2.Thesis 2
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PublishedOnlineJune2014 inSciRes. http://www.scirp.org/journal/jseahttp://dx.doi.org/10.4236/jsea.2014.77053
Comparative Study of Different Representations in Genetic Algorithms for JobShop Scheduling Problem
Vedavyasrao Jorapur*,V.S. Puranik,A.S.Deshpande,M.R. Sharma
Visvesvaraya Technological University, Belgaum, India
Email:*jorapur@fragnel.ac.in
Received 13February 2014;revised 10March2014;accepted 18March2014
Copyright ©2014by authorsand ScientificResearch PublishingInc.
Thiswork islicensed underthe Creative Commons Attribution International License(CCBY).
http://creativecommons.org/licenses/by/4.0/
Abstract
Dueto NP-Hardnature of the JobShop Scheduling Problems(JSP),exactmethods failtoprovide the optimalsolutions in quiter easonable computational time.Dueto this natureof theproblem, so many heuristicsand meta-heuristics havebeen proposed in the pasttogetop timalornear- op- timalsolutions foreasy to toughJSP instances in lesser computational time comparedto exact methods.One of suchheuristicsis geneticalgorithm(GA). Representationsin GA will havea direct impact on computationa timeittakesin providing optimalornearoptimalsolutions.Different representationschemesarepossibleincaseofJobScheduling Problems.Theseschemesinturn willhavea higher impacton the performanceof GA.It isintendedto show throughthis paper, how these representationswill perform,bya comparativeanalysisbased on average deviation,evolu- tionof solution over entiregenerationsetc.
Keywords
JobShopScheduling,GeneticAlgorithm, GeneticRepresentation, Conceptual Model
1.Introduction
Scheduling isadecision-making processwhich dealswithallocation of resourcestotasksovergiventime-pe- riodsanditsgoalistooptimizeoneormoreobjectivefunctions.Aschedulingproblem isrepresentedbytriplet α/β/γ.αfielddescribesmachineenvironment;βfieldprovides details of processing characteristicsandcon- straintsandγfielddescribes theobjective functiontobeminimized.Being essentially acombinatorial optimiza- tionproblem,jobshopschedulinghascaughttheattentionof researchersinthelastsomany yearsforoptimized
*Correspondingauthor.
How tocite this paper:Jorapur,V.,Puranik,V.S.,Deshpande,A.S.andSharma,M.R.(2014)ComparativeStudyofDifferent
RepresentationsinGeneticAlgorithmsforJobShopSchedulingProblem.JournalofSoftwareEngineeringandApplications,
7,571-580. http://dx.doi.org/10.4236/jsea.2014.77053
performance.Combinatorialoptimization problemscanbeclassifiedas easyandhard. Problemswhicharepoly- nomialysolvablewithlimitednumberofvariablesaretreatedeasy andarecalledP.Thenotionpolynomial solvabledependsonthetypeofencoding.Itisassumedthatproblemsdescribingnumericaldataarebinary en- codedandthenumberof stepsinvolvedin solvingthese increasesexponentiallywithincreasein length of string andhencecomputationaltimewill beenormously large andtreatedto behardproblems.Jobscheduling prob- lemsbelongtothiscategory andaretermedNP-Hard[1].Inthepracticalmanufacturingenvironment,thescale ofjobshopsisgenerallymuch largerthan thatof JSSPbenchmarkinstancesconsideredintheoreticalresearch. Optimizationalgorithmsforjobshopschedulingusually proceed by BranchandBoundandamongthemostre- centandsuccessful,onesarethoseofCarlierandPinson(1989) andApplegateandCook(1991)[2].Approxi- mationproceduresorheuristicswereinitiallydevelopedonthebasisofpriority rulesordispatching rules.The quality ofsolutionsgeneratedbytheseprocedures leaveplenty of room forimprovement(1998)[3].Therefore, traditionalormeta-heuristicalgorithmscanhardly beabletosolvesuchproblems satisfactorily.Inmanufactur- ingworkshops,availabilityof computationalresourcesismuch lessthanthelaboratorieswhichleadtodifficulty inexploring all possiblefeasiblesolutions.Undersuchcircumstances,itis reasonabletoreducethesearchspace andrangetoonlypromisingareas.Theveryideaof usingconstructiveheuristicsandheuristicsearchalgorithms forlargerproblem sizesof JSSPisthecomputationalexpensivenatureofenumerativetechniques andLagrann- gianalgorithms.AccordingtoOsman(1996),aheuristicsearch“isaniterativegenerationprocesswhichguides asubordinateheuristicby combining intelligently differentconceptsforexploring andexploiting thesearch spaces”.
