数据、模型与决策(运筹学)课后习题和案例答案008

Chapter8

Pert/cpmmodelsforProjectManagementReviewQuestions8.1-1 Thebidisfor$5.4millionwithapenaltyof$300,000ifthedeadlineof47weeksisnotmet.Inaddition,abonusof$150,000willbepaidiftheplantiscompletedwithin40weeks.8.1-2 Hehasdecidedtofocusonmeetingthedeadlineof47weeks.8.1-3 Animmediatepredecessorisanactivitythatmustbecompletedjustpriortostartingthegivenactivity.Giventheimmediatepredecessorsofanactivity,thisactivitythenbecomestheimmediatesuccessorofeachoftheseimmediatepredecessors.8.1-4 (1)theactivitiesoftheproject;(2)theimmediatepredecessorsoftheactivities;and(3)theestimateddurationoftheactivities8.2-1 (1)activityinformation;(2)precedencerelationships;and(3)timeinformation(duration)8.2-2 InanAOAnetwork,eachactivityisrepresentedbyanarc,whileinanAONnetwork,eachactivityisrepresentedbyanode.AONnetworksarebeingusedhere.8.2-3 ThebarsinaGanttchartshowthescheduledstartandfinishtimesforactivitiesinaproject.8.3-1 (a)Apaththroughaprojectnetworkisoneoftheroutesfollowingthearrows(arcs)fromthestartnodetothefinishnode;(b)thelengthofapathisthesumoftheestimateddurationsoftheactivitiesonthepath;(c)thelongestpathiscalledthecriticalpath.8.3-2 (1)Theactualdurationofeachactivitymustbethesameasitsestimatedduration;and(2)eachactivitymustbeginassoonasallitsimmediatepredecessorsarefinished.8.3-3 Theearlieststarttimeofanactivityisequaltothelargestoftheearliestfinishtimesofitsimmediatepredecessors.8.3-4 Aforwardpassistheprocessofstartingwiththeinitialactivitiesandworkingforwardintimetowardthefinalactivities.8.3-5 Itisalastchanceschedulebecauseanythinglaterwilldelaythecompletionoftheproject.8.3-6 Thelatestfinishtimeofanactivityisequaltothesmallestofthelateststarttimesofitsimmediatesuccessors.8.3-7 Abackwardpassstartswiththefinalactivitiesandworksbackwardintimetowardtheinitialactivitiesinsteadofstartingwiththeinitialactivities.8.3-8 Anydelayalongthecriticalpathwilldelayprojectcompletion.8.3-9 (1)Identifythelongestpaththroughtheprojectnetwork;and(2)identifytheactivitieswithzeroslack—theyareonthecriticalpath.8.4-1 Thethreeestimatesarethemostlikelyestimate,optimisticestimate,andpessimisticestimate.8.4-2 Theoptimisticandpessimisticestimatesaremeanttolieattheextremesofwhatispossible,whereasthemostlikelyestimateprovidesthehighestpointoftheprobabilitydistribution.8.4-3 Itisassumedthatthemeancriticalpathwillturnouttobethelongestpaththroughtheprojectnetwork.8.4-4 Itisassumedthatthedurationsoftheactivitiesonthemeancriticalpatharestatisticallyindependent.8.4-5 p=sumofthemeansofthedurationsfortheactivitiesonthemeancriticalpath.8.4-6 p2=sumofthevariancesofthedurationsfortheactivitiesonthemeancriticalpath.8.4-7 Itisassumedthattheformoftheprobabilitydistributionofprojectdurationisthenormaldistribution.8.4-8 Itisusuallyhigherthanthetrueprobability.8.5-1 Usingovertime,hiringadditionallabor,andusingspecialmaterialsorequipmentareallwaysofcrashinganactivity.8.5-2 Thetwokeypointsarelabelednormalandcrash.Thenormalpointshowsthetimeandcostoftheactivitywhenitisperformedinthenormalway.Thecrashpointshowsthetimeandcostwhentheactivityisfullycrashed.8.5-3 No,onlycrashingactivitiesonthecriticalpathwillreducethedurationoftheproject.8.5-4 Crashcostsperweeksavedarebeingexamined.8.5-5 Thedecisionstobemadearethestarttimeofeachactivity,thereductioninthedurationofeachactivityduetocrashing,andthefinishtimeoftheproject.8.5-6 Anactivitycannotstartuntilitsimmediatepredecessorstartsandthencompletesitsduration.8.5-7 Becauseofuncertainty,theplanforcrashingtheprojectonlyprovidesa50%chanceofactuallyfinishingwithin40weeks,sotheextracostoftheplanisnotjustified.8.6-1 PERT/Costisasystematicproceduretohelpthemanagerplan,schedule,andcontrolprojectcosts.8.6-2 Itbeginsbydevelopinganestimateofthecostofeachactivitywhenitisperformedintheplannedway.8.6-3 Acommonassumptionisthatthecostsofperforminganactivityareincurredataconstantratethroughoutitsduration.8.6-4 Aworkpackageisagroupofrelatedactivities.8.6-5 PERT/Costusesearlieststarttimeandlateststarttimeschedulesasabasisfordevelopingcostschedules.8.6-6 APERT/Costscheduleofcostsshowstheweeklyprojectcostandcumulativeprojectcostforeachtimeperiod.8.6-7 APERT/Costreportshowsthebudgetedvalueoftheworkcompletedofeachactivityandthecostoverrunstodate.8.6-8 Sincedeviationsfromtheplannedworkschedulemayoccur,aPERT/Costreportisneededtoevaluatethecostperformanceofindividualactivities.8.7-1 Planning,scheduling,dealingwithuncertainty,time-costtrade-offs,andcontrollingcostsareaddressedbyPERT/CPM.8.7-2 Computerimplementationhasallowedforapplicationtolargerprojects,fasterrevisionsinprojectplansandeffortlessupdatesandchangesinschedules.8.7-3 Theaccuracyandreliabilityofend-pointestimatesarenotasgoodforpointsthatarenotattheextremesoftheprobabilitydistribution.8.7-4 ThetechniqueofcomputersimulationtoapproximatetheprobabilitythattheprojectwillmeetitsdeadlineisanalternativeforimprovingonPERT/CPM.8.7-5 ThePrecedenceDiagrammingMethodhasbeendevelopedasanextensionofPERT/CPMtodealwithoverlappingactivities.8.7-6 PERT/CPMassumesthateachactivityhasavailablealltheresourcesneededtoperformtheactivityinanormalway.8.7-7 Itencourageseffectiveinteractionbetweentheprojectmanagerandsubordinatesthatleadstosettingmutualgoalsfortheproject.8.7-8 Newimprovementsandextensionsarestillbeingdevelopedbuthavenotbeenincorporatedmuchintopracticeyet.Problems8.1 a)

