第九章网络计划技术(MBA讲义)

Chapter12PERT/CPMModelsfor

ProjectManagement§1ProjectNetworks方案网络图§2SchedulingaProjectwithPERT/CPM网络时间与关键道路§3DealingwithUncertainActivityDurations§4网络优化1工程管理ProjectManagement工程特征CharacteristicsofProjectsUnique,one-timeoperationsInvolvealargenumberofactivitiesthatmustbeplannedandcoordinatedLongtime-horizonGoalsofmeetingcompletiondeadlinesandbudgetsExamplesBuildingahousePlanningameetingIntroducinganewproductPERT—ProjectEvaluationandReviewTechnique方案评审技术

CPM—CriticalPathMethod关键道路法Agraphicalornetworkapproachforplanningandcoordinatinglarge-scaleprojects.可信建筑公司工程ReliableConstructionCompanyProjectP285TheReliableConstructionCompanyhasjustmadethewinningbidof$5.4milliontoconstructanewplantforamajormanufacturer.可信建筑公司中标540万美圆的工程Thecontractincludesthefollowingprovisions:Apenaltyof$300,000ifReliablehasnotcompletedconstructionwithin47weeks.假设不能在47周内完成工程，将被罚款30万美圆。假设能在40周内完成，将获得15万美圆的奖励Abonusof$150,000ifReliablehascompletedtheplantwithin40weeks.Questions:286Howcantheprojectbedisplayedgraphicallytobettervisualizetheactivities?如何用图来表示工程中活动流程Whatisthetotaltimerequiredtocompletetheprojectifnodelaysoccur?Whendotheindividualactivitiesneedtostartandfinish?Whatarethecriticalbottleneckactivities?对于非关键的活动Forotheractivities,howmuchdelaycanbetolerated接受?Whatistheprobabilitytheprojectcanbecompletedin47weeks?Whatistheleastexpensivewaytocompletetheprojectwithin40weeks?Howshouldongoingcostsbemonitoredtotrytokeeptheprojectwithinbudget?如何对本钱进展实时监控，使工程本钱控制在预算内活动明细表ActivityListforReliableConstructionActivityActivityDescription紧前Immediate

Predecessors估计工期(Weeks)AExcavate挖掘—2BLaythefoundation打地基A4CPutuptheroughwall承重墙施工B10DPutuptheroof封顶C6EInstalltheexteriorplumbing安装外部管道C4FInstalltheinteriorplumbing安装内部管道E5GPutuptheexteriorsiding外墙施工D7HDotheexteriorpainting外部粉刷E,G9IDotheelectricalwork电路铺设C7JPutupthewallboard竖墙板F,I8KInstalltheflooring铺地板J4LDotheinteriorpainting内部粉刷J5MInstalltheexteriorfixtures设施H2NInstalltheinteriorfixtures安装内部设备K,L6工程网络ProjectNetworks表示整个工程的网络称为工程网络Anetworkusedtorepresentaprojectiscalledaprojectnetwork.Aprojectnetworkconsistsofanumberof节点nodesandanumberofarcs.Twotypesofprojectnetworks:Activity-on-arc(AOA弧表示活动):eachactivityisrepresentedbyanarc.Anodeisusedtoseparateanactivityfromitspredecessors.Thesequencingofthearcsshowstheprecedencerelationships.Activity-on-node(AON节点表示活动):eachactivityisrepresentedbyanode.Thearcsareusedtoshowtheprecedence优先relationships.AdvantagesofAON(usedinthistextbook):P288considerablyeasiertoconstruct易于构建easiertounderstand易于了解easiertorevisewhentherearechanges更便于修正ReliableConstruction

ProjectNetworkMicrosoftProjectGanttChart甘特图W1/1表示第一周第一天；W2/5表示第二周第五天；MicrosoftProject