Extensiveuseofgeneticalgorithmstosolvejobshopschedulingproblemscan beseen throughliteraturesur- vey[4].However,how effectivelygeneticalgorithmscanbeusedinJSSPcaseisnotcompletelyexplored.In this context, some directionis provided byTamer F.Abdelmaguid[5]in hispaper. Ga’sare based onanabstract modelof naturalevolution,suchthatquality of individualsbuildstothehighestlevelcompatiblewith theenvi- ronment (constraintsofthe problem). (Holland, 1975;Goldberg, 1989)
Representations inGA environmentappliedsofarin jobshopschedulingcanbeclassifiedintoninecatego- ries asgivenbyChengetal. (1996):
1) Operationbased 2)Jobbased 3)Jobpairrelationbased 4) Completiontime based
5)Randomkeys 6)Preferencelistbased 7)Priority rulebased 8)Disjunctivegraphbased.
9) Machinebased.
Ninecategoriesmentionedabovecan begroupedintotwobasicencodingapproaches—directandindirect encoding.Indirectapproach,aΠj scheduleisencodedasachromosomeandgeneticoperatorsareused to evolvebetterindividualones.Categories1to5areexamplesofthiscategory.Incaseofindirectapproach,a sequenceofdecisionpreferenceswillbeencodedintoachromosome.Inthis,encoding,geneticoperatorsareappliedtoimprovetheorderingofvariouspreferencesandaΠj scheduleisthengeneratedfromthesequenceofpreferences.Categories6to9areexamplesof thiscategory[6].Theserepresentationsneedtobestudiedin case ofjobshopschedulingproblemstocomparetheirperformancecriteriatogenerateoptimalornearoptimalsolu- tions,eventhoughcomputationalcomparisonofdifferentrepresentationsisreportedina tutorialpaperby Cheng,GenandTsujimura[6].AreportbyAnderson,GlassandPotts[7],conductedwithdifferentmetaheu- risticsapproachesincludingfourdifferentGA implementations,lacksinconsistencyaswellascoherenceasre- gards numberoftestproblemsbeingtestedwithrequisitenumber ofruns.
Therestof this paperisorganizedas follows: Wewillstartwithmathematicalmodels withcertain assump- tions thathavebeenusedinnextsectionfollowedbythe literaturereview onthedifferentGA representations usedin thecaseof JSP.Followedby reviewofGArepresentations,wewilldiscussregardingdifferentGAop- eratorsfrequentlyusedby researchersandourownviewsonaddingotheroperatorsnotdiscussedsofar.Now, wewillanalyzetheexperimentalresultsconductedfollowedbytheconclusionprovidedin thefinalpartofthis paper.
2.Problem Formulation
Sinceitisan importantpracticalproblem,someauthorshaveformulatedvarious JSPmodels basedon different production situationsand problemassumptions. Themostcommonassumptionsincaseof JSPare:
1) A machinemayprocessmorethanone jobata time;
2) Nojobmaybe processedbymorethanonemachineatatime;
3)The sequence of machines whicha jobvisitsiscompletelyspecifiedandhasa linear precedence structure;
4) Processingtimesare known. Allthe processingtimesareassumedto be integers;
5) Eachjobmustbe processed oneachmachineonlyonce. There is no recirculation;
6) Set-uptimesare assumedzero;
7) Pre-emptionis notallowed.
Let“J” representasetofjobsandeachjobwillbe processedona setof machines ina particular order. LetI=
(1…..v)representtheoperationindexes.Theoperationindexesareassignedsuchthatfora job
k ∈ J,the subset
ofconsecutiveindexes Ik =
βk,βk + 1,βk + 2,ωk
⊆I,isasubsetcontainingindexesforthatjob.Nowfrom
thesubsetIk dependingonthepriorityoperationwithhigherorlowervalueisprocessedfirst.Letpi bethe
processingtimeofithoperation,thejobwhichitbelongstoisj(i)andthemachineonwhichithoperationcar-
riedism(i).
Now theobjectiveofschedulingprocessistodeterminethestarttimesti ofanoperation
i∈I.Whileassign-
ingajobtoamachinebasedonabovecalculationsfollowingconstraintsshouldbetakenintoconsiderationviz.
