research article minimization of delay and travel time of yard...

Research ArticleMinimization of Delay and Travel Time ofYard Trucks in Container Terminals Using an ImprovedGA with Guidance Search

Z X Wang1 Felix T S Chan1 S H Chung1 and Ben Niu2

1Department of Industrial and Systems Engineering The Hong Kong Polytechnic University Hung Hom Hong Kong2College of Management Shenzhen University Shenzhen 518060 China

Correspondence should be addressed to Ben Niu drniubengmailcom

Received 4 June 2014 Revised 21 August 2014 Accepted 21 August 2014

Academic Editor Yan-WuWang

Copyright copy 2015 Z X Wang 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

Yard truck scheduling and storage allocation problems (YTS-SAP) are two important issues that influence the efficiency of acontainer terminal These two problems aim to determine the routing of trucks and proper storage locations for dischargingcontainers from incoming vessels This paper integrates YTS and SAP as a whole and tries to minimize the weighted summationof total delay and total yard trucks travel time A genetic algorithm (GA) is proposed to deal with the problem In the proposedGA guidance mutation approach and exhaustive heuristic for local searching are used in order to force the GA to converge fasterand be steadier To test the performance of the proposed GA both small scale and large scale cases are studied The results of thesecases are compared with CPLEX for the small scale cases Since this problem is an NP-hard problem which CPLEX cannot solvea simple GA is studied for comparison in large scale cases The comparison demonstrates that the proposed GA can obtain nearoptimal solutions in much shorter computational time for small scale cases In addition the proposed GA can obtain better resultsthan other methods in reasonable time for large scale cases

1 Introduction

Container terminal plays a crucial role in logistics networksunder the rapid development of globalization trade trans-porting goods from one country to another Efficiency ofa terminal determines its competitiveness in the terminalindustries Accordingly terminals have been devoting mucheffort in shortening the vessel staying time meanwhileincreasing the turnaround time by developing various deci-sion support systems [1] Terminal operations are usuallyclassified into quay operations (eg berth allocation quaycrane scheduling) and yard operations (eg yard truckscheduling yard crane scheduling and storage allocation)[2] The objective of this paper is to study yard truckscheduling and storage allocation problems (YTS-SAP)

YTS refers to the scheduling of yard trucks to servethe transportation of exportimport containers between thequay side and the yard side and the corresponding routeadoption Meanwhile SAP refers to the allocation of storage

locations for the import containers which are dischargedfrom incoming vessels These two problems are known to beone of the most critical terminal operations as they directlyinfluence the efficiency of the terminals [3] For exampleany delay of transporting an export container from the yardside to the quay side for the uploading operation definitelymay cause delay of vessel departure Thus this must beavoided Similarly an inadequate storage plan will inducelong transportation time overload certain yard crane whileidling others and so forth [4 5]

From the literature it is known that YTS and SAP arehighly interrelated Back to 2008 Lee et al [6] firstly proposedan integratedmodel simultaneously dealingwith theYTS andSAP However in that model loading operations from yardside to quay side are not being considered Later on Lee etal [2] further improved the integrated model by proposinga two-step approach in which the first step deals with thetruck-job assignment by using a hybrid insertion algorithmwhile the second step deals with the storage allocation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 710565 12 pageshttpdxdoiorg1011552015710565

2 Mathematical Problems in Engineering

problem In this model it is assumed that the number ofimport containers must be equal to the reserved storagelocation However this assumption may limit the practicalapplicationsTherefore in this paper we will further enhancethe model by bringing in the consideration of all availablestorage locations in the terminals In otherwords the numberof storage locations can be larger than the number of importcontainers This modified model will be more complicatedthan the previous one as the number of possible solutioncombinations increased dramatically

YTS is proved to be NP-hard by Bish et al [7] Thus theintegrated problem is also an NP-hard problem Accordinglythis paper proposed a new hybrid genetic algorithm (GA) todeal with the new integrated model which can enable us tohandle up to 300 containers with a reasonable computationaltime greatly improved from the 20 containers in Lee et alrsquos [2]modelThe paper is organized as follows Section 2 gives a lit-erature review Section 3 provides a mathematical model andproblem description Section 4 presents the proposed hybridGA Section 5 presents the computational experiments resultsand Section 6 concludes the paper

2 Literature Review

Operation research in the area of storage allocation has beenstudied by many researchers In the field of storage allocationof containers K H Kim and H B Kim [8] addressed theproblem of allocating storage space for import containersand a segregation policy was considered in which the authoranalysed cyclic and dynamic arrival rates The objectiveof this problem was to minimize the expected number ofrehandled cases K H Kim and H B Kim [9] discusseda method in which the optimal number of storage spaceand yard cranes for handling import containers could bedetermined The authors proposed a cost model for decisionmaking A deterministic model and a stochastic model wereproposed for the solution Zhang et al [10] studied thestorage allocation problem To solve the problem a rolling-horizon approach was proposed in which the problem wasdecomposed into two levels The first level aimed to balancetwo types of workloads among the blocks while the secondlevel focused on minimizing the total distance to transportthe containers between their storage blocks and the vesselberthing locations Lee et al [11] studied the integration ofterminal and yard allocation problem The objective was tominimize the handling cost of transhipment containers inmultiterminal systems A two-level heuristic algorithm wasproposed to deal with the problem

In the area of scheduling problems in the yard side Kimand Bae [12] discussed how to dispatch automated guidedvehicles (AGVs) by utilizing information on locations andtimes of future delivery tasks Amixed-integer programmingmodel was proposed and the problem was solved by aheuristic algorithm Ng et al [3] considered the problemof scheduling a fleet of trucks in a container terminalto minimize the makespan The loading and dischargingjobs had sequence-dependent processing times and differentready times The problem was formulated as a mixed integerprogram (MIP) and solved by using a GA Nguyen and Kim

[13] discussed the problem of dispatching automated liftingvehicles (ALV) and the problem was formulated in an MIPmodel similar to multiple travelling salesman problems withprecedence constraints and time windows The problem wassolved by a heuristic algorithm Hu et al [14] studied theperformance of three types of transporting machines whichare ground trolleys (GTs) transfer platforms (TPs) andframe trolleys (FTs) Zhao and Goodchild [15] explored thetruck travel time reliability and the predictability The truckrouting choices were analysed by examining the relationshipbetween the routing choice and route attributes Yan et al[16] investigated a knowledge-based system for yard cranescheduling problem The proposed system was capable ofmaking off-line planning and real-time scheduling Javanshirand Seyedalizadeh Ganji [17] studied the problem of yardcrane scheduling with noninterference constraints for singleblocksThe problemwas formulated as amathematicalmodeland solved by GA Javanshir et al [18] studied the problem ofyard crane scheduling ofmultiple blocks inmultiple planningperiods The problem was formulated as MIP model andsolved by Lingo He et al [19] proposed a model to solve theyard crane scheduling problem based on the rolling-horizontechnique A hybrid parallel GA was proposed to solve theproblem Cao et al [20] studied the integrated model foryard truck and yard crane scheduling problemsThe handlingsequence of outbound containers was determined in thismodel The model was formulated as a MIP programmingHomayouni and Tang [21] investigated the coordination ofcrane scheduling and vehicle routing A genetic algorithm isproposed for solving the mathematical model