—ProjectNetwork活动名称编号工期最早开始时间最早结束时间P290W1/1第一周第一天开场W2/5第二周第五天完成二周AOAProjectNetwork1235ABCD6GEI4871011139H12FJKLNM关于AOA方案网络图统筹方法的第一步任务就是绘制方案网络图，也就是将工序〔或称为活动〕进度表转换为统筹方法的网络图。下面用一个例子来阐明例3、某公司研制新产品的部分工序与所需时间以及它们之间的相互关系都显示在其工序进度表如表12-8所示，请画出其统筹方法网络图。工序代号工序内容所需时间（天)紧前工序abcde产品设计与工艺设计外购配套零件外购生产原料自制主件主配可靠性试验601513388-aacb,d2表12-8网络图中的节点表示一个事件，是一个或假设干个工序的开场或终了，是相邻工序在时间上的分界点，点圆圈表示，数字表示点的编号。弧表示一个工序〔或活动〕，弧的方向是从工序开场指向工序的终了，弧上是各工序的代号，下面标以完成此工序所需的时间〔或资源〕等数据，即为对此弧所赋的权数．虚工序：只表示相邻工序的前后逻辑关系。需求人力、物力等资源与时间。画网络图的规那么：两节点之间只能有一条弧〔箭线〕。两事件一工序网络图只能有一个开场节点〔事项〕和一个终点事项不允许有回路，不能有缺口

abcb,c的紧前工序为aabcdc,d的紧前工序为a,babcdc的紧前工序为a。d的紧前工序为a和b3绘制网络图分3步：(1)义务分解WBSWorkbreakdownStructure任务工程明细表：工序称号，代号；前后逻辑关系，耗费资源。(2)按表绘制网络图。〔3〕节点一致编号。采用平行和交叉作业技术例４、把例３的工序进度表做一些扩展，如表12-9，请画出其统筹方法的网络图。表12-9工序代号所需时间（天）紧前工序工序代号所需时间（天）紧前工序abcd60151338－aacefgh810165b，ｄdde，ｆ，ｇ１２４５３abcde601383815图12-44解：我们把工序ｆ扩展到图12-4发生了问题，由于ｆ的紧前工序是ｄ，故ｄ的终了应该是ｆ的开场，所以代表ｆ的弧的起点应该是④，由于工序ｂ的终了也是④，所以工序ｂ也成了工序ｆ的紧前工序，与题意不符。为此我们利用虚工序。由于e的紧前工序是b和ｄ，在④和⑤之间参与虚工序

8e15243a60b151013dc38f图12-55在网络图上添加ｇ、ｈ工序得网络图12-6。

在统筹方法的网络图中不允许两个点之间多于一条弧，因此添加了一个点和虚工序如图12-7。615243a60b158e1013dc38fg16h515243a60b158e1013dc38fg16h57681ActivityTime(Weeks)Immediate

PredecessorsActivityTime(Weeks)Immediate

PredecessorsABCDE241064－ABCCHIJKL97845E,GCF,IJJF5EM2HG7DN6K,L2A23B4C4105D67EI46G87F7510J89H9KLM542111312N6关键道路TheCriticalPathApaththroughanetworkisoneoftheroutesfollowingthearrows(arcs)fromthestartnodetothefinishnode.路是指沿着箭头从开场点到终点的一条道路Thelengthofapath

