research article simulation-based optimization for storage...

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


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


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


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


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


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


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


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


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


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


Table 5 Color relations of tokens in partial model of QCs


1011mdash mdash mdash 119889

1310az 119889

1310qz mdash mdash

1198891015840


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


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


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


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)


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


End condition met

Crossover and variation


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


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)


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


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


[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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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


C

A B

P1T1 P2

P3

T2 P4










1

1198752 1198753 1198754 and 119875


1and 119879

2) and five arcs


3


2and transaction


3and 119875

4are in two



QC

B

IO pointA







2

























119894119895119896= 0


119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896

isin 1 2 119870

(1)


sum

119894

119909119894119895119896le 119862119895119896 forall119895 119896 (2)


sum

119895119896

119909119894119895119896= 1 forall119894 (3)






1198891015840



as 1199111and 1199112



1199112




to simulation



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


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



M(11)998400d9984001311

M(11)998400d1113

B times A

B times A[d76aid =

d96aid]

[d88cid =d118 cid]

M(1 d1bid =

d41bid)998400d99840012




119905119901rdquo for






2is used to






18are for the QC








Min 120575 = Min 1119876sum

119902

(

119862119873119902

sum

119899=2

(119905119904119902119899minus 119905119890119902(119899minus1)

)) (4)


20is adjusted





1with the same



12) are created


11 and a new token (1198891015840

13) is


2are fired









2with




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


1998400d99840036

1998400d65 1998400d99840057T5 P7

T6

M(1 d1bid =

d41bid)998400d99840012

1998400d96

1998400d99840068

1998400d998400620P9

P20

P8

B times A

B times A

[d76aid = d96aid]




1198751


1198752


1198753


1198754


1198755



11987512


11987513



11987520





35) is







6at the


















8are adjusted





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


M(11)998400d9984001311

M(11)998400d1113

B times A

B times A

[d88cid = d118 cid]




1211988911cid mdash 119889

11aid 119889

11qid mdash 119889

11qz

1198891015840

13mdash 119889


1198891015840

24mdash 119889


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


1198891015840



1198891015840

62011988976cid mdash 119889

76aid 119889

76qid mdash mdash


(11987511not empty)119879



11and creating








8and







7 after which





1and 119879











in 11987518 11987515 and 119875



















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


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


1113pos 119889

1113az) 119889

1113az mdash

1198891015840


1198891015840





1310az 119889

1310qz mdash mdash

1198891015840



1198891015840

11181198891911

qid

Q

Q

P18

1998400d1810

1998400d1911

1998400d9984001118

1998400d9984001019T11 P19

V

V

1998400d1310

1998400d2010

1998400d9984001011

P11

T10

P20

P15

[d1810 pid = d1310pid

















real number


elements





generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










C

A B

P1T1 P2

P3

T2 P4










1

1198752 1198753 1198754 and 119875


1and 119879

2) and five arcs


3


2and transaction


3and 119875

4are in two



QC

B

IO pointA







2

























119894119895119896= 0


119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896

isin 1 2 119870

(1)


sum

119894

119909119894119895119896le 119862119895119896 forall119895 119896 (2)


sum

119895119896

119909119894119895119896= 1 forall119894 (3)






1198891015840



as 1199111and 1199112



1199112




to simulation



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


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



M(11)998400d9984001311

M(11)998400d1113

B times A

B times A[d76aid =

d96aid]

[d88cid =d118 cid]

M(1 d1bid =

d41bid)998400d99840012




119905119901rdquo for






2is used to






18are for the QC








Min 120575 = Min 1119876sum

119902

(

119862119873119902

sum

119899=2

(119905119904119902119899minus 119905119890119902(119899minus1)

)) (4)


20is adjusted





1with the same



12) are created


11 and a new token (1198891015840

13) is


2are fired









2with




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


1998400d99840036

1998400d65 1998400d99840057T5 P7

T6

M(1 d1bid =

d41bid)998400d99840012

1998400d96

1998400d99840068

1998400d998400620P9

P20

P8

B times A

B times A

[d76aid = d96aid]