The yard truck scheduling problem and the storage allo-cation problem were studied separately in the past Bish et al[7] firstly studied the two problems but solved each problemseparately Bish et al [7] considered the assumptions foreach container with a number of potential locations in theyard where it could be stored and the container was movedfrom the vessel to the yard by using a fleet of vehicleseach of which could carry one container at a time Theproblem was solved in two steps The first step was todetermine the location assignments by ignoring the vehicleschedule and the second step determined the vehicle schedulefor the location arrangements obtained from the first stepThe problem was solved by using a heuristic algorithmBish [22] further studied the problem of determining astorage location for each discharging container dispatchingvehicles to the containers and scheduling the unloadingand loading operations on each quay crane The objectivewas to minimize the maximum turnaround time of a setof vessels Bish et al [23] developed easily implementableheuristic algorithms for solving the problem studied in 2005Han et al [24] studied the yard truck scheduling and storageallocation problems as a whole in transhipment terminals Amathematical model was proposed and the model was solvedby dedicated heuristic algorithms Lee et al [6] proposedan integer programming model to deal with the problemof yard truck scheduling and storage allocation This paperconsidered the two problems as a whole instead of solvingeach aspect separately The objective is to reduce congestionand idling time of the yard trucks in order to decrease the

Mathematical Problems in Engineering 3

Sea sideSea side

Quay sideQuay crane

Quay side

Vessel

Empty truck

Loaded truck

Yard crane

Travel path Blocks

Storage block

Yard sideYard side

(a)(b)

Figure 1 Outline of a container terminal

makespan of the discharging containers Later on Lee etal [2] further extended the previous study and proposedanother integrated model for the yard truck scheduling andstorage allocation problem A hybrid insertion algorithmwasproposed for the solution and 20 containers are considered inthe computational experiments

From the literature reviews we can clearly know thatlittle work has been done on YTS-SAP considering all emptystorage locations in the yard side and no literature study YTS-SAP for large scale instances

3 Problem Description and Formulation

In this paper the problem is how to schedule a fleet oftrucks to load or discharge all the containers and determinethe storage location for the discharging containers First ofall we define the movement of a container from its originto its destination as a request denoted by 119894 and 119895 Thereare two types of requests considered in this study loadingrequests and discharging requests For loading request theorigin is the location where a container is loaded onto atruck by a yard crane from a storage block in the yard sidewhile the destination is the location of the quay crane bywhich a container is loaded onto the vessel For dischargingrequest the origin is the location of the quay crane by whicha container is unloaded from a vessel while the destination isthe location where a container is unloaded from a truck to astorage block by a yard crane in the yard side as in Figure 1

In terminal practice a soft time window for each request[119886119894119887119894) is already predetermined by the terminal operator as a

given data for YTS The soft time window is a period of timewhich consists of the earliest possible time 119886

119894and the due time

119887119894 A container can only be served after the earliest possible

time and 119887119894can be viewed as penaltyWe define the processing

time (loaded travel time) 119905119894as the period of time that a truck

processes a request 119894 from its origin to its destination and thesetup time (empty travel time) 119904

119894119895is the period of time that a

truck spends from the destination of the current request 119894 tothe origin of next request 119895 We also define the starting time119908119894of request 119894 as the time when it starts and the completion

time 119888119894of request 119894 is the time when it finishes The difference

between the completion time 119888119894and 119887119894of request 119894 is the delay

of request 119894 Completion time of request 119894 is 119888119894= 119908119894+ 119905119894 Delay

of request 119894 is 119889119894= max0 119888

119894minus119887119894 If request 119895 is the successive

request of request 119894 served by the same truck the starting timeof request 119895 is 119908

119895= max 119908

119894+ 119905119894+ 119904119894119895 119886119895

The following assumptions are made in this study(1) There are limited numbers of trucks and one truck

serves only one route We use dummy requests 119897119903and

119896119903to represent the initial and finial status of each

route(2) The trucks travel between any pair of locations along

the shortest and same path so the travel times aresymmetric For example in Figure 1 the truck travelsfrom location (a) to location (b) and location (b) tolocation (a) along the same and shortest travel path


(3) The number of storage locations is more than or equalto the number of discharging containers

(4) The yard crane and quay crane can serve the yardtruck once the yard truck arrives at the yard crane orquay crane This means that the yard crane and quaycrane are always available

(5) Congestions among yard trucks on a guide route arenot considered

The following notations are used to describe the problemstudied in this paper

Indices

119894 119895 Index of request 119894 = 119895

119903 Index of route

119901 119902 Index of location

119896 Index of storage location

Problem Data

120591119901119902 The travel time between each pair of locations (119901 119902)

119900119894 The origin of request 119894

119890119894 The destination of request 119894

120577119896 The location of storage location 119896

1205721 The weight of total delay of requests

1205722 The weight of total travel time of yard trucks

Set of Indices

119869minus The set of discharging requests with cardinality of|119869minus| = 119899minus

119869+ The set of loading requests with cardinality of |119869+| =119899+

119869 The set of all requests 119869 = 119869minus cup 119869+ with cardinality of|119869| = 119899

1198691015840 The union set of all requests and initial status 1198691015840 =119869 cup 119897119903

11986910158401015840 The union set of all requests and final status 11986910158401015840 = 119869 cup119896119903

119877 The set of routes |119877| = 119898

119872 The set of locations of the loading containers

119873 The set of locations of the discharging containers

119870 The set of the storage locations

119871 The union set of the locations of the loading contain-ers the locations of the discharging containers andthe storage locations 119871 = 119872 cup119873 cup 119870

Decision Variables

119909119894119896= 1 if container 119894 is allocated to storage location 119896

= 0 otherwise

119910119894119895= 1 if request 119894 is connected to request 119895 in the sameroute

= 0 otherwise

119908119894 The starting time of request 119894

119888119894 The completion time of request 119894

119889119894 The delay of request 119894

119905119894 The processing time of the yard trucks from the originof request 119894 to the destination of request 119894 119905

119894= 120591119900119894119890119894

ifrequest 119894 is a loading request 119905

119894= 120591119900119894120577119896

if request 119894 isa discharging request and allocated to storage location119896

119904119894119895 The setup time of the yard trucks from the destinationof request 119894 to the origin of request 119895 119904

119894119895= 120591119890119894119900119895


119894119895= 120591120577119896119900119895


The objective is to schedule the yards trucks and allo-cation of the loading and discharging containers aiming atminimizing the weighted summation of the total delay andthe total yard trucks travel time as model in (1) The problemformulation is modified based on the model provided by Leeet al [2] In our model we consider all the available storagelocation in the yard side however Lee et al [2] only considerthe reserved storage locations for discharging containerswhich means storage locations and discharging containersare equal in amount The revised model is as shown in thefollowing

Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119904119894119895119910119894119895) (1)

subject to

sum

119894isin119869minus

119909119894119896le 1 forall119896 isin 119870 (2)

sum

119896isin119870

119909119894119896= 1 forall119894 isin 119869

minus

(3)

sum

119895isin11986910158401015840

119910119894119895= 1 forall119894 isin 119869

1015840

(4)

sum

119894isin1198691015840

119910119894119895= 1 forall119895 isin 119869

10158401015840

(5)


Table 1 Sample data of containers

Container ID Origin Destination Time window 119886 (unit second) Time window 119887 (unit second) Type1 (1035 971) (60 665) 1362 1639 L2 (108 895) (1464 336) 716 1214 L3 (359 689) (748 1353) 284 634 L4 (148 391) (1246 312) 1320 1745 L5 (800 1180) (113 1287) 1201 1522 L6 (767 1015) 8 293 D7 (496 1210) 490 855 D8 (1485 414) 1160 1486 D9 (99 1440) 107 325 D10 (130 1498) 323 610 D

119908119894ge 119886119894

forall119894 isin 1198691015840cup 11986910158401015840 (6)

119889119894ge 119908119894+ 119905119894minus 119887119894

forall119894 isin 1198691015840cup 11986910158401015840 (7)

119908119895+119872(1 minus 119910

119894119895) ge 119908

119894+ 119905119894+ 119878119894119895

forall119894 isin 1198691015840 forall119895 isin 119869

10158401015840 (8)

