DONT DECAY THE LEARNING RATE INCREASE THE BATCH SIZE
Smith & Le (2017) observed an optimal batch size Bopt which maximized the test set accuracy at We suspect a similar intuition may hold in deep learning.
Courier: Real-Time Optimal Batch Size Prediction for Latency SLOs
23 avr. 2021 ABSTRACT. Distributed machine learning has seen immense rise in popularity in recent years. Many companies and universities are utilizing.
Adaptive Learning of the Optimal Mini-Batch Size of SGD
19 nov. 2021 [12] demonstrated the effect of increasing the batch size instead of decreasing the learning rate in training a deep neural network. However ...
arXiv:2108.11713v1 [math.OC] 26 Aug 2021
26 août 2021 the number of steps needed for training deep neural networks. ... Comparisons of Optimal Batch Sizes for Different Learning Rate Rules.
An Empirical Model of Large-Batch Training
14 déc. 2018 A very common source of parallelism in deep learning has been data ... learning: in reinforcement learning batch sizes of over a million ...
A Bayesian Perspective on Generalization and Stochastic Gradient
14 févr. 2018 (2016) who showed deep neural networks can easily memorize randomly ... This optimal batch size is proportional to the learning rate and ...
Effective Elastic Scaling of Deep Learning Workloads
24 juin 2020 2) A fast dynamic programming based optimizer that uses the job scalability analyzer to allocate optimal resources and batch sizes to DL jobs ...
Training Tips for the Transformer Model Martin Popel Ond?ej Bojar
This article describes our experiments in neural machine translation using the recent proved training regarding batch size learning rate
Micro Batch Streaming: Allowing the Training of DNN models Using
24 oct. 2021 is to allow deep learning models to train using mathematically determined optimal batch sizes that cannot fit into the memory.
Dont Decay the Learning Rate Increase the Batch Size
24 févr. 2018 Smith & Le (2017) observed an optimal batch size Bopt which ... We suspect a similar intuition may hold in deep learning.
A arXiv:171100489v2 [csLG] 24 Feb 2018
“Increasing batch size” also uses a learning rate of0 1 initial batch size of 128 and momentum coef?cient of 0 9 but the batch size increases by afactor of 5 at each step These schedules are identical to “Decaying learning rate” and “Increasingbatch size” in section 5 1 above
On The Importance of Batch Size for Deep Learning
the batch size hyperparameter directly in?uences parameter movements and the properties of the minima by controlling gradient noise during optimization [1 2 3] Determining an opti-mal batch size is necessary for generalization but remains a challenging problem within deep learning
arXiv:180309820v2 [csLG] 24 Apr 2018
Smith and Le (Smith & Le 2017) explore batch sizes and correlate the optimal batch size to the learning rate size of the dataset and momentum This report is more comprehensive and more practical in its focus In addition Section 4 2 recommends a larger batch size than this paper
A arXiv:180407612v1 [csLG] 20 Apr 2018
The presented results con?rm that using small batch sizes achieves the best training stability and generalization performance for a given computational cost across a wide range of experiments In all cases the best results have been obtained with batch sizes m= 32 or smaller often as small as m= 2 or m= 4
Online Evolutionary Batch Size Orchestration for Scheduling
Online Evolutionary Batch Size Orchestration for Scheduling Deep Learning Workloads in GPU Clusters Zhengda Bian1 Shenggui Li1 Wei Wang2 Yang You1 National University of Singapore1 ByteDance2 Singapore ABSTRACT Efficient GPU resource scheduling is essential to maximize re-source utilization and save training costs for the increasing amount
Searches related to optimal batch size deep learning filetype:pdf
batch size and learning rate using the gradient similarity measurement •We integrate SimiGrad into mainstream machine learning frameworks and open-source it 1 • SimiGrad enables a record breaking large batch size of 77k for BERT-Large pretraining
(PDF) Impact of Training Set Batch Size on the Performance of
7 mar 2023 · A problem of improving the performance of convolutional neural networks is considered A parameter of the training set is investigated
[PDF] Study on the Large Batch Size Training of Neural Networks - arXiv
16 déc 2020 · Large batch size training in deep neural networks (DNNs) possesses a well-known 'generalization gap' that remarkably induces generalization
The Number of Steps Needed for Nonconvex Optimization of a Deep
26 août 2021 · The first fact is that there exists an optimal batch size such that the number of steps needed for nonconvex optimization is minimized
[PDF] DONT DECAY THE LEARNING RATE INCREASE THE BATCH SIZE
Smith Le (2017) observed an optimal batch size Bopt which maximized the test set accuracy at We suspect a similar intuition may hold in deep learning
[PDF] Control Batch Size and Learning Rate to Generalize Well
In this paper we present both theoretical and empirical evidence for a training strategy for deep neural networks: When employing SGD to train deep neural
[PDF] Which Algorithmic Choices Matter at Which Batch Sizes? Insights