Thetechnological constraintswilltakecareof order of operations tobecarriedout on a jobandasecondsetof constraintwilltakecare of conflictof twojobstobeprocessedonthesamemachinesimultaneously.Accor- dingly:
and
sti +pi ≤sti+1..Is the equation to satisfytechnologicalconstraints
(1)
sti ≥stj +pj Or
stj ≥sti +pi
(2)
Isthe equationto satisfythe conflictoftwojobsonthe samemachineatthe same time.
∀i, j∈ Iwherem(i ) =m( j ) and j(i ) ≠ j( j )
Differenttotalcostfunctionsthatcanbe studiedare
Fmax.(C) :=max.{ fi (Ci) i=1,,n} …IscalledBottleneckobjective and
i=n
∑ fi (C) =∑ fi (Ci) …IscalledSumObjective.
i=1
The most common objective functions are the make span max
{(Ci) i= 1,,n}
and total flow time
n
∑(Ci ) ,andweighted(total)flowtime
i =1
n
∑ wi ⋅ Ci .Wehaveconsideredtheminimizationofmakespanas
i=1
ourobjectivefunction.Mannes’ [8]proposed anintegerlinearprogrammingmodel(ILP)whichuses different formsof binary variables.Thismodelhasgainedlargerinterestintheresearch community duetosmallnumber ofvariablesconsideredin themodel.Thetechnological constraintsof Equation (1)areanalogoustoaseriesof consecutiveactivities thatarecarried outin projectscheduling.Thisanalogyhasmotivatedimportingproject networksintoJSPenvironment.Torepresentdisjunctiveconstraints as in Equation (2),additionalsets ofarcs required.Thisisachieved ina disjunctive graphmodel[9]andPIANmodel[10].
In thedisjunctivegraphmodel,adisjunctivearc is definedbetweenapairof operationsthatsharethema- chine.Eachdisjunctivearcisassignedabinary decisionvariablesuchthatselectiononthevaluethatvariable definesthelength anddirectionofeach disjunctivearc. ThisistotheMannes’model.Very efficientalgorithms like immediateselectionsandshifting bottleneckheuristicswere proposedby Carlier[11]andAdams [12]and Lars Monch[13], whichare derivedfromdisjunctive graphmodel.
A variable notationofthe type
m
xi,t
= 1…ifoperation ‘i’isprocessed onmachine‘m’ inunittime ‘t’
=0...otherwise.
In ILPmodelwas proposed byBowman[14]. Wagner[15]proposedamodelwhereavariablenotation of the
|
m
i,l
= 1…ifoperation‘i’takes‘ith’positioninthe processingsequence onmachine ‘m’
=0...otherwise.
|
processedpriorto operation ‘j’onmachinem.
m
i,j
=1…ifoperation‘i’is
= 0...otherwise.
3.Representationofthe Problem inGA andGA Operators
Darwin’sprinciple“survivalofthefittest”can beusedasastartingpointinintroducingevolutionary computa- tion.Theproblemsofchaos,chance,nonlinearinteractivitiesandtemporality beingsolvedby biologicalspe- cies are provedto beinequivalencewithclassicmethod ofoptimization[15].
Evolutionarycomputationstechniquesthatcontainalgorithmsbasedonevolutionaryprinciplesareusedto
searchforan optimalorbestpossiblesolutionforagivenproblem.Inasearchalgorithm,numberof possible solutionsisavailableandthetask istofindthebestpossiblesolutionin afixedamountoftime. Traditional searchalgorithmsrandomlysearch(e.g.randomwalk)orheuristicallysearch(e.g.gradientdescent),explore onesolution at atime in thesearch spacetofindbest possibleoroptimalsolution,whichiscomputationally in- efficient asthesearchspacegrowsin size. Whereasevolutionaryalgorithmsfromsuchtraditionalalgorithmsare populationbased.Evolutionary algorithmperformsadirectedefficientsearchby adaptationofsuccessivegen- erationsofalargernumberofindividuals.GeneticAlgorithmsisonesuchevolutionary algorithminfindingan optimalornear optimalsolutiontoa problem.Inatraditionalgeneticalgorithm,therepresentationis bitlength string.Itsapproachistogenerateasetof random solutionsfrom theexistingsolutions, sothat there isan im- provement inthequality ofsolutionsthroughoutthegenerations.Thisimplementationis achieved throughmain GAoperators’viz.random selectionoftwosolutionsfrom individualsintheparentgeneration;performing crossoveroperation onthesetwosolutionstogeneratetwonew childsolutions.Crossoveroperation isper- formedby exchangingspecificelementsof thetwosolutionsselected;andmutationoperationisconductedon childsolutionstofurtherexplorethesearchspaceforbettersolutions.Differentvariations insimpleGA ap- proachcanbefoundinliterature surveyto improveitssearchcapabilities [16]. Representation of solutions of an optimizationproblem istobedoneinasuitableformatin GAtodealwithreproductionandmutation operators. Thisformatorstructurereferredasgenotype,needstobeeasilyinterpretabletoasolutionoftheproblem under study.Inacombinatorialoptimization problem,representationofasolution inGA isdifficultaswellasachal- lengingtask.Theseareproblemscontainingdiscretedecision variablesandare interrelatedby logical relation- ships.Asaresult,differentmathematicalmodelsmayexistforthesamecombinatorialoptimizationproblem andthismayleadto differentrepresentationsfor the same problem.