1198751


1198752


1198753


1198754


1198755



11987512


11987513



11987520





35) is







6at the


















8are adjusted





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


M(11)998400d9984001311

M(11)998400d1113

B times A

B times A

[d88cid = d118 cid]




1211988911cid mdash 119889

11aid 119889

11qid mdash 119889

11qz

1198891015840

13mdash 119889


1198891015840

24mdash 119889


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


1198891015840



1198891015840

62011988976cid mdash 119889

76aid 119889

76qid mdash mdash


(11987511not empty)119879



11and creating








8and







7 after which





1and 119879











in 11987518 11987515 and 119875



















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


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


1113pos 119889

1113az) 119889

1113az mdash

1198891015840


1198891015840





1310az 119889

1310qz mdash mdash

1198891015840



1198891015840

11181198891911

qid

Q

Q

P18

1998400d1810

1998400d1911

1998400d9984001118

1998400d9984001019T11 P19

V

V

1998400d1310

1998400d2010

1998400d9984001011

P11

T10

P20

P15

[d1810 pid = d1310pid

















real number


elements





generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










QC

B

IO pointA







2

























119894119895119896= 0


119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896

isin 1 2 119870

(1)


sum

119894

119909119894119895119896le 119862119895119896 forall119895 119896 (2)


sum

119895119896

119909119894119895119896= 1 forall119894 (3)






1198891015840



as 1199111and 1199112



1199112




to simulation



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


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



M(11)998400d9984001311

M(11)998400d1113

B times A

B times A[d76aid =

d96aid]

[d88cid =d118 cid]

M(1 d1bid =

d41bid)998400d99840012




119905119901rdquo for






2is used to






18are for the QC








Min 120575 = Min 1119876sum

119902

(

119862119873119902

sum

119899=2

(119905119904119902119899minus 119905119890119902(119899minus1)

)) (4)


20is adjusted





1with the same



12) are created


11 and a new token (1198891015840

13) is


2are fired









2with




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


1998400d99840036

1998400d65 1998400d99840057T5 P7

T6

M(1 d1bid =

d41bid)998400d99840012

1998400d96

1998400d99840068

1998400d998400620P9

P20

P8

B times A

B times A

[d76aid = d96aid]




1198751


1198752


1198753


1198754


1198755



11987512


11987513



11987520





35) is







6at the


















8are adjusted





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


M(11)998400d9984001311

M(11)998400d1113

B times A

B times A

[d88cid = d118 cid]




1211988911cid mdash 119889

11aid 119889

11qid mdash 119889

11qz

1198891015840

13mdash 119889


1198891015840

24mdash 119889


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


1198891015840



1198891015840

62011988976cid mdash 119889

76aid 119889

76qid mdash mdash


(11987511not empty)119879



11and creating








8and







7 after which





1and 119879











in 11987518 11987515 and 119875



















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


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


1113pos 119889

1113az) 119889

1113az mdash

1198891015840


1198891015840





1310az 119889

1310qz mdash mdash

1198891015840



1198891015840

11181198891911

qid

Q

Q

P18

1998400d1810

1998400d1911

1998400d9984001118

1998400d9984001019T11 P19

V

V

1998400d1310

1998400d2010

1998400d9984001011

P11

T10

P20

P15

[d1810 pid = d1310pid

















real number


elements





generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119894119895119896= 0


119909119894119895119896| 119894 isin 1 2 119868 119895 isin 1 2 119869 119896

isin 1 2 119870

(1)


sum

119894

119909119894119895119896le 119862119895119896 forall119895 119896 (2)


sum

119895119896

119909119894119895119896= 1 forall119894 (3)






1198891015840



as 1199111and 1199112



1199112




to simulation



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


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



M(11)998400d9984001311

M(11)998400d1113

B times A

B times A[d76aid =

d96aid]

[d88cid =d118 cid]

