cool-down of this
allowance In case that the cool-down ends and a token 119889 isplaced in 119875
4 all the tokens (containers) in 119875
1with the same
batch number (bid) as token119889 are fired (11988911) as token119889 itself
(11988941) At the same time one new tokens (1198891015840
12) are created
immediately into 1198752for every 119889
11 and a new token (1198891015840
13) is
also created to 1198753 On this creation transactions 119879
2are fired
creating a new token (119889101584024) into 119875
4after a fixed interval With
the help of 1198792 transaction 119879
1brings periodically batches of
containers that are allowed to be retrieved by ASC followingthe handling schedule in a real ACT
Transaction 1198793is fired when an ASC is ready for a
next container (11987510
not empty) and there is some containerallowed to be retrieved by it already (token exists in 119875
2with
the same aid) Considering that theremay bemultiple tokensin 1198752and they could go to different QC an ASC always
retrieve a container to the QC with minimum inventory
Mathematical Problems in Engineering 7
A
AA
A
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
1998400d99840024 1998400d32
1998400d411998400d99840013 1998400d103
M(1 d1bid = d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d76
1998400d99840035
1998400d998400610
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65 1998400d99840057T5 P7
T6
M(1 d1bid =
d41bid)998400d99840012
1998400d96
1998400d99840068
1998400d998400620P9
P20
P8
B times A
B times A
[d76aid = d96aid]
Figure 5 Partial model of ASC
Table 1 Meanings of places
Place Practical meaning
1198751
Outbound containers in blocks notallowed to be retrieved
1198752
Outbound containers in blocks allowedto be retrieved
1198753
A batch of containers that has beenallowed to be retrieved
1198754
Time to allow a new batch of containersto be retrieved
1198755
Information of containers that ASCs havedecided to retrieve but no AGV isallocated
1198756 Containers that ASCs start to retrieve1198757 Loaded ASCs at IO points1198758 Containers stacked at IO points1198759 Residual capacities of IO points11987510 ASCs ready for a next container11987511 Unloaded free AGVs
11987512
Unloaded AGVs allocated to somecontainer
11987513
Unloaded AGVs allocated to somecontainer at IO point
11987514 Loaded AGVs at IO point11987515 Loaded AGVs at the foot of QC11987516 Ready to update positions of AGVs11987517 Position of AGVs updated11987518 Unloaded QCs at landside11987519 Loaded QCs at landside
11987520
Containers that have been placed into IOpoint but not handled by QCs
as shown in the firing condition QC inventory means thenumber of containers which have been placed in the IOpoint but not handled by the QC equal to the number oftokens to the QC in 119875
20 This strategy is also used in [38]
in which it is believed that this strategy helps in preventingQC from waiting for AGV leading to a good productivity ofthe terminal On firing of this transaction a token (1198891015840
35) is
created into 1198755immediately as an AGV task to be allocated
Table 2 Initial marking of the TCPN
Place Number oftokens cid bid aid qid aim pos az qz
1198751 119862 radic radic radic radic mdash mdash mdash radic
1198754 1 mdash mdash mdash mdash mdash mdash mdash mdash11987510 119860 mdash mdash radic mdash mdash mdash radic mdash1198759 119861 times 119860 mdash mdash radic mdash mdash mdash mdash mdash11987518 119876 mdash mdash mdash radic mdash mdash mdash mdash11987511 119881 mdash mdash mdash mdash mdash radic radic mdash11987517 1 mdash mdash mdash mdash mdash mdash mdash mdash
Moreover another token (119889101584036) is also created into 119875
6at the
same time as the next container to be handledTransaction 119879
5is for the cycle time of ASC When
deciding to retrieve a container (1198756not empty) anASCmoves
to its stacking location pick it up and carry it to the IOpoint Time length of this process could be calculated sincethe stacking position of container and the moving speed ofASC are both knownOn the arrival of ASC to the IO point anew token is created to119875
7 Notice that since stacking positions
of containers are all different firing time of 1198795is not fixed
Transaction 1198796is to add a new container to the IO point
of some ASC In case that an ASC is loaded at the IO point(1198757not empty) and the IO point is not filled (at least one
token exists in 1198758with the same ASC number aid) this
transaction consumes a token in 1198757and1198758 respectively After
a time a token is created to 1198758as a container stacked in
IO point and another token is also created to 11987520
as QCinventory Firing time of 119879
6is fixed which equals the time
an ASC spends in putting a container down to the IO pointColor relations of tokens to and from the same transac-
tion in this partial model are listed in Table 3 Notice thattokens from or to the same place always get the same colorwhile the color of tokens created in a firing is not alwaysinherited from the consumed tokens Attributes not involvedin this partial model are excluded
Partial model of AGV is shown in Figure 6 Positions of1198758 1198759 11987513 and 119879
8are adjusted
There are in total 6 transactions in the partial model ofAGV Transaction 119879
4is to allocate a container task to an
AGV with no task In case that there is some tasks to be
8 Mathematical Problems in Engineering
VV
11
A V
V V
P5
1998400d114
1998400d1612
1998400d129
1998400d1713
1998400d127
1998400d118
1998400d88
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d99840089
1998400d9984008141998400d998400713
1998400d54 1998400d998400412T4
P11
P9
P8
P17
P13
T8
P15
P16
P12 T7
T12
T13
T9 P14
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A
[d88cid = d118 cid]
Figure 6 Partial model of AGV
Table 3 Color relations of tokens in partial model of ASCs
Token cid bid aid qid az qz1198891015840
1211988911cid mdash 119889
11aid 119889
11qid mdash 119889
11qz
1198891015840
13mdash 119889
41bid + 1 mdash mdash mdash mdash
1198891015840
24mdash 119889
32bid mdash mdash mdash mdash
1198891015840
3511988923cid mdash 119889
23aid 119889
23qid 119889
23az 119889
23qz
1198891015840
3611988923cid mdash 119889
23aid 119889
23qid mdash mdash
1198891015840
5711988965cid mdash 119889
65aid 119889
65qid mdash mdash
1198891015840
6811988976cid mdash 119889
76aid mdash mdash mdash
1198891015840
610mdash mdash 119889
76aid mdash mdash mdash
1198891015840
62011988976cid mdash 119889
76aid 119889
76qid mdash mdash
allocated (1198755not empty) and some AGV are with no tasks
(11987511not empty)119879
4fires to allocate a task to a nearest AGV by
consuming corresponding tokens in 1198755and 119875
11and creating
a new token into 11987512at once
Transaction 1198797is used to simulate the travel time of
an unloaded AGV with some task to the IO point of thecontainer Firing time of the transaction equals travel time ofAGV which could be determined since the current zone anddestination zone are both known for the AGV
Transaction 1198798is for the process in which an AGV picks
up a container at IO point In case that an AGV has reachedthe IO point (119875
13) and the container is placed there as well
(1198758) this transaction fires consuming the tokens in 119875
8and
11987513 and creating one new token in119875
9and11987514after a fixed time
interval respectively hence the AGV receives the container(11987514) and the number of containers that could be placed in
the IO point is added by 1 (1198758)
Transaction 1198799is used to simulate the travel time of a
loaded AGV to the foot of the destination QC Firing time ofthis transaction is calculated as in transaction 119879
7 after which
a new token is created into 11987515
Transactions 11987912and 119879
13are used to update the locations
of the AGVwith no tasks periodicallyThey work in a similarway as Transactions 119879
1and 119879
2 For all the AGV with no
task they are moved in a fixed interval to a neighboring zonenearer to the ASC which they are bound to if possible
Color relations of tokens in this partial model are listedin Table 4 Attributes not involved in this partial model areexcluded
Partial model of QC is shown in Figure 7 Positions of 11987511
are adjustedThere are only 2 transactions in this partial model
Transaction 11987910
is for the process that a QC picks up acontainer from an AGV In case that a QC is ready for anext container loading (119875
18not empty) and a loaded AGV
has reached the foot of that QC (token with the same QCnumber in 119875
15) this transaction fires consuming one token
in 11987518 11987515 and 119875
20 following firing conditions as shown in
Figure 7 In case that there are multiple loaded AGV at thefoot of the same QC the AGV carrying the container with aminimal container number will be consumed Firing time oftransaction 119879
10is fixed When the firing ends a new token is
created into 11987511
as an AGV with no task allocated to it andanother new token is created into 119875
19indicating that the QC
has received the containerTransaction 119879
11is used to simulate the time a QC trolley
spends in putting a container onto a vessel and turn backempty to the quayside Firing time of 119879
11is also fixed When
the firing ends a new token is created to 11987518 hence the QC
is ready for a next loadColor relations of tokens in this partial model are listed in
Table 5 Also attributes not involved in this partial model areexcluded
4 Optimization Approaches
The TCPN model proposed in the last section could beused as fitness functions of optimizationmethods evaluating
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
Shoreline
QC
AGV
Block
IO points
ASC
Figure 1 A schematic diagram of an ACT
container handling with no time wasted in other actionssuch as moving itself or waiting for an overdue AGV Conse-quently to optimize the performance of container handlingsystem of ACT a key factor is to prevent QC from waitinghence the average QC efficiency could be maximized
Unreasonable distribution of outbound containersamong blocks is a fundamental reason for QC waiting Forevery outbound container stacked in some block the QCand ASC to handle it could be predetermined the QC isdetermined since the stowage position of the container onthe vessel is already known while there is only one ASC ina block retrieving containers to the seaside IO point Inhandling containers a QC works continuously on conditionthat there is always some loaded AGV at its foot everytime the trolley moves back empty to the quayside Owingto the fact that capacity of ASC is commonly lower thanthat of QC the condition for continuous QC working iskept by receiving containers from multiple ASC Howeverthis condition could be failed in case that outboundcontainers of one QC are all stacked in the same blockand QC waiting is unavoidable Still time of QC waitingcould be even longer in case that the ASC of that block isretrieving outbound containers for another QC at the sametime
This paper presents a research on the storage allocationproblem for outbound containers in ACT (SAPOBA) whichoutputs a storage plan allocating storage spaces to outboundcontainers before they arrive at the terminal It is intendedthat outbound containers are stacked following the storageplan resulting in a container distribution that leads tominimalQCwaiting hence the performance of the containerhandling system could be optimized The rest of the paperis organized as follows Section 2 presents a review of therelated works and an introduction to the methodology usedin this paper A Timed-Colored-Petri-Net (TCPN) model isproposed in Section 3 for simulation of container handlingsystem in ACT and two optimization approaches ParticleSwarm Optimization (PSO) and Genetic Algorithm (GA)are proposed in Section 4 Experiments are conducted inSection 5 to verify the effectiveness of the simulation-based
optimization method and to compare the two optimizationapproaches A conclusion is driven in the last section
2 Related Works and Methodology
21 Related Works SAPOBA has received hardly any atten-tion for the past few years In the field of ACT existingresearch is mainly on problems of inbound containers whichpass through container terminals in a reversed directionSome works are on block selection problem in which everyinbound container is assigned to a block to when they areunloaded from vessel [3 4] Some others are on containerstacking problem in which every inbound container isassigned to a stacking position in a block [5ndash8] Differentfrom outbound containers that are already stacked in someblock in container handling inbound containers could beplaced in either block in the storage yard hence the candidateASC to handle an inbound container is not uniqueThereforethese research results are not applicable to SAPOBA
As a counterpart problem of SAPOBA in another kind ofcontainer terminals inwhich cranes and vehicles are operatedby human labor storage allocation problem of outbound con-tainers inmanual container terminals (SAPOBM) is a hotspotin recent years [9] The idea of SAPOBM is quite similar tothat of SAPOBA With storage space properly assigned tooutbound containers before they arrive at the terminal cycletime of storage yard operations could be reduced Howbeitin manual container terminals yard cranes are not boundto blocks and multiple yard cranes are allowed to work inthe same block hence the SAPOBM is a bit different fromSAPOBA Moreover a general defect of current researchon this problem is that no clear description could be givenof the impact of outbound containersrsquo storage plan on theperformance of container handling system Existing researchpropose mathematical models for their problem in whichobjective functions are constructed for optimal storage planThe aim of these functions could be minimizing total traveldistance of vehicles [10ndash19] maximizing the balance level ofworkload among blocks [11 14ndash17 20ndash27] or maximizing theutilization of storage space [13 15 19 24 26] Since none of
Mathematical Problems in Engineering 3
C
A B
P1T1 P2
P3
T2 P4
Figure 2 A simple Petri Net example
thesemodels is tominimize the QCwaiting time it could notbe guaranteed that the optimal storage plans of these modelswill lead to best performances of container handling systemsin the terminals
It is not an easy work to describe the numerical relation-ship between QC waiting time and storage plan of outboundcontainers using mathematical models while discrete eventsimulation (simulation hereinafter) is a suitable tool toachieve this purpose In addition to the fact that QC waitingis a result of many factors related to QC AGV and ASCactivities AGV are dispatched in real-time and the AGVto transport each container is not determined before vesselhandling starts For the two reasons above evaluation ofstorage plans by mathematical modeling in SAPOBA is quitea difficult task In contrast simulation is a process-basedmethod which could go through the working process of thesystem to bemodeledUsing thismethodQCAGV andASCin the container handling system are treated as individualobjects and the working process could be described as statechanges in events In this way QCwaiting could be expressedby time lags of some events and AGV dispatching couldbe executed following dispatching rules implemented in thesimulation model
In consideration of the facts mentioned above this paperis to solve the SAPOBA using simulation-based optimizationmethod This method is hybrid of simulation and optimiza-tion in which simulation is used to evaluate candidates gen-erated during the optimization procedurewhile optimizationis used to search for a best solution for the problem Theonly difference between simulation-based optimization andpure optimization is the evaluation method of candidates Insimulation-based optimization candidates are evaluated onthe basis of simulation results while in pure optimizationthese evaluations are done by math expressions Just recentlysimulation-based optimizationmethod has been successfullyused to optimize ambulance fleet allocation and base stationlocations [28] to determine the best design and controlfactors of a paint shop production line in an automotivecompany [29] and to decide the best vehicle schedules forhousekeeping in a container transshipment terminal [30] Tothe extent of our knowledge this method has not been usedin ACT
22 Methodology Container handling systems in ACT arefull of concurrency and asynchrony Concurrency is a con-cept meaning that multiple independent processes in thissystem may go on simultaneously In the container handlingsystem consisting of multiple QC ASC and AGV any two
objects of them could operate concurrently as long as they arenot handling or transporting the same container Asynchronymeans that processes of different objects upon the sameactivity may occur in different times For example whentransferring a container from anASC to anAGV theASC justmove to the IOpoint andput the container down there whiletheAGV just pick the container up once the container and theAGV itself are both at the IO point In this example the ASCand AGV are for the same container (activity) however theydo not have to be at the IO point at the same time
Flowchart is a traditional formalism to describe theprocesses of a simulation model in concept yet it is notsuitable for simulation modeling of the container handlingsystem in ACT A flowchart describes a process by dividingit into subprocesses and connecting them with certain orderWhen a subprocess ends the next subprocess according toorder starts immediately Thus flowchart could only be usedto describe sequential motions of one object as processesbut not to express the concurrency and asynchrony amongthese processes andmake up a model for the whole containerhandling system
Known for the applicability to express asynchrony andconcurrency Petri Net is a modeling formalism for dis-tributed parallel systems [31] Basic Petri Nets consist ofplaces transitions and directed arcs signified by circlesrectangles and arrows respectively An arc may run froma place to a transaction or in a reversed direction form atransaction to a place In the former case the place is calledthe input place of the transaction while in the latter case theplace is called the output place A number of tokens signifiedby dots may be located in some places and move to someother places following regulations of the Petri Net Supposethat there is a transaction and some input and output placesconnected to it In case that there are enough tokens locatedin the input places of a transaction this transaction firesconsuming a required number of tokens in the input placesand creating new tokens in the output places When used tobuild up a simulation model places in a Petri Net indicatestates of an object while transactions indicate events of thesystem
Petri Net is a formalism which is able to express theconcurrency and asynchrony of a system in nature Figure 2is a simple example of Petri Net consisting of five places (119875
1
1198752 1198753 1198754 and 119875
5) two transactions (119879
1and 119879
2) and five arcs
Three token A B and C are now located in 1198751 1198752 and 119875
3
respectively In this figure places 1198751and 119875
2and transaction
1198791are in the same process while places 119875
3and 119875
4are in two
processes each TokensA andB are independent of each other
4 Mathematical Problems in Engineering
QC
B
IO pointA
Figure 3 Illustration on grids and AGV routing
in the process between 1198751and 119875
2 hence the processes of the
two tokens could go on concurrently Tokens B and C are intwo asynchronous processes since it is not required that theseprocesses should start or end in the same time Howeverthe following process of them (119875
4) will not start till the two
processes both end and the two tokens are in the input placesof transaction 119879
2