AsexplainedaboveCheng, GenandTsujimura[6]intheirpaperrepresentationofJSPinGA intodirectand indirecttype.Furthertothat,T.F.Abdelmaguid[17]inhispaperclassifiedGArepresentationsintoModel basedandAlgorithm based.In ouropinion,all representationsarealgorithm basedthoughtheyappeartobe model based.
InPriorityRuleBased(PR)representation, achromosomeisrepresentedasastringof(n−1)entries(p1, p2…pn)wheren−1isthenumberofoperationsintheproblem instance.Anentryp1 representsapriority rule selectedbeforehand.Accordingly,aconflictintheith iterationofGifflerandThompsonalgorithm[18]should be resolved usingpriorityrulerepresentedbypi. Itmeans an operation fromtheconflictset hasto be selectedby thepi tiesarebrokenrandomly.InGA domain,abestsetofpriority rulesshouldbeselected.Heresimplecros- soveryieldsfeasible schedules.
InRandom KeysRepresentation (RK) wasfirstproposedbyBean[19]. Inthisrepresentation,eachgeneis representedwithrandom numbersgeneratedbetween0 and1.These random numbersinagivenchromosome aresortedoutandarereplacedby integers andnow theresultingorderistheorderofoperationsinachromo- some.Thisstringis then interpretedintoafeasibleschedule.Anyviolationofprecedenceconstraintscan be correctedbya correctionalgorithmincorporated.
InOperationbasedrepresentation,each generepresentsanoperation.Achromosomecontainsasmany genesasthenumber of operations.For example, annxm JSPtherewillbenxmgenes in the chromosome. Beirwirth proposedatechnique“permutationwith repetition”[20]which issimilartooperation basedrepresen- tation.Fang[21]alsoproposed akindoperation basedrepresentationwherestringcontainsnxmchunkswhich arelargeenoughtoholdthelargestjobnumberforthenxm JSP.Whereas BeirwirthusedaspecialGOXcros- sovertechnique togeneratefeasibleschedule,Fang usedaspecialdecoding approach todecodeachromosome intoa validschedule always.
ThePreferenceListbasedrepresentation(PL)usesastring of operationsforeach machineinsteadof a singlestringforall operationswhichisadirectrepresentation of processingsequencedecisionvariables. Quite oftenviolationofconstraintsisencounteredwhichcanbe overcome byrepair algorithm.
In theMachinebasedrepresentation,[21]thechromosomecontainsastring of lengthequaltothenumber ofmachines.Thesequenceofmachinesinthestringistheorderbywhichamachineistreatedasabottleneck
machineinthe shiftingbottleneckalgorithm[12].
IntheJobbasedrepresentation[22] achromosomeisastringoflengthequaltothenumber ofjobsinthe problem understudy.Usingthisrepresentation,asimple algorithmcangenerateafeasibleschedulegivense- quence ofthe jobsonto differentmachines.
4.Methodology
Thereproductionandmutation operatorsappliedtoJSPmodelaregenerallyadoptedfromTravelling Salesman Problembecause of thesimilarity inrepresentations.Reproductionoperatorsaregenerally requiredinGAto conducttheneighborhoodsearchandamutation operatorgenerallyensuresthatthesolution isnottrappedin localminima.Thedesign of both operatorsiscrucialforthesuccess of GA.Amongthereproduction operators reportedin theliterature, PMX(partiallymatchedcrossover)[23],OX (orderedcrossover)[24]anduniform crossover[25]areextensivelyusedin JSSP.PMXandOX crossovertechniquesuseeithersinglepointortwo pointcrossover.Differentmutation operatorsusedareswapmutation, inversionmutation andinsertion orshift mutationreportedinthe literature[17].