M(1 d1bid =

d41bid)998400d99840012




119905119901rdquo for






2is used to






18are for the QC








Min 120575 = Min 1119876sum

119902

(

119862119873119902

sum

119899=2

(119905119904119902119899minus 119905119890119902(119899minus1)

)) (4)


20is adjusted





1with the same



12) are created


11 and a new token (1198891015840

13) is


2are fired









2with




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


1998400d99840036

1998400d65 1998400d99840057T5 P7

T6

M(1 d1bid =

d41bid)998400d99840012

1998400d96

1998400d99840068

1998400d998400620P9

P20

P8

B times A

B times A

[d76aid = d96aid]




1198751


1198752


1198753


1198754


1198755



11987512


11987513



11987520





35) is







6at the


















8are adjusted





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


M(11)998400d9984001311

M(11)998400d1113

B times A

B times A

[d88cid = d118 cid]




1211988911cid mdash 119889

11aid 119889

11qid mdash 119889

11qz

1198891015840

13mdash 119889


1198891015840

24mdash 119889


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


1198891015840



1198891015840

62011988976cid mdash 119889

76aid 119889

76qid mdash mdash


(11987511not empty)119879



11and creating








8and







7 after which





1and 119879











in 11987518 11987515 and 119875



















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


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


1113pos 119889

1113az) 119889

1113az mdash

1198891015840


1198891015840





1310az 119889

1310qz mdash mdash

1198891015840



1198891015840

11181198891911

qid

Q

Q

P18

1998400d1810

1998400d1911

1998400d9984001118

1998400d9984001019T11 P19

V

V

1998400d1310

1998400d2010

1998400d9984001011

P11

T10

P20

P15

[d1810 pid = d1310pid

















real number


elements





generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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


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



M(11)998400d9984001311

M(11)998400d1113

B times A

B times A[d76aid =

d96aid]

[d88cid =d118 cid]

M(1 d1bid =

d41bid)998400d99840012




119905119901rdquo for






2is used to






18are for the QC








Min 120575 = Min 1119876sum

119902

(

119862119873119902

sum

119899=2

(119905119904119902119899minus 119905119890119902(119899minus1)

)) (4)


20is adjusted





1with the same



12) are created


11 and a new token (1198891015840

13) is


2are fired









2with




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


1998400d99840036

1998400d65 1998400d99840057T5 P7

T6

M(1 d1bid =

d41bid)998400d99840012

1998400d96

1998400d99840068

1998400d998400620P9

P20

P8

B times A

B times A

[d76aid = d96aid]




1198751


1198752


1198753


1198754


1198755



11987512


11987513



11987520





35) is







6at the


















8are adjusted





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


M(11)998400d9984001311

M(11)998400d1113

B times A

B times A

[d88cid = d118 cid]




1211988911cid mdash 119889

11aid 119889

11qid mdash 119889

11qz

1198891015840

13mdash 119889


1198891015840

24mdash 119889


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


1198891015840



1198891015840

62011988976cid mdash 119889

76aid 119889

76qid mdash mdash


(11987511not empty)119879



11and creating








8and







7 after which





1and 119879











in 11987518 11987515 and 119875



















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


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


1113pos 119889

1113az) 119889

1113az mdash

1198891015840


1198891015840





1310az 119889

1310qz mdash mdash

1198891015840



1198891015840

11181198891911

qid

Q

Q

P18

1998400d1810

1998400d1911

1998400d9984001118

1998400d9984001019T11 P19

V

V

1998400d1310

1998400d2010

1998400d9984001011

P11

T10

P20

P15

[d1810 pid = d1310pid

















real number


elements





generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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


1998400d99840036

1998400d65 1998400d99840057T5 P7

T6

M(1 d1bid =

d41bid)998400d99840012

1998400d96

1998400d99840068

1998400d998400620P9

P20

P8

B times A

B times A

[d76aid = d96aid]