Colored Petri Net (CPN) [30] and Timed Petri Net(TPN) [32] are two extensions of Petri Net In a CPNtokens have properties and data values attached to themcalled token color after which distinction between tokens ismade possible [33ndash35] In a TPN transactions are no longerfinished instantaneously On firing of a transaction requiredtokens in the input places are consumed immediately whiletokens in the output places are generated some time later[36 37] In simulating the container handling system in ACTtoken colors could be used to distinguish different QC ASCAGV and containers while transaction times could be usedto indicate the time length of equipment motions
The possible contributions of this paper are listed asfollows
(1) A simulation-based optimizationmethod is proposedto solve the SAPOBA which outputs a storage plan ofoutbound containers that leads to a best performanceof the container handling system in ACT
(2) TCPN is used to build up the simulation modelwhich establishes directly the relationship betweenstorage plans of outbound containers and QCwaitingtime of container handling system
(3) Two optimization methods PSO and GA are com-pared when used in a simulation-based optimizationfor optimal solution
3 TCPN Model for Simulation
31 Model Assumptions Assumptions of the model are listedbelow
(1) Only outbound containers are considered in thismodel
(2) Only 40 ft container are considered in this model20 ft containers could be converted into 40 ft ones inpairs
(3) The capacity of IO points as buffers between ASCand AGV is fixed and is the same No such bufferexists between QC and AGV
(4) Moving range of AGV could be divided into two-dimensional grids of equal square zones Positions ofAGV could be represented by coordinate of the gridwhich they are in as positions of QC and IO points
(5) Capacity of each grid is large enough to hold all theAGV at the same time and traffic congestions arenot considered in this model All AGV move in aconstant speed and they always spend the same timein crossing a current grid
(6) AGV routes are decided step by step following rulesWhen travelling to a destination grid at first an AGVtries to run into a next grid nearer the destination gridin a direction parallel to the shoreline In case that nosuch grid exists the AGV runs into a next grid nearerto the destination grid in a direction perpendicular tothe shoreline
Assumptions (4) to (6) could be illustrated in Figure 3which shows a two-dimensional grid of 6 times 4 square zonesAGV could run into a neighboring grid crossing a dash linewhile full lines are unbridgeable There are 6 IO points and2 QC in these grids Two AGV A and B are traveling to theirdestinations Routes of the two AGV are shown as directedfold lines intersecting at the same grid Following the routesthe two AGV will run into the same grid simultaneouslyhowever their travelling times will never be lengthened
(7) Outbound containers are allowed to be retrieved inbatches Containers in the same batch get the sameearliest time for retrieval before which they are notallowed to be retrieved
(8) Travelling speeds of ASC and AGV are treated asconstants in this model while QC are assumed to stayin a fixed grid during handling process
(9) An AGV is always bound to some ASC In case thatan AGV gets no task at the moment it runs back tothe ASC it is bound to and waits there for a next task
(10) Storage spaces are allocated in slots (spaces in thesame bay and same row of a block) and outbound
Mathematical Problems in Engineering 5
containers are divided into stacks (a group of contain-ers that could be placed in one slot) On this basisstorage plans are treated as one-to-one matching ofslots and stacks
32 Constraints of Storage Plans Storage plan is an allocationof storage spaces to containers a one-to-one matching planin nature Definition and constraints of these plans could beexpressed by math formulas Notations used in the formulasare listed below
119868 total number of stacks indexed by 119894 or 1198941015840 1 le 119894 1198941015840 le119868119869 total number of blocks in the storage yard indexedby 119895 1 le 119895 le 119869119870 total number of bays in a block indexed by 119896 1 le119896 le 119870 The smaller the bay number is the closer thisbay is to the shoreline119862119895119896 number of free slots in the 119896th bay of block 119895
119909119894119895119896 if stack 119894 is allocated to the slot in the 119896th bay of
block 119895 then 119909119894119895119896= 1 otherwise 119909
119894119895119896= 0
A storage plan could be described by a set of variables119909119894119895119896
119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896
isin 1 2 119870
(1)
The constraints are shown in the equations below
sum
119894
119909119894119895119896le 119862119895119896 forall119895 119896 (2)
Constraint (3) means that the total number of stacksallocated to the same bay in the same block should neverexceeds the capacity of that bay
sum
119895119896
119909119894119895119896= 1 forall119894 (3)
Equation (4) means that each stack could be allocatedonly once Outbound containers are to be stacked on theirarrivals in locations as determined in the storage plan
33 Overall Model Notations used in the simulation modelare listed below
119876 total number of QC119860 total number of ASC119862 total number of containers119881 total number of AGV119861 capacity of an IO point119866 total number of zones in themoving range of AGV119875 total number of places indexed by 119901 In this model119875 = 20119879 total number of transactions indexed by 119905 In thismodel 119879 = 13
119867 total number of containers to be handled119889119901 a token in place 119901
119889119901119905 a token that is fired from place 119901 to transaction 119905
1198891015840
119905119901 a token that is fired out of transaction 119905 into place
119901CS color set of tokens an 8-tuple of integers as⟨119888119894119889 119887119894119889 119886119894119889 119902119894119889 119886119894119898 119901119900119904 119886119911 119902119911⟩cid attribute of tokens indicating the container num-berbid attribute of tokens indicating the batch numberaid attribute of tokens indicating the ASC numberqid attribute of tokens indicating the QC numberaim attribute of tokens indicating the destinationzone number of an AGVpos attribute of tokens indicating the current zonenumber of an AGVaz attribute of tokens indicating the zone number ofIO point of an ASC to which an AGV is bound AnAGV moves back to the zone az when it is freeqz attribute of tokens indicating the zone number ofQC joint point119871(1199111 1199112) travel distance between two zones written
as 1199111and 1199112
119872(119901 exp) number of tokens in places 119901 as definedby expressions exp in the bracket If there is noexpression in the brackets this natation means thetotal number of tokens in places 119901119873(1199111 1199112) the next zone of 119911
1when running to zone
1199112
120575 average QC waiting time as output of the model119862119873119902 total number of containers handled by QC 119902
indexed by 119899119905119904119902119899 start time of the 119899th handling of QC 119902 according
to simulation119905119890119902119899 end time of the 119899th handling of QC 119902 according
to simulation
The overall TCPN model is shown in Figure 4 Circlesin this figure are for places in which dashed circles indicateboundaries between different kinds of equipment As forplaces with limited capacity the maximal number of tokensallowed is marked at the upper left of the circle Meanings ofplaces in Figure 1 are listed in Table 1 Squares and rectanglesare for transactions the former fire instantaneously whilethe latter fire with some time Square brackets are for firingconditions of transactions In case that there is a squarebracket over or under a squarerectangle the transactionwill not fire till the conditions are met Arrows are for arcseach with the number of tokens to or from a transactionmarked Marks of arrows could be divided into two parts Forarrows from a place to a transaction the first part of the markindicates number of tokens consumed by the transaction and
6 Mathematical Problems in Engineering
VV
11
Q
Q
A
AA
V
V
A
V
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
P18
1998400d99840024 1998400d32
1998400d41 1998400d99840013 1998400d103
M(1 d1bid =
d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d1810
1998400d114
1998400d1612
1998400d1310
1998400d2010
1998400d129
1998400d1713
1998400d127
1998400d96
1998400d118
1998400d88
1998400d76
1998400d1911
1998400d99840035
1998400d9984001118
1998400d9984001019
1998400d9984001011
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d998400610 1998400d99840068
1998400d99840089
1998400d998400814
1998400d998400620
1998400d998400713
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65
1998400d54
1998400d99840057
1998400d998400412
T5
T4
P11
T10T11 P19
P9
P8
P17
P13
T8
P20
P15
P16
P12
P7
T6
T7
T13
T12
T9 P14
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A[d76aid =
d96aid]
[d88cid =d118 cid]
M(1 d1bid =
d41bid)998400d99840012
Figure 4 TCPN model for a handling system in ACT
the second part is written as ldquo119889119901119905rdquo with 119901 for the origin place
and 119905 for the destination transaction indicating every tokengoing through the arc The two parts are put together to be awhole mark with a back single quotation set between themMarks of arcs in an opposed direction are formed in a similarway while in these marks the tokens are written as ldquo1198891015840
119905119901rdquo for
distinctionMeanings of places in Figure 1 are listed in Table 1 while
the transactions and arcs are to be explained in partialmodels
In the beginning some colored tokens are scatteredinitially in some places in the TCPN model which is calledthe initial marking of the model as is shown in Table 2Tokens located in 119875
1are for containers to be handled Each
container gets a unique container number and the batchnumbers are known as well In addition the ASC and QC tohandle the containers are known as well (aid and qid) as thezonenumber ofQC joint point (qz)The token in119875
2is used to
allow container retrievals in batches and no color is assignedto it Tokens in 119875
10are for ASC hence the ASC number of
each token (aid) is known and unique as the zone number ofIO points (az) Tokens in 119875
9are used to indicate the residual
capacity of IO points hence tokens in this place all get anASC number (aid) and the amount of tokens that have thesame ASC number is fixed Tokens in 119875
18are for the QC
hence each of these tokens gets a unique QC number (qid)Tokens in 119875
11are for the AGV which are always located in
some zone (pos) and bound to some ASC (az) There is onlyone token in 119875
17with no color used to update the positions
of AGV periodically The TCPN model ends with a similarmarking as in the beginning yet no container is left in place1198751When simulation ends the total QC waiting time could
be calculated as sum of time intervals between the time a QC
finishes a current container handling and the time the QCstarts a next container handling As a result the average QCwaiting time as the output of the simulation runs and theoptimization objective of the SAPOBA could be calculatedfollowing (1) as follows
Min 120575 = Min 1119876sum
119902
(
119862119873119902
sum
119899=2
(119905119904119902119899minus 119905119890119902(119899minus1)
)) (4)
34 Partial Models The overall TCPN model could bedivided into three parts with those dashed circles in Figure 4as boundaries corresponding to ASC AGV and QC respec-tively Partial model of ASC is shown in Figure 5 Notice thatthe position of 119875
20is adjusted
There are 5 transactions in this partial model Transition1198791is to allow a new batch of outbound containers to be
retrieved by ASC and transaction 1198792is the cool-down of this
allowance In case that the cool-down ends and a token 119889 isplaced in 119875
4 all the tokens (containers) in 119875
1with the same
batch number (bid) as token119889 are fired (11988911) as token119889 itself
(11988941) At the same time one new tokens (1198891015840
12) are created
immediately into 1198752for every 119889
11 and a new token (1198891015840
13) is
also created to 1198753 On this creation transactions 119879
2are fired
creating a new token (119889101584024) into 119875
4after a fixed interval With
the help of 1198792 transaction 119879
1brings periodically batches of
containers that are allowed to be retrieved by ASC followingthe handling schedule in a real ACT
Transaction 1198793is fired when an ASC is ready for a
next container (11987510
not empty) and there is some containerallowed to be retrieved by it already (token exists in 119875
2with
the same aid) Considering that theremay bemultiple tokensin 1198752and they could go to different QC an ASC always
retrieve a container to the QC with minimum inventory
Mathematical Problems in Engineering 7
A
AA
A
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
1998400d99840024 1998400d32
1998400d411998400d99840013 1998400d103
M(1 d1bid = d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d76
1998400d99840035
1998400d998400610
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65 1998400d99840057T5 P7
T6
M(1 d1bid =
d41bid)998400d99840012
1998400d96
1998400d99840068
1998400d998400620P9
P20
P8
B times A
B times A
[d76aid = d96aid]
Figure 5 Partial model of ASC
Table 1 Meanings of places
Place Practical meaning
1198751
Outbound containers in blocks notallowed to be retrieved
1198752
Outbound containers in blocks allowedto be retrieved
1198753
A batch of containers that has beenallowed to be retrieved
1198754
Time to allow a new batch of containersto be retrieved
1198755
Information of containers that ASCs havedecided to retrieve but no AGV isallocated
1198756 Containers that ASCs start to retrieve1198757 Loaded ASCs at IO points1198758 Containers stacked at IO points1198759 Residual capacities of IO points11987510 ASCs ready for a next container11987511 Unloaded free AGVs
11987512
Unloaded AGVs allocated to somecontainer
11987513
Unloaded AGVs allocated to somecontainer at IO point
11987514 Loaded AGVs at IO point11987515 Loaded AGVs at the foot of QC11987516 Ready to update positions of AGVs11987517 Position of AGVs updated11987518 Unloaded QCs at landside11987519 Loaded QCs at landside
11987520
Containers that have been placed into IOpoint but not handled by QCs
as shown in the firing condition QC inventory means thenumber of containers which have been placed in the IOpoint but not handled by the QC equal to the number oftokens to the QC in 119875
20 This strategy is also used in [38]
in which it is believed that this strategy helps in preventingQC from waiting for AGV leading to a good productivity ofthe terminal On firing of this transaction a token (1198891015840
35) is
created into 1198755immediately as an AGV task to be allocated
Table 2 Initial marking of the TCPN
Place Number oftokens cid bid aid qid aim pos az qz
1198751 119862 radic radic radic radic mdash mdash mdash radic
1198754 1 mdash mdash mdash mdash mdash mdash mdash mdash11987510 119860 mdash mdash radic mdash mdash mdash radic mdash1198759 119861 times 119860 mdash mdash radic mdash mdash mdash mdash mdash11987518 119876 mdash mdash mdash radic mdash mdash mdash mdash11987511 119881 mdash mdash mdash mdash mdash radic radic mdash11987517 1 mdash mdash mdash mdash mdash mdash mdash mdash
Moreover another token (119889101584036) is also created into 119875
6at the
same time as the next container to be handledTransaction 119879
5is for the cycle time of ASC When
deciding to retrieve a container (1198756not empty) anASCmoves
to its stacking location pick it up and carry it to the IOpoint Time length of this process could be calculated sincethe stacking position of container and the moving speed ofASC are both knownOn the arrival of ASC to the IO point anew token is created to119875
7 Notice that since stacking positions
of containers are all different firing time of 1198795is not fixed
Transaction 1198796is to add a new container to the IO point
of some ASC In case that an ASC is loaded at the IO point(1198757not empty) and the IO point is not filled (at least one
token exists in 1198758with the same ASC number aid) this
transaction consumes a token in 1198757and1198758 respectively After
a time a token is created to 1198758as a container stacked in
IO point and another token is also created to 11987520
as QCinventory Firing time of 119879
6is fixed which equals the time
an ASC spends in putting a container down to the IO pointColor relations of tokens to and from the same transac-
tion in this partial model are listed in Table 3 Notice thattokens from or to the same place always get the same colorwhile the color of tokens created in a firing is not alwaysinherited from the consumed tokens Attributes not involvedin this partial model are excluded
Partial model of AGV is shown in Figure 6 Positions of1198758 1198759 11987513 and 119879
8are adjusted
There are in total 6 transactions in the partial model ofAGV Transaction 119879
4is to allocate a container task to an
AGV with no task In case that there is some tasks to be
8 Mathematical Problems in Engineering
VV
11
A V
V V
P5
1998400d114
1998400d1612
1998400d129
1998400d1713
1998400d127
1998400d118
1998400d88
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d99840089
1998400d9984008141998400d998400713
1998400d54 1998400d998400412T4
P11
P9
P8
P17
P13
T8
P15
P16
P12 T7
T12
T13
T9 P14
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A
[d88cid = d118 cid]
Figure 6 Partial model of AGV
Table 3 Color relations of tokens in partial model of ASCs
Token cid bid aid qid az qz1198891015840
1211988911cid mdash 119889
11aid 119889
11qid mdash 119889
11qz
1198891015840
13mdash 119889
41bid + 1 mdash mdash mdash mdash
1198891015840
24mdash 119889
32bid mdash mdash mdash mdash
1198891015840
3511988923cid mdash 119889
23aid 119889
23qid 119889
23az 119889
23qz
1198891015840
3611988923cid mdash 119889
23aid 119889
23qid mdash mdash
1198891015840
5711988965cid mdash 119889
65aid 119889
65qid mdash mdash
1198891015840
6811988976cid mdash 119889
76aid mdash mdash mdash
1198891015840
610mdash mdash 119889
76aid mdash mdash mdash
1198891015840
62011988976cid mdash 119889
76aid 119889
76qid mdash mdash
allocated (1198755not empty) and some AGV are with no tasks
(11987511not empty)119879
4fires to allocate a task to a nearest AGV by
consuming corresponding tokens in 1198755and 119875
11and creating
a new token into 11987512at once
Transaction 1198797is used to simulate the travel time of
an unloaded AGV with some task to the IO point of thecontainer Firing time of the transaction equals travel time ofAGV which could be determined since the current zone anddestination zone are both known for the AGV
Transaction 1198798is for the process in which an AGV picks
up a container at IO point In case that an AGV has reachedthe IO point (119875
13) and the container is placed there as well
(1198758) this transaction fires consuming the tokens in 119875
8and
11987513 and creating one new token in119875
9and11987514after a fixed time
interval respectively hence the AGV receives the container(11987514) and the number of containers that could be placed in
the IO point is added by 1 (1198758)
Transaction 1198799is used to simulate the travel time of a
loaded AGV to the foot of the destination QC Firing time ofthis transaction is calculated as in transaction 119879
7 after which
a new token is created into 11987515
Transactions 11987912and 119879
13are used to update the locations
of the AGVwith no tasks periodicallyThey work in a similarway as Transactions 119879
1and 119879
2 For all the AGV with no
task they are moved in a fixed interval to a neighboring zonenearer to the ASC which they are bound to if possible
Color relations of tokens in this partial model are listedin Table 4 Attributes not involved in this partial model areexcluded
Partial model of QC is shown in Figure 7 Positions of 11987511