Ingeneral, theflowchartfor GAcanbe representedasshown.
5.Results andAnalysis
Inourexperiment,fourrepresentationsareusedviz.Operationbased(OB),Jobbased(JB),Machinebased (MB),Priorityrule based (PR). Allexperimentsare conductedwith50generationsanda populationsize of1000. Mutationprobabilityvarieswith 0.1to0.9valuesdynamicallyandelite populationsizeis20%.Reproduction probabilityusedinourexperimentis0.1Parentsinourexperimentareselectedfromtwogroupssortedout basedon fitnessvalue (i.e.minimum makespan). Eachparentisselectedfromthesegroupsprobabilistically.
Inour experimentation,GA isprogrammedwith differentreproductionandmutation operators’.Insteadof selectingoperators randomlyas in [17],wehavebuilt-in reproduction operatorsandare beingusedacross the representationsandthebenchmark instances.Thebenchmarkproblemsusedinthispaperaretakenfrom ORli- brary [26]availablein World Wide Web.AlltheexperimentsareconductedwithaPentium-4dualcoreproces- sorwithclockspeed of2.06GHzandRAM of512 Mbs.68benchmarkinstancesare takenandinthe single run, thebestand averagevalues areobtainedandcomparedwith lowerboundoroptimumvalue ofthebenchmark instance.Resultsare showninTable 1. Differentgraphsgeneratedare alsoshownbelow.
Table 1.Results ofbenchmarkinstances underdifferentrepresentations.
Problem Size No.of
Operations
BestKnown
Solution
OB OB JB JB MB MB PR PR Best Avg. BestAvg. Best Avg. Best Avg.
mt06 6×6 36 55 55 64.889 55 65.712 55 61.822 55 66.648 mt10 10 × 10 100 971 989 1116.02 971 1100.9 992 1145.06 958 1100.42 mt20 5× 20 100 1206 1220 1394.47 1206 1383.081245 1427.24 1242 1426.54 abz05 10 × 10 100 1259 1275 1394.79 1259 1386.12 1287 1409.87 12671390.94 abz06 10 × 10 100 971 958 1072.19971 1075.97 996 1096.13 978 1080.3 abz07 15 × 20 300 742 734 821.16 742 804.892 751 817.937 730 807.128 abz08 15 × 20 300 758 751 833.362758 825.982 763 838.59 755 826.954 abz09 15 × 20 300 752 784 877.468 752 849.838 773 873.541 764859.258 car01 5× 11 55 7038 7038 8747.847038 8694.017038 8707.83 7038 8782.28 car02 4× 13 52 7376 7378 8788.38 7376 8738.23 7221 8817 7166 8881.94 car03 5× 12 60 7725 7590 9219.197725 9195.367725 9293.86 7725 9272.51 car04 4× 14 56 8072 8003 9620.16 8072 9452.62 8276 9697.218132 9643.3 car05 6× 10 60 7835 7873 9207.147835 9130.267862 9251.68 7862 9407 car06 9×8 72 8505 8505 10017.7 8505 9886.82 8505 10229.58485 9830.33 car07 7×7 49 6558 6576 7673.646558 7782.766627 7751.89 6632 7738.75 car08 8×8 64 8407 8407 9436.29 8407 9500.61 8458 9470.578366 9470.64 la01 5× 10 50 666 666 783.616666 796.901 674 746.506 666 782.789
Continued |
|
||||||||||
la02 |
5× 10 |
50 |
655 |
665 |
748.122 |
655 |
774.664 |
660 |
745.747 |
667 |
757.79 |
la03 |
5× 10 |
50 |
617 |
620 |
688.729 |
617 |
687.389 |
626 |
690.773 |
620 |
699.527 |
la04 |
5× 10 |
50 |
607 |
595 |