isthesumofthe(estimated)durationsoftheactivitiesonthepath.路长度为路上一切活动时间总和The(estimated)projectdurationequalsthelengthofthelongestpaththroughtheprojectnetwork.工程工期等于工程网络中最长路的长度Thislongestpathiscalledthecriticalpath.(Ifmorethanonepathtieforthelongest,theyallarecriticalpaths.)工程网络中最长路称为关键路ThePathsforReliable’sProjectNetworkPathLength(Weeks)StartABCDGHMFinish2+4+10+6+7+9+2=40StartABCEHMFinish2+4+10+4+9+2=31StartABCEFJKNFinish2+4+10+4+5+8+4+6=43StartABCEFJLNFinish2+4+10+4+5+8+5+6=44StartABCIJKNFinish2+4+10+7+8+4+6=41StartABCIJLNFinish2+4+10+7+8+5+6=42关键道路criticalpathP292最早开场时间和最早终了时间Thestartingandfinishingtimesofeachactivityifnodelaysoccuranywhereintheprojectarecalledtheearlieststarttimeandtheearliestfinishtime.用ES表示一个活动最早开场时间；EF表示一个活动最早终了时间。最早能够开场时间和终了时间。EarliestStartTimeRule:ES=LargestEFoftheimmediatepredecessors.最早开场时间=一切紧前工序EF的最大者ABCiES=Max{20,23,19}=23EF=20EF=23EF=19HCBAHEF=20EF=23EF=19ES=23计算一切活动的ES时间:Foreachactivitythatstartstheproject(includingthestartnode),setitsES=0.工程的最早开场时间为0最早终了时间EF=ES+活开工期ForeachactivitywhoseEShasjustbeenobtained,calculateEF=ES+duration对于每个新活动，用最早开场时间法那么求ESForeachnewactivitywhoseimmediatepredecessorsnowhaveEFvalues,obtainitsESbyapplyingtheearlieststarttimerule.Applystep2tocalculateEF.反复上述3个步骤Repeatstep3untilESandEFhavebeenobtainedforallactivities.最早终了时间EF=最早开场时间ES+活开工期从开场(左)向后计算ES只需单个紧前工序的工序ES与EF的计算ESandEFTimesforReliableConstruction多个紧前工序的ES与EF的计算ES=Max{25,23}=25ES=Max{29,20}=29最迟开场时间和最迟终了时间Thelateststarttimeforanactivityisthelatestpossibletimethatitcanstartwithoutdelayingthecompletionoftheproject(sothefinishnodestillisreachedatitsearliestfinishtime).在不影响工程完成总时间的前提下，一个活动最晚必需开场的时间。对于终点，最晚时间依然是最早终了时间,LF=EFLS=Lateststarttimeforaparticularactivity最晚开场时间LF=Latestfinishtimeforaparticularactivity最晚终了时间LatestFinishTimeRule:LF=SmallestLSoftheimmediatesuccessors.最晚时间等于其一切紧后活动最晚开场时间的最小者ABCLF=Min{30,28,27}=27LS=30LS=28LS=27HCBAHLS=30LS=28LS=27LF=27LSandLFTimesforReliable’sProjectMin{20,33}=20F[20,25][20,25]EFLSLFESH[29,38][33,42]EFLSLFESTheCompleteProjectNetworkCriticalpath:StartABCEFJLNFinish{ES，LS}{EF，LF}TimesforReliable’sProjectAOA1235ABCD6GEI4871011139H12FJKLNM[0,2][0,2]2[2,6][2,6]4[6,16][6,16]107[16,23][16,20]4[16,22]67[22,29][29,38]9[38,40]25[20,25][25,33]85[33,38][33,37]4[38,44]6[38,44][42,44][34,38][33,38][31,42][25,33][20,25][18,25][16,20][24,31][18,24]SlackforReliable’sActivities活动〔工序〕的时差活动LSESLFEF总时差(LS–ES)是否关键工序A00220YesB22440YesC4416160YesD201626224NoE161620200YesF202025250YesG262233294NoH332942384NoI181625232NoJ252533330YesK343338371NoL333338380YesM423844404NoN383844440Yes总时差=最迟开场时间－最早开场时间=最迟终了时间－最早终了时间TS=LS－ES=LF－EFTheslackofanactivity例、某公司装配一条新的消费线，详细过程如表12-10,求：完成此工程的最少时间，关键道路及相应的关键工序，各工序的最早开场时间和非关键工序在不影响工程完成时间的前提下，其开场时间与终了时间可以推迟多久。7工序代号工序内容所需时间（天）紧前工序abcdefghij生产线设计外购零配件下料、锻件工装制造1木模、铸件机械加工1工装制造2机械加工2机械加工3装配调试60451020401830152535/aaaacdd,egb,i,f,h解：据表12-10,绘制网络图如图12-8。图12-8关键道路：如图12-8,①-②-③-⑦-⑧就是一条关键道路，我们要干完一切的工序就必需走完一切这样的道路，由于很多工序可以同时进展，所以网络中最长的道路就决议了完成整个工程所需的最少时间，这条道路称为关键道路。12346785a60b45echj35ig1030d204025f18159

图12-10其次,从网络的收点开场计算出在不影响整个工程最早终了时间的情况下各个工序的最晚开场时间(缩写为LS)和最晚终了时间〔缩写为LF),显然对同一工序有LS=LF-t1236785a[0,60]60b[60,105]45e[60.100]c[60,70]h[100,115]j[,170]35i[110.]g[80,110]30d[60.80]204025f[70,88]184101511