are adjustedThere are only 2 transactions in this partial model
Transaction 11987910
is for the process that a QC picks up acontainer from an AGV In case that a QC is ready for anext container loading (119875
18not empty) and a loaded AGV
has reached the foot of that QC (token with the same QCnumber in 119875
15) this transaction fires consuming one token
in 11987518 11987515 and 119875
20 following firing conditions as shown in
Figure 7 In case that there are multiple loaded AGV at thefoot of the same QC the AGV carrying the container with aminimal container number will be consumed Firing time oftransaction 119879
10is fixed When the firing ends a new token is
created into 11987511
as an AGV with no task allocated to it andanother new token is created into 119875
19indicating that the QC
has received the containerTransaction 119879
11is used to simulate the time a QC trolley
spends in putting a container onto a vessel and turn backempty to the quayside Firing time of 119879
11is also fixed When
the firing ends a new token is created to 11987518 hence the QC
is ready for a next loadColor relations of tokens in this partial model are listed in
Table 5 Also attributes not involved in this partial model areexcluded
4 Optimization Approaches
The TCPN model proposed in the last section could beused as fitness functions of optimizationmethods evaluating
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
C
A B
P1T1 P2
P3
T2 P4
Figure 2 A simple Petri Net example
thesemodels is tominimize the QCwaiting time it could notbe guaranteed that the optimal storage plans of these modelswill lead to best performances of container handling systemsin the terminals
It is not an easy work to describe the numerical relation-ship between QC waiting time and storage plan of outboundcontainers using mathematical models while discrete eventsimulation (simulation hereinafter) is a suitable tool toachieve this purpose In addition to the fact that QC waitingis a result of many factors related to QC AGV and ASCactivities AGV are dispatched in real-time and the AGVto transport each container is not determined before vesselhandling starts For the two reasons above evaluation ofstorage plans by mathematical modeling in SAPOBA is quitea difficult task In contrast simulation is a process-basedmethod which could go through the working process of thesystem to bemodeledUsing thismethodQCAGV andASCin the container handling system are treated as individualobjects and the working process could be described as statechanges in events In this way QCwaiting could be expressedby time lags of some events and AGV dispatching couldbe executed following dispatching rules implemented in thesimulation model
In consideration of the facts mentioned above this paperis to solve the SAPOBA using simulation-based optimizationmethod This method is hybrid of simulation and optimiza-tion in which simulation is used to evaluate candidates gen-erated during the optimization procedurewhile optimizationis used to search for a best solution for the problem Theonly difference between simulation-based optimization andpure optimization is the evaluation method of candidates Insimulation-based optimization candidates are evaluated onthe basis of simulation results while in pure optimizationthese evaluations are done by math expressions Just recentlysimulation-based optimizationmethod has been successfullyused to optimize ambulance fleet allocation and base stationlocations [28] to determine the best design and controlfactors of a paint shop production line in an automotivecompany [29] and to decide the best vehicle schedules forhousekeeping in a container transshipment terminal [30] Tothe extent of our knowledge this method has not been usedin ACT
22 Methodology Container handling systems in ACT arefull of concurrency and asynchrony Concurrency is a con-cept meaning that multiple independent processes in thissystem may go on simultaneously In the container handlingsystem consisting of multiple QC ASC and AGV any two
objects of them could operate concurrently as long as they arenot handling or transporting the same container Asynchronymeans that processes of different objects upon the sameactivity may occur in different times For example whentransferring a container from anASC to anAGV theASC justmove to the IOpoint andput the container down there whiletheAGV just pick the container up once the container and theAGV itself are both at the IO point In this example the ASCand AGV are for the same container (activity) however theydo not have to be at the IO point at the same time
Flowchart is a traditional formalism to describe theprocesses of a simulation model in concept yet it is notsuitable for simulation modeling of the container handlingsystem in ACT A flowchart describes a process by dividingit into subprocesses and connecting them with certain orderWhen a subprocess ends the next subprocess according toorder starts immediately Thus flowchart could only be usedto describe sequential motions of one object as processesbut not to express the concurrency and asynchrony amongthese processes andmake up a model for the whole containerhandling system
Known for the applicability to express asynchrony andconcurrency Petri Net is a modeling formalism for dis-tributed parallel systems [31] Basic Petri Nets consist ofplaces transitions and directed arcs signified by circlesrectangles and arrows respectively An arc may run froma place to a transaction or in a reversed direction form atransaction to a place In the former case the place is calledthe input place of the transaction while in the latter case theplace is called the output place A number of tokens signifiedby dots may be located in some places and move to someother places following regulations of the Petri Net Supposethat there is a transaction and some input and output placesconnected to it In case that there are enough tokens locatedin the input places of a transaction this transaction firesconsuming a required number of tokens in the input placesand creating new tokens in the output places When used tobuild up a simulation model places in a Petri Net indicatestates of an object while transactions indicate events of thesystem
Petri Net is a formalism which is able to express theconcurrency and asynchrony of a system in nature Figure 2is a simple example of Petri Net consisting of five places (119875
1
1198752 1198753 1198754 and 119875
5) two transactions (119879
1and 119879
2) and five arcs
Three token A B and C are now located in 1198751 1198752 and 119875
3
respectively In this figure places 1198751and 119875
2and transaction
1198791are in the same process while places 119875
3and 119875
4are in two
processes each TokensA andB are independent of each other
4 Mathematical Problems in Engineering
QC
B
IO pointA
Figure 3 Illustration on grids and AGV routing
in the process between 1198751and 119875
2 hence the processes of the
two tokens could go on concurrently Tokens B and C are intwo asynchronous processes since it is not required that theseprocesses should start or end in the same time Howeverthe following process of them (119875
4) will not start till the two
processes both end and the two tokens are in the input placesof transaction 119879
2
Colored Petri Net (CPN) [30] and Timed Petri Net(TPN) [32] are two extensions of Petri Net In a CPNtokens have properties and data values attached to themcalled token color after which distinction between tokens ismade possible [33ndash35] In a TPN transactions are no longerfinished instantaneously On firing of a transaction requiredtokens in the input places are consumed immediately whiletokens in the output places are generated some time later[36 37] In simulating the container handling system in ACTtoken colors could be used to distinguish different QC ASCAGV and containers while transaction times could be usedto indicate the time length of equipment motions
The possible contributions of this paper are listed asfollows
(1) A simulation-based optimizationmethod is proposedto solve the SAPOBA which outputs a storage plan ofoutbound containers that leads to a best performanceof the container handling system in ACT
(2) TCPN is used to build up the simulation modelwhich establishes directly the relationship betweenstorage plans of outbound containers and QCwaitingtime of container handling system
(3) Two optimization methods PSO and GA are com-pared when used in a simulation-based optimizationfor optimal solution
3 TCPN Model for Simulation
31 Model Assumptions Assumptions of the model are listedbelow
(1) Only outbound containers are considered in thismodel
(2) Only 40 ft container are considered in this model20 ft containers could be converted into 40 ft ones inpairs
(3) The capacity of IO points as buffers between ASCand AGV is fixed and is the same No such bufferexists between QC and AGV
(4) Moving range of AGV could be divided into two-dimensional grids of equal square zones Positions ofAGV could be represented by coordinate of the gridwhich they are in as positions of QC and IO points
(5) Capacity of each grid is large enough to hold all theAGV at the same time and traffic congestions arenot considered in this model All AGV move in aconstant speed and they always spend the same timein crossing a current grid
(6) AGV routes are decided step by step following rulesWhen travelling to a destination grid at first an AGVtries to run into a next grid nearer the destination gridin a direction parallel to the shoreline In case that nosuch grid exists the AGV runs into a next grid nearerto the destination grid in a direction perpendicular tothe shoreline
Assumptions (4) to (6) could be illustrated in Figure 3which shows a two-dimensional grid of 6 times 4 square zonesAGV could run into a neighboring grid crossing a dash linewhile full lines are unbridgeable There are 6 IO points and2 QC in these grids Two AGV A and B are traveling to theirdestinations Routes of the two AGV are shown as directedfold lines intersecting at the same grid Following the routesthe two AGV will run into the same grid simultaneouslyhowever their travelling times will never be lengthened
(7) Outbound containers are allowed to be retrieved inbatches Containers in the same batch get the sameearliest time for retrieval before which they are notallowed to be retrieved
(8) Travelling speeds of ASC and AGV are treated asconstants in this model while QC are assumed to stayin a fixed grid during handling process
(9) An AGV is always bound to some ASC In case thatan AGV gets no task at the moment it runs back tothe ASC it is bound to and waits there for a next task
(10) Storage spaces are allocated in slots (spaces in thesame bay and same row of a block) and outbound
Mathematical Problems in Engineering 5
containers are divided into stacks (a group of contain-ers that could be placed in one slot) On this basisstorage plans are treated as one-to-one matching ofslots and stacks
32 Constraints of Storage Plans Storage plan is an allocationof storage spaces to containers a one-to-one matching planin nature Definition and constraints of these plans could beexpressed by math formulas Notations used in the formulasare listed below
119868 total number of stacks indexed by 119894 or 1198941015840 1 le 119894 1198941015840 le119868119869 total number of blocks in the storage yard indexedby 119895 1 le 119895 le 119869119870 total number of bays in a block indexed by 119896 1 le119896 le 119870 The smaller the bay number is the closer thisbay is to the shoreline119862119895119896 number of free slots in the 119896th bay of block 119895
119909119894119895119896 if stack 119894 is allocated to the slot in the 119896th bay of
block 119895 then 119909119894119895119896= 1 otherwise 119909
119894119895119896= 0
A storage plan could be described by a set of variables119909119894119895119896
119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896
isin 1 2 119870
(1)
The constraints are shown in the equations below
sum
119894
119909119894119895119896le 119862119895119896 forall119895 119896 (2)
Constraint (3) means that the total number of stacksallocated to the same bay in the same block should neverexceeds the capacity of that bay
sum
119895119896
119909119894119895119896= 1 forall119894 (3)
Equation (4) means that each stack could be allocatedonly once Outbound containers are to be stacked on theirarrivals in locations as determined in the storage plan
33 Overall Model Notations used in the simulation modelare listed below
119876 total number of QC119860 total number of ASC119862 total number of containers119881 total number of AGV119861 capacity of an IO point119866 total number of zones in themoving range of AGV119875 total number of places indexed by 119901 In this model119875 = 20119879 total number of transactions indexed by 119905 In thismodel 119879 = 13
119867 total number of containers to be handled119889119901 a token in place 119901
119889119901119905 a token that is fired from place 119901 to transaction 119905
1198891015840
119905119901 a token that is fired out of transaction 119905 into place
119901CS color set of tokens an 8-tuple of integers as⟨119888119894119889 119887119894119889 119886119894119889 119902119894119889 119886119894119898 119901119900119904 119886119911 119902119911⟩cid attribute of tokens indicating the container num-berbid attribute of tokens indicating the batch numberaid attribute of tokens indicating the ASC numberqid attribute of tokens indicating the QC numberaim attribute of tokens indicating the destinationzone number of an AGVpos attribute of tokens indicating the current zonenumber of an AGVaz attribute of tokens indicating the zone number ofIO point of an ASC to which an AGV is bound AnAGV moves back to the zone az when it is freeqz attribute of tokens indicating the zone number ofQC joint point119871(1199111 1199112) travel distance between two zones written
as 1199111and 1199112
119872(119901 exp) number of tokens in places 119901 as definedby expressions exp in the bracket If there is noexpression in the brackets this natation means thetotal number of tokens in places 119901119873(1199111 1199112) the next zone of 119911
1when running to zone
1199112
120575 average QC waiting time as output of the model119862119873119902 total number of containers handled by QC 119902
indexed by 119899119905119904119902119899 start time of the 119899th handling of QC 119902 according
to simulation119905119890119902119899 end time of the 119899th handling of QC 119902 according
to simulation
The overall TCPN model is shown in Figure 4 Circlesin this figure are for places in which dashed circles indicateboundaries between different kinds of equipment As forplaces with limited capacity the maximal number of tokensallowed is marked at the upper left of the circle Meanings ofplaces in Figure 1 are listed in Table 1 Squares and rectanglesare for transactions the former fire instantaneously whilethe latter fire with some time Square brackets are for firingconditions of transactions In case that there is a squarebracket over or under a squarerectangle the transactionwill not fire till the conditions are met Arrows are for arcseach with the number of tokens to or from a transactionmarked Marks of arrows could be divided into two parts Forarrows from a place to a transaction the first part of the markindicates number of tokens consumed by the transaction and
6 Mathematical Problems in Engineering
VV
11
Q
Q
A
AA
V
V
A
V
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
P18
1998400d99840024 1998400d32
1998400d41 1998400d99840013 1998400d103
M(1 d1bid =
d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d1810
1998400d114
1998400d1612
1998400d1310
1998400d2010
1998400d129
1998400d1713
1998400d127
1998400d96
1998400d118
1998400d88
1998400d76
1998400d1911
1998400d99840035
1998400d9984001118
1998400d9984001019
1998400d9984001011
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d998400610 1998400d99840068
1998400d99840089
1998400d998400814
1998400d998400620
1998400d998400713
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65
1998400d54
1998400d99840057
1998400d998400412
T5
T4
P11
T10T11 P19
P9
P8
P17
P13
T8
P20
P15
P16
P12
P7
T6
T7
T13
T12
T9 P14
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A[d76aid =
d96aid]
[d88cid =d118 cid]
M(1 d1bid =
d41bid)998400d99840012
Figure 4 TCPN model for a handling system in ACT
the second part is written as ldquo119889119901119905rdquo with 119901 for the origin place
and 119905 for the destination transaction indicating every tokengoing through the arc The two parts are put together to be awhole mark with a back single quotation set between themMarks of arcs in an opposed direction are formed in a similarway while in these marks the tokens are written as ldquo1198891015840
119905119901rdquo for
distinctionMeanings of places in Figure 1 are listed in Table 1 while
the transactions and arcs are to be explained in partialmodels
In the beginning some colored tokens are scatteredinitially in some places in the TCPN model which is calledthe initial marking of the model as is shown in Table 2Tokens located in 119875
1are for containers to be handled Each
container gets a unique container number and the batchnumbers are known as well In addition the ASC and QC tohandle the containers are known as well (aid and qid) as thezonenumber ofQC joint point (qz)The token in119875
2is used to
allow container retrievals in batches and no color is assignedto it Tokens in 119875
10are for ASC hence the ASC number of
each token (aid) is known and unique as the zone number ofIO points (az) Tokens in 119875
9are used to indicate the residual
capacity of IO points hence tokens in this place all get anASC number (aid) and the amount of tokens that have thesame ASC number is fixed Tokens in 119875
18are for the QC
hence each of these tokens gets a unique QC number (qid)Tokens in 119875
11are for the AGV which are always located in
some zone (pos) and bound to some ASC (az) There is onlyone token in 119875
17with no color used to update the positions
of AGV periodically The TCPN model ends with a similarmarking as in the beginning yet no container is left in place1198751When simulation ends the total QC waiting time could
be calculated as sum of time intervals between the time a QC
finishes a current container handling and the time the QCstarts a next container handling As a result the average QCwaiting time as the output of the simulation runs and theoptimization objective of the SAPOBA could be calculatedfollowing (1) as follows
Min 120575 = Min 1119876sum
119902
(
119862119873119902
sum
119899=2
(119905119904119902119899minus 119905119890119902(119899minus1)
)) (4)
34 Partial Models The overall TCPN model could bedivided into three parts with those dashed circles in Figure 4as boundaries corresponding to ASC AGV and QC respec-tively Partial model of ASC is shown in Figure 5 Notice thatthe position of 119875
20is adjusted
There are 5 transactions in this partial model Transition1198791is to allow a new batch of outbound containers to be
retrieved by ASC and transaction 1198792is the cool-down of this
allowance In case that the cool-down ends and a token 119889 isplaced in 119875
4 all the tokens (containers) in 119875
1with the same
batch number (bid) as token119889 are fired (11988911) as token119889 itself
(11988941) At the same time one new tokens (1198891015840
12) are created
immediately into 1198752for every 119889
11 and a new token (1198891015840
13) is
also created to 1198753 On this creation transactions 119879
2are fired
creating a new token (119889101584024) into 119875
4after a fixed interval With
the help of 1198792 transaction 119879
1brings periodically batches of
containers that are allowed to be retrieved by ASC followingthe handling schedule in a real ACT
Transaction 1198793is fired when an ASC is ready for a
next container (11987510
not empty) and there is some containerallowed to be retrieved by it already (token exists in 119875