optimal learning rates and large batch training making it a useful tool to generate testable predictions about neural network optimization 1 Introduction
Analysis on the Selection of the Appropriate Batch Size in CNN
29 avr 2022 · Batch Size is an essential hyper-parameter in deep learning Choosing an optimal batch size is crucial when training a neural network
[PDF] ONLINE BATCH SELECTION FOR FASTER TRAINING OF NEURAL
Deep neural networks (DNNs) are currently the best-performing method for many Instead it is common in SGD to fix the batch size and iteratively sweep
The effect of batch size on the generalizability of the convolutional
Lowering the learning rate and decreasing the batch size will allow the network regarding the batch size is which is the optimal batch size for training
[PDF] Interactive Effect of Learning Rate and Batch Size to Implement
15 fév 2023 · ResNet architecture-based DL models proved to be the best models for brain tumor classification in comparison to other pre-trained DL mod- els
What is the role of batch size in deep learning?
- Batch size has an important role in this optimization technique. Because, parallelization of stochastic gradient descent in deep learning is limited in case of small batch sizes and sequential iterations.To solve this problem, large batch sizes are used. However, increasing the batch size increases computational cost.
What is a good batch size?
- The batch size affects some indicators such as overall training time, training time per epoch, quality of the model, and similar. Usually, we chose the batch size as a power of two, in the range between 16 and 512. But generally, the size of 32 is a rule of thumb and a good initial choice. 4. Relation Between Learning Rate and Batch Size
What is the difference between large and small batch training?
- This is one of those areas where we see clear differences. There has been a lot of research into the difference in generalization between large and small batch training methods. The conventional wisdom states the following: increasing batch size drops the learners' ability to generalize.
Can batch size and learning rate match the same learning curve?
- First of all, our goal is not to match the same accuracy and learning curve using two sets of batch size and learning rate but to achieve as good as possible results. For instance, if we increase the batch size and our accuracy increases, there’s no sense to modify the learning rate to achieve the prior results.
ControlBatchSizeandLear ningRateto Generalize
Well:TheoreticalandEmpirical Evidence
FengxiangHeTongliangLiu DachengT ao
UBTECHSydney AICentre,SchoolofComputer Science,Faculty ofEngineering TheUniv ersityofSydney,Darlington,NSW2008,Australia {fengxiang.he,t ongliang.liu,dacheng.tao}@sydney.edu.auAbstract
Deepneuralnetw orkshav ereceiveddramatic successbasedontheoptimization methodofstochastic gradientdescent (SGD).Howe ver, itisstill notclearho w totuneh yper-parameters,especially batchsizeandlearningrate,toensuregood generalization.Thispaper reportsboththeoretical andempiricale videnceofa trainingstrategy thatweshouldcontroltherati oofbatch sizetolearning ratenot too largetoachieve agoodgeneralization ability.Specifically,weprovea PA C-Bayes generalizationbound forneuralnetw orkstrainedby SGD,whichhas apositive correlationwith theratioof batchsizeto learningrate.This correlationbuilds the theoreticalfoundationof thetrainingstrate gy.Furthermore, weconducta large- scaleexperiment toverifythecorrelationand trainingstrategy .Wetrained1,600 modelsbasedon architecturesResNet-110,and VGG-19with datasetsCIFAR -10 andCIF AR-100whilestrictlycontrolunrelatedv ariables.Accuracieson thetest setsarecollected forthe evaluation. Spearman'srank-order correlationcoeffi cients andthecorresponding pvalueson164groupsofthecollected datademonstrate thatthecorrelation isstatistically significant,whichfully supportsthetraining strategy.1Introduction
Therecentdecade sawdramatic successofdeep neuralnetworks[9]basedon theoptimization methodofs tochasticgradientdescent (SGD)[2,32].Itis aninterestingand importantproblem that howtotunethe hyper-parameters ofSGDto make neuralnetworksgeneralizewell.Someworks havebeenaddressingthestrategiesof tuninghyper -parameters[5,10,14,15]andthe generalization abilityofSGD [4,11,19,26,27].Howe ver,therestilllackssolidevidenceforthetrainingstrategies regardingthehyper-parametersforneural networks. Inthispaper ,wepresent boththeoreticalandempiricalevidence foratraining strategy fordeep neuralnetworks: Whenemploying SGDtotraindeepneuralnetw orks,weshould controlthebatch sizenottoo largeand learningratenot toosmall,inordertomak ethenetw orks generalizewell. Thisstrate gygivesaguidetotune thehyper-parametersthathelpsneuralnetworks achieve goodtest performancewhenthe trainingerrorhas beensmall.It isderiv edfromthe following property: Thegeneralizationability ofdeep neuralnetworks hasane gativ ecorrelationwith theratioof batchsize tolearningrate. Asre gardsthetheoreticalevidence,weprov eano velP AC-Bayes[24,25]upperbound forthe generalizationerrorof deepneural networkstrained bySGD.The proposedgeneralizationbound33rdConferenceon NeuralInformation ProcessingSystems(NeurIPS 2019),Vancouv er,Canada.