119905119894= 120591119900119894119890119894

forall119894 isin 119869+ (9)

119905119894= sum

119896isin119870

120591119900119894120577119896

119909119894119896

forall119894 isin 119869minus

(10)

119878119894119895= 120591119890119894119900119895

forall119894 isin 119869+ forall119895 isin 119869 (11)

119878119894119895= sum

119896isin119870

120591119900119894120585119894

119909119894119896

forall119894 isin 119869minus forall119895 isin 119869 (12)

119909119894119896 119910119894119895isin 0 1 forall119894 isin 119869

1015840 forall119895 isin 119869

10158401015840 forall119896 isin 119870 (13)

119908119894isin R forall119894 isin 119869

1015840cup 11986910158401015840


119878119894119895isin R forall119894 isin 119869 forall119895 isin 119869

119889119894ge 0 forall119894 isin 119869

1015840cup 11986910158401015840

(14)

Constraints (2) ensure that each storage location will beassigned with at most one discharging container Constraints(3) ensure that each discharging container will be assignedwith one storage location Constraints (4) ensure that 119910

119894119895=

1 if the yard truck processes request 119895 after request 119894Constraints (5) ensure that 119910

119894119895= 1 if the yard truck processes

request 119894 before request 119895 Constraints (6) ensure that requestscan only be served after the earliest possible time Constraints(7) calculate the delay of each request Constraints (8) givethe relationship of the starting time of a request and that ofits successor Constraints (9) calculate the travel time of theloading requests Constraints (10) calculate the travel timeof the discharging requests Constraints (11) calculate thesetup time of the loading requests Constraints (12) calculatethe setup time of the discharging requests Constraints (13)ensure that 119909

119894119896and 119910

119894119895are binary variables Constraints (14)

define the range of values for 119908119894 119905119894 119904119894119895 and 119889

119894

We define one more decision variable 119897119894119895to linearize

the nonlinear form in the objective that is 119904119894119895119910119894119895 Then the

objective function can be rewritten as

Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119897119894119895) (15)

We also need to add two more constraints

119897119894119895ge 119910119894119895+ 119878119894119895minus 1 minus119872(1 minus 119910

119894119895) forall119894 isin 119869 forall119895 isin 119869

119897119894119895le 119872 sdot 119910

119894119895forall119894 isin 119869 forall119895 isin 119869

119897119894119895ge 0 forall119894 isin 119869 forall119895 isin 119869

(16)

Then the model can be formulated as a mixed integerlinear program as objective (15) subject to constraints (2)ndash(14) and (16)

4 Methodology

This paper proposes a hybrid GA to solve the yard truckscheduling and storage allocation problems

41 Chromosome Representation The chromosome repre-sents a potential solution of the yard truck scheduling andstorage allocation problems A gene represents a requestwhich contains the information of container ID time win-dow origin and destination of the request as shown inTable 1 and Figure 2 Each chromosome consists of |119869| + |119877|genes Each gene may be a positive number or a negativenumber A positive number represents a request and thesequence of the request prioritized from the left to the rightA negative number represents a route number Moreover therequests which are between two successive negative genesare allocated to the same truck

A chromosome of the proposed GA can be generatedusing the following steps

Step 1 Randomly allocate different storage locations for eachdischarging requestThen each gene contains information onthe origin destination and sequence of each request


Table 2 Sample data of storage locations for discharging containers

1 2 3 4 5 6 7(1039 592) (395 686) (18 1263) (635 357) (143 789) (113 1323) (321 563)

Table 3 An example of chromosome encoding

Request 9 6 10 7 minus1 3 2 8 5 1 4 minus2Sequence 1 2 3 4 1 2 3 4 5 6Truck Truck 1 Truck 2

Table 4 Decoding of chromosome illustrated in Table 3

Route 1 1198971rarr 9 rarr 6 rarr 10 rarr 7 rarr 119896

1

Route 2 1198972rarr 3 rarr 2 rarr 8 rarr 5 rarr 1 rarr 4 rarr 119896

2

Step 2 Randomly allocate all negative number genes into thechromosome and then the number of request in each routecan be calculated

Step 3 Randomly allocate all the requests to all the routesThen the requests and the requestsrsquo sequence in each routecan be obtained

Table 3 is an example of a representation of the proposedGA for scheduling two yard trucks (|119877| = 2) to processten requests (|119869| = 10) with a total length of |119877| + |119869|

of a chromosome The |119869| requests are represented by apermutation of the integers from 1 to |119869| The |119877| routes arerepresented by the integers from minus|119877| to minus1 The decodingprocedure is in the reverse order of encoding In the exampleshown in Table 3 the first yard truck would sequentiallyprocess requests 9 6 10 7 the second truck would processrequests 3 2 8 5 1 4 as shown in Table 4 As the sampledata shown in Tables 1 and 2 if the discharging container8 is allocated to storage location 1 the second truck maytravel the coordinates (359 689) (748 1353) (108 895)(1464 336) (1485 414) (1039 592) (800 1180) (113 1287)(1035 971) (60 665) (148 391) and (1246 312) one byone

42 Generation of Initial Pool In this paper the initial pool(with pool size 119875) will be generated by heuristic rules andrandom generation To increase the quality of the initial poolone of the chromosomes is generated according to the earliestpossible time combining with the nearest storage locationOne of the chromosomes will be generated by the earliest duetime combining with the nearest storage location The rest ofthe chromosomes are randomly generated

43 Mating Pool and Elitist Strategy The commonly usedroulette wheel selection approach is applied for forming amating pool Furthermore an elitist strategy is used to keepthe best chromosome(s) The stored best chromosome foundduring the evolutionwill replace the chromosomewith lowestfitness value

6

ID a6 b6 o6 e6

S1 S2 S3

Figure 2 An example of guidance mutation method one

Table 5 An example of crossover operation

Parent 1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198752

9 2 5 minus1 10 1 4 3 6 7 8 minus2After first timestep 2

Ω1

6 8 5 4 7 3 2 9Ω2

6 8 5 4 7 3 2 9After secondtime step 2

Ω1

1 4 10Ω2

1 2 4 5 7 8 10

Offspring 1198741

6 9 3 7 2 8 5 minus1 10 4 1 minus21198742

6 9 3 minus1 10 7 2 8 5 1 4 minus2

44 Fitness Value The objective is to minimize the weightedsummation of the total delay and the total yard trucks traveltime Thus the fitness value of a chromosome can be thereciprocal of its objective function value as shown in (17)In this way the best chromosome which corresponds to thescheduling of the trucks and the allocation of the dischargingcontainers withminimumweighted summation of total delayand total travel time can be found

Fitness =1

119885 (17)

45 Crossover Operation Many studies (eg [25ndash28]) haveshown that instance-specified information can make theGA searching process more effective In the YTS-SAP theinstance-specified information is the requestrsquos earliest startingtime the requestrsquos due time the requestrsquos processing time andthe setup time between the two requests In the proposedGAthis instance-specified information tries to be inherited withthe crossover operation Consider the crossover operation oftwo parents 119875

1and 1198752to reproduce two offspring 119874

1and 119874

2

The procedure of the proposed crossover operation is shownin the following steps Table 5 shows an example of crossoverand the example uses the data shown in Table 1

Step 1 Add all the requests in route one of both parent1198751and

parent1198752into an empty requests setΩ

1 Delete the duplicated

requests in Ω1 Let setΩ

2be the same asΩ

1

Step 2 Rank the requests in Ω1in nondecreasing order of

their earliest starting time and let set Φ be the ranked set


9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 5 7 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10