2with
the same aid) Considering that theremay bemultiple tokensin 1198752and they could go to different QC an ASC always
retrieve a container to the QC with minimum inventory
Mathematical Problems in Engineering 7
A
AA
A
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
1998400d99840024 1998400d32
1998400d411998400d99840013 1998400d103
M(1 d1bid = d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d76
1998400d99840035
1998400d998400610
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65 1998400d99840057T5 P7
T6
M(1 d1bid =
d41bid)998400d99840012
1998400d96
1998400d99840068
1998400d998400620P9
P20
P8
B times A
B times A
[d76aid = d96aid]
Figure 5 Partial model of ASC
Table 1 Meanings of places
Place Practical meaning
1198751
Outbound containers in blocks notallowed to be retrieved
1198752
Outbound containers in blocks allowedto be retrieved
1198753
A batch of containers that has beenallowed to be retrieved
1198754
Time to allow a new batch of containersto be retrieved
1198755
Information of containers that ASCs havedecided to retrieve but no AGV isallocated
1198756 Containers that ASCs start to retrieve1198757 Loaded ASCs at IO points1198758 Containers stacked at IO points1198759 Residual capacities of IO points11987510 ASCs ready for a next container11987511 Unloaded free AGVs
11987512
Unloaded AGVs allocated to somecontainer
11987513
Unloaded AGVs allocated to somecontainer at IO point
11987514 Loaded AGVs at IO point11987515 Loaded AGVs at the foot of QC11987516 Ready to update positions of AGVs11987517 Position of AGVs updated11987518 Unloaded QCs at landside11987519 Loaded QCs at landside
11987520
Containers that have been placed into IOpoint but not handled by QCs
as shown in the firing condition QC inventory means thenumber of containers which have been placed in the IOpoint but not handled by the QC equal to the number oftokens to the QC in 119875
20 This strategy is also used in [38]
in which it is believed that this strategy helps in preventingQC from waiting for AGV leading to a good productivity ofthe terminal On firing of this transaction a token (1198891015840
35) is
created into 1198755immediately as an AGV task to be allocated
Table 2 Initial marking of the TCPN
Place Number oftokens cid bid aid qid aim pos az qz
1198751 119862 radic radic radic radic mdash mdash mdash radic
1198754 1 mdash mdash mdash mdash mdash mdash mdash mdash11987510 119860 mdash mdash radic mdash mdash mdash radic mdash1198759 119861 times 119860 mdash mdash radic mdash mdash mdash mdash mdash11987518 119876 mdash mdash mdash radic mdash mdash mdash mdash11987511 119881 mdash mdash mdash mdash mdash radic radic mdash11987517 1 mdash mdash mdash mdash mdash mdash mdash mdash
Moreover another token (119889101584036) is also created into 119875
6at the
same time as the next container to be handledTransaction 119879
5is for the cycle time of ASC When
deciding to retrieve a container (1198756not empty) anASCmoves
to its stacking location pick it up and carry it to the IOpoint Time length of this process could be calculated sincethe stacking position of container and the moving speed ofASC are both knownOn the arrival of ASC to the IO point anew token is created to119875
7 Notice that since stacking positions
of containers are all different firing time of 1198795is not fixed
Transaction 1198796is to add a new container to the IO point
of some ASC In case that an ASC is loaded at the IO point(1198757not empty) and the IO point is not filled (at least one
token exists in 1198758with the same ASC number aid) this
transaction consumes a token in 1198757and1198758 respectively After
a time a token is created to 1198758as a container stacked in
IO point and another token is also created to 11987520
as QCinventory Firing time of 119879
6is fixed which equals the time
an ASC spends in putting a container down to the IO pointColor relations of tokens to and from the same transac-
tion in this partial model are listed in Table 3 Notice thattokens from or to the same place always get the same colorwhile the color of tokens created in a firing is not alwaysinherited from the consumed tokens Attributes not involvedin this partial model are excluded
Partial model of AGV is shown in Figure 6 Positions of1198758 1198759 11987513 and 119879
8are adjusted
There are in total 6 transactions in the partial model ofAGV Transaction 119879
4is to allocate a container task to an
AGV with no task In case that there is some tasks to be
8 Mathematical Problems in Engineering
VV
11
A V
V V
P5
1998400d114
1998400d1612
1998400d129
1998400d1713
1998400d127
1998400d118
1998400d88
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d99840089
1998400d9984008141998400d998400713
1998400d54 1998400d998400412T4
P11
P9
P8
P17
P13
T8
P15
P16
P12 T7
T12
T13
T9 P14
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A
[d88cid = d118 cid]
Figure 6 Partial model of AGV
Table 3 Color relations of tokens in partial model of ASCs
Token cid bid aid qid az qz1198891015840
1211988911cid mdash 119889
11aid 119889
11qid mdash 119889
11qz
1198891015840
13mdash 119889
41bid + 1 mdash mdash mdash mdash
1198891015840
24mdash 119889
32bid mdash mdash mdash mdash
1198891015840
3511988923cid mdash 119889
23aid 119889
23qid 119889
23az 119889
23qz
1198891015840
3611988923cid mdash 119889
23aid 119889
23qid mdash mdash
1198891015840
5711988965cid mdash 119889
65aid 119889
65qid mdash mdash
1198891015840
6811988976cid mdash 119889
76aid mdash mdash mdash
1198891015840
610mdash mdash 119889
76aid mdash mdash mdash
1198891015840
62011988976cid mdash 119889
76aid 119889
76qid mdash mdash
allocated (1198755not empty) and some AGV are with no tasks
(11987511not empty)119879
4fires to allocate a task to a nearest AGV by
consuming corresponding tokens in 1198755and 119875
11and creating
a new token into 11987512at once
Transaction 1198797is used to simulate the travel time of
an unloaded AGV with some task to the IO point of thecontainer Firing time of the transaction equals travel time ofAGV which could be determined since the current zone anddestination zone are both known for the AGV
Transaction 1198798is for the process in which an AGV picks
up a container at IO point In case that an AGV has reachedthe IO point (119875
13) and the container is placed there as well
(1198758) this transaction fires consuming the tokens in 119875
8and
11987513 and creating one new token in119875
9and11987514after a fixed time
interval respectively hence the AGV receives the container(11987514) and the number of containers that could be placed in
the IO point is added by 1 (1198758)
Transaction 1198799is used to simulate the travel time of a
loaded AGV to the foot of the destination QC Firing time ofthis transaction is calculated as in transaction 119879
7 after which
a new token is created into 11987515
Transactions 11987912and 119879
13are used to update the locations
of the AGVwith no tasks periodicallyThey work in a similarway as Transactions 119879
1and 119879
2 For all the AGV with no
task they are moved in a fixed interval to a neighboring zonenearer to the ASC which they are bound to if possible
Color relations of tokens in this partial model are listedin Table 4 Attributes not involved in this partial model areexcluded
Partial model of QC is shown in Figure 7 Positions of 11987511
are adjustedThere are only 2 transactions in this partial model
Transaction 11987910
is for the process that a QC picks up acontainer from an AGV In case that a QC is ready for anext container loading (119875
18not empty) and a loaded AGV
has reached the foot of that QC (token with the same QCnumber in 119875
15) this transaction fires consuming one token
in 11987518 11987515 and 119875
20 following firing conditions as shown in
Figure 7 In case that there are multiple loaded AGV at thefoot of the same QC the AGV carrying the container with aminimal container number will be consumed Firing time oftransaction 119879
10is fixed When the firing ends a new token is
created into 11987511
as an AGV with no task allocated to it andanother new token is created into 119875
19indicating that the QC
has received the containerTransaction 119879
11is used to simulate the time a QC trolley
spends in putting a container onto a vessel and turn backempty to the quayside Firing time of 119879
11is also fixed When
the firing ends a new token is created to 11987518 hence the QC
is ready for a next loadColor relations of tokens in this partial model are listed in
Table 5 Also attributes not involved in this partial model areexcluded
4 Optimization Approaches
The TCPN model proposed in the last section could beused as fitness functions of optimizationmethods evaluating
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
QC
B
IO pointA
Figure 3 Illustration on grids and AGV routing
in the process between 1198751and 119875
2 hence the processes of the
two tokens could go on concurrently Tokens B and C are intwo asynchronous processes since it is not required that theseprocesses should start or end in the same time Howeverthe following process of them (119875
4) will not start till the two
processes both end and the two tokens are in the input placesof transaction 119879
2
Colored Petri Net (CPN) [30] and Timed Petri Net(TPN) [32] are two extensions of Petri Net In a CPNtokens have properties and data values attached to themcalled token color after which distinction between tokens ismade possible [33ndash35] In a TPN transactions are no longerfinished instantaneously On firing of a transaction requiredtokens in the input places are consumed immediately whiletokens in the output places are generated some time later[36 37] In simulating the container handling system in ACTtoken colors could be used to distinguish different QC ASCAGV and containers while transaction times could be usedto indicate the time length of equipment motions
The possible contributions of this paper are listed asfollows
(1) A simulation-based optimizationmethod is proposedto solve the SAPOBA which outputs a storage plan ofoutbound containers that leads to a best performanceof the container handling system in ACT
(2) TCPN is used to build up the simulation modelwhich establishes directly the relationship betweenstorage plans of outbound containers and QCwaitingtime of container handling system
(3) Two optimization methods PSO and GA are com-pared when used in a simulation-based optimizationfor optimal solution
3 TCPN Model for Simulation
31 Model Assumptions Assumptions of the model are listedbelow
(1) Only outbound containers are considered in thismodel
(2) Only 40 ft container are considered in this model20 ft containers could be converted into 40 ft ones inpairs
(3) The capacity of IO points as buffers between ASCand AGV is fixed and is the same No such bufferexists between QC and AGV
(4) Moving range of AGV could be divided into two-dimensional grids of equal square zones Positions ofAGV could be represented by coordinate of the gridwhich they are in as positions of QC and IO points
(5) Capacity of each grid is large enough to hold all theAGV at the same time and traffic congestions arenot considered in this model All AGV move in aconstant speed and they always spend the same timein crossing a current grid
(6) AGV routes are decided step by step following rulesWhen travelling to a destination grid at first an AGVtries to run into a next grid nearer the destination gridin a direction parallel to the shoreline In case that nosuch grid exists the AGV runs into a next grid nearerto the destination grid in a direction perpendicular tothe shoreline
Assumptions (4) to (6) could be illustrated in Figure 3which shows a two-dimensional grid of 6 times 4 square zonesAGV could run into a neighboring grid crossing a dash linewhile full lines are unbridgeable There are 6 IO points and2 QC in these grids Two AGV A and B are traveling to theirdestinations Routes of the two AGV are shown as directedfold lines intersecting at the same grid Following the routesthe two AGV will run into the same grid simultaneouslyhowever their travelling times will never be lengthened
(7) Outbound containers are allowed to be retrieved inbatches Containers in the same batch get the sameearliest time for retrieval before which they are notallowed to be retrieved
(8) Travelling speeds of ASC and AGV are treated asconstants in this model while QC are assumed to stayin a fixed grid during handling process
(9) An AGV is always bound to some ASC In case thatan AGV gets no task at the moment it runs back tothe ASC it is bound to and waits there for a next task
(10) Storage spaces are allocated in slots (spaces in thesame bay and same row of a block) and outbound
Mathematical Problems in Engineering 5
containers are divided into stacks (a group of contain-ers that could be placed in one slot) On this basisstorage plans are treated as one-to-one matching ofslots and stacks
32 Constraints of Storage Plans Storage plan is an allocationof storage spaces to containers a one-to-one matching planin nature Definition and constraints of these plans could beexpressed by math formulas Notations used in the formulasare listed below
119868 total number of stacks indexed by 119894 or 1198941015840 1 le 119894 1198941015840 le119868119869 total number of blocks in the storage yard indexedby 119895 1 le 119895 le 119869119870 total number of bays in a block indexed by 119896 1 le119896 le 119870 The smaller the bay number is the closer thisbay is to the shoreline119862119895119896 number of free slots in the 119896th bay of block 119895
119909119894119895119896 if stack 119894 is allocated to the slot in the 119896th bay of
block 119895 then 119909119894119895119896= 1 otherwise 119909
119894119895119896= 0
A storage plan could be described by a set of variables119909119894119895119896
119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896
isin 1 2 119870
(1)
The constraints are shown in the equations below
sum
119894
119909119894119895119896le 119862119895119896 forall119895 119896 (2)
Constraint (3) means that the total number of stacksallocated to the same bay in the same block should neverexceeds the capacity of that bay
sum
119895119896
119909119894119895119896= 1 forall119894 (3)
Equation (4) means that each stack could be allocatedonly once Outbound containers are to be stacked on theirarrivals in locations as determined in the storage plan
33 Overall Model Notations used in the simulation modelare listed below
119876 total number of QC119860 total number of ASC119862 total number of containers119881 total number of AGV119861 capacity of an IO point119866 total number of zones in themoving range of AGV119875 total number of places indexed by 119901 In this model119875 = 20119879 total number of transactions indexed by 119905 In thismodel 119879 = 13
119867 total number of containers to be handled119889119901 a token in place 119901
119889119901119905 a token that is fired from place 119901 to transaction 119905
1198891015840
119905119901 a token that is fired out of transaction 119905 into place
119901CS color set of tokens an 8-tuple of integers as⟨119888119894119889 119887119894119889 119886119894119889 119902119894119889 119886119894119898 119901119900119904 119886119911 119902119911⟩cid attribute of tokens indicating the container num-berbid attribute of tokens indicating the batch numberaid attribute of tokens indicating the ASC numberqid attribute of tokens indicating the QC numberaim attribute of tokens indicating the destinationzone number of an AGVpos attribute of tokens indicating the current zonenumber of an AGVaz attribute of tokens indicating the zone number ofIO point of an ASC to which an AGV is bound AnAGV moves back to the zone az when it is freeqz attribute of tokens indicating the zone number ofQC joint point119871(1199111 1199112) travel distance between two zones written
as 1199111and 1199112
119872(119901 exp) number of tokens in places 119901 as definedby expressions exp in the bracket If there is noexpression in the brackets this natation means thetotal number of tokens in places 119901119873(1199111 1199112) the next zone of 119911
1when running to zone
1199112
120575 average QC waiting time as output of the model119862119873119902 total number of containers handled by QC 119902
indexed by 119899119905119904119902119899 start time of the 119899th handling of QC 119902 according
to simulation119905119890119902119899 end time of the 119899th handling of QC 119902 according
to simulation
The overall TCPN model is shown in Figure 4 Circlesin this figure are for places in which dashed circles indicateboundaries between different kinds of equipment As forplaces with limited capacity the maximal number of tokensallowed is marked at the upper left of the circle Meanings ofplaces in Figure 1 are listed in Table 1 Squares and rectanglesare for transactions the former fire instantaneously whilethe latter fire with some time Square brackets are for firingconditions of transactions In case that there is a squarebracket over or under a squarerectangle the transactionwill not fire till the conditions are met Arrows are for arcseach with the number of tokens to or from a transactionmarked Marks of arrows could be divided into two parts Forarrows from a place to a transaction the first part of the markindicates number of tokens consumed by the transaction and
6 Mathematical Problems in Engineering
VV
11
Q
Q
A
AA
V
V
A
V
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
P18
1998400d99840024 1998400d32
1998400d41 1998400d99840013 1998400d103
M(1 d1bid =
d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d1810
1998400d114
1998400d1612
1998400d1310
1998400d2010
1998400d129
1998400d1713
1998400d127
1998400d96
1998400d118
1998400d88
1998400d76
1998400d1911
1998400d99840035
1998400d9984001118
1998400d9984001019
1998400d9984001011
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d998400610 1998400d99840068
1998400d99840089
1998400d998400814
1998400d998400620
1998400d998400713
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65
1998400d54
1998400d99840057
1998400d998400412
T5
T4
P11
T10T11 P19
P9
P8
P17
P13
T8
P20
P15
P16
P12
P7
T6
T7
T13
T12
T9 P14
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A[d76aid =
d96aid]
[d88cid =d118 cid]
M(1 d1bid =
d41bid)998400d99840012
Figure 4 TCPN model for a handling system in ACT
the second part is written as ldquo119889119901119905rdquo with 119901 for the origin place
and 119905 for the destination transaction indicating every tokengoing through the arc The two parts are put together to be awhole mark with a back single quotation set between themMarks of arcs in an opposed direction are formed in a similarway while in these marks the tokens are written as ldquo1198891015840
119905119901rdquo for
distinctionMeanings of places in Figure 1 are listed in Table 1 while
the transactions and arcs are to be explained in partialmodels
In the beginning some colored tokens are scatteredinitially in some places in the TCPN model which is calledthe initial marking of the model as is shown in Table 2Tokens located in 119875
1are for containers to be handled Each
container gets a unique container number and the batchnumbers are known as well In addition the ASC and QC tohandle the containers are known as well (aid and qid) as thezonenumber ofQC joint point (qz)The token in119875
2is used to
allow container retrievals in batches and no color is assignedto it Tokens in 119875
10are for ASC hence the ASC number of
each token (aid) is known and unique as the zone number ofIO points (az) Tokens in 119875
9are used to indicate the residual
capacity of IO points hence tokens in this place all get anASC number (aid) and the amount of tokens that have thesame ASC number is fixed Tokens in 119875
18are for the QC
hence each of these tokens gets a unique QC number (qid)Tokens in 119875
11are for the AGV which are always located in
some zone (pos) and bound to some ASC (az) There is onlyone token in 119875