b) StartACFinish Length=4weeks

StartADEFinish Length=7weeks

StartBCFinish Length=5weeks

StartBDEFinish Length=8weeks *criticalpath c)ActivityESEFLSLFSlackCriticalPathStart00000YesA02131NoB03030YesC35683NoD35350YesE58580YesFinish88880Yes

CriticalPath: StartBDEFinish d) No,thiswillnotshortenthelengthoftheprojectbecausetheactivityisnotonthecriticalpath.8.2 a)

b) StartADFinish Length=4weeks

StartAEFinish Length=5weeks

StartAFKFinis Length=8weeks *criticalpath

StartAGHIJFinish Length=8weeks *criticalpath

StartBDFinish Length=3weeks

StartBCEFinish Length=6weeks

StartBCHIJFinish Length=8weeks *criticalpath

StartBCKFinish Length=7weeks c)ActivityESEFLSLFSlackCriticalPathStart00000YesA02020YesB01010YesC13130YesD24684NoE36582NoF24240YesG23230YesH34340YesI45450YesJ58580YesK48480YesFinish88880Yes

CriticalPaths: StartAFKFinish

StartAGHIJFinish

StartBCHIJFinish d) No,thiswillnotshortenthelengthoftheprojectbecauseAisnotonallofthecriticalpaths.8.3 a)

b,c,&d)ActivityESEFLSLFSlackCriticalPathStart00000YesA01010YesB13130YesC37370YesD36360YesE35461NoF7107100YesG6116110YesH101110110YesI111511150YesJ151715170YesK151816191NoL172017200YesM182319241NoN202420240YesFinish242424240Yes