Figure 3 An example of guidance mutation method two

Rank the requests in Ω2in nondecreasing order of their due

time and let set Ψ be the ranked set

Step 3 Insert the requests in the corresponding route in 1198741

according to their order in set Φ and delete the insertedrequests from set Ω

1 Then insert the requests in the

corresponding route in 1198742according to their order in set Ψ

and delete the inserted requests from setΩ2

Step 4 Add all the requests in the next route for both parent1198751and parent 119875

2into set Ω

1 Then delete the duplicated

requests and delete the requests which have been inserted in1198741 Add all the requests in the next route for both parent 119875

1

and parent1198752into setΩ

2Then delete the duplicated requests

and delete the requests which have been inserted in 1198742

Step 5 Repeat Steps 2ndash4 until all the routes are assigned

46 Fine Local Searching To make the GA converge fasterand be steadier an exhaustive heuristic method [29 30] isadoptedThe exhaustive heuristic method is used to reinforcethe GArsquos local searching ability In one part of a chromosomea set of continuous genes is selected as a segment and thenumber of genes formed in the segment is set to be 5 asadopted by Chung et al [29] This method is adopted in eachchromosome part such that each chromosome for each of thetrucks in the exhaustive searching process will be executedonce Take the chromosome shown in Table 3 for examplethe first part of the chromosome contains four genes that arenot enough to form a segment and then the local searchingwill not be employed for the first part of the chromosomeIf the genes 2 8 5 1 and 4 which are in the second partof the chromosome are randomly selected as a segment allcombinations of the containers sequences will be tested andthen the one with best fitness value will be recorded

47 Mutation Method (1)-Simple Mutation Operation Muta-tion operation can help the GA prevent premature con-vergence and find the global optimal solution In order toevaluate the performance of the proposed hybrid GA asimple GA is used as a comparison In the proposed simpleGA each chromosome contains three types of informationstorage locations of the discharging containers the sequenceof requests in each route and the amount of requests in eachrouteThus each chromosome can be mutated in three waysThe first way is to randomly choose a discharging request andchange the requestrsquos storage location into another one which

Table 6 An example of mutation of the second way

1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 9 3 2 minus1 1 7 10 minus2

Table 7 An example of mutation of the third way

1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 3 2 minus1 1 9 7 10 minus2

is an empty storage location The second way is to randomlyselect two positions then swap the requests on these positionsas shown in the example in Table 6 The gene 7 and gene9 are swapped The third way is to change the amount ofrequests in the two routes which randomly selects a requestin a truck and inserts the request in another truck as shown inan example in Table 7 Gene 7 is inserted between gene 9 andgene 10 Each of the three mutation methods will be appliedonce during one mutation operation

48 Mutation Method (2)-Mutation Ways with Guidance Inthe proposed hybrid GA new mutation ways with guidanceinstead of the simple mutation ways will be adopted Duringthe mutation of the storage location a discharging requestis randomly selected first Then all the storage locations atwhich the requestrsquos travel time is within the requestrsquos due timeare selected as a set for example storage locations 1 2 and3 as shown in Figure 2 Finally randomly choose a storagelocation in the set to replace the origin storage location

For themutationway of changing two request srsquo positionsa request is randomly selected in one truck recording theposition 119898 of the request Then randomly select anotherrequest in the range of119898+119899 to119898minus119899 in another truck where 119899is a positive integer Finally swap these two requests Figure 3is an example of this guidance mutation method request 10is selected to swap with another request As request 10 is thethird request in the first part of the chromosome 119898 is equalto 3 If 119899 is set to be 2 another request is randomly selectedbetween request 3 and request 1 In this paper 119899 is set to be 3

For the mutation way of changing the amount of request119904 in the two trucks a request is randomly selected in onetruck recording the position m of the request Then insertthe request in the range of 119898 + 119899 to 119898 minus 119899 in another truckwhere 119899 is a positive integer Figure 4 is an example of thisguidance mutation method request 10 is selected to swapwith anther request As request 10 is the third request in the


9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 57 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10

Figure 4 An example of guidance mutation method three

Start

Generate initial pool

Generate mating pool

Crossover operation

Roulette wheel selection

Mutation operation

Fine local search

Elitist strategy

Check if the number of generations is equal to the

upper bound

End

Yes

No

Figure 5 The flowchart of the proposed GA

first part of the chromosome119898 is equal to 3 If 119899 is set to be 2request 10 is randomly inserted between request 3 and request1 In this paper 119899 is set to be 3 in order to avoid large changeof chromosomes The details of the proposed hybrid GA aregiven as shown in Figure 5

5 Computational Experiments

In this section a series of computational experiments are usedto evaluate the performance of the proposed GA The GA iscoded by using Java Language and executed on a PC withIntel Core i7 34GHz and 8GB RAM Instances used in theexperiments are created based on the following criteria

(1) Both the origin and destination of the loading con-tainers the origin of the discharging containersand the storage locations are generated througha two-dimensional uniform distribution in the squarefrom (0 0) to (1500 1500) (unit meter)

(2) The earliest start time of the requests is randomlygenerated from a uniform distribution of 119880(0 1500)(unit second) and the length of time window ofrequests is generated from a uniform distribution of119880(200 500) (unit second)

(3) The trucks travel at the speed of 1111ms (40 kmh)We also assume the twoweight parameters 120572

1and 1205722have the

relation of 1205721+ 1205722= 1 and 120572

1is equal to 06 as described by

Lee et al [2]

51 Small Scale Problems For small scale problems a simpleGA which is the hybrid GA without exhaustive heuristicand guidance mutation is used for comparison with theMIP model solved by CPLEX The parameters of the simpleGA are set as population size 10 crossover rate 119875

11986208

mutation rate 119875119872

1 and maximum number of generations2000 The number of routes is set as two The hybrid GA isalso compared with the MIP model solved by CPLEX The


Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size

(loadingtimesdischarging

timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours


Table 9 Number of containers and storage locations used in theinstances

Number ofloading

containers

Number ofdischargingcontainers

Number ofstoragelocations

100 containers 60 40 100200 containers 100 100 140300 containers 160 140 200

Table 10 Criterion of generating earliest possible time and due timefor instances in large scale

Number ofdistributions Earliest possible time Due time

1 Uniform distribution Uniform distribution2 Normal distribution Uniform distribution3 Exponential distribution Uniform distribution4 Uniform distribution Normal distribution

Table 11 Computational time and generation GA used

Simple GA Hybrid GACPU (s) Generation CPU (s) Generation

100 containers 22 10000 74 1000200 containers 178 30000 228 1300300 containers 375 60000 382 1500

parameters of the hybrid GA are the same as the simpleGA except that maximum number of generations is set to200

As is shown in Table 8 it is evident that the simpleGA can obtain the optimal solution in reasonable time inthe first four cases Due to the interacting of yard truckscheduling problem and storage allocation problem CPLEXrequires hours to solve each single instance but the simpleGA as a comparison only uses a few seconds to solve theproblem For the last six instances the simple GA can obtainthe near optimal solution and the average gap between thesimple GA and the optimal solution obtained by Branch andBound coded in CPLEX is computed at about 435 Withthe instances size becoming larger the gap also becomeslarger The simple GA performs poorly with the increasingof instance size However the performance of the simple GAis acceptable from the practical point of view On the otherhand the hybrid GA can always obtain optimal solutionsbecause of guidance mutation and exhaustive heuristic forlocal searching As the maximum number of generations issmaller than the simple GA the hybrid GA is faster than thesimple GA in the first six instances However the hybrid GAneeds more time than the simple GA when the instance scalebecomes larger

52 Large Scale Problems To evaluate the performance of theproposed hybrid GA in large scale problems the simple GAis applied as a comparison for the hybrid GA