17with no color used to update the positions
of AGV periodically The TCPN model ends with a similarmarking as in the beginning yet no container is left in place1198751When simulation ends the total QC waiting time could
be calculated as sum of time intervals between the time a QC
finishes a current container handling and the time the QCstarts a next container handling As a result the average QCwaiting time as the output of the simulation runs and theoptimization objective of the SAPOBA could be calculatedfollowing (1) as follows
Min 120575 = Min 1119876sum
119902
(
119862119873119902
sum
119899=2
(119905119904119902119899minus 119905119890119902(119899minus1)
)) (4)
34 Partial Models The overall TCPN model could bedivided into three parts with those dashed circles in Figure 4as boundaries corresponding to ASC AGV and QC respec-tively Partial model of ASC is shown in Figure 5 Notice thatthe position of 119875
20is adjusted
There are 5 transactions in this partial model Transition1198791is to allow a new batch of outbound containers to be
retrieved by ASC and transaction 1198792is the cool-down of this
allowance In case that the cool-down ends and a token 119889 isplaced in 119875
4 all the tokens (containers) in 119875
1with the same
batch number (bid) as token119889 are fired (11988911) as token119889 itself
(11988941) At the same time one new tokens (1198891015840
12) are created
immediately into 1198752for every 119889
11 and a new token (1198891015840
13) is
also created to 1198753 On this creation transactions 119879
2are fired
creating a new token (119889101584024) into 119875
4after a fixed interval With
the help of 1198792 transaction 119879
1brings periodically batches of
containers that are allowed to be retrieved by ASC followingthe handling schedule in a real ACT
Transaction 1198793is fired when an ASC is ready for a
next container (11987510
not empty) and there is some containerallowed to be retrieved by it already (token exists in 119875
2with
the same aid) Considering that theremay bemultiple tokensin 1198752and they could go to different QC an ASC always
retrieve a container to the QC with minimum inventory
Mathematical Problems in Engineering 7
A
AA
A
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
1998400d99840024 1998400d32
1998400d411998400d99840013 1998400d103
M(1 d1bid = d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d76
1998400d99840035
1998400d998400610
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65 1998400d99840057T5 P7
T6
M(1 d1bid =
d41bid)998400d99840012
1998400d96
1998400d99840068
1998400d998400620P9
P20
P8
B times A
B times A
[d76aid = d96aid]
Figure 5 Partial model of ASC
Table 1 Meanings of places
Place Practical meaning
1198751
Outbound containers in blocks notallowed to be retrieved
1198752
Outbound containers in blocks allowedto be retrieved
1198753
A batch of containers that has beenallowed to be retrieved
1198754
Time to allow a new batch of containersto be retrieved
1198755
Information of containers that ASCs havedecided to retrieve but no AGV isallocated
1198756 Containers that ASCs start to retrieve1198757 Loaded ASCs at IO points1198758 Containers stacked at IO points1198759 Residual capacities of IO points11987510 ASCs ready for a next container11987511 Unloaded free AGVs
11987512
Unloaded AGVs allocated to somecontainer
11987513
Unloaded AGVs allocated to somecontainer at IO point
11987514 Loaded AGVs at IO point11987515 Loaded AGVs at the foot of QC11987516 Ready to update positions of AGVs11987517 Position of AGVs updated11987518 Unloaded QCs at landside11987519 Loaded QCs at landside
11987520
Containers that have been placed into IOpoint but not handled by QCs
as shown in the firing condition QC inventory means thenumber of containers which have been placed in the IOpoint but not handled by the QC equal to the number oftokens to the QC in 119875
20 This strategy is also used in [38]
in which it is believed that this strategy helps in preventingQC from waiting for AGV leading to a good productivity ofthe terminal On firing of this transaction a token (1198891015840
35) is
created into 1198755immediately as an AGV task to be allocated
Table 2 Initial marking of the TCPN
Place Number oftokens cid bid aid qid aim pos az qz
1198751 119862 radic radic radic radic mdash mdash mdash radic
1198754 1 mdash mdash mdash mdash mdash mdash mdash mdash11987510 119860 mdash mdash radic mdash mdash mdash radic mdash1198759 119861 times 119860 mdash mdash radic mdash mdash mdash mdash mdash11987518 119876 mdash mdash mdash radic mdash mdash mdash mdash11987511 119881 mdash mdash mdash mdash mdash radic radic mdash11987517 1 mdash mdash mdash mdash mdash mdash mdash mdash
Moreover another token (119889101584036) is also created into 119875
6at the
same time as the next container to be handledTransaction 119879
5is for the cycle time of ASC When
deciding to retrieve a container (1198756not empty) anASCmoves
to its stacking location pick it up and carry it to the IOpoint Time length of this process could be calculated sincethe stacking position of container and the moving speed ofASC are both knownOn the arrival of ASC to the IO point anew token is created to119875
7 Notice that since stacking positions
of containers are all different firing time of 1198795is not fixed
Transaction 1198796is to add a new container to the IO point
of some ASC In case that an ASC is loaded at the IO point(1198757not empty) and the IO point is not filled (at least one
token exists in 1198758with the same ASC number aid) this
transaction consumes a token in 1198757and1198758 respectively After
a time a token is created to 1198758as a container stacked in
IO point and another token is also created to 11987520
as QCinventory Firing time of 119879
6is fixed which equals the time
an ASC spends in putting a container down to the IO pointColor relations of tokens to and from the same transac-
tion in this partial model are listed in Table 3 Notice thattokens from or to the same place always get the same colorwhile the color of tokens created in a firing is not alwaysinherited from the consumed tokens Attributes not involvedin this partial model are excluded
Partial model of AGV is shown in Figure 6 Positions of1198758 1198759 11987513 and 119879
8are adjusted
There are in total 6 transactions in the partial model ofAGV Transaction 119879
4is to allocate a container task to an
AGV with no task In case that there is some tasks to be
8 Mathematical Problems in Engineering
VV
11
A V
V V
P5
1998400d114
1998400d1612
1998400d129
1998400d1713
1998400d127
1998400d118
1998400d88
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d99840089
1998400d9984008141998400d998400713
1998400d54 1998400d998400412T4
P11
P9
P8
P17
P13
T8
P15
P16
P12 T7
T12
T13
T9 P14
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A
[d88cid = d118 cid]
Figure 6 Partial model of AGV
Table 3 Color relations of tokens in partial model of ASCs
Token cid bid aid qid az qz1198891015840
1211988911cid mdash 119889
11aid 119889
11qid mdash 119889
11qz
1198891015840
13mdash 119889
41bid + 1 mdash mdash mdash mdash
1198891015840
24mdash 119889
32bid mdash mdash mdash mdash
1198891015840
3511988923cid mdash 119889
23aid 119889
23qid 119889
23az 119889
23qz
1198891015840
3611988923cid mdash 119889
23aid 119889
23qid mdash mdash
1198891015840
5711988965cid mdash 119889
65aid 119889
65qid mdash mdash
1198891015840
6811988976cid mdash 119889
76aid mdash mdash mdash
1198891015840
610mdash mdash 119889
76aid mdash mdash mdash
1198891015840
62011988976cid mdash 119889
76aid 119889
76qid mdash mdash
allocated (1198755not empty) and some AGV are with no tasks
(11987511not empty)119879
4fires to allocate a task to a nearest AGV by
consuming corresponding tokens in 1198755and 119875
11and creating
a new token into 11987512at once
Transaction 1198797is used to simulate the travel time of
an unloaded AGV with some task to the IO point of thecontainer Firing time of the transaction equals travel time ofAGV which could be determined since the current zone anddestination zone are both known for the AGV
Transaction 1198798is for the process in which an AGV picks
up a container at IO point In case that an AGV has reachedthe IO point (119875
13) and the container is placed there as well
(1198758) this transaction fires consuming the tokens in 119875
8and
11987513 and creating one new token in119875
9and11987514after a fixed time
interval respectively hence the AGV receives the container(11987514) and the number of containers that could be placed in
the IO point is added by 1 (1198758)
Transaction 1198799is used to simulate the travel time of a
loaded AGV to the foot of the destination QC Firing time ofthis transaction is calculated as in transaction 119879
7 after which
a new token is created into 11987515
Transactions 11987912and 119879
13are used to update the locations
of the AGVwith no tasks periodicallyThey work in a similarway as Transactions 119879
1and 119879
2 For all the AGV with no
task they are moved in a fixed interval to a neighboring zonenearer to the ASC which they are bound to if possible
Color relations of tokens in this partial model are listedin Table 4 Attributes not involved in this partial model areexcluded
Partial model of QC is shown in Figure 7 Positions of 11987511
are adjustedThere are only 2 transactions in this partial model
Transaction 11987910
is for the process that a QC picks up acontainer from an AGV In case that a QC is ready for anext container loading (119875
18not empty) and a loaded AGV
has reached the foot of that QC (token with the same QCnumber in 119875
15) this transaction fires consuming one token
in 11987518 11987515 and 119875
20 following firing conditions as shown in
Figure 7 In case that there are multiple loaded AGV at thefoot of the same QC the AGV carrying the container with aminimal container number will be consumed Firing time oftransaction 119879
10is fixed When the firing ends a new token is
created into 11987511
as an AGV with no task allocated to it andanother new token is created into 119875
19indicating that the QC
has received the containerTransaction 119879
11is used to simulate the time a QC trolley
spends in putting a container onto a vessel and turn backempty to the quayside Firing time of 119879
11is also fixed When
the firing ends a new token is created to 11987518 hence the QC
is ready for a next loadColor relations of tokens in this partial model are listed in
Table 5 Also attributes not involved in this partial model areexcluded
4 Optimization Approaches
The TCPN model proposed in the last section could beused as fitness functions of optimizationmethods evaluating
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
containers are divided into stacks (a group of contain-ers that could be placed in one slot) On this basisstorage plans are treated as one-to-one matching ofslots and stacks
32 Constraints of Storage Plans Storage plan is an allocationof storage spaces to containers a one-to-one matching planin nature Definition and constraints of these plans could beexpressed by math formulas Notations used in the formulasare listed below
119868 total number of stacks indexed by 119894 or 1198941015840 1 le 119894 1198941015840 le119868119869 total number of blocks in the storage yard indexedby 119895 1 le 119895 le 119869119870 total number of bays in a block indexed by 119896 1 le119896 le 119870 The smaller the bay number is the closer thisbay is to the shoreline119862119895119896 number of free slots in the 119896th bay of block 119895
119909119894119895119896 if stack 119894 is allocated to the slot in the 119896th bay of
block 119895 then 119909119894119895119896= 1 otherwise 119909
119894119895119896= 0
A storage plan could be described by a set of variables119909119894119895119896
119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896
isin 1 2 119870
(1)
The constraints are shown in the equations below
sum
119894
119909119894119895119896le 119862119895119896 forall119895 119896 (2)
Constraint (3) means that the total number of stacksallocated to the same bay in the same block should neverexceeds the capacity of that bay
sum
119895119896
119909119894119895119896= 1 forall119894 (3)
Equation (4) means that each stack could be allocatedonly once Outbound containers are to be stacked on theirarrivals in locations as determined in the storage plan
33 Overall Model Notations used in the simulation modelare listed below
119876 total number of QC119860 total number of ASC119862 total number of containers119881 total number of AGV119861 capacity of an IO point119866 total number of zones in themoving range of AGV119875 total number of places indexed by 119901 In this model119875 = 20119879 total number of transactions indexed by 119905 In thismodel 119879 = 13
119867 total number of containers to be handled119889119901 a token in place 119901
119889119901119905 a token that is fired from place 119901 to transaction 119905
1198891015840
119905119901 a token that is fired out of transaction 119905 into place
119901CS color set of tokens an 8-tuple of integers as⟨119888119894119889 119887119894119889 119886119894119889 119902119894119889 119886119894119898 119901119900119904 119886119911 119902119911⟩cid attribute of tokens indicating the container num-berbid attribute of tokens indicating the batch numberaid attribute of tokens indicating the ASC numberqid attribute of tokens indicating the QC numberaim attribute of tokens indicating the destinationzone number of an AGVpos attribute of tokens indicating the current zonenumber of an AGVaz attribute of tokens indicating the zone number ofIO point of an ASC to which an AGV is bound AnAGV moves back to the zone az when it is freeqz attribute of tokens indicating the zone number ofQC joint point119871(1199111 1199112) travel distance between two zones written
as 1199111and 1199112
119872(119901 exp) number of tokens in places 119901 as definedby expressions exp in the bracket If there is noexpression in the brackets this natation means thetotal number of tokens in places 119901119873(1199111 1199112) the next zone of 119911
1when running to zone
1199112
120575 average QC waiting time as output of the model119862119873119902 total number of containers handled by QC 119902
indexed by 119899119905119904119902119899 start time of the 119899th handling of QC 119902 according
to simulation119905119890119902119899 end time of the 119899th handling of QC 119902 according
to simulation
The overall TCPN model is shown in Figure 4 Circlesin this figure are for places in which dashed circles indicateboundaries between different kinds of equipment As forplaces with limited capacity the maximal number of tokensallowed is marked at the upper left of the circle Meanings ofplaces in Figure 1 are listed in Table 1 Squares and rectanglesare for transactions the former fire instantaneously whilethe latter fire with some time Square brackets are for firingconditions of transactions In case that there is a squarebracket over or under a squarerectangle the transactionwill not fire till the conditions are met Arrows are for arcseach with the number of tokens to or from a transactionmarked Marks of arrows could be divided into two parts Forarrows from a place to a transaction the first part of the markindicates number of tokens consumed by the transaction and
6 Mathematical Problems in Engineering
VV
11
Q
Q
A
AA
V
V
A
V
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
P18
1998400d99840024 1998400d32
1998400d41 1998400d99840013 1998400d103
M(1 d1bid =
d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d1810
1998400d114
1998400d1612
1998400d1310
1998400d2010
1998400d129
1998400d1713
1998400d127
1998400d96
1998400d118
1998400d88
1998400d76
1998400d1911
1998400d99840035
1998400d9984001118
1998400d9984001019
1998400d9984001011
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d998400610 1998400d99840068
1998400d99840089
1998400d998400814
1998400d998400620
1998400d998400713
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65
1998400d54
1998400d99840057
1998400d998400412
T5
T4
P11
T10T11 P19
P9
P8
P17
P13
T8
P20
P15
P16
P12
P7
T6
T7
T13
T12
T9 P14
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A[d76aid =
d96aid]
[d88cid =d118 cid]
M(1 d1bid =
d41bid)998400d99840012
Figure 4 TCPN model for a handling system in ACT
the second part is written as ldquo119889119901119905rdquo with 119901 for the origin place
and 119905 for the destination transaction indicating every tokengoing through the arc The two parts are put together to be awhole mark with a back single quotation set between themMarks of arcs in an opposed direction are formed in a similarway while in these marks the tokens are written as ldquo1198891015840
119905119901rdquo for
distinctionMeanings of places in Figure 1 are listed in Table 1 while
the transactions and arcs are to be explained in partialmodels
In the beginning some colored tokens are scatteredinitially in some places in the TCPN model which is calledthe initial marking of the model as is shown in Table 2Tokens located in 119875
1are for containers to be handled Each
container gets a unique container number and the batchnumbers are known as well In addition the ASC and QC tohandle the containers are known as well (aid and qid) as thezonenumber ofQC joint point (qz)The token in119875
2is used to
allow container retrievals in batches and no color is assignedto it Tokens in 119875
10are for ASC hence the ASC number of
each token (aid) is known and unique as the zone number ofIO points (az) Tokens in 119875
9are used to indicate the residual
capacity of IO points hence tokens in this place all get anASC number (aid) and the amount of tokens that have thesame ASC number is fixed Tokens in 119875
18are for the QC
hence each of these tokens gets a unique QC number (qid)Tokens in 119875
11are for the AGV which are always located in
some zone (pos) and bound to some ASC (az) There is onlyone token in 119875
17with no color used to update the positions
of AGV periodically The TCPN model ends with a similarmarking as in the beginning yet no container is left in place1198751When simulation ends the total QC waiting time could
be calculated as sum of time intervals between the time a QC
finishes a current container handling and the time the QCstarts a next container handling As a result the average QCwaiting time as the output of the simulation runs and theoptimization objective of the SAPOBA could be calculatedfollowing (1) as follows
Min 120575 = Min 1119876sum
119902
(
119862119873119902
sum
119899=2
(119905119904119902119899minus 119905119890119902(119899minus1)
)) (4)
34 Partial Models The overall TCPN model could bedivided into three parts with those dashed circles in Figure 4as boundaries corresponding to ASC AGV and QC respec-tively Partial model of ASC is shown in Figure 5 Notice thatthe position of 119875
20is adjusted
There are 5 transactions in this partial model Transition1198791is to allow a new batch of outbound containers to be
retrieved by ASC and transaction 1198792is the cool-down of this
allowance In case that the cool-down ends and a token 119889 isplaced in 119875
4 all the tokens (containers) in 119875
1with the same
batch number (bid) as token119889 are fired (11988911) as token119889 itself
(11988941) At the same time one new tokens (1198891015840
12) are created
immediately into 1198752for every 119889
11 and a new token (1198891015840
13) is
also created to 1198753 On this creation transactions 119879
2are fired
creating a new token (119889101584024) into 119875
4after a fixed interval With
the help of 1198792 transaction 119879
1brings periodically batches of
containers that are allowed to be retrieved by ASC followingthe handling schedule in a real ACT
Transaction 1198793is fired when an ASC is ready for a
next container (11987510
not empty) and there is some containerallowed to be retrieved by it already (token exists in 119875
2with