whereall(X i ,Y i )constitutethetraining sampleT.Equivalently,theempiricalrisk
Ristheerror ofthe algorithmontraining data,whilethe expected riskRisthee xpectationof theerrorontestdataor unseendata.Therefore, thedifference between themisan informative indexto expressthegeneralizationabilityofthealgorithm,which iscalled generalizationerror .Theupperboundofthe generalizationerror(usually calledgeneralizationbound) expresseshowlar gethegeneralizationerrorispossible tobe.Therefore,generalizationboundis also animportantinde xtosho wthegeneralizationabilityofan algorithm. Stochasticgradient descent.Tooptimizetheexpected risk(eq.1), anaturaltool isthegradient descent(GD).Specifically ,thegradient ofeq.(1)intermsof theparameter ✓andthecorresponding updateequationare definedasfollo ws, g(✓(t)),r ✓(t)R(✓(t))= r
✓(t) E (X,Y) l(F ✓(t) (X),Y),(5) ✓(t+1)= ✓(t)⌘g(✓(t)),(6) where✓(t)istheparameter attheinteration tand⌘>0isthe learningrate. Stochasticgradientdescent (SGD)estimatesthe gradientfrommini batchesof thetrainingsample settoestimate thegradientg(✓).LetSbetheindices ofamini batch,inwhich allindicesare independentlyand identically(i.i.d.)dra wnfrom{1,2,...,N},whereNisthetraining samplesize. Thensimilarto thegradient,the iterationofSGD onthemini-batch Sisdefinedas follows, ˆg S (✓(t))=r ✓(t)R(✓(t))=
1 |S| X n2S r ✓(t) l(F ✓(t) (X n ),Y n ),(7) ✓(t+1)= ✓(t)⌘ˆg(✓(t)),(8) whereR(✓)=
1 |S| P n2S l(F (X n ),Y n )istheempirical riskonthe minibatchand |S|isthe cardinalityofthe setS.F orbrevity,were writel(F (X n ),Y n )=l n (✓)intherest ofthispaper . Also,supposethat instepi,thedistrib utionofparameter isQ i ,theinitial distributionis Q 0 ,andthe convergentdistributionisQ.ThenSGD isusedto findQfromQ 0 throughaseries ofQ i3Theoretical Evidence
Inthissection, weexplore anddev elopthetheoretical foundationsforthetrainingstrate gy.The main ingredientis aPA C-Bayesgeneralizationbound ofdeepneuralnetworksbasedontheoptimization methodSGD.The generalizationbound hasapositi vecorrelationwiththe ratioofbatch sizeto learningrate.This correlationsuggests thepresentedtraining strategy.3.1AGeneralization Boundfor SGD
Apparently,bothl
n (✓)and R(✓)areun-biasedestimations ofthe expectedrisk R(✓),whiler l n andˆg S (✓)arebothun-biased estimationsof thegradientg(✓)=rR(✓):
E[l n (✓)]=E hR(✓)
i =R(✓),(9) E[r l n (✓)]= E[ˆg S (✓)]=g(✓)=rR(✓),(10)
wherethee xpectationsarein termsofthecorrespondinge xamples(X,Y). Anassumption(see, e.g.,[ 7,23])forthe estimationsisthat allthe gradients{r l n (✓)}calculated fromindividual datapointsarei.i.d.drawn fromaGaussian distributioncentred atg(✓)=rR(✓):
r l n (✓)⇠N(g(✓),C).(11) whereCistheco variancematrix andisaconstantmatrixforall✓.Asco variancematrices are(semi) positive-definite,forbrevity,we supposethatCcanbef actorizedasC=BB .Thisassumption can bejustifiedby thecentrallimit theoremwhenthe samplesizeNislarge enoughcomparedwiththe batchsize S.Consideringdeep learningis usuallyutilizedto processlarge-scale data,thisassumption approximatelyholds inthereal-life cases. 3 Therefore,thestochastic gradientisalso drawnfrom aGaussiandistrib utioncentredat g(✓): ˆg S 1 |S| X n2S r l n (✓)⇠N g(✓), 1 |S| C .(12)SGDusesthe stochasticgradientˆg
S (✓)toiterativ elyupdatetheparameter✓inorderto minimizethe functionR(✓): S (✓(t))=⌘g(✓)+ p |S|BW,W⇠N(0,I).(13)
Inthispaper ,weonly considerthecasethatthebatch size|S|andlearningrate ⌘areconstant.Eq. (13)expresses astochasticprocesswhichiswell-kno wnasOrnstein-Uhlenbeck process[33]. Furthermore,weassume thattheloss functioninthe localregion aroundthe minimumiscon vex and2-orderdif ferentiable:
R(✓)=