Table 12 Computational results of random instances in large scale

Number of containersCriteria offorminginstances

SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17

Four different kinds of distribution combinations of theearliest possible time and the due time are applied as thecriteria of generating instances to increase the variety of the


instancesWewill also change the number of available trucksThe number of trucks is set as 3 at first and then the numberof trucks is set as 6 for the fifth kind of criteriaThree differentkinds of instances with different sizes are formed by usingeach of the five criteria The criteria of the instances arecreated as shown in Tables 9 and 10 The parameters of theproposed hybrid GA for large scale are set as population size10 crossover rate 119875

11986208 and mutation rate 119875

11987209 Table 10

also shows the number of generations which is long enoughto attain a steady solution and the computational time of theGA

As shown in Tables 11 and 12 the proposed hybrid GAcan obtain the best results and the computational time is alittle longer than the simple GA The lowest gaps betweenthe simple GA and the new hybrid GA are 1 1 and3 respectively for 100 containers 200 containers and 300containers The highest gaps between the simple GA and thenew hybrid GA are 35 44 and 17 respectively for 100containers 200 containers and 300 containers The averagegaps between the simple GA and the hybrid GA are 11 11and 7 respectively for 100 containers 200 containers and300 containers Since the hybrid GA has stronger local searchability and themutation operation will not be totally randomthe results of the hybrid GA are better than the simple oneHowever the exhaustive heuristic is time consuming and itwill take the hybrid genetic more time to find a solution

6 Conclusions and Future Work

Yard truck scheduling and storage allocation are two impor-tant problems for container terminals to enhance theiroperation efficiency In recent year Lee et al [6] proposed anintegrated model simultaneously solving the two problemsand later on they further enhanced themodel in Lee et al [6]We base on themodel in Lee et al [6] and further improve themodel by considering the situation that the number of avail-able storage locations is not equal to the number of importcontainers Such improvement can make the model morepractical As the problem complexity increases dramaticallya new hybrid GA with exhaustive heuristic and guidancemutation is proposed The crossover operation of proposedGA is based on the information of a requestrsquos ready time anddue timeThemutation operator combines three new ways ofmutation approach To evaluate and demonstrate the qualityof the proposed hybrid GA both a simple GA and the hybridGA are compared with the MIP model solved by CPLEX insmall scale problems and then the proposed hybrid genetic iscompared with the simple GA by using large scale instancesIt is proven that the simple GA and the hybrid GA can obtainnear optimal solutions in reasonable time by using a series ofcomputational experiments in small size problems For largescale problems 100 200 and 300 containers with differentnumbers of storage locations and trucks are studied Theresults demonstrated that the proposed hybridGA can obtainthe best solutions compared to the simple GA method

In this paper the number of vehicles and storage locationsare assumed to be given Given this information yard truckrouting and storage location for discharging containers aredetermined However in practical situation the number of

trucks can be flexible and the number of storage locationsmay dynamically change throughout the operating horizonTherefore the amount of trucks and storage locations can beconsidered as variables in the future work Another potentialfurther research topic is to incorporate multilayer containerstorage in yard side Combined with the current model theseare expected to give a more realistic description of containerterminal operations

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Teaching CompanyScheme project (Project no ZW1H (TCS162)) The HongKong Polytechnic University Research Committee (Projectno G-UB03) for financial and technical support a grantfrom The Hong Kong Scholars Program Mainland-HongKong Joint Postdoctoral Fellows Program (Project no G-YZ24) and The National Natural Science Foundation ofChina (Grants nos 71471158 and 71271140) The authors alsowould like to thank The Hong Kong Polytechnic UniversityResearch Committee for financial and technical support

References

[1] D Steenken S Voszlig and R Stahlbock ldquoContainer terminaloperation and operations research a classification and literaturereviewrdquo OR Spectrum vol 26 no 1 pp 3ndash49 2004

[2] D-H Lee J X Cao Q Shi and J H Chen ldquoA heuristicalgorithm for yard truck scheduling and storage allocationproblemsrdquo Transportation Research E Logistics and Transporta-tion Review vol 45 no 5 pp 810ndash820 2009

[3] W C Ng K L Mak and Y X Zhang ldquoScheduling trucksin container terminals using a genetic algorithmrdquo EngineeringOptimization vol 39 no 1 pp 33ndash47 2007

[4] C Zhang Y-W Wan J Liu and R J Linn ldquoDynamiccrane deployment in container storage yardsrdquo TransportationResearch Part B Methodological vol 36 no 6 pp 537ndash5552002

[5] 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

[6] D H Lee J X Cao and Q Shi ldquoIntegrated model for truckscheduling and storage allocation problem at contain termi-nalsrdquo in Proceeding of TRB 87th Annual Meeting Compendiumof Papers DVD 2008

[7] E K Bish T Leong C Li J W C Ng and D Simchi-LevildquoAnalysis of a new vehicle scheduling and location problemrdquoNaval Research Logistics vol 48 no 5 pp 363ndash385 2001

[8] K H Kim and H B Kim ldquoSegregating space allocationmodels for container inventories in port container terminalsrdquoInternational Journal of Production Economics vol 59 no 1ndash3pp 415ndash423 1999

[9] K H Kim and H B Kim ldquoThe optimal sizing of the storagespace and handling facilities for import containersrdquoTransporta-tion Research BMethodological vol 36 no 9 pp 821ndash835 2002


[10] C Zhang J Liu Y-W Wan K G Murty and R J LinnldquoStorage space allocation in container terminalsrdquo Transporta-tion Research B vol 37 no 10 pp 883ndash903 2003

[11] D-H Lee J G Jin and J H Chen ldquoTerminal and yardallocation problem for a container transshipment hub withmultiple terminalsrdquo Transportation Research E Logistics andTransportation Review vol 48 no 2 pp 516ndash528 2012

[12] K H Kim and J W Bae ldquoA look-ahead dispatching methodfor automated guided vehicles in automated port containerterminalsrdquo Transportation Science vol 38 no 2 pp 224ndash2342004

[13] V D Nguyen and K H Kim ldquoA dispatching method for auto-mated lifting vehicles in automated port container terminalsrdquoComputers and Industrial Engineering vol 56 no 3 pp 1002ndash1020 2009

[14] H Hu B K Lee Y Huang L H Lee and E P ChewldquoPerformance analysis on transfer platforms in frame bridgebased automated container terminalsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 593847 8 pages 2013

[15] W Zhao and A V Goodchild ldquoTruck travel time reliability andprediction in a port drayage networkrdquoMaritime Economics andLogistics vol 13 no 4 pp 387ndash418 2011

[16] W Yan Y Huang D Chang and J He ldquoAn investigationinto knowledge-based yard crane scheduling for containerterminalsrdquo Advanced Engineering Informatics vol 25 no 3 pp462ndash471 2011

[17] H Javanshir and S R SeyedalizadehGanji ldquoYard crane schedul-ing in port container terminals using genetic algorithmrdquo Journalof Industrial Engineering International vol 6 no 11 pp 39ndash502010

[18] H Javanshir S Ghomi and M Ghomi ldquoInvestigating trans-portation system in container terminals and developing a yardcrane schedulingmodelrdquoManagement Science Letters vol 2 no1 pp 171ndash180 2012

[19] J He D Chang W Mi and W Yan ldquoA hybrid parallel geneticalgorithm for yard crane schedulingrdquo Transportation ResearchE Logistics and Transportation Review vol 46 no 1 pp 136ndash155 2010

[20] J X Cao D-H Lee J H Chen and Q Shi ldquoThe inte-grated yard truck and yard crane scheduling problem bendersrsquodecomposition-based methodsrdquo Transportation Research PartE Logistics and Transportation Review vol 46 no 3 pp 344ndash353 2010

[21] S M Homayouni and S H Tang ldquoMulti objective optimizationof coordinated scheduling of cranes and vehicles at containerterminalsrdquo Mathematical Problems in Engineering vol 2013Article ID 746781 9 pages 2013