the same aid) Considering that theremay bemultiple tokensin 1198752and they could go to different QC an ASC always
retrieve a container to the QC with minimum inventory
Mathematical Problems in Engineering 7
A
AA
A
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
1998400d99840024 1998400d32
1998400d411998400d99840013 1998400d103
M(1 d1bid = d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d76
1998400d99840035
1998400d998400610
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65 1998400d99840057T5 P7
T6
M(1 d1bid =
d41bid)998400d99840012
1998400d96
1998400d99840068
1998400d998400620P9
P20
P8
B times A
B times A
[d76aid = d96aid]
Figure 5 Partial model of ASC
Table 1 Meanings of places
Place Practical meaning
1198751
Outbound containers in blocks notallowed to be retrieved
1198752
Outbound containers in blocks allowedto be retrieved
1198753
A batch of containers that has beenallowed to be retrieved
1198754
Time to allow a new batch of containersto be retrieved
1198755
Information of containers that ASCs havedecided to retrieve but no AGV isallocated
1198756 Containers that ASCs start to retrieve1198757 Loaded ASCs at IO points1198758 Containers stacked at IO points1198759 Residual capacities of IO points11987510 ASCs ready for a next container11987511 Unloaded free AGVs
11987512
Unloaded AGVs allocated to somecontainer
11987513
Unloaded AGVs allocated to somecontainer at IO point
11987514 Loaded AGVs at IO point11987515 Loaded AGVs at the foot of QC11987516 Ready to update positions of AGVs11987517 Position of AGVs updated11987518 Unloaded QCs at landside11987519 Loaded QCs at landside
11987520
Containers that have been placed into IOpoint but not handled by QCs
as shown in the firing condition QC inventory means thenumber of containers which have been placed in the IOpoint but not handled by the QC equal to the number oftokens to the QC in 119875
20 This strategy is also used in [38]
in which it is believed that this strategy helps in preventingQC from waiting for AGV leading to a good productivity ofthe terminal On firing of this transaction a token (1198891015840
35) is
created into 1198755immediately as an AGV task to be allocated
Table 2 Initial marking of the TCPN
Place Number oftokens cid bid aid qid aim pos az qz
1198751 119862 radic radic radic radic mdash mdash mdash radic
1198754 1 mdash mdash mdash mdash mdash mdash mdash mdash11987510 119860 mdash mdash radic mdash mdash mdash radic mdash1198759 119861 times 119860 mdash mdash radic mdash mdash mdash mdash mdash11987518 119876 mdash mdash mdash radic mdash mdash mdash mdash11987511 119881 mdash mdash mdash mdash mdash radic radic mdash11987517 1 mdash mdash mdash mdash mdash mdash mdash mdash
Moreover another token (119889101584036) is also created into 119875
6at the
same time as the next container to be handledTransaction 119879
5is for the cycle time of ASC When
deciding to retrieve a container (1198756not empty) anASCmoves
to its stacking location pick it up and carry it to the IOpoint Time length of this process could be calculated sincethe stacking position of container and the moving speed ofASC are both knownOn the arrival of ASC to the IO point anew token is created to119875
7 Notice that since stacking positions
of containers are all different firing time of 1198795is not fixed
Transaction 1198796is to add a new container to the IO point
of some ASC In case that an ASC is loaded at the IO point(1198757not empty) and the IO point is not filled (at least one
token exists in 1198758with the same ASC number aid) this
transaction consumes a token in 1198757and1198758 respectively After
a time a token is created to 1198758as a container stacked in
IO point and another token is also created to 11987520
as QCinventory Firing time of 119879
6is fixed which equals the time
an ASC spends in putting a container down to the IO pointColor relations of tokens to and from the same transac-
tion in this partial model are listed in Table 3 Notice thattokens from or to the same place always get the same colorwhile the color of tokens created in a firing is not alwaysinherited from the consumed tokens Attributes not involvedin this partial model are excluded
Partial model of AGV is shown in Figure 6 Positions of1198758 1198759 11987513 and 119879
8are adjusted
There are in total 6 transactions in the partial model ofAGV Transaction 119879
4is to allocate a container task to an
AGV with no task In case that there is some tasks to be
8 Mathematical Problems in Engineering
VV
11
A V
V V
P5
1998400d114
1998400d1612
1998400d129
1998400d1713
1998400d127
1998400d118
1998400d88
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d99840089
1998400d9984008141998400d998400713
1998400d54 1998400d998400412T4
P11
P9
P8
P17
P13
T8
P15
P16
P12 T7
T12
T13
T9 P14
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A
[d88cid = d118 cid]
Figure 6 Partial model of AGV
Table 3 Color relations of tokens in partial model of ASCs
Token cid bid aid qid az qz1198891015840
1211988911cid mdash 119889
11aid 119889
11qid mdash 119889
11qz
1198891015840
13mdash 119889
41bid + 1 mdash mdash mdash mdash
1198891015840
24mdash 119889
32bid mdash mdash mdash mdash
1198891015840
3511988923cid mdash 119889
23aid 119889
23qid 119889
23az 119889
23qz
1198891015840
3611988923cid mdash 119889
23aid 119889
23qid mdash mdash
1198891015840
5711988965cid mdash 119889
65aid 119889
65qid mdash mdash
1198891015840
6811988976cid mdash 119889
76aid mdash mdash mdash
1198891015840
610mdash mdash 119889
76aid mdash mdash mdash
1198891015840
62011988976cid mdash 119889
76aid 119889
76qid mdash mdash
allocated (1198755not empty) and some AGV are with no tasks
(11987511not empty)119879
4fires to allocate a task to a nearest AGV by
consuming corresponding tokens in 1198755and 119875
11and creating
a new token into 11987512at once
Transaction 1198797is used to simulate the travel time of
an unloaded AGV with some task to the IO point of thecontainer Firing time of the transaction equals travel time ofAGV which could be determined since the current zone anddestination zone are both known for the AGV
Transaction 1198798is for the process in which an AGV picks
up a container at IO point In case that an AGV has reachedthe IO point (119875
13) and the container is placed there as well
(1198758) this transaction fires consuming the tokens in 119875
8and
11987513 and creating one new token in119875
9and11987514after a fixed time
interval respectively hence the AGV receives the container(11987514) and the number of containers that could be placed in
the IO point is added by 1 (1198758)
Transaction 1198799is used to simulate the travel time of a
loaded AGV to the foot of the destination QC Firing time ofthis transaction is calculated as in transaction 119879
7 after which
a new token is created into 11987515
Transactions 11987912and 119879
13are used to update the locations
of the AGVwith no tasks periodicallyThey work in a similarway as Transactions 119879
1and 119879
2 For all the AGV with no
task they are moved in a fixed interval to a neighboring zonenearer to the ASC which they are bound to if possible
Color relations of tokens in this partial model are listedin Table 4 Attributes not involved in this partial model areexcluded
Partial model of QC is shown in Figure 7 Positions of 11987511
are adjustedThere are only 2 transactions in this partial model
Transaction 11987910
is for the process that a QC picks up acontainer from an AGV In case that a QC is ready for anext container loading (119875
18not empty) and a loaded AGV
has reached the foot of that QC (token with the same QCnumber in 119875
15) this transaction fires consuming one token
in 11987518 11987515 and 119875
20 following firing conditions as shown in
Figure 7 In case that there are multiple loaded AGV at thefoot of the same QC the AGV carrying the container with aminimal container number will be consumed Firing time oftransaction 119879
10is fixed When the firing ends a new token is
created into 11987511
as an AGV with no task allocated to it andanother new token is created into 119875
19indicating that the QC
has received the containerTransaction 119879
11is used to simulate the time a QC trolley
spends in putting a container onto a vessel and turn backempty to the quayside Firing time of 119879
11is also fixed When
the firing ends a new token is created to 11987518 hence the QC
is ready for a next loadColor relations of tokens in this partial model are listed in
Table 5 Also attributes not involved in this partial model areexcluded
4 Optimization Approaches
The TCPN model proposed in the last section could beused as fitness functions of optimizationmethods evaluating
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
VV
11
Q
Q
A
AA
V
V
A
V
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
P18
1998400d99840024 1998400d32
1998400d41 1998400d99840013 1998400d103
M(1 d1bid =
d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d1810
1998400d114
1998400d1612
1998400d1310
1998400d2010
1998400d129
1998400d1713
1998400d127
1998400d96
1998400d118
1998400d88
1998400d76
1998400d1911
1998400d99840035
1998400d9984001118
1998400d9984001019
1998400d9984001011
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d998400610 1998400d99840068
1998400d99840089
1998400d998400814
1998400d998400620
1998400d998400713
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65
1998400d54
1998400d99840057
1998400d998400412
T5
T4
P11
T10T11 P19
P9
P8
P17
P13
T8
P20
P15
P16
P12
P7
T6
T7
T13
T12
T9 P14
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A[d76aid =
d96aid]
[d88cid =d118 cid]
M(1 d1bid =
d41bid)998400d99840012
Figure 4 TCPN model for a handling system in ACT
the second part is written as ldquo119889119901119905rdquo with 119901 for the origin place
and 119905 for the destination transaction indicating every tokengoing through the arc The two parts are put together to be awhole mark with a back single quotation set between themMarks of arcs in an opposed direction are formed in a similarway while in these marks the tokens are written as ldquo1198891015840
119905119901rdquo for
distinctionMeanings of places in Figure 1 are listed in Table 1 while
the transactions and arcs are to be explained in partialmodels
In the beginning some colored tokens are scatteredinitially in some places in the TCPN model which is calledthe initial marking of the model as is shown in Table 2Tokens located in 119875
1are for containers to be handled Each
container gets a unique container number and the batchnumbers are known as well In addition the ASC and QC tohandle the containers are known as well (aid and qid) as thezonenumber ofQC joint point (qz)The token in119875
2is used to
allow container retrievals in batches and no color is assignedto it Tokens in 119875
10are for ASC hence the ASC number of
each token (aid) is known and unique as the zone number ofIO points (az) Tokens in 119875
9are used to indicate the residual
capacity of IO points hence tokens in this place all get anASC number (aid) and the amount of tokens that have thesame ASC number is fixed Tokens in 119875
18are for the QC
hence each of these tokens gets a unique QC number (qid)Tokens in 119875
11are for the AGV which are always located in
some zone (pos) and bound to some ASC (az) There is onlyone token in 119875
17with no color used to update the positions
of AGV periodically The TCPN model ends with a similarmarking as in the beginning yet no container is left in place1198751When simulation ends the total QC waiting time could
be calculated as sum of time intervals between the time a QC
finishes a current container handling and the time the QCstarts a next container handling As a result the average QCwaiting time as the output of the simulation runs and theoptimization objective of the SAPOBA could be calculatedfollowing (1) as follows
Min 120575 = Min 1119876sum
119902
(
119862119873119902
sum
119899=2
(119905119904119902119899minus 119905119890119902(119899minus1)
)) (4)
34 Partial Models The overall TCPN model could bedivided into three parts with those dashed circles in Figure 4as boundaries corresponding to ASC AGV and QC respec-tively Partial model of ASC is shown in Figure 5 Notice thatthe position of 119875
20is adjusted
There are 5 transactions in this partial model Transition1198791is to allow a new batch of outbound containers to be
retrieved by ASC and transaction 1198792is the cool-down of this
allowance In case that the cool-down ends and a token 119889 isplaced in 119875
4 all the tokens (containers) in 119875
1with the same
batch number (bid) as token119889 are fired (11988911) as token119889 itself
(11988941) At the same time one new tokens (1198891015840
12) are created
immediately into 1198752for every 119889
11 and a new token (1198891015840
13) is
also created to 1198753 On this creation transactions 119879
2are fired
creating a new token (119889101584024) into 119875
4after a fixed interval With
the help of 1198792 transaction 119879
1brings periodically batches of
containers that are allowed to be retrieved by ASC followingthe handling schedule in a real ACT
Transaction 1198793is fired when an ASC is ready for a
next container (11987510
not empty) and there is some containerallowed to be retrieved by it already (token exists in 119875
2with
the same aid) Considering that theremay bemultiple tokensin 1198752and they could go to different QC an ASC always
retrieve a container to the QC with minimum inventory
Mathematical Problems in Engineering 7
A
AA
A
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
1998400d99840024 1998400d32
1998400d411998400d99840013 1998400d103
M(1 d1bid = d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d76
1998400d99840035
1998400d998400610
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65 1998400d99840057T5 P7
T6
M(1 d1bid =
d41bid)998400d99840012
1998400d96
1998400d99840068
1998400d998400620P9
P20
P8
B times A
B times A
[d76aid = d96aid]
Figure 5 Partial model of ASC
Table 1 Meanings of places
Place Practical meaning
1198751
Outbound containers in blocks notallowed to be retrieved
1198752
Outbound containers in blocks allowedto be retrieved
1198753
A batch of containers that has beenallowed to be retrieved
1198754
Time to allow a new batch of containersto be retrieved
1198755
Information of containers that ASCs havedecided to retrieve but no AGV isallocated
1198756 Containers that ASCs start to retrieve1198757 Loaded ASCs at IO points1198758 Containers stacked at IO points1198759 Residual capacities of IO points11987510 ASCs ready for a next container11987511 Unloaded free AGVs
11987512
Unloaded AGVs allocated to somecontainer
11987513
Unloaded AGVs allocated to somecontainer at IO point
11987514 Loaded AGVs at IO point11987515 Loaded AGVs at the foot of QC11987516 Ready to update positions of AGVs11987517 Position of AGVs updated11987518 Unloaded QCs at landside11987519 Loaded QCs at landside
11987520
Containers that have been placed into IOpoint but not handled by QCs
as shown in the firing condition QC inventory means thenumber of containers which have been placed in the IOpoint but not handled by the QC equal to the number oftokens to the QC in 119875
20 This strategy is also used in [38]
in which it is believed that this strategy helps in preventingQC from waiting for AGV leading to a good productivity ofthe terminal On firing of this transaction a token (1198891015840
35) is
created into 1198755immediately as an AGV task to be allocated
Table 2 Initial marking of the TCPN
Place Number oftokens cid bid aid qid aim pos az qz
1198751 119862 radic radic radic radic mdash mdash mdash radic
1198754 1 mdash mdash mdash mdash mdash mdash mdash mdash11987510 119860 mdash mdash radic mdash mdash mdash radic mdash1198759 119861 times 119860 mdash mdash radic mdash mdash mdash mdash mdash11987518 119876 mdash mdash mdash radic mdash mdash mdash mdash11987511 119881 mdash mdash mdash mdash mdash radic radic mdash11987517 1 mdash mdash mdash mdash mdash mdash mdash mdash
Moreover another token (119889101584036) is also created into 119875
6at the
same time as the next container to be handledTransaction 119879
5is for the cycle time of ASC When
deciding to retrieve a container (1198756not empty) anASCmoves
to its stacking location pick it up and carry it to the IOpoint Time length of this process could be calculated sincethe stacking position of container and the moving speed ofASC are both knownOn the arrival of ASC to the IO point anew token is created to119875
7 Notice that since stacking positions
of containers are all different firing time of 1198795is not fixed
Transaction 1198796is to add a new container to the IO point
of some ASC In case that an ASC is loaded at the IO point(1198757not empty) and the IO point is not filled (at least one
token exists in 1198758with the same ASC number aid) this
transaction consumes a token in 1198757and1198758 respectively After
a time a token is created to 1198758as a container stacked in
IO point and another token is also created to 11987520
as QCinventory Firing time of 119879
6is fixed which equals the time
an ASC spends in putting a container down to the IO pointColor relations of tokens to and from the same transac-
tion in this partial model are listed in Table 3 Notice thattokens from or to the same place always get the same colorwhile the color of tokens created in a firing is not alwaysinherited from the consumed tokens Attributes not involvedin this partial model are excluded
Partial model of AGV is shown in Figure 6 Positions of1198758 1198759 11987513 and 119879
8are adjusted
There are in total 6 transactions in the partial model ofAGV Transaction 119879
4is to allocate a container task to an
AGV with no task In case that there is some tasks to be
8 Mathematical Problems in Engineering
VV
11
A V
V V
P5
1998400d114
1998400d1612
1998400d129
1998400d1713
1998400d127
1998400d118
1998400d88
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d99840089
1998400d9984008141998400d998400713
1998400d54 1998400d998400412T4
P11
P9
P8
P17
P13
T8
P15
P16
P12 T7
T12
T13
T9 P14
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A
[d88cid = d118 cid]
Figure 6 Partial model of AGV
Table 3 Color relations of tokens in partial model of ASCs
Token cid bid aid qid az qz1198891015840
1211988911cid mdash 119889
11aid 119889
11qid mdash 119889
11qz
1198891015840
13mdash 119889
41bid + 1 mdash mdash mdash mdash
1198891015840
24mdash 119889
32bid mdash mdash mdash mdash
1198891015840
3511988923cid mdash 119889
23aid 119889
23qid 119889
23az 119889
23qz
1198891015840
3611988923cid mdash 119889
23aid 119889
23qid mdash mdash
1198891015840
5711988965cid mdash 119889
65aid 119889
65qid mdash mdash
1198891015840
6811988976cid mdash 119889
76aid mdash mdash mdash
1198891015840
610mdash mdash 119889
76aid mdash mdash mdash
1198891015840
62011988976cid mdash 119889
76aid 119889
76qid mdash mdash
allocated (1198755not empty) and some AGV are with no tasks
(11987511not empty)119879
4fires to allocate a task to a nearest AGV by
consuming corresponding tokens in 1198755and 119875
11and creating
a new token into 11987512at once
Transaction 1198797is used to simulate the travel time of
an unloaded AGV with some task to the IO point of thecontainer Firing time of the transaction equals travel time ofAGV which could be determined since the current zone anddestination zone are both known for the AGV
Transaction 1198798is for the process in which an AGV picks
up a container at IO point In case that an AGV has reachedthe IO point (119875
13) and the container is placed there as well
(1198758) this transaction fires consuming the tokens in 119875
8and
11987513 and creating one new token in119875
9and11987514after a fixed time
interval respectively hence the AGV receives the container(11987514) and the number of containers that could be placed in
the IO point is added by 1 (1198758)
Transaction 1198799is used to simulate the travel time of a