运用此法那么，可以从首点开场计算出每个工序的LF与LS，如图12-11所示。

接着，可以计算出每一个工序的时差，把在不影响工程最早终了时间的条件下，工序最早开场〔或终了〕的时间可以推迟的时间，成为该工序的时差，对每个工序来说其时差记为Ts有Ts=LS-ES=LF-EF1236785a[0,60]60[0,60]b[60,105]45[90,]e[60.100]c[60,70]h[100,115]j[,170]35[,170]i[110.]g[80,110]30[80,110]d[60.80]20[60,80]40[80,120]25[110,]f[70,88]18[117,]410[107,117]15[120,]12最后将各工序的时差，以及其他信息构成工序时间表如表12-11所示。这样就找到了一条由关键工序a,d,g,i和j依次衔接成的从发点到收点的关键道路。13ThePERT三种估计方式Mostlikelyestimate(m)=Estimateofmostlikelyvalueoftheduration最大能够估计

Optimisticestimate(o)=Estimateofdurationundermostfavorableconditions.乐观估计时间

Pessimisticestimate(p)=Estimateofdurationundermostunfavorableconditions.悲观估计时间处置不确定活开工期分布时间均值和规范差MeanandStandardDeviationAnapproximateformulaforthevariance活动时间方差(2)ofanactivityis Anapproximateformulaforthemean活动时间的均值(m)ofanactivityis

乐观估计P和悲观估计时间O的差大致为6个规范差可信公司活动的均值和方差ActivityompMeanVarianceA12321/9B23.5841C6918104D45.51061E14.5544/9F441051G56.51171H581794I37.5971J39981K44440L15.5751M12321/9N55.5964/9P306PessimisticPathLengthsforReliable’sProject悲观估计时间的途径与途径长度PathPessimisticLength(Weeks)采用悲观估计时间StartABCDGHMFinish3+8+18+10+11+17+3=70StartABCEHMFinish3+8+18+5+17+3=54StartABCEFJKNFinish3+8+18+5+10+9+4+9=66StartABCEFJLNFinish3+8+18+5+10+9+7+9=69StartABCIJKNFinish3+8+18+9+9+4+9=60StartABCIJLNFinish3+8+18+9+9+7+9=63三个简化的近似PERT/CPMP306Themeancriticalpathwillturnouttobethelongestpaththroughtheprojectnetwork.假设均值关键路是工程中最长的一条路统计独立性Thedurationsoftheactivitiesonthemeancriticalpatharestatisticallyindependent.Thus,thethreeestimatesofthedurationofanactivitywouldneverchangeafterlearningthedurationsofsomeoftheotheractivities.正太分布假设Theformoftheprobabilitydistributionofprojectdurationisthenormaldistribution.Byusingsimplifyingapproximations1and2,thereissomestatisticaltheory(oneversionofthecentrallimittheorem)thatjustifiesthisasbeingareasonableapproximationifthenumberofactivitiesonthemeancriticalpathisnottoosmall〔活动数至少5个〕.计算工程的均值和方差ActivitiesonMeanCriticalPathMeanVarianceA21/9B41C104E44/9F51J81L51N64/9Projectdurationmp=44s2p=9ProbabilityofMeetingDeadlinez为d超越μp的规范差个数，查统计表可得概率值ProbabilityofMeetingaDeadlineP(T≤

d)P(T≤d)–3.00.001400.50–2.50.00620.250.60–2.00.0230.50.69–1.750.0400.750.77–1.50.0671.00.84–1.250.111.250.89–1.00.161.50.933–0.750.231.750.960–0.50.312.00.977–0.250.402.50.993800.503.00.9986SpreadsheetforPERTThree-EstimateApproach时间和本钱权衡ConsideringTime-CostTrade-OffsQuestion:Ifextramoneyisspenttoexpeditetheproject,whatistheleastexpensivewayofattemptingtomeetthetargetcompletiontime(40weeks)?CPMMethodofTime-CostTrade-Offs:Crashinganactivity赶工完成一项活动referstotakingspecialcostlymeasurestoreducethedurationofanactivitybelowitsnormalvalue.Specialmeasuresmightincludeovertime,hiringadditionaltemporaryhelp,usingspecialtime-savingmaterials,obtainingspecialequipment,etc.Crashingtheproject赶工完成工程referstocrashinganumberofactivitiestoreducethedurationoftheprojectbelowitsnormalvalue.Time-CostGraphforanActivity赶工常规赶工本钱正常本钱活动本钱常规常规时间赶工时间活动时间Time-CostTrade-OffDataforReliable’sProject

Time(weeks)