1198751


1198752


1198753


1198754


1198755



11987512


11987513



11987520





35) is







6at the


















8are adjusted





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


M(11)998400d9984001311

M(11)998400d1113

B times A

B times A

[d88cid = d118 cid]




1211988911cid mdash 119889

11aid 119889

11qid mdash 119889

11qz

1198891015840

13mdash 119889


1198891015840

24mdash 119889


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


1198891015840



1198891015840

62011988976cid mdash 119889

76aid 119889

76qid mdash mdash


(11987511not empty)119879



11and creating








8and







7 after which





1and 119879











in 11987518 11987515 and 119875



















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


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


1113pos 119889

1113az) 119889

1113az mdash

1198891015840


1198891015840





1310az 119889

1310qz mdash mdash

1198891015840



1198891015840

11181198891911

qid

Q

Q

P18

1998400d1810

1998400d1911

1998400d9984001118

1998400d9984001019T11 P19

V

V

1998400d1310

1998400d2010

1998400d9984001011

P11

T10

P20

P15

[d1810 pid = d1310pid

















real number


elements





generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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


M(11)998400d9984001311

M(11)998400d1113

B times A

B times A

[d88cid = d118 cid]




1211988911cid mdash 119889

11aid 119889

11qid mdash 119889

11qz

1198891015840

13mdash 119889


1198891015840

24mdash 119889


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


1198891015840



1198891015840

62011988976cid mdash 119889

76aid 119889

76qid mdash mdash


(11987511not empty)119879



11and creating








8and







7 after which





1and 119879











in 11987518 11987515 and 119875



















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


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


1113pos 119889

1113az) 119889

1113az mdash

1198891015840


1198891015840





1310az 119889

1310qz mdash mdash

1198891015840



1198891015840

11181198891911

qid

Q

Q

P18

1998400d1810

1998400d1911

1998400d9984001118

1998400d9984001019T11 P19

V

V

1998400d1310

1998400d2010

1998400d9984001011

P11

T10

P20

P15

[d1810 pid = d1310pid

















real number


elements





generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












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


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


1113pos 119889

1113az) 119889

1113az mdash

1198891015840


1198891015840





1310az 119889

1310qz mdash mdash

1198891015840



1198891015840

11181198891911

qid

Q

Q

P18

1998400d1810

1998400d1911

1998400d9984001118

1998400d9984001019T11 P19

V

V

1998400d1310

1998400d2010

1998400d9984001011

P11

T10

P20

P15

[d1810 pid = d1310pid

















real number


elements





generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













generation 1199031205751015840







119901119903119892= 119890119903119892119904| 1 le 119904 le 119878 119904 isin 119873

lowast 1 lt 119890

119903119892119904lt 119860 + 1 119890

119903119892119894

isin 119877

(5)




119901119887119892=

119901119903119892 119903 = 1 or 120575

119903119892lt 1205751015840

119892

119901119887119892 otherwise

119904119887 =

119901119903119892 119903 = 1 or (120575

119903119892lt 12057510158401015840 120575119903119892= Min1198921015840

(1205751199031198921015840))


(6)


V119903119892= 120596VV119903119892 + 120596119901 (119901119887119892 minus 119901119903119892) + 120596119904 (119904119887 minus 119901119903119892) (7)

119901119903119892= 119901119903119892+ V119903119892 (8)




Start

Result output


End condition met

Iteration


End

No

Yes







11989013119904

=

11989011119904 119904 = 5

1199033 otherwise

11989014119904

11989012119904 119894 = 5

1199034 otherwise

(9)





1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1199033+ 1199034= 119890135

+ 119890145

1 lt 1199033 1199034lt 119860 + 1 119903

3 1199034isin 119877

(10)


3and 1199034are





5 Experiment





Start

Result output


End condition met



End

No

Yes

Selection








PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













PSOGA


0100200300400500600

Fitn

ess v

alue

(s)




0







PSOGA

0100200300400500600700800900

Fitn

ess v

alue

(s)








6 Conclusion






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article simulation-based optimization for storage...

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