loaded AGV to the foot of the destination QC Firing time ofthis transaction is calculated as in transaction 119879
7 after which
a new token is created into 11987515
Transactions 11987912and 119879
13are used to update the locations
of the AGVwith no tasks periodicallyThey work in a similarway as Transactions 119879
1and 119879
2 For all the AGV with no
task they are moved in a fixed interval to a neighboring zonenearer to the ASC which they are bound to if possible
Color relations of tokens in this partial model are listedin Table 4 Attributes not involved in this partial model areexcluded
Partial model of QC is shown in Figure 7 Positions of 11987511
are adjustedThere are only 2 transactions in this partial model
Transaction 11987910
is for the process that a QC picks up acontainer from an AGV In case that a QC is ready for anext container loading (119875
18not empty) and a loaded AGV
has reached the foot of that QC (token with the same QCnumber in 119875
15) this transaction fires consuming one token
in 11987518 11987515 and 119875
20 following firing conditions as shown in
Figure 7 In case that there are multiple loaded AGV at thefoot of the same QC the AGV carrying the container with aminimal container number will be consumed Firing time oftransaction 119879
10is fixed When the firing ends a new token is
created into 11987511
as an AGV with no task allocated to it andanother new token is created into 119875
19indicating that the QC
has received the containerTransaction 119879
11is used to simulate the time a QC trolley
spends in putting a container onto a vessel and turn backempty to the quayside Firing time of 119879
11is also fixed When
the firing ends a new token is created to 11987518 hence the QC
is ready for a next loadColor relations of tokens in this partial model are listed in
Table 5 Also attributes not involved in this partial model areexcluded
4 Optimization Approaches
The TCPN model proposed in the last section could beused as fitness functions of optimizationmethods evaluating
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
A
AA
A
P4 T2 P3
P1 T1 P2
P10
P6
P5T3
1998400d99840024 1998400d32
1998400d411998400d99840013 1998400d103
M(1 d1bid = d41bid)998400d11
[d11bid = d41bid]
1998400d23
1998400d76
1998400d99840035
1998400d998400610
[d23aid = d103 aid
minM(20 d20qid = d23qid)]
1998400d99840036
1998400d65 1998400d99840057T5 P7
T6
M(1 d1bid =
d41bid)998400d99840012
1998400d96
1998400d99840068
1998400d998400620P9
P20
P8
B times A
B times A
[d76aid = d96aid]
Figure 5 Partial model of ASC
Table 1 Meanings of places
Place Practical meaning
1198751
Outbound containers in blocks notallowed to be retrieved
1198752
Outbound containers in blocks allowedto be retrieved
1198753
A batch of containers that has beenallowed to be retrieved
1198754
Time to allow a new batch of containersto be retrieved
1198755
Information of containers that ASCs havedecided to retrieve but no AGV isallocated
1198756 Containers that ASCs start to retrieve1198757 Loaded ASCs at IO points1198758 Containers stacked at IO points1198759 Residual capacities of IO points11987510 ASCs ready for a next container11987511 Unloaded free AGVs
11987512
Unloaded AGVs allocated to somecontainer
11987513
Unloaded AGVs allocated to somecontainer at IO point
11987514 Loaded AGVs at IO point11987515 Loaded AGVs at the foot of QC11987516 Ready to update positions of AGVs11987517 Position of AGVs updated11987518 Unloaded QCs at landside11987519 Loaded QCs at landside
11987520
Containers that have been placed into IOpoint but not handled by QCs
as shown in the firing condition QC inventory means thenumber of containers which have been placed in the IOpoint but not handled by the QC equal to the number oftokens to the QC in 119875
20 This strategy is also used in [38]
in which it is believed that this strategy helps in preventingQC from waiting for AGV leading to a good productivity ofthe terminal On firing of this transaction a token (1198891015840
35) is
created into 1198755immediately as an AGV task to be allocated
Table 2 Initial marking of the TCPN
Place Number oftokens cid bid aid qid aim pos az qz
1198751 119862 radic radic radic radic mdash mdash mdash radic
1198754 1 mdash mdash mdash mdash mdash mdash mdash mdash11987510 119860 mdash mdash radic mdash mdash mdash radic mdash1198759 119861 times 119860 mdash mdash radic mdash mdash mdash mdash mdash11987518 119876 mdash mdash mdash radic mdash mdash mdash mdash11987511 119881 mdash mdash mdash mdash mdash radic radic mdash11987517 1 mdash mdash mdash mdash mdash mdash mdash mdash
Moreover another token (119889101584036) is also created into 119875
6at the
same time as the next container to be handledTransaction 119879
5is for the cycle time of ASC When
deciding to retrieve a container (1198756not empty) anASCmoves
to its stacking location pick it up and carry it to the IOpoint Time length of this process could be calculated sincethe stacking position of container and the moving speed ofASC are both knownOn the arrival of ASC to the IO point anew token is created to119875
7 Notice that since stacking positions
of containers are all different firing time of 1198795is not fixed
Transaction 1198796is to add a new container to the IO point
of some ASC In case that an ASC is loaded at the IO point(1198757not empty) and the IO point is not filled (at least one
token exists in 1198758with the same ASC number aid) this
transaction consumes a token in 1198757and1198758 respectively After
a time a token is created to 1198758as a container stacked in
IO point and another token is also created to 11987520
as QCinventory Firing time of 119879
6is fixed which equals the time
an ASC spends in putting a container down to the IO pointColor relations of tokens to and from the same transac-
tion in this partial model are listed in Table 3 Notice thattokens from or to the same place always get the same colorwhile the color of tokens created in a firing is not alwaysinherited from the consumed tokens Attributes not involvedin this partial model are excluded
Partial model of AGV is shown in Figure 6 Positions of1198758 1198759 11987513 and 119879
8are adjusted
There are in total 6 transactions in the partial model ofAGV Transaction 119879
4is to allocate a container task to an
AGV with no task In case that there is some tasks to be
8 Mathematical Problems in Engineering
VV
11
A V
V V
P5
1998400d114
1998400d1612
1998400d129
1998400d1713
1998400d127
1998400d118
1998400d88
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d99840089
1998400d9984008141998400d998400713
1998400d54 1998400d998400412T4
P11
P9
P8
P17
P13
T8
P15
P16
P12 T7
T12
T13
T9 P14
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A
[d88cid = d118 cid]
Figure 6 Partial model of AGV
Table 3 Color relations of tokens in partial model of ASCs
Token cid bid aid qid az qz1198891015840
1211988911cid mdash 119889
11aid 119889
11qid mdash 119889
11qz
1198891015840
13mdash 119889
41bid + 1 mdash mdash mdash mdash
1198891015840
24mdash 119889
32bid mdash mdash mdash mdash
1198891015840
3511988923cid mdash 119889
23aid 119889
23qid 119889
23az 119889
23qz
1198891015840
3611988923cid mdash 119889
23aid 119889
23qid mdash mdash
1198891015840
5711988965cid mdash 119889
65aid 119889
65qid mdash mdash
1198891015840
6811988976cid mdash 119889
76aid mdash mdash mdash
1198891015840
610mdash mdash 119889
76aid mdash mdash mdash
1198891015840
62011988976cid mdash 119889
76aid 119889
76qid mdash mdash
allocated (1198755not empty) and some AGV are with no tasks
(11987511not empty)119879
4fires to allocate a task to a nearest AGV by
consuming corresponding tokens in 1198755and 119875
11and creating
a new token into 11987512at once
Transaction 1198797is used to simulate the travel time of
an unloaded AGV with some task to the IO point of thecontainer Firing time of the transaction equals travel time ofAGV which could be determined since the current zone anddestination zone are both known for the AGV
Transaction 1198798is for the process in which an AGV picks
up a container at IO point In case that an AGV has reachedthe IO point (119875
13) and the container is placed there as well
(1198758) this transaction fires consuming the tokens in 119875
8and
11987513 and creating one new token in119875
9and11987514after a fixed time
interval respectively hence the AGV receives the container(11987514) and the number of containers that could be placed in
the IO point is added by 1 (1198758)
Transaction 1198799is used to simulate the travel time of a
loaded AGV to the foot of the destination QC Firing time ofthis transaction is calculated as in transaction 119879
7 after which
a new token is created into 11987515
Transactions 11987912and 119879
13are used to update the locations
of the AGVwith no tasks periodicallyThey work in a similarway as Transactions 119879
1and 119879
2 For all the AGV with no
task they are moved in a fixed interval to a neighboring zonenearer to the ASC which they are bound to if possible
Color relations of tokens in this partial model are listedin Table 4 Attributes not involved in this partial model areexcluded
Partial model of QC is shown in Figure 7 Positions of 11987511
are adjustedThere are only 2 transactions in this partial model
Transaction 11987910
is for the process that a QC picks up acontainer from an AGV In case that a QC is ready for anext container loading (119875
18not empty) and a loaded AGV
has reached the foot of that QC (token with the same QCnumber in 119875
15) this transaction fires consuming one token
in 11987518 11987515 and 119875
20 following firing conditions as shown in
Figure 7 In case that there are multiple loaded AGV at thefoot of the same QC the AGV carrying the container with aminimal container number will be consumed Firing time oftransaction 119879
10is fixed When the firing ends a new token is
created into 11987511
as an AGV with no task allocated to it andanother new token is created into 119875
19indicating that the QC
has received the containerTransaction 119879
11is used to simulate the time a QC trolley
spends in putting a container onto a vessel and turn backempty to the quayside Firing time of 119879
11is also fixed When
the firing ends a new token is created to 11987518 hence the QC
is ready for a next loadColor relations of tokens in this partial model are listed in
Table 5 Also attributes not involved in this partial model areexcluded
4 Optimization Approaches
The TCPN model proposed in the last section could beused as fitness functions of optimizationmethods evaluating
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
VV
11
A V
V V
P5
1998400d114
1998400d1612
1998400d129
1998400d1713
1998400d127
1998400d118
1998400d88
1998400d9984001316
1998400d998400913
1998400d9984001217
1998400d99840089
1998400d9984008141998400d998400713
1998400d54 1998400d998400412T4
P11
P9
P8
P17
P13
T8
P15
P16
P12 T7
T12
T13
T9 P14
[Min L(d54pos d164pos)]
M(11)998400d9984001311
M(11)998400d1113
B times A
B times A
[d88cid = d118 cid]
Figure 6 Partial model of AGV
Table 3 Color relations of tokens in partial model of ASCs
Token cid bid aid qid az qz1198891015840
1211988911cid mdash 119889
11aid 119889
11qid mdash 119889
11qz
1198891015840
13mdash 119889
41bid + 1 mdash mdash mdash mdash
1198891015840
24mdash 119889
32bid mdash mdash mdash mdash
1198891015840
3511988923cid mdash 119889
23aid 119889
23qid 119889
23az 119889
23qz
1198891015840
3611988923cid mdash 119889
23aid 119889
23qid mdash mdash
1198891015840
5711988965cid mdash 119889
65aid 119889
65qid mdash mdash
1198891015840
6811988976cid mdash 119889
76aid mdash mdash mdash
1198891015840
610mdash mdash 119889
76aid mdash mdash mdash
1198891015840
62011988976cid mdash 119889
76aid 119889
76qid mdash mdash
allocated (1198755not empty) and some AGV are with no tasks
(11987511not empty)119879
4fires to allocate a task to a nearest AGV by
consuming corresponding tokens in 1198755and 119875
11and creating
a new token into 11987512at once
Transaction 1198797is used to simulate the travel time of
an unloaded AGV with some task to the IO point of thecontainer Firing time of the transaction equals travel time ofAGV which could be determined since the current zone anddestination zone are both known for the AGV
Transaction 1198798is for the process in which an AGV picks
up a container at IO point In case that an AGV has reachedthe IO point (119875
13) and the container is placed there as well
(1198758) this transaction fires consuming the tokens in 119875
8and
11987513 and creating one new token in119875
9and11987514after a fixed time
interval respectively hence the AGV receives the container(11987514) and the number of containers that could be placed in
the IO point is added by 1 (1198758)
Transaction 1198799is used to simulate the travel time of a
loaded AGV to the foot of the destination QC Firing time ofthis transaction is calculated as in transaction 119879
7 after which
a new token is created into 11987515
Transactions 11987912and 119879
13are used to update the locations
of the AGVwith no tasks periodicallyThey work in a similarway as Transactions 119879
1and 119879
2 For all the AGV with no
task they are moved in a fixed interval to a neighboring zonenearer to the ASC which they are bound to if possible
Color relations of tokens in this partial model are listedin Table 4 Attributes not involved in this partial model areexcluded
Partial model of QC is shown in Figure 7 Positions of 11987511
are adjustedThere are only 2 transactions in this partial model
Transaction 11987910
is for the process that a QC picks up acontainer from an AGV In case that a QC is ready for anext container loading (119875
18not empty) and a loaded AGV
has reached the foot of that QC (token with the same QCnumber in 119875
15) this transaction fires consuming one token
in 11987518 11987515 and 119875
20 following firing conditions as shown in
Figure 7 In case that there are multiple loaded AGV at thefoot of the same QC the AGV carrying the container with aminimal container number will be consumed Firing time oftransaction 119879
10is fixed When the firing ends a new token is
created into 11987511
as an AGV with no task allocated to it andanother new token is created into 119875
19indicating that the QC
has received the containerTransaction 119879
11is used to simulate the time a QC trolley
spends in putting a container onto a vessel and turn backempty to the quayside Firing time of 119879
11is also fixed When
the firing ends a new token is created to 11987518 hence the QC
is ready for a next loadColor relations of tokens in this partial model are listed in
Table 5 Also attributes not involved in this partial model areexcluded
4 Optimization Approaches
The TCPN model proposed in the last section could beused as fitness functions of optimizationmethods evaluating
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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
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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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 4 Color relations of tokens in partial model of AGVs
Token cid aid qid aim pos az qz1198891015840
41211988954cid 119889
54aid 119889
54qid 119889
54az 119889
114pos 119889
114az 119889
54qz
1198891015840
713119889127
cid 119889127
aid 119889127
qid 119889127
aim 119889127
aim 119889127
az 119889127
qz1198891015840
89mdash 119889
88aid mdash mdash mdash mdash mdash
1198891015840
814119889118
cid 119889118
aid 119889118
qid 119889118
qz 11988988pos 119889
118az mdash
1198891015840
913119889129
cid mdash 119889129
qid 119889129
aim 119889129
aim 119889129
az 119889129
qz1198891015840
1311mdash mdash mdash mdash 119873(119889
1113pos 119889
1113az) 119889
1113az mdash
1198891015840
1316mdash mdash mdash mdash mdash mdash mdash
1198891015840
1217mdash mdash mdash mdash mdash mdash mdash
Table 5 Color relations of tokens in partial model of QCs
Token cid aid qid aim pos az qz1198891015840
1011mdash mdash mdash 119889
1310az 119889
1310qz mdash mdash
1198891015840
1019mdash mdash 119889
1810qid mdash mdash mdash mdash
1198891015840
11181198891911
qid
Q
Q
P18
1998400d1810
1998400d1911
1998400d9984001118
1998400d9984001019T11 P19
V
V
1998400d1310
1998400d2010
1998400d9984001011
P11
T10
P20
P15
[d1810 pid = d1310pid
d1310cid = d2010 cidd2010 cid = mind20cid]
Figure 7 Partial model of QC
the influences of candidate storage plans on the performanceof container handling system In this section two intelligentoptimization approaches based onParticle SwarmOptimiza-tion (PSO) and Genetic Algorithm (GA) are proposed tocomplete the simulation-based optimization method
41 Encoding Intelligent optimization is a category of meth-ods used to search for a solution of a problem as good aspossible These methods go through new candidate solutionscontinuously in a self-defined direction till an end conditionis fulfilled and the best candidate solution found is output asthe result Encoding is a first step of intelligent optimizationUsing some set of regulations solutions of a problem could beexpressed in a form convenient for optimization operations
Storage plans are encoded as arrays of real numbers ascandidate solutions of the SAPOBA Each number in thearray is for a stack to be allocated while the value of thisnumber indicates a slot for the stack For every real number ina code array the integer part is for the number of block thatthe stack is to be allocated in while the decimal part is forthe priority of this stack to get a slot in the block Stacks witha smaller decimal part tends to get a slot nearer to the ASChence the ASC cycle time of containers in this stack could beshorter In case that there is no slot left in a block stacks to be
Table 6 A coding example
Stack number 1 2 3 4 5 6 7Value 133 124 108 166 159 227 295
Table 7 Storage spaces available for the code
Block number Bay number Number of slots available1 1 21 2 22 1 22 2 2
allocated in this block will be reallocated to some other blockand the number of these stacks will also be changed
Table 6 is a coding example of 7 stacks Storage spaces arescattered in 2 blocks as shown in Table 7 In this examplestacks 2 and 3 are allocated to the 1st bay in block 1 stacks 1and 5 are allocated to the 2nd bay in block 1 and stacks 6 and7 are allocated to the 1st bay in block 2 There are no slot leftfor stack 4 hence it is allocated to the 2nd bay in block 2 andthe value of this stack is to be changed into 266 accordingly
42 PSOApproach Particle SwarmOptimizationworks witha fixed number of particles (called swarm) iteratively Aparticle moves to a possible solution in each iteration whilein a next iteration this particle moves to another The bestsolution of the swarm and best solutions of particles arekept over iterations which are used to guide the searchingdirection of particles Notations used in the PSO approachare listed below
119866 scale of the swarm total number of particlesindexed by 119892 1 le 119892 le 119866
119877 maximum number of iterations indexed by 119903 1 le119903 le 119872
119878 length of a particle equal to the number of stacksto be allocated
119890119903119892119904 the 119904th element of 119892th particle in generation 119903 119886
real number
119901119903119892 the 119892th particle in generation 119903 an array of 119868
elements
10 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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 Mathematical Problems in Engineering
V119903119892 speed of the 119892th particle in generation 119903 an array
of 119868 real numbers119901119887119892 best historical solution of particle 119892
119904119887 best historical solution of the swarm120575119903119892 fitness value of candidate solution of particle 119892 in
generation 1199031205751015840
119892 fitness value of best historical candidate solution
of particle 11989212057510158401015840 fitness value of best historical candidate solution
of the swarm120575119864 maximal fitness value of a satisfactory candidate