1 2 A ,(14) whereAistheHessian matrixaroundthe minimumand isa(semi) positive-definite matrix.This assumptionhasbeen primarilydemonstratedby empiricalworks (see[18,p.1, Figures1(a)and 1(b) andp. 6,Figures4(a) and4(b)]).W ithoutlossof generality,we assumethatthe globalminimum of theobjectiv efunctionR(✓)is0andachiev esat✓=0.Generalcases canbe obtainedbytranslation operations,which wouldnot changethegeometryofobjectiv efunctionand thecorresponding generalizationability. FromtheresultsofOrnstein-Uhlenbeckprocess, eq.(13)has ananalytic stationarydistribution: q(✓)=Mexp 1 2 1 ,(15) whereMisthe normalizer[8]. EstimatingSGDas acontinuous-timestochastic processdatesback toworks by[ 17,21].For a detailedjustification,please referto arecentw ork[see23, pp.6-8,Section 3.2]. Wethenobtaina generalizationboundfor SGDasfollo ws. Theorem1.Foranypositivereal 2(0,1),withpr obabilityatleast 1overatrainingsample setofsize N,wehave thefollowinginequality forthe distribution Qoftheoutput hypothesisfunction ofSGD:R(Q)
R(Q)+ s |S| tr(CA 1 )2log(det(⌃))2d+4l og 1 +4l ogN+8 8N4 ,(16) and ⌃A+A⌃= |S|C,(17)
whereAistheHessian matrixof thelossfunction aroundthe localminimum,Bisfrom thecovariance matrixoft hegradients calculatedbysinglesamplepoints, anddisthedimens ionof theparameter✓ (networksize). Theprooffor thisgeneralizationbound hastwo parts:(1)utilize resultsfrom stochasticdifferential equation(SDE)to findthe stationarysolutionof thelatentOrnstein-Uhlenbeck process(eq.13) whiche xpressestheiterativeupdate ofSGD;and (2)adaptthePAC-Bayesframework toobtain the generalizationbound basedonthe stationarydistribution. Adetailedproof isomittedhere andis giveninAppendixB.1.3.2ASpecial Caseofthe GeneralizationBound
Inthissubsection, westudya specialcase withtwo moreassumptionsfor furtherunderstatingthe influenceofthe gradientfluctuationon ourproposedgeneralization bound. 4 Assumption1can betranslatedas thatboththe localgeometryaround theglobalminima and thestationarydistrib utionarehomogeneous toeverydimensionof theparameter space.Similar assumptionsarealso usedbya recentwork [14].Thisassumption indicatesthatthe product⌃Aof matricesAand⌃isalsosymmetric. Basedon Assumptions1,we canfurtherget thefollowing theorem. Theorem2.WhenAssumptions1 holds,underall theconditionsof Theorem1, thestationary distributionofSGDhasthe followinggener alizationbound,R(Q)
R(Q) v u u t 2|S| tr(CA 1 )+dlog 2|S| log(det(CA 1 ))d+2l og 1 +2l ogN+4 4N2 (18) Adetailedproof isomitted hereandis given inAppendixB.2 inthesupplementary materials. Intuitively,ourgeneralizationboundlinksthegeneralizationabilityof thedeepneural networks trainedbySGD withthreef actors: Localgeometryar oundminimum.Thedeterminant oftheHessianmatrix Aexpressesthelocal geometryofthe objective functionaroundthe localminimum.Specifically,themagnitudeofdet(A) expressesthesharpnessofthe localminima. Manyw orkssuggestthat sharplocalminima relateto poorgeneralizationability [15,10]. Gradientfluctuation. Thecov ariancematrixC(orequiv alentlythematrixB)expresses thefluctua- tionofthe estimationtothe gradientfrom individualdata pointswhichis thesourceof gradientnoise. Arecentintuition fortheadv antageofSGD isthatit introducesnoiseinto thegradient,sothatit can jumpoutof badlocalminima. Hyper-parameters.Batchsize|S|andlearningrate ⌘,adjustt hefluctuationof gradient.Specifically, underthefollo wingassumption,our generalizationboundhasapositi vecorrelation withthe ratioof batchsizeto learningrate.Assumption2.Thenetworksize islarg eenough:
d> tr(CA 1 2|S| ,(19) wheredisthenumber ofthe parameters, Cexpressesthemagnitudeofindividualgr adientnoise, A istheHessian matrixaround theglobalminima, ⌘istheleaning rate, and|S|isthebatc hsize . Thisassumptioncan bejustifiedthat thenetwork sizesofneural networks areusuallye xtremelylarge. Thispropertyis alsocalledoverparametrization[6,3,1].We canobtainthefollowingcorollary by combiningTheorem2 andAssumption2. Corollary1.Whenallconditions ofTheorem 2andAssumption 2hold,the generalizationboundof thenetworkhas apositivecorr elationwiththe ratioof batchsize tolearningrate. Theproofis omittedfrom themainte xtandgi venin AppendixB.3 Itrev ealsthenegativecorrelation betweenthegeneralization abilityandtheratio.Thisproperty furtherderiv esthetrainingstrategythatweshouldcontrol therationot toolar getoachievea good generalizationwhentraining deepneuralnetw orksusingSGD.4EmpiricalEvidence
Toevaluatethe trainingstrategyfromtheempiricalaspect,weconduct extensiv esystematic experi- mentstoin vestigate theinfluenceofthebatchsizeandlearningrateonthegenerali zationabilityof deepneural networkstrained bySGD.Todeliv errigorousresults, ourexperiments strictlycontrol allunrelatedv ariables.Theempirical resultsshowthatthereis astatisticallysignificant neg ative correlationbetween thegeneralizationa bilityofthe networksand theratioofthebatch sizeto the learningrate,which buildsa solidempiricalfoundation forthetrainingstrategy. 5 (a)Test accuracy-batchsize(b)Test accuracy-learningrate Figure2:Curv esof testaccuracytobatchsize andlearningrate. Thefourro wsarerespectivelyfor (1)ResNet-110 trainedonCIF AR-10,(2)ResNet-110 trainedonCIF AR-100,(3)VGG-19trainedon CIFAR-10,and(4)VGG-19 trainedonCIF AR-10.Eachcurv eisbasedon20 networks.4.1ImplementationDetails
Toguaranteethattheempiric alresultsgenerally applytoan ycase,our experimentsare conductedbasedon twopopular architectures,ResNet-110[ 12,13]andV GG-19[28],on twostandarddatasets,CIF AR-10andCIF AR-100[16],whichcan bedo wnloadedfrom https://www.cs.toronto.edu/kriz/cifar.html. Theseparationsofthetrainingsetsandthetest setsare thesame astheof ficialversion.Wetrained1,600modelswith 20batch sizes,S
BS ={16,32,48,64,80,96,112,128,144,160,176,192,208,224,240,256,272,288,304,320},and20 learningrates,S
LR ={0.01,0.02,0.03,0.04, ingtechniques ofSGD,suchasmomentum,are disabled.Also,both batchsizeand learningrate areconstantin ourexperiments. Every modelwith aspecificpairofbatchsize andlearningrateis trainedfor200 epochs.Thetest accuraciesofall 200epochsare collectedforanalysis. We selectthe highestaccurac yonthetestsetto expressthe generalizationabilityof eachmodel,since thetraining errorisalmost thesameacross allmodels (theyare allnearly0). Thecollecteddata isthen utilizedtoin vestigate threecorrelations:(1) thecorrelationbetween the generalizationabilityof networksand thebatchsize, (2)thecorrelationbetweenthe generalization abilityofnetw orksandthe learningrate,and(3)the correlationbetweenthe generalizationability ofnetworks andtheratioofbatch sizetolearning rate,wherethe firsttwo arepreparationsfor thefinalone. Specifically,we calculatetheSpearman' srank-ordercorrelationcoefficients (SCCs) andthecorresponding pvalueof164groupsof thecollecteddata toinv estigatethe statistically significanceofthe correlations.Almostall resultsdemonstratethe correlationsarestatisticall y significant(p<0.005) 1 .Thepvaluesofthecorrelationbetween thetestaccurac yandthe ratioare alllower than10 180(seeTable 3). Thearchitecturesof ourmodelsare similartoa popularimplementationof ResNet-110and VGG-19 2 Additionally,ourexperiments areconductedon acomputingclusterwithGPUsof NVIDIA Tesla
V10016GBand CPUsofIntel
XeonGold6140CPU @2.30GHz.