[22] E K Bish ldquoA multiple-crane-constrained scheduling problemin a container terminalrdquo European Journal of OperationalResearch vol 144 no 1 pp 83ndash107 2003

[23] E K Bish F Y Chen Y T Leong B L Nelson J W C Ngand D Simchi-Levi ldquoDispatching vehicles in a mega containerterminalrdquo OR Spectrum vol 27 no 4 pp 491ndash506 2005

[24] Y Han L H Lee E P Chew and K C Tan ldquoA yard storagestrategy forminimizing traffic congestion in amarine containertransshipment hubrdquo OR Spectrum vol 30 no 4 pp 697ndash7202008

[25] J L Blanton Jr and R L Wainwright ldquoMultiple vehicle routingwith time and capacity constraints using genetic algorithmsrdquoin Proceedings of the 5th International Conference on GeneticAlgorithms pp 452ndash459 1993

[26] P W Poon and J N Carter ldquoGenetic algorithm crossoveroperators for ordering applicationsrdquo Computers and OperationsResearch vol 22 no 1 pp 135ndash147 1995

[27] P Larranaga C M H Kuijpers R H Murga I Inza andS Dizdarevic ldquoGenetic algorithms for the travelling salesmanproblem a review of representations and operatorsrdquo ArtificialIntelligence Review vol 13 no 2 pp 129ndash170 1999

[28] C Moon J Kim G Choi and Y Seo ldquoAn efficient geneticalgorithm for the traveling salesman problem with precedenceconstraintsrdquo European Journal of Operational Research vol 140no 3 pp 606ndash617 2002

[29] S H Chung F T S Chan and W H Ip ldquoMinimization oforder tardiness through collaboration strategy in multifactoryproduction systemrdquo Systems Journal IEEE vol 5 no 1 pp 40ndash49 2011

[30] M Palpant C Artigues and P Michelon ldquoLSSPER solvingthe resource-constrained project scheduling problemwith largeneighbourhood searchrdquo Annals of Operations Research vol 131no 1ndash4 pp 237ndash257 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


problem In this model it is assumed that the number ofimport containers must be equal to the reserved storagelocation However this assumption may limit the practicalapplicationsTherefore in this paper we will further enhancethe model by bringing in the consideration of all availablestorage locations in the terminals In otherwords the numberof storage locations can be larger than the number of importcontainers This modified model will be more complicatedthan the previous one as the number of possible solutioncombinations increased dramatically

YTS is proved to be NP-hard by Bish et al [7] Thus theintegrated problem is also an NP-hard problem Accordinglythis paper proposed a new hybrid genetic algorithm (GA) todeal with the new integrated model which can enable us tohandle up to 300 containers with a reasonable computationaltime greatly improved from the 20 containers in Lee et alrsquos [2]modelThe paper is organized as follows Section 2 gives a lit-erature review Section 3 provides a mathematical model andproblem description Section 4 presents the proposed hybridGA Section 5 presents the computational experiments resultsand Section 6 concludes the paper

2 Literature Review

Operation research in the area of storage allocation has beenstudied by many researchers In the field of storage allocationof containers K H Kim and H B Kim [8] addressed theproblem of allocating storage space for import containersand a segregation policy was considered in which the authoranalysed cyclic and dynamic arrival rates The objectiveof this problem was to minimize the expected number ofrehandled cases K H Kim and H B Kim [9] discusseda method in which the optimal number of storage spaceand yard cranes for handling import containers could bedetermined The authors proposed a cost model for decisionmaking A deterministic model and a stochastic model wereproposed for the solution Zhang et al [10] studied thestorage allocation problem To solve the problem a rolling-horizon approach was proposed in which the problem wasdecomposed into two levels The first level aimed to balancetwo types of workloads among the blocks while the secondlevel focused on minimizing the total distance to transportthe containers between their storage blocks and the vesselberthing locations Lee et al [11] studied the integration ofterminal and yard allocation problem The objective was tominimize the handling cost of transhipment containers inmultiterminal systems A two-level heuristic algorithm wasproposed to deal with the problem

In the area of scheduling problems in the yard side Kimand Bae [12] discussed how to dispatch automated guidedvehicles (AGVs) by utilizing information on locations andtimes of future delivery tasks Amixed-integer programmingmodel was proposed and the problem was solved by aheuristic algorithm Ng et al [3] considered the problemof scheduling a fleet of trucks in a container terminalto minimize the makespan The loading and dischargingjobs had sequence-dependent processing times and differentready times The problem was formulated as a mixed integerprogram (MIP) and solved by using a GA Nguyen and Kim

[13] discussed the problem of dispatching automated liftingvehicles (ALV) and the problem was formulated in an MIPmodel similar to multiple travelling salesman problems withprecedence constraints and time windows The problem wassolved by a heuristic algorithm Hu et al [14] studied theperformance of three types of transporting machines whichare ground trolleys (GTs) transfer platforms (TPs) andframe trolleys (FTs) Zhao and Goodchild [15] explored thetruck travel time reliability and the predictability The truckrouting choices were analysed by examining the relationshipbetween the routing choice and route attributes Yan et al[16] investigated a knowledge-based system for yard cranescheduling problem The proposed system was capable ofmaking off-line planning and real-time scheduling Javanshirand Seyedalizadeh Ganji [17] studied the problem of yardcrane scheduling with noninterference constraints for singleblocksThe problemwas formulated as amathematicalmodeland solved by GA Javanshir et al [18] studied the problem ofyard crane scheduling ofmultiple blocks inmultiple planningperiods The problem was formulated as MIP model andsolved by Lingo He et al [19] proposed a model to solve theyard crane scheduling problem based on the rolling-horizontechnique A hybrid parallel GA was proposed to solve theproblem Cao et al [20] studied the integrated model foryard truck and yard crane scheduling problemsThe handlingsequence of outbound containers was determined in thismodel The model was formulated as a MIP programmingHomayouni and Tang [21] investigated the coordination ofcrane scheduling and vehicle routing A genetic algorithm isproposed for solving the mathematical model

The yard truck scheduling problem and the storage allo-cation problem were studied separately in the past Bish et al[7] firstly studied the two problems but solved each problemseparately Bish et al [7] considered the assumptions foreach container with a number of potential locations in theyard where it could be stored and the container was movedfrom the vessel to the yard by using a fleet of vehicleseach of which could carry one container at a time Theproblem was solved in two steps The first step was todetermine the location assignments by ignoring the vehicleschedule and the second step determined the vehicle schedulefor the location arrangements obtained from the first stepThe problem was solved by using a heuristic algorithmBish [22] further studied the problem of determining astorage location for each discharging container dispatchingvehicles to the containers and scheduling the unloadingand loading operations on each quay crane The objectivewas to minimize the maximum turnaround time of a setof vessels Bish et al [23] developed easily implementableheuristic algorithms for solving the problem studied in 2005Han et al [24] studied the yard truck scheduling and storageallocation problems as a whole in transhipment terminals Amathematical model was proposed and the model was solvedby dedicated heuristic algorithms Lee et al [6] proposedan integer programming model to deal with the problemof yard truck scheduling and storage allocation This paperconsidered the two problems as a whole instead of solvingeach aspect separately The objective is to reduce congestionand idling time of the yard trucks in order to decrease the