solution119873119864 maximum allowable number of successive itera-
tions in which best solution is not changed
Since candidate solutions are encoded as one-dimension-al arrays of real numbers a particle 119892 in iteration 119903 could bedefined as in the following equation
119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873
lowast 1 lt 119890
119903119892119904lt 119860 + 1 119890
119903119892119894
isin 119877
(5)
Notation 119860 is for the total number of ASC in ACTwhich is already defined in Section 32 In the beginningof optimization particles and their speeds are initialized inrandom In each iteration the current candidate solution ofeach particle is first evaluated by simulation output 120575 thenthe best solutions 119901119887
119892and 119904119887 are updated in case that better
solutions are found in the particles as shown in the followingtwo equations
119901119887119892=
119901119903119892 119903 = 1 or 120575
119903119892lt 1205751015840
119892
119901119887119892 otherwise
119904119887 =
119901119903119892 119903 = 1 or (120575
119903119892lt 12057510158401015840 120575119903119892= Min1198921015840
(1205751199031198921015840))
119904119887 otherwise
(6)
Iterations are used to generate new the candidate solutionof all the particles This operation is conducted followingequations below
V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)
119901119903119892= 119901119903119892+ V119903119892 (8)
Notations 120596V 120596119901 and 120596119904 in (7) are all coefficients valuedbetween 0 and 1Three end conditions are set in the approachas listed belowThePSOapproach ends if that one condition issatisfied in an iteration and the best candidate solution foundis output as a result of PSO approach
(1) Fitness value of the best candidate solution found isno worse than 120575119864
Swarm initialization
Start
Result output
Simulation evaluation
End condition met
Iteration
Simulation evaluation
End
No
Yes
Figure 8 Flowchart of simulation-based optimization using PSOapproach
(2) Current iteration number 119903 reaches the maximumiteration number 119877
(3) Best historical solutions has not been changed overthe past119873119864 iterations
The procedure of simulation-based optimization methodusing PSO approach is shown in Figure 8
43 GAApproach InGeneticAlgorithm candidate solutionsare treated as chromosomes of which the basic units aretreated as genes just as the real numbers in the code of a stor-age plan In the beginning a fixed number of chromosomesare created and treated as the initial group New chromo-somes are created from current chromosomes by crossoverand variation after which a number of chromosomes areselected into a new group with the size unchanged in a nextgeneration Due to the fact that definitions of chromosomesin GA approach and candidate solutions in PSO approach arethe same notations for swarms and particles are used here forgroups and chromosomes
Crossover is executed between two chromosomes asparents With the values of some genes in same positionschanged in pairs two new chromosomes are created fromtheir parents For example chromosomes 1 and 2 in genera-tion 1 crossover at the 5th gene to form twonew chromosomes3 and 4 This crossover could be expressed by formulas asfollows
11989013119904
=
11989011119904 119904 = 5
1199033 otherwise
11989014119904
11989012119904 119894 = 5
1199034 otherwise
(9)
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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
Mathematical Problems in Engineering 11
Table 8 Parameters of equipment1
Parameter Value Parameter ValueQC efficiency 40 movesh AGV moving speed 6msASC hoisting speed (unloaded) 1ms ASC hoisting speed (loaded) 05msASC moving speed 35ms
Time of an AGV picking up a container at IO point 10 s1These parameters have been used in the simulation program for the Next Generation Container Port Challenge (httpwwwisenusedusgnewsaward-mpa-winnershtml) Three authors of this paper participated in the challenge and their team is the final winner
1199033+ 1199034= 119890135
+ 119890145
1 lt 1199033 1199034lt 119860 + 1 119903
3 1199034isin 119877
(10)
Commas in the formulas above are used to separate digitsof different notations in subscripts Real numbers 119903
3and 1199034are
randomly generated under constraints as in formula (10)Variations are executed with single chromosomes In a
variation values of some genes in a chromosome will bereplaced by a random number hence a new chromosome iscreated
Selection is executed to the whole group of chromosomesof which the fitness values are already known In this paperthe best chromosome will always be selected while the restare selected following roulette strategy The better the fitnessvalue of a chromosome the larger the chance it is selected intoa next generation
End conditions in the GA approach is same as in the PSOapproach The procedure of simulation-based optimizationmethod using GA approach is shown in Figure 9
5 Experiment
The simulation-based optimization method is applied forrandom examples in two scenariosThere are 4QC 10 blocksand 10 ASC in the small scenario and 6 QC 15 blocks and15 ASC in the large scenario Three AGV are bound to thesame ASC in both scenarios A 15 times 4 grid of square zoneswith side-lengths of 36 meters (with reference to the satellitephoto of Maasvlakte Container terminal in Rotterdam) isestablished to hold the QC AGV and IO points of ASC asshown in Figure 4 QC 5 and QC 6 in this figure only existin scenario 2 as ASC 11ndash15 Containers are stored in blocksin the same size of 20 bays 6 rows and 5 tiers each yet theblocks are not included in Figure 10 ASC are all dedicated toone block Containers in the same block are always retrievedby the same ASC while containers in different blocks are not
Equipment parameters used in the experiment are listedin Table 8
Instances of outbound containers are generated as fol-lows In each scenario there are 40 containers (divided into8 stacks) to be handled by each QC per hour in the coming12 hours In case that the storage plan is properly made QCwaiting could be eliminated entirely and the storage plan isconsidered satisfactory Consequently there are 384 stacks ofcontainers to be allocated into 10 blocks in the small scenarioand 576 stacks to be allocated into 15 blocks in the largescenario as are the code lengths of them
Group initialization
Start
Result output
Simulation evaluation
End condition met
Crossover and variation
Simulation evaluation
End
No
Yes
Selection
Figure 9 Flowchart of simulation-based optimization using GAapproach
The simulation program of the TCPN model is realizedin Tecnomatix Plant Simulation 11 a professional software fordiscrete event simulations The PSO and GA approaches arerealized in the same environment as well When evaluating acandidate solution the code of the solution is first convertedinto a corresponding storage plan On this basis simulationruns and QC waiting times are calculated as the fitness valueof the candidate solution For the PSO approach parameters120596V 120596119901 120596119904 119877 and 119873119864 in the algorithm are assigned to 0908 08 100 and 20 respectively For the GA approach scaleof the group is set the same as the scale of swarm in PSOapproach Crossover is executed 5 times every generationand the probability of a gene to be selected to cross is 08Theprobability of a chromosome to be selected and varied is 015as the probability of a gene
Experiments are conducted separately in two scenariosFor each scenario 20 instances are generated in random Eachinstance is solved 5 timeswith simulation-based optimizationmethod using PSO approach first and then solved another 5times using GA approach The best fitness values of the 100
12 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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 Mathematical Problems in Engineering
QC 1 QC 2 QC 3 QC 4 QC 5 QC 6
ASC 1 ASC 2 ASC 3 ASC 4 ASC 5 ASC 6 ASC 7 ASC 8 ASC 9 ASC 10 ASC 11 ASC 12 ASC 13 ASC 14 ASC 15
Figure 10 Layout of two scenarios
PSOGA
5 9 13 17 21 25 29 33 37 41 45 49 53 57 611Iterationgeneration
0100200300400500600
Fitn
ess v
alue
(s)
Figure 11 Average best fitness values in the small scenario
Table 9 Rest results in both scenarios
Scenario 119879119860 (s) Approach 119873
0
Small 273 PSO 89GA 73
Large 386 PSO 82GA 69
optimization runs in every iterationgeneration are recordedand the mean values of average best fitness values in everyiterationgeneration (if not ended) of the two approaches areshown in Figures 11 and 12 Moreover attentions are paidto the number of satisfactory solutions found in the 100optimization runs using PSO and GA approaches (noted as1198730) and the average time span of simulation runs (119879119860)Figure 11 is the average best fitness values in the small
scenario in which the 119883-axis is for the iterationgenerationnumbers and the 119884-axis is for the fitness values (in seconds)The figure tells that GA approach outperforms PSO approachin the earlier stage of optimization for fast convergence whilein the later stage PSO approaches are more likely to find asatisfactory storage plan
Figure 12 is the results in the large scenario It couldbe told form the figure that GA approach outperformsPSO approach in the earlier stage of optimization for fastconvergence while in the later stage PSO approaches aremore likely to find a satisfactory storage plan as found in thesmall scenario
Number of satisfactory solutions (1198730) and average timespan of simulation runs (119879119860) are listed in Table 9 In the 100
PSOGA
0100200300400500600700800900
Fitn
ess v
alue
(s)
7 13 19 25 31 37 43 49 55 61 67 73 79 85 911Iterationgeneration
Figure 12 Average best fitness values in the large scenario
optimization runs of the small scenario PSO approach finds89 satisfactory solutions while GA approach only finds 73 Inthe 100 optimization runs of the large scenario PSO approachfinds 82 satisfactory solutions while GA approach only finds69 Average time span of simulation runs is 273 seconds inthe small scenario and 386 seconds in the large scenario
The following could be concluded from Table 9 andFigure 12
(1) Simulation-based optimizationmethod could be usedto solve the SAPOBA while in this paper a satisfac-tory solution could not always be guaranteed
(2) Due to the fact that simulation is quite time-consuming in evaluating candidate solutions fewersimulation runs are preferred for optimization runs
(3) PSO approach is more suitable than GA approach insimulation-based optimization for SAPOBA since itis more likely to output a satisfactory storage plan
6 Conclusion
Thispaper proposes a simulation-based optimizationmethodfor storage allocation problems of outbound containers inautomated container terminals which has rarely been men-tioned in the literature Considering that there are quitea number of concurrent and asynchronous processes inthe container handling system Timed-Colored-Petri-Netmethod is used to build up the simulation model which isused to obtain the average QCwaiting time in container han-dling in case that outbound containers are stacked accordingto storage plan Two optimization approaches using Genetic
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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
Mathematical Problems in Engineering 13
Algorithm and Particle Swarm Optimization respectivelyare proposed to form the complete simulation-based opti-mization method Experiments are conducted to verify theeffectiveness of the simulation-based optimization methodproposed and to get the PSO and GA approaches comparedResults of the experiments shows that PSO approach is morelikely to output a satisfactory solution althoughGA approachconverge faster in the early stage Further work could beverifying the simulation model in real automated containerterminals shortening the time span of simulation runs andimproving the performance of optimization approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of Min-istry of Education of the Peoplersquos Republic of China (no20133121110005) the Innovation Projects for Graduate Stu-dent at Shanghai Maritime University (no 2013ycx050) ldquoSci-entific Research Innovation Projectrdquo of Shanghai MunicipalEducation Commission (no 14ZZ140) the ldquoPhD InnovationProgramrdquo of ShanghaiMaritimeUniversity (no 2014ycx040)and the ldquoYang Fan Plan for Shanghai Youth Science and Tech-nology Talentrdquo of the Science andTechnologyCommission ofShanghai Municipality (no 15YF1404900)
References
[1] C Mi X He H Liu Y Huang and W Mi ldquoResearch on a fasthuman-detection algorithm for unmanned surveillance area inbulk portsrdquo Mathematical Problems in Engineering vol 2014Article ID 386764 17 pages 2014
[2] C Mi Y Shen W Mi and Y Huang ldquoShip identificationalgorithm based on 3D point cloud for automated ship loadersrdquoJournal of Coastal Research vol 73 pp 28ndash34 2015
[3] R Gujjula and H-O Gunther ldquoThe impact of storage blockassignment for import containers onAGVdispatching in highlyautomated seaport container terminalsrdquo in Proceedings of theIEEE International Conference on Industrial Engineering andEngineering Management (IEEM rsquo08) pp 1739ndash1743 SingaporeDecember 2008
[4] J Luo andYWu ldquoModelling of dual-cycle strategy for containerstorage and vehicle scheduling problems at automated containerterminalsrdquo Transportation Research Part E Logistics and Trans-portation Review vol 79 pp 49ndash64 2015
[5] R Dekker P Voogd and E van Asperen ldquoAdvanced methodsfor container stackingrdquo in Container Terminals and CargoSystems pp 131ndash154 Springer Berlin Germany 2007
[6] B Borgman E van Asperen and R Dekker ldquoOnline rules forcontainer stackingrdquo OR Spectrum vol 32 no 3 pp 687ndash7162010
[7] T Park R Choe Y Hun Kim and K Ryel Ryu ldquoDynamicadjustment of container stacking policy in an automated con-tainer terminalrdquo International Journal of Production Economicsvol 133 no 1 pp 385ndash392 2011
[8] M Yu and X Qi ldquoStorage space allocation models for inboundcontainers in an automatic container terminalrdquo European Jour-nal of Operational Research vol 226 no 1 pp 32ndash45 2013
[9] H J Carlo I F A Vis andK J Roodbergen ldquoStorage yard oper-ations in container terminals literature overview trends andresearch directionsrdquo European Journal of Operational Researchvol 235 no 2 pp 412ndash430 2014
[10] K H Kim and K T Park ldquoA note on a dynamic space-allocation method for outbound containersrdquo European Journalof Operational Research vol 148 no 1 pp 92ndash101 2003
[11] W Mi W Yan J He and D Chang ldquoAn investigation into yardallocation for outbound containersrdquoCOMPEL vol 28 no 6 pp1442ndash1457 2009
[12] Y Zhang Q Zhou Z Zhu and W Hu ldquoStorage planningfor outbound container on maritime container terminalsrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 320ndash325 IEEE Shenyang ChinaAugust 2009
[13] M Li and S Li ldquoAn effective heuristic for yard template designin land-scarce container terminalsrdquo in IEEE International Con-ference on Industrial Engineering and Engineering Management(IEEM rsquo11) pp 908ndash912 Singapore December 2011
[14] L Chen and Z Lu ldquoThe storage location assignment problemfor outbound containers in a maritime terminalrdquo InternationalJournal of Production Economics vol 135 no 1 pp 73ndash80 2012
[15] Y H Jeong K H Kim Y J Woo et al ldquoA simulation studyon a workload-based operation planning method in containerterminalsrdquo Industrial Engineeering ampManagement Systems vol11 no 1 pp 103ndash113 2012
[16] D-H Lee J G Jin and J H Chen ldquoSchedule template designand storage allocation for cyclically visiting feeders in containertransshipment hubsrdquo Transportation Research Record vol 2273no 1 pp 87ndash95 2012
[17] O Sharif and N Huynh ldquoStorage space allocation at marinecontainer terminals using ant-based controlrdquo Expert Systemswith Applications vol 40 no 6 pp 2323ndash2330 2013
[18] D T Eliiyi G Mat and B Ozmen ldquoStorage optimization forexport containers in the port of izmirrdquo PROMETmdashTraffic ampTransportation vol 25 no 4 pp 359ndash367 2013
[19] W Hu H Wang and Z Min ldquoA storage allocation algorithmfor outbound containers based on the outer-inner cellularautomatonrdquo Information Sciences vol 281 pp 147ndash171 2014
[20] L H Lee E P Chew K C Tan and Y Han ldquoAn optimizationmodel for storage yardmanagement in transshipment hubsrdquoORSpectrum vol 28 no 4 pp 539ndash561 2006
[21] Y Han L H Lee E P Chew and K Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008
[22] L P Ku L H Lee E P Chew and K C Tan ldquoAn optimisationframework for yard planning in a container terminal case withautomated rail-mounted gantry cranesrdquo OR Spectrum vol 32no 3 pp 519ndash541 2010
[23] L P Ku E P Chew L H Lee and K C Tan ldquoA novelapproach to yard planning under vessel arrival uncertaintyrdquoFlexible Services and Manufacturing Journal vol 24 no 3 pp274ndash293 2012
[24] X Jiang L H Lee E P Chew Y Han and K C Tan ldquoAcontainer yard storage strategy for improving land utilizationand operation efficiency in a transshipment hub portrdquoEuropeanJournal of Operational Research vol 221 no 1 pp 64ndash73 2012
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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
14 Mathematical Problems in Engineering
[25] S H Won X Zhang and K H Kim ldquoWorkload-based yard-planning system in container terminalsrdquo Journal of IntelligentManufacturing vol 23 no 6 pp 2193ndash2206 2012
[26] X Jiang E P Chew L H Lee and K Tan ldquoFlexible space-sharing strategy for storage yard management in a transship-ment hub portrdquo OR Spectrum vol 35 no 2 pp 417ndash439 2013
[27] L Zhen ldquoYard template planning in transshipment hubs underuncertain berthing time and positionrdquo Journal of the Opera-tional Research Society vol 64 no 9 pp 1418ndash1428 2013
[28] R McCormack and G Coates ldquoA simulation model to enablethe optimization of ambulance fleet allocation and base stationlocation for increased patient survivalrdquo European Journal ofOperational Research vol 247 no 1 pp 294ndash309 2015
[29] B Dengiz Y T Ic and O Belgin ldquoA meta-model based simula-tion optimization using hybrid simulation-analytical modelingto increase the productivity in automotive industryrdquo Mathe-matics and Computers in Simulation 2015
[30] J-F Cordeau P Legato R M Mazza and R Trunfio ldquoSim-ulation-based optimization for housekeeping in a containertransshipment terminalrdquo Computers and Operations Researchvol 53 pp 81ndash95 2015
[31] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989
[32] W M Zuberek ldquoTimed Petri nets definitions properties andapplicationsrdquoMicroelectronics Reliability vol 31 no 4 pp 627ndash644 1991
[33] J Liu X Ye and J Zhou ldquoTest purpose oriented IO confor-mance test selection with colored petri netsrdquo Journal of AppliedMathematics vol 2014 Article ID 645235 10 pages 2014
[34] H Zhang H Zhang M Gu and J Sun ldquoModeling a hetero-geneous embedded system in coloured Petri netsrdquo Journal ofAppliedMathematics vol 2014 Article ID 943094 8 pages 2014
[35] J Zhou C Sun W Fu J Liu L Jia and H Tan ldquoModelingdesign and implementation of a cloud workflow engine basedon anekardquo Journal of Applied Mathematics vol 2014 Article ID512476 9 pages 2014
[36] Y-S Weng Y-S Huang Y-L Pan and M Jeng ldquoDesign oftraffic safety control systems for railroads and roadways usingtimed Petri netsrdquoAsian Journal of Control vol 17 no 2 pp 626ndash635 2015
[37] C Li WWu Y Feng and G Rong ldquoScheduling FMS problemswith heuristic search function and transition-timed Petri netsrdquoJournal of Intelligent Manufacturing 2014
[38] D Briskorn A Drexl and S Hartmann ldquoInventory-baseddispatching of automated guided vehicles on container termi-nalsrdquo in Container Terminals and Cargo Systems pp 195ndash214Springer Berlin Germany 2007
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