成本Cost

Maximum

Reduction

inTime(weeks)CrashCost

perWeek

SavedActivityNormalCrashNormalCrashA21$180,000$280,0001$100,000B42320,000420,000250,000C107620,000860,000380,000D64260,000340,000240,000E43410,000570,0001160,000F53180,000260,000240,000G74900,0001,020,000340,000H96200,000380,000360,000I75210,000270,000230,000J86430,000490,000230,000K43160,000200,000140,000L53250,000350,000250,000M21100,000200,0001100,000N63330,000510,000360,000Sumofnormalcost=4550,000;Sumofcrashcost=6150,000P400MarginalCostAnalysisforReliable’sProject

InitialTableLengthofPathActivity

to

CrashCrash

CostABCDGHMABCEHMABCEFJKNABCEFJLNABCIJKNABCIJLN403143444142最长路MarginalCostAnalysisforReliable’sProject

TableAfterCrashingOneWeekLengthofPathActivity

to

CrashCrash

CostABCDGHMABCEHMABCEFJKNABCEFJLNABCIJKNABCIJLN403143444142J$30,000403142434041MarginalCostAnalysisforReliable’sProject

TableAfterCrashingTwoWeeksLengthofPathActivity

to

CrashCrash

CostABCDGHMABCEHMABCEFJKNABCEFJLNABCIJKNABCIJLN403143444142J$30,000403142434041J$30,000403141423940J只大只能赶工2天MarginalCostAnalysisforReliable’sProject

TableAfterCrashingThreeWeeksLengthofPathActivity

to

CrashCrash

CostABCDGHMABCEHMABCEFJKNABCEFJLNABCIJKNABCIJLN403143444142J$30,000403142434041J$30,000403141423940F$40,000403140413940MarginalCostAnalysisforReliable’sProject

FinalTableAfterCrashingFourWeeksLengthofPathActivity

to

CrashCrash

CostABCDGHMABCEHMABCEFJKNABCEFJLNABCIJKNABCIJLN403143444142J$30,000403142434041J$30,000403141423940F$40,000403140413940F$40,000403139403940ProjectNetworkAfterCrashingUsingLPtoMakeCrashingDecisionsRestatmentoftheproblem:Considerthetotalcostoftheproject,includingtheextracostofcrashingactivities.Theproblemthenistominimizethistotalcost,subjecttotheconstraintthatprojectdurationmustbelessthanorequaltothetimedesiredbytheprojectmanager.Thedecisionstobemadearethefollowing:Thestarttimeofeachactivity.每一个活动的开场时间Thereductioninthedurationofeachactivityduetocrashing.每个活开工期的减少量Thefinishtimeoftheproject(mustnotexceed40weeks).Theconstraintsare:TimeReduction≤MaxReduction(foreachactivity).ProjectFinishTime≤DesiredFinishTime.ActivityStartTime≥ActivityFinishTimeofallpredecessors(foreachactivity).ProjectFinishTime≥FinishTimeofallimmediatepredecessorsoffinishnode.SpreadsheetModelMr.Perty’sConclusionsTheplanforcrashingtheprojectonlyprovidesa50percentchanceofactuallyfinishingtheprojectwithin40weeks,sotheextracostoftheplan($140,000)isnotjustified.Therefore,Mr.Pertyrejectsanycrashingatthisstage.Theextracostofthecrashingplancanbejustifiedifitalmostcertainlywouldearnthebonusof$150,000forfinishingtheprojectwithin40weeks.Therefore,Mr.PertywillholdtheplaninreservetobeimplementediftheprojectisrunningwellaheadofschedulebeforereachingactivityF.Theextracostofpartorallofthecrashingplancanbeeasilyjustifiedifitlikelywouldmakethedifferenceinavoidingthepenaltyof$300,000fornotfinishingtheprojectwithin47weeks.Therefore,Mr.PertywillholdthecrashingplaninreservetobepartiallyorwhollyimplementediftheprojectisrunningfarbehindschedulebeforereachingactivityForactivityJ.SchedulingandControllingProjectCostsPERT/Costisasystematicprocedure(normallycomputerized)tohelptheprojectmanagerplan,schedule,andcontrolcosts.Assumption:AcommonassumptionwhenusingPERT/Costisthatthecostsofperforminganactivityareincurredataconstantratethroughoutitsduration.BudgetforReliable’sProjectActivityEstimated

Duration(weeks)Estimated

CostCostperWeek

o