CriticalPaths: StartABCFHIJLNFinish

StartABDGIJLNFinish8.4 a)

b) StartABJLFinish Length=75minutes *criticalpath

StartCDJLFinish Length=45minutes

StartEFJLFinish Length=72minutes

StartGHIJLFinish Length=67minutes

StartKLFinish Length=45minutes c,d&e)ActivityESEFLSLFSlackCriticalPathStart00000YesA0300300YesB303530350YesC02303230NoD25323530NoE073103NoF73210353NoG0158238NoH152523338NoI252733358NoJ354535450YesK015304530NoL457545750YesFinish757575750Yes

CriticalPath: StartABJLFinish f) Dinnerwillbedelayed3minutesbecauseofthephonecall.Ifthefoodprocessorisusedthendinnerwillnotbedelayedbecausetherewas3minutesofslackand5minutesofcuttingtimesavedandthecallonlyused6minutesofthe8total.8.5 a) StartADHMFinish Length=19weeks

StartBEJMFinish Length=20weeks *criticalpath

StartCFKNFinish Length=16weeks

StartAIMFinish Length=17weeks

StartCGLNFinish Length=20weeks *criticalpath b)ActivityESEFLSLFSlackCriticalPathStart00000YesA06171NoB03030YesC04040YesD6107111NoE3103100YesF488124NoG4104100YesH101311141NoI6119143NoJ101410140YesK81112154NoL101510150YesM142014200YesN152015200YesFinish202020200Yes

Kenwillbeabletomeethisdeadlineifnodelaysoccur. c) CriticalPaths: StartBEJMFinish

StartCGLNFinish

Focusattentiononactivitieswith0slack(thoseinthecriticalpaths). d) IfactivityItakes2extraweekstherewillbenodelaybecauseitsslackis3.IfactivityHtakes2extraweeksthentherewillbeadelayof1weekbecauseitsslackis1.IfactivityJtakes2extraweekstherewillbeadelayof2weeksbecauseithasnoslack.8.6 a)

b)ActivityESEFLSLFSlackCriticalPathStart00000YesA05050YesB01111211NoC13121411NoD597112NoE5115110YesF111411140YesFinish141414140Yes

CriticalPath: StartAEFFinish Length=14months c) 6months8.7 ActivityESEFLSLFSlackCriticalPathStart00000YesA03030YesB3113110YesC112911290YesD293929390YesE293430351NoF344435451NoG395039500YesH506750670YesI293836457NoJ445345541NoK535757614NoL536054611NoM677067700YesN606961701NoFinish707070700Yes

CriticalPath: StartABCDGHMFinish

Totalduration=70weeks8.8ActivityESEFLSLFSlackCriticalPathStart00000YesA01010YesB13130YesC39390YesD91311152NoE9109100YesF101410140YesG131815202NoH182320252NoI91211142NoJ141714170YesK172117210YesL171820213NoM232425262NoN212621260YesFinish262626260Yes

CriticalPath: StartABCEFJKNFinish

Totalduration=26weeks8.9ActivityESEFLSLFSlackCriticalPathStart00000YesA01010YesB13130YesC3103100YesD101413173NoE101310130YesF131613160YesG141817213NoH182421273NoI101511161NoJ162216220YesK222522250YesL222522250YesM242527283NoN252825280YesFinish282828280Yes

CriticalPaths: StartABCEFJKNFinish

StartABCEFJLNFinish

Totalduration=28weeks8.10 =(o+4m+p)/6=[30+(4)(36)+48]/6=37

=[(p–o)/6]2=[(48–30)/6]2=98.11 a) StartAEIFinish Length=17months

StartACFIFinish Length=17months

StartBDGJFinish Length=17months

StartBHJFinish Length=18months *criticalpath b)

ByTable8.7,P(T≤22months)isjustlessthan0.77. c) StartAEIFinish

ByTable8.7,P(T≤22months)=0.84.

StartACFIFinish

ByTable8.7,P(T≤22months)isjustlessthan0.84.