Sea sideSea side

Quay sideQuay crane

Quay side

Vessel

Empty truck

Loaded truck

Yard crane

Travel path Blocks

Storage block

Yard sideYard side

(a)(b)


















of request 119894 is 119889119894= max0 119888



119895= max 119908

119894+ 119905119894+ 119904119894119895 119886119895











Indices

119894 119895 Index of request 119894 = 119895




Problem Data







Set of Indices






119877 The set of routes |119877| = 119898





Decision Variables


= 0 otherwise


= 0 otherwise





119894= 120591119900119894119890119894


119894= 120591119900119894120577119896



119894119895= 120591119890119894119900119895


119894119895= 120591120577119896119900119895



Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119904119894119895119910119894119895) (1)

subject to

sum


119909119894119896le 1 forall119896 isin 119870 (2)

sum

119896isin119870

119909119894119896= 1 forall119894 isin 119869

minus

(3)

sum

119895isin11986910158401015840

119910119894119895= 1 forall119894 isin 119869

1015840

(4)

sum

119894isin1198691015840

119910119894119895= 1 forall119895 isin 119869

10158401015840

(5)




119908119894ge 119886119894

forall119894 isin 1198691015840cup 11986910158401015840 (6)

119889119894ge 119908119894+ 119905119894minus 119887119894

forall119894 isin 1198691015840cup 11986910158401015840 (7)

119908119895+119872(1 minus 119910

119894119895) ge 119908

119894+ 119905119894+ 119878119894119895


10158401015840 (8)

119905119894= 120591119900119894119890119894

forall119894 isin 119869+ (9)

119905119894= sum

119896isin119870

120591119900119894120577119896

119909119894119896


(10)

119878119894119895= 120591119890119894119900119895


119878119894119895= sum

119896isin119870

120591119900119894120585119894

119909119894119896


119909119894119896 119910119894119895isin 0 1 forall119894 isin 119869

1015840 forall119895 isin 119869

10158401015840 forall119896 isin 119870 (13)


1015840cup 11986910158401015840



119889119894ge 0 forall119894 isin 119869

1015840cup 11986910158401015840

(14)


119894119895=




119894119896and 119910



119894




Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119897119894119895) (15)




119897119894119895le 119872 sdot 119910



(16)


4 Methodology







1 2 3 4 5 6 7(1039 592) (395 686) (18 1263) (635 357) (143 789) (113 1323) (321 563)





1


2







6

ID a6 b6 o6 e6

S1 S2 S3



Parent 1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198752


Ω1

6 8 5 4 7 3 2 9Ω2


Ω1

1 4 10Ω2

1 2 4 5 7 8 10

Offspring 1198741

6 9 3 7 2 8 5 minus1 10 4 1 minus21198742

6 9 3 minus1 10 7 2 8 5 1 4 minus2


Fitness =1

119885 (17)



1and 119874

2






2be the same asΩ

1




9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 5 7 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10










2into set Ω



1








1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 9 3 2 minus1 1 7 10 minus2


1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 3 2 minus1 1 9 7 10 minus2






9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 57 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10


Start



Crossover operation


Mutation operation

Fine local search

Elitist strategy


upper bound

End

Yes

No









relation of 1205721+ 1205722= 1 and 120572


Lee et al [2]


11986208




Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size


timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours



Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sea sideSea side

Quay sideQuay crane

Quay side

Vessel

Empty truck

Loaded truck

Yard crane

Travel path Blocks

Storage block

Yard sideYard side

(a)(b)


















of request 119894 is 119889119894= max0 119888



119895= max 119908

119894+ 119905119894+ 119904119894119895 119886119895











Indices

119894 119895 Index of request 119894 = 119895




Problem Data







Set of Indices






119877 The set of routes |119877| = 119898





Decision Variables


= 0 otherwise


= 0 otherwise





119894= 120591119900119894119890119894


119894= 120591119900119894120577119896



119894119895= 120591119890119894119900119895


119894119895= 120591120577119896119900119895



Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119904119894119895119910119894119895) (1)

subject to

sum


119909119894119896le 1 forall119896 isin 119870 (2)

sum

119896isin119870

119909119894119896= 1 forall119894 isin 119869

minus

(3)

sum

119895isin11986910158401015840

119910119894119895= 1 forall119894 isin 119869

1015840

(4)

sum

119894isin1198691015840

119910119894119895= 1 forall119895 isin 119869

10158401015840

(5)




119908119894ge 119886119894

forall119894 isin 1198691015840cup 11986910158401015840 (6)

119889119894ge 119908119894+ 119905119894minus 119887119894

forall119894 isin 1198691015840cup 11986910158401015840 (7)

119908119895+119872(1 minus 119910

119894119895) ge 119908

119894+ 119905119894+ 119878119894119895


10158401015840 (8)

119905119894= 120591119900119894119890119894

forall119894 isin 119869+ (9)

119905119894= sum

119896isin119870

120591119900119894120577119896

119909119894119896


(10)

119878119894119895= 120591119890119894119900119895


119878119894119895= sum

119896isin119870

120591119900119894120585119894

119909119894119896


119909119894119896 119910119894119895isin 0 1 forall119894 isin 119869

1015840 forall119895 isin 119869

10158401015840 forall119896 isin 119870 (13)


1015840cup 11986910158401015840



119889119894ge 0 forall119894 isin 119869

1015840cup 11986910158401015840

(14)


119894119895=




119894119896and 119910



119894




Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119897119894119895) (15)




119897119894119895le 119872 sdot 119910



(16)


4 Methodology







1 2 3 4 5 6 7(1039 592) (395 686) (18 1263) (635 357) (143 789) (113 1323) (321 563)





1


2







6

ID a6 b6 o6 e6

S1 S2 S3



Parent 1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198752


Ω1

6 8 5 4 7 3 2 9Ω2


Ω1

1 4 10Ω2

1 2 4 5 7 8 10

Offspring 1198741

6 9 3 7 2 8 5 minus1 10 4 1 minus21198742

6 9 3 minus1 10 7 2 8 5 1 4 minus2


Fitness =1

119885 (17)



1and 119874

2






2be the same asΩ

1




9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 5 7 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10










2into set Ω



1








1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 9 3 2 minus1 1 7 10 minus2


1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 3 2 minus1 1 9 7 10 minus2






9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 57 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10


Start



Crossover operation


Mutation operation

Fine local search

Elitist strategy


upper bound

End

Yes

No









relation of 1205721+ 1205722= 1 and 120572


Lee et al [2]


11986208




Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size


timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours



Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Indices

119894 119895 Index of request 119894 = 119895




Problem Data







Set of Indices






119877 The set of routes |119877| = 119898





Decision Variables


= 0 otherwise


= 0 otherwise





119894= 120591119900119894119890119894


119894= 120591119900119894120577119896



119894119895= 120591119890119894119900119895


119894119895= 120591120577119896119900119895



Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119904119894119895119910119894119895) (1)

subject to

sum


119909119894119896le 1 forall119896 isin 119870 (2)

sum

119896isin119870

119909119894119896= 1 forall119894 isin 119869

minus

(3)

sum

119895isin11986910158401015840

119910119894119895= 1 forall119894 isin 119869

1015840

(4)

sum

119894isin1198691015840

119910119894119895= 1 forall119895 isin 119869

10158401015840

(5)




119908119894ge 119886119894

forall119894 isin 1198691015840cup 11986910158401015840 (6)

119889119894ge 119908119894+ 119905119894minus 119887119894

forall119894 isin 1198691015840cup 11986910158401015840 (7)

119908119895+119872(1 minus 119910

119894119895) ge 119908

119894+ 119905119894+ 119878119894119895


10158401015840 (8)

119905119894= 120591119900119894119890119894

forall119894 isin 119869+ (9)

119905119894= sum

119896isin119870

120591119900119894120577119896

119909119894119896


(10)

119878119894119895= 120591119890119894119900119895


119878119894119895= sum

119896isin119870

120591119900119894120585119894

119909119894119896


119909119894119896 119910119894119895isin 0 1 forall119894 isin 119869

1015840 forall119895 isin 119869

10158401015840 forall119896 isin 119870 (13)