695.259 |
607 |
690.822 |
619 |
699.926 |
602 |
688.268 |
la05 |
5× 10 |
50 |
593 |
593 |
640.494 |
593 |
658.885 |
593 |
606.404 |
593 |
699.114 |
la06 |
5× 15 |
75 |
926 |
926 |
1000.79 |
926 |
1021.85 |
926 |
958.039 |
926 |
1075.5 |
la07 |
5× 15 |
75 |
890 |
890 |
998.253 |
890 |
1015.05 |
893 |
983.784 |
890 |
994.044 |
la08 |
5× 15 |
75 |
863 |
863 |
981.109 |
863 |
985.795 |
863 |
959.264 |
863 |
995.054 |
la09 |
5× 15 |
75 |
951 |
951 |
1051.15 |
951 |
1084.34 |
951 |
988.331 |
951 |
1167.81 |
la10 |
5× 15 |
75 |
958 |
958 |
1017.01 |
958 |
1045.73 |
958 |
971.19 |
958 |
1089.97 |
la11 |
5× 20 |
100 |
1222 |
1222 |
1308.89 |
1222 |
1334.96 |
1222 |
1264.12 |
1222 |
1389.85 |
la12 |
5× 20 |
100 |
1039 |
1039 |
1132.34 |
1039 |
1157.8 |
1039 |
1104.58 |
1039 |
1226.58 |
la13 |
5× 20 |
100 |
1150 |
1150 |
1248.7 |
1150 |
1278.51 |
1150 |
1191.37 |
1150 |
1314.64 |
la14 |
5× 20 |
100 |
1292 |
1292 |
1320.59 |
1292 |
1348.95 |
1292 |
1295.72 |
1292 |
1388.82 |
la15 |
5× 20 |
100 |
1207 |
1207 |
1336.66 |
1207 |
1352.88 |
1227 |
1368.04 |
1207 |
1352.41 |
la16 |
10 × 10 |
100 |
979 |
982 |
1083.26 |
979 |
1066.88 |
988 |
1088.68 |
987 |
1071.61 |
la17 |
10 × 10 |
100 |
797 |
793 |
890.389 |
797 |
885.073 |
832 |
905.275 |
807 |
888.012 |
la18 |
10 × 10 |
100 |
861 |
861 |
962.052 |
861 |
967.819 |
885 |
976.877 |
883 |
977.685 |
la19 |
10 × 10 |
100 |
875 |
875 |
970.966 |
875 |
972.302 |
899 |
983.686 |
877 |
976.925 |
la20 |
10 × 10 |
100 |
936 |
907 |
1022.37 |
936 |
1040.62 |
944 |
1039.52 |
914 |
1041.05 |
la21 |
10 × 15 |
150 |
1105 |
1098 |
1252.76 |
1105 |
1247.82 |
1115 |
1264.74 |
1111 |
1281.85 |
la22 |
10 × 15 |
150 |
972 |
988 |
1146.21 |
972 |
1125.68 |
1031 |
1161.32 |
990 |
1133.83 |
la23 |
10 × 15 |
150 |
1035 |
1045 |
1188.52 |
1035 |
1168.18 |
1037 |
1180.52 |
1068 |
1187.63 |
la24 |
10 × 15 |
150 |
1004 |
1006 |
1135.91 |
1004 |
1134.02 |
1029 |
1155.7 |
995 |
1149.69 |
la25 |
10 × 15 |
150 |
1040 |
1055 |
1177.18 |
1040 |
1170.37 |
1036 |
1175.69 |
1058 |
1178.36 |
la26 |
10 × 20 |
200 |
1269 |
1279 |
1457.86 |
1269 |
1424.04 |
1304 |
1466.35 |
1310 |
1446.12 |
la27 |
10 × 20 |
200 |
1341 |
1363 |
1529.94 |
1341 |
1500.85 |
1421 |
1539.77 |
1374 |
1538.38 |
la28 |
10 × 20 |
200 |
1301 |
1295 |
1454.95 |
1301 |
1456.94 |
1334 |
1463.16 |
1284 |
1453.35 |
la29 |
10 × 20 |
200 |
1274 |
1302 |
1441.65 |
1274 |
1416.42 |
1307 |
1429.08 |
1270 |
1425.34 |
la30 |
10 × 20 |
200 |
1418 |
1429 |
1576.32 |
1418 |
1554.81 |
1444 |
1592.45 |
1432 |
1591.74 |
la31 |
10 × 30 |
300 |
1784 |
1784 |
1927.56 |
1784 |
1938.51 |
1785 |
1934.69 |
1784 |
1933.39 |
la32 |
10 × 30 |
300 |
1850 |
1850 |
2019.59 |
1850 |
2029.95 |
1855 |
2024.57 |
1853 |
2031.31 |
la33 |
10 × 30 |
300 |
1719 |
1725 |
1890.07 |
1719 |
1873.95 |
1719 |
1871.39 |
1725 |