StartBDGJFinish

ByTable8.7,P(T≤

22months)isjustlessthan0.84. d) Thereissomewhatlessthana77%chancethatthedrugwillbereadyin22weeks.8.12 StartAEIFinish

StartACFIFinish

StartBDGJFinish

StartBHJFinish

Then,basedonthesespreadsheets,theanswersto(a),(b),(c),and(d)wouldbe a) StartAEIFinish Length=17.08months

StartACFIFinish Length=17.58months

StartBDGJFinish Length=17.83months

StartBHJFinish Length=18.42months *criticalpath b) P(StartBHJFinish≤22months)=0.7394 c) P(StartAEIFinish≤22)=0.8356

P(StartACFIFinish≤22)=0.8007

P(StartBDGJFinish≤22)=0.7843 d) Thereisapproximatelya73%chancethatthedrugwillbereadyin22weeks.8.13 a)

b)ActivityA40.111B20C4.830.25D30.444E3.170.25 c) StartABCFinish Length=10.83weeks *criticalpath

StartABEFinish Length=9.17weeks

StartADEFinish Length=10.17weeks d) .

Then,byAppendixA,P(T≤11weeks)=0.512,

orbyTable8.7,P(T≤11weeks)=slightlymorethan0.5. e) Itisaborderlinecall.BythePERTanalysis,thereisbarelymorethana50%chanceofmeetingthedeadline,butPERTtendstoprovideoptimisticestimates.8.14 a)ActivityA120B2316C151D279E184F64 b) StartACEFFinish Length=51days *criticalpath

StartBDFinish Length=50days c) .

ByAppendixA,P(T≤57days)=0.9772.

ByTable8.7,P(T≤

57days)=0.997. d) .

ByAppendixA,P(T≤57days)=0.9192.

ByTable8.7,P(T≤

57days)isbetween0.89and0.933. e) (0.9772)(0.9192)=0.8982.

Thisanswertellsusthattheprocedureusedinpart(c)overestimatestheprobabilityofcompletingwithin57days.8.15 a)ActivityA321.78B27.72.78C3611.1D160.444E320F53.732.1G16.74H20.32.78I347.11J17.79 b) StartACJFinish Length=85.7weeks

StartBFJFinish Length=99.1weeks *criticalpath

StartBEHFinish Length=80weeks

StartBEIFinish Length=93.7weeks

StartBDGHFinish Length=80.7weeks

StartBDGIFinish Length=94.4weeks c)

ByAppendixA,P(T≤100weeks)=0.5557.

ByTable8.7,P(T≤

100weeks)isbetween0.5and0.6. d) Higher8.16 a) False.Theoptimisticestimateisthedurationunderthemostfavorableconditions.Thereforeactivitydurationsareassumedtobenosmallerthantheoptimisticestimate.Similarly,activitydurationsareassumedtobenolargerthanthepessimisticestimate.(Pg.340) b) False.PERTalsoassumesthattheformoftheprobabilitydistributionisabetadistribution.(Pg.340) c) False.Itisassumedthatthemeancriticalpathwillturnouttobethelongestpaththroughtheprojectnetwork.(Pg.343)8.17ActivityCrashLengthofPathtoCrashCostA–CB-D1416B$5,0001415B$5,0001414C,D$4,000+$6,0001313C,D$4,000+$6,00012128.18 a) Let A=reductioninAduetocrashing

C=reductioninCduetocrashing

MinimizeCrashingCost=$5,000A+$4,000C

subjectto A≤

3months

C≤

2months

A+C≥2months

and A≥

0,C≥0.

Optimalsolution:(x1,x2)=(A,C)=(0,2)andCrashingCost=$8,000. b) Let B=reductioninBduetocrashing

D=reductioninDduetocrashing

MinimizeCrashingCost=$5,000B+$6,000D

subjectto B≤

2months

D≤

3months

B+D≥4months

and B≥

0,D≥0.

Optimalsolution:(x1,x2)=(B,D)=(2,2)andCrashingCost=$22,000. c) Let A=reductioninAduetocrashing

B=reductioninBduetocrashing

C=reductioninCduetocrashing

D=reductioninDduetocrashing

MinimizeCrashingCost=$5,000A+$5,000B+$4,000C+$6,000D

subjectto A≤

3months

B≤

2months

C≤

2months

D≤

3months

A+C≥2months

B+D≥4months

and A≥

0,B≥0,C≥0,D≥0.