1015840cup 11986910158401015840



119889119894ge 0 forall119894 isin 119869

1015840cup 11986910158401015840

(14)


119894119895=




119894119896and 119910



119894




Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119897119894119895) (15)




119897119894119895le 119872 sdot 119910



(16)


4 Methodology







1 2 3 4 5 6 7(1039 592) (395 686) (18 1263) (635 357) (143 789) (113 1323) (321 563)





1


2







6

ID a6 b6 o6 e6

S1 S2 S3



Parent 1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198752


Ω1

6 8 5 4 7 3 2 9Ω2


Ω1

1 4 10Ω2

1 2 4 5 7 8 10

Offspring 1198741

6 9 3 7 2 8 5 minus1 10 4 1 minus21198742

6 9 3 minus1 10 7 2 8 5 1 4 minus2


Fitness =1

119885 (17)



1and 119874

2






2be the same asΩ

1




9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 5 7 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10










2into set Ω



1








1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 9 3 2 minus1 1 7 10 minus2


1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 3 2 minus1 1 9 7 10 minus2






9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 57 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10


Start



Crossover operation


Mutation operation

Fine local search

Elitist strategy


upper bound

End

Yes

No









relation of 1205721+ 1205722= 1 and 120572


Lee et al [2]


11986208




Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size


timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours



Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119908119894ge 119886119894

forall119894 isin 1198691015840cup 11986910158401015840 (6)

119889119894ge 119908119894+ 119905119894minus 119887119894

forall119894 isin 1198691015840cup 11986910158401015840 (7)

119908119895+119872(1 minus 119910

119894119895) ge 119908

119894+ 119905119894+ 119878119894119895


10158401015840 (8)

119905119894= 120591119900119894119890119894

forall119894 isin 119869+ (9)

119905119894= sum

119896isin119870

120591119900119894120577119896

119909119894119896


(10)

119878119894119895= 120591119890119894119900119895


119878119894119895= sum

119896isin119870

120591119900119894120585119894

119909119894119896


119909119894119896 119910119894119895isin 0 1 forall119894 isin 119869

1015840 forall119895 isin 119869

10158401015840 forall119896 isin 119870 (13)


1015840cup 11986910158401015840



119889119894ge 0 forall119894 isin 119869

1015840cup 11986910158401015840

(14)


119894119895=




119894119896and 119910



119894




Min 119885 = 1205721sum

119894isin119869

119889119894+ 1205722(sum

119894isin119869

119905119894+ sum

119894119895isin119869

119897119894119895) (15)




119897119894119895le 119872 sdot 119910



(16)


4 Methodology







1 2 3 4 5 6 7(1039 592) (395 686) (18 1263) (635 357) (143 789) (113 1323) (321 563)





1


2







6

ID a6 b6 o6 e6

S1 S2 S3



Parent 1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198752


Ω1

6 8 5 4 7 3 2 9Ω2


Ω1

1 4 10Ω2

1 2 4 5 7 8 10

Offspring 1198741

6 9 3 7 2 8 5 minus1 10 4 1 minus21198742

6 9 3 minus1 10 7 2 8 5 1 4 minus2


Fitness =1

119885 (17)



1and 119874

2






2be the same asΩ

1




9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 5 7 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10










2into set Ω



1








1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 9 3 2 minus1 1 7 10 minus2


1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 3 2 minus1 1 9 7 10 minus2






9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 57 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10


Start



Crossover operation


Mutation operation

Fine local search

Elitist strategy


upper bound

End

Yes

No









relation of 1205721+ 1205722= 1 and 120572


Lee et al [2]


11986208




Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size


timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours



Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 2 3 4 5 6 7(1039 592) (395 686) (18 1263) (635 357) (143 789) (113 1323) (321 563)





1


2







6

ID a6 b6 o6 e6

S1 S2 S3



Parent 1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198752


Ω1

6 8 5 4 7 3 2 9Ω2


Ω1

1 4 10Ω2

1 2 4 5 7 8 10

Offspring 1198741

6 9 3 7 2 8 5 minus1 10 4 1 minus21198742

6 9 3 minus1 10 7 2 8 5 1 4 minus2


Fitness =1

119885 (17)



1and 119874

2






2be the same asΩ

1




9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 5 7 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10










2into set Ω



1








1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 9 3 2 minus1 1 7 10 minus2


1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 3 2 minus1 1 9 7 10 minus2






9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 57 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10


Start



Crossover operation


Mutation operation

Fine local search

Elitist strategy


upper bound

End

Yes

No









relation of 1205721+ 1205722= 1 and 120572


Lee et al [2]


11986208




Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size


timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours



Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 5 7 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10










2into set Ω



1








1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 9 3 2 minus1 1 7 10 minus2


1198751

6 8 5 4 7 3 2 minus1 1 9 10 minus21198741

6 8 5 4 3 2 minus1 1 9 7 10 minus2






9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 57 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10


Start



Crossover operation


Mutation operation

Fine local search

Elitist strategy


upper bound

End

Yes

No









relation of 1205721+ 1205722= 1 and 120572


Lee et al [2]


11986208




Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size


timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours



Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










9 6 10 7 minus1 3 2 8 1 4

minus2

minus2

9 6 57 minus1 3 2 8 1 4

m = 3 n = 2

5Chromosome

Chromosome 10


Start



Crossover operation


Mutation operation

Fine local search

Elitist strategy


upper bound

End

Yes

No









relation of 1205721+ 1205722= 1 and 120572


Lee et al [2]


11986208




Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size


timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours



Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table8Com

putatio

nalresultsof

rand

ominsta

nces

insm

allscale

Experim

ent

number

Size


timessto

rage

locatio

ns)

CPLE

XSimpleG

AGap

()b

etween

CPLE

Xandsim

pleG

AHybrid

GA

Gap

()b

etweenCP

LEX

andhybrid

GA

Value

CPU(s)

Value

CPU(s)

Value

CPU(s)

13times3times3

1776

777

1776

311

01776

183

02

3times3times5

1572

2731

1572

297

01572

201

03

4times4times4

2096

14822

2096

353

02096

261

04

4times4times5

2096

713348

2096

367

02096

260

05

5times4times4

214

46350

2185

370

21

214

317

06

5times5times5

2836

97612

2924

381

31

2836

364

07

7times5times5

3726

18935

3846

386

32

3726

410

08

7times7times9

365

lowast3822

404

47

365

478

09

9times7times10

385

lowast40

69

427

57

385

495

010

10times10times20

4435

lowast4761

438

73

4435

559

0lowastTh

ecom

putatio

naltim

eislon

gerthan10

hours



Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Number ofloading

containers















SimpleGA

HybridGA

Gap()

100 containers

1 34324 32302 51 2836 26952 51 29978 28868 42 34704 26044 252 39972 36396 92 57528 5032 123 6206 54072 133 122392 111108 93 110132 81226 264 45074 29452 354 29168 27092 74 3068 29334 55 2740 26604 35 26656 26374 15 28572 27574 3

200 containers

1 248636 215668 131 420802 35506 161 299266 264212 122 50892 508752 12 545594 514872 62 519532 472416 93 57543 509172 123 75370 601928 203 523036 415236 214 304706 169524 444 367088 349622 54 344274 320316 75 48938 46454 55 48618 48124 15 5020 49434 2

300 containers

1 120613 1175776 31 1599312 1518258 51 1522314 145829 42 140016 1331998 52 1287972 1230562 42 151134 1414346 63 1818946 1742536 43 183754 1769558 43 2000728 1913796 44 1491586 1237188 174 1549768 1438308 74 155862 151624 35 110034 98646 105 194598 18050 75 200054 165382 17





11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11987209 Table 10









Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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