4.2EmpiricalResults onthe Correlation
Correlationbetweengeneralizationabilityand batchsize.Whenthelearning rateisfix edasan elementofS LR ,we trainResNet-110and VGG-19on CIFAR10and CIFAR100with 20batchsizes ofS BS .Theplots oftest accuracyto batchsizeare illustratedinFigure 2a.Welist1/4ofallplots duetospace limitation.Therest oftheplots areinthe supplementarymaterials. Wethen calculatethe SCCsand thepvaluesasTable 1,wherebold pvaluesrefertothestatistically significantobservations, whileunderlinedones refertothose notsignificant(as wellasT able2). Theresultsclearly showthat thereisa statisticallysignificantne gativ ecorrelationbetween generalizationabilityand batchsize. 1 Thedefinitionof "statisticallysignificant" hasvarious versions,such asp<0.05andp<0.01.This paper usesamore rigorousone( p<0.005). 2 SeeWei Yang,https://github.com/bearpaw/pytorch-classification, 2017. 6 Table1:SCCand pvaluesofbatchsize totestaccurac yfordif ferentlearningrate (LR). LRResNet-110on CIFAR-10 ResNet-110onCIF AR-100VGG-19onCIFAR-10 VGG-19on CIFAR-100SCCpSCCpSCCpSCCp
0.010.962.6⇥10
110.925.6⇥10
80.983.7⇥10
140.997.1⇥10
180.020.961.2⇥10
110.941.5⇥10
90.993.6⇥10
170.997.1⇥10
180.030.963.4⇥10
110.991.5⇥10
160.997.1⇥10
181.001.9⇥10
210.040.981.8⇥10
140.987.1⇥10
140.999.6⇥10
190.993.6⇥10
170.050.983.7⇥10
140.981.3⇥10
130.997.1⇥10
180.991.4⇥10
150.060.961.8⇥10
110.976.7⇥10
131.001.9⇥10
210.991.4⇥10
150.070.985.9⇥10
150.945.0⇥10
100.988.3⇥10
150.971.7⇥10
120.080.971.7⇥10
120.971.7⇥10
120.982.4⇥10
130.971.7⇥10
120.090.974.0⇥10
130.983.7⇥10
140.981.8⇥10
140.961.2⇥10
110.100.971.9⇥10
120.968.7⇥10
120.988.3⇥10
150.932.2⇥10
90.110.971.1⇥10
120.981.3⇥10
130.992.2⇥10
160.932.7⇥10
90.120.974.4⇥10
120.962.5⇥10
110.987.1⇥10
130.907.0⇥10
80.130.941.5⇥10
90.981.3⇥10
130.971.7⇥10
120.891.2⇥10
70.140.972.6⇥10
120.913.1⇥10
80.976.7⇥10
130.861.1⇥10
60.150.964.6⇥10
110.981.3⇥10
130.958.3⇥10
110.793.1⇥10
50.160.953.1⇥10
10quotesdbs_dbs8.pdfusesText_14[PDF] optimal solution example
[PDF] optimal solution in lpp
[PDF] optimal solution in transportation problem
[PDF] optimal solution of linear programming problem
[PDF] optimal work week hours
[PDF] optimise b2 workbook answers pdf
[PDF] optimise workbook b2 answers
[PDF] optimistic words
[PDF] optimum basic feasible solution in transportation problem
[PDF] optimum camera
[PDF] optimum channel guide ct
[PDF] optimum dental insurance
[PDF] optimum google
[PDF] optimum portal