Optimalsolution(A,B,C,D)=(0,2,2,2)andCrashingCost=$30,000. d)

e) Deadlineof11months:

Deadlineof13months:

8.19 a) ActivityCrashLengthofPathtoCrashCostB–D50BorD$10,00049BorD$10,00048BorD$10,00047 b) ActivityCrashLengthofPathtoCrashCostA–C–E–F51C$10,00050C$10,00049C$10,00048E$15,00047 c)

8.20 a)ActivityESEFLSLFSlackCriticalPathStart00000YesA03030YesB37481NoC38380YesD7109122NoE8128120YesFinish121212120Yes

CriticalPath: StartACEFinish

Totalduration=12weeks b)ActivityCrashLengthofPathtoCrashCostA–B–DA–B–EA–C-E101112C$1,333101111E$2,500101010D&E$4,000999B&C$4,333888

NewPlan:ActivityDurationCostA3weeks$54,000B3weeks$65,000C3weeks$68,666D2weeks$41,500E2weeks$80,000

$7,834issavedbythiscrashingschedule. c)

ExtraDirectCost=($309,167–$297,000)=$12,167

IndirectCostsSaved=4($5,000)=$20,000

NetSavings=$20,000–$12,167=7,834.8.21

8.22

8.23 a)ActivityESEFStart00A03B36C36D68E68Finish88

Totalduration=8weeks b)

c)

d)

e)ActivityBudgetedCostPercentCompletedValueCompletedActualCostToDateCostOverrunToDateA$54,000100%$54,000$65,000$11,000B$65,000100%$65,000$55,000–$10,000C$68,66633%$22,660$44,000$21,340Total$187,666$141,660$164,000$22,340

MichaelshouldconcentratehiseffortsonactivityCsinceitisnotyetcompleted.8.24 a)ActivityESEFLSLFSlackStart00000A06060B02464C6109133D6116110E101713203F112011200Finish202020200

Theearliestfinishtimefortheprojectis20weeks. b)

c)

LateststarttimescheduleEarlieststarttimeschedule d)

LateststarttimescheduleEarlieststarttimeschedule e)ActivityBudgetedCostPercentCompletedValueCompletedActualCostToDateCostOverrunToDateA$420,00050%$210,000$200,000-$10,000B$180,000100%$180,000$200,000$20,000D$360,00050%$180,000$210,000$30,000Total$960,000$570,000$610,000$40,000

TheprojectmanagershouldfocusattentiononactivityDsinceitisnotyetfinishedandisrunningoverbudget.8.25 a)

b)

LateststarttimescheduleEarlieststarttimeschedule c)

LateststarttimescheduleEarlieststarttimeschedule d)ActivityBudgetedCostPercentCompletedValueCompletedActualCostToDateCostOverrunToDateA$180,000100%$180,000$190,000$10,000B$75,000100%$75,000$70,000-$5,000C$120,000100%$120,000$150,000$30,000D$140,00040%$56,000$70,000$14,000E$175,00050%$87,500$100,000$12,500F$80,00060%$48,000$45,000-$3,000G$210,00025%$52,500$50,000-$2,500I$125,00020%$25,000$35,000$10,000Total$1,105,000$644,000$710,000$66,000

TheprojectmanagershouldinvestigateactivitiesD,EandIsincetheyarenotyetfinishedandrunningoverbudget.Cases8.1 a) Adiagramoftheprojectnetworkappearsbelow.

Todeterminetheprojectscheduleandwhichactivitiesarecritical,wecalculatetheearlystart,latestart,earlyfinish,latefinish,andslackbelow.

Theinitialpublicofferingprocessis27.5weekslong.Thecriticalpathis:

STARTABDEFJKLOPFINISH b) Weexploreeachchangeindependently.i) Negotiatingthecommitment(stepC)isperformedparalleltonegotiatingthespread(stepD).Inpart(a)above,negotiatingthespreadisonthecriticalpathsinceittakesthreedaystocompletewhilenegotiatingthecommitmenttakesonlytwodaystocomplete.Wenowincreasethetimetonegotiatethecommitmentfromtwodaystothreedays,andnegotiatingthecommitmentnowtakesasmuchtimeasnegotiatingthespread.Thus,therearenowtwocriticalpathsthroughthenetwork:

STARTABCEFJKLOPFINISH

STARTABDEFJKLOPFINISH

Theprojectdurationisstill27.5weeks.ii) Inpart(a)above,calculatingtheissuepriceisnotonthecriticalpath.Thus,decreasingthetimetocalculatethepricedoesnotchangethesolutionfoundinpart(a).Thecriticalpathremainsthesame,andtheprojectdurationisstill27.5weeks.iii) Inpart(a)above,thesteptoamendthestatementandresubmitittotheSEC(stepK)isonthecriticalpath.Therefore,increasingthetimeforthestepincreasestheprojectduration.Theprojectdurationincreasesto29weeks,andthecriticalpathremainsthesame.iv) Inpart(a)above,thesteptoconfirmthatthenewissuecomplieswith“bluesky〞laws(stepM)occursinparalleltoappointingaregistrar(stepN)andtoappointingatransferagent(stepO).StepMisnotonthecriticalpathsinceitonlytakesoneweekwhilestepOtakes3.5weeks.WhenweincreasethetimetocompletestepMfromoneweektofourweeks,wechangethecriticalpathsincestepMnowtakeslongerthanstepO.Wealsochangetheprojectduration.Theprojectdurationisnow28weeks.Twonewcriticalpathsappear:

STARTABDEFGMPFINISH

STARTABDEFJKLMPFINISH c)

Theconstraintsinthelinearprogrammingspreadsheetmodelwereasfollows:

TimeReduction≤MaxTimeReduction

ProjectFinishTime≤MaxTime

BStart≥AFinish

CStart≥BFinish

DStart≥BFinish

EStart≥CFinish

EStart≥DFinish

FStart≥EFinish

GStart≥FFinish

HStart≥FFinish

IStart≥FFinish

JStart≥FFinish

KStart≥JFinish

LStart≥KFinish

MStart≥GFinish

MStart≥HFinish

MStart≥IFinish

MStart≥LFinish

NStart≥LFinish

OStart≥LFinish

PStart≥MFinish

PStart≥

NFinish

PStart≥OFinish

QStart≥MFinish

QStart≥NFinish

QStart≥OFinish

ProjectFinishTime≥PFinish

ProjectFinishTime≥QFinish

JanetandGilbertshouldreducethetimeforstepA(evaluatingtheprestigeofeachpotentialunderwriter)by1.5weeks,thetimeforstepB(selectingasyndicateofunderwriters)by1week,thetimeforstepK(amendingstatementandresubmittingittotheSEC)by0.5weeks,thetimeforstepN(appointingaregistrar)by1.5weeks,thetimeforstepO(appointingatransferagent)bytwoweeks,andthetimeforstepP(issuingfinalprospectus)by0.5weeks.JanetandGilbertcannowmeetthenewdeadlineof22weeksatatotalcostof$260,800. d) Weusethesamemodelformulationthatwasusedinpart(c).Wechangeoneconstraint,however.Theprojectdurationnowhastobeless-than-or-equalto24weeksinsteadof22weeks.Weobtainthefollowingsolution.

JanetandGilbertshouldreducethetimeforstepA(evaluatingtheprestigeofeachpotentialunderwriter)by1.5weeks,thetimeforstepB(selectingasyndicateofunderwriters)by1week,thetimeforstepK(amendingstatementandresubmittingittotheSEC)by0.5weeks,andthetimeforstepO(appointingatransferagent)by0.5weeks.JanetandGilbertcannowmeetthenewdeadlineof24weeksatatotalcostof$236,000.8.2 a) Theprojectnetworkappearsbelow.

Todeterminetheprojectscheduleandwhichactivitiesarecritical,wecalculatetheearlystart,latestart,earlyfinish,latefinish,andslackbelow.

Brentcanstarttheinterviewsin49days.Thecriticalstepsintheprocessare:

STARTEOPQRFINISH b) Wesubstitutefirstthepessimisticandthentheoptimisticestimatesforthetimevaluesusedinthepart(a)spreadsheet.

Thespreadsheetshowingtheprojectschedulewiththepessimistictimeestimatesappearsbelow.

Undertheworst-casescenario,Brentwillrequire74daysbeforeheisreadytobegininterviewing.Thecriticalpathchangesto:

STARTBFLMOPQRFINISH

Thespreadshee