Probabilistic Time EstimatesExample Beta DistributionsPERT ExampleActivity InformationEarly And Late TimesNetwork With TimesProject VarianceProbabilistic Network AnalysisNormal Distribution Of Project TimeProbabilistic Analysis ExamplePowerPoint PresentationSlide 12Project CrashingHouse-building NetworkNormal Activity And Crash DataNetwork With Crashing CostsNormal And Crash RelationshipsCrashing SolutionCrashed ProjectTime-Cost RelationshipTime-Cost TradeoffCh 17 - 23© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eProbabilistic Time Estimates•Reflect uncertainty of activity times•Beta distribution is used in PERTb - a6( )Variance: 2 =a = optimistic estimatem = most likely time estimateb = pessimistic time estimateWhere,2Mean (expected time):a + 4m + b6t =Ch 17 - 24© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eExample Beta Distributionsm = tbaP (time)baP (time)tmbmtaP (time)Ch 17 - 25© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/ePERT Example124673 5 98Manual TestingDummySystem TrainingDummySystem TestingOrientationPosition recruitingSystem developmentEquipment installationEquipment testing and modificationFinal debuggingSystem changeover Job trainingCh 17 - 26© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eActivity Information1 - 2 6 8 10 8 .441 - 3 3 6 9 6 1.001 - 4 1 3 5 3 .442 - 5 0 0 0 0 .002 - 6 2 4 12 5 2.783 - 5 2 3 4 3 .114 - 5 3 4 5 4 .114 - 8 2 2 2 2 .005 - 7 3 7 11 7 1.785 - 8 2 4 6 4 .447 - 8 0 0 0 0 .006 - 9 1 4 7 4 1.007 - 9 1 10 13 9 4.00 Time estimates (wks) Mean Time VarianceActivity a b c t 2Ch 17 - 27© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eEarly And Late Times1 - 2 8 0.44 08 19 11 - 3 6 1.00 06 06 01 - 4 3 0.44 03 25 22 - 5 0 0.00 88 99 12 - 6 5 2.78 813 1621 83 - 5 3 0.11 69 69 04 - 5 4 0.11 37 59 24 - 8 2 0.00 35 1416 115 - 7 7 1.78 916 916 05 - 8 4 0.44 913 1216 37 - 8 0 0.00 1313 1616 36 - 9 4 1.00 1317 2125 87 - 9 9 4.00 1625 1625 0Activity t 2ES EF LS LF SCh 17 - 28© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eNetwork With Times124673 5 98( )ES=8, EF=8LS=9, LF=9( )ES=6, EF=9LS=6, LF=9( )ES=3, EF=5LS=14, LF=16( )ES=0, EF=3LS=2, LF=5( )ES=0, EF=6LS=0, LF=6( )ES=0, EF=8LS=1, LF=9380544702936( )ES=3, EF=7LS=5, LF=94( )ES=9, EF=13LS=12, LF=16( )ES=9, EF=13LS=9, LF=16( )ES=13, EF=13LS=16 LF=16( )ES=13, EF=25LS=16 LF=25( )ES=13, EF=25LS=16 LF=25( )ES=8, EF=13LS=16 LF=21Ch 17 - 29© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eProject Variance•Project variance is the sum of variances on the critical pathweeksweeks * 89.600.478.111.000.12792572352132Ch 17 - 30© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eProbabilistic Network AnalysisDetermine probability that project is completed within specified timewhere = tp = project mean time = project standard deviationx = proposed project timeZ = number of standard deviations x is from meanZ = x - Ch 17 - 31© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eNormal Distribution Of Project Time = tpTimexZProbabilityCh 17 - 32© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eProbabilistic Analysis ExampleWhat is the probability that the project is completed within 30 weeks?26 896 89 2 6230 252 62191191 0 9719 .. ...( . ) .weeksweeksZxP ZCh 17 - 33© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eDetermining Probability From Z ValueZ 0.00 0.01 ... 0.091.9 0.4713 0.4719 … 0.4767............ = 25 Time (weeks)x = 30P( x<= 30 weeks) = 0.9719Ch 17 - 34© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eWhat is the probability that the project is completed within 22 weeks? = 25 Time (weeks)x = 22P( x<= 22 weeks) =0.1271ZP Z 22 252 6232 62114114 0 1271. ..( . ) .Ch 17 - 35© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eProject Crashing•Crashing is reducing project time by expending additional resources•Crash time is an amount of time an activity is reduced•Crash cost is the cost of reducing the activity time•Goal is to reduce project duration at minimum costCh 17 - 36© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eHouse-building NetworkActivity times in weeks1 2 467351280412444Ch 17 - 37© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eNormal Activity And Crash Data1-2 12 7 $3,000 $5,000 5 $4002-3 8 5 2,000 3,500 3 5002-4 4 3 4,000 7,000 1 3,0003-4 0 0 0 0 0 04-5 4 1 500 1,100 3 2004-6 12 9 50,000 71,000 3 7,0005-6 4 1 500 1,100 3 2006-7 4 3 15,000 22,000 1 7,000$75,000 $110,700TotalNormal Crash Allowable CrashTime Time Normal Crash Crash Time Cost perActivity (wks) (wks) Cost Cost (wks) WeekCh 17 - 38© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eNetwork With Crashing Costs1 2 467351280412444$7,000$7,000$500$3,000$400$200$200Activity 1-2 can be crashed a total of 5 weeks for $2000Crash cost per week = Total crash cost/Total crash time= $2,000/5 = $400 per weekCh 17 - 39© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eNormal And Crash Relationships1242 6 8 10 1401,0003,0004,0005,0007,0002,0006,000$WeeksCrashed activityNormal activityCrash costNormal costCrash time Normal timeSlope = crash cost per weekCh 17 - 40© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eCrashing Solution1-2 12 7 5 $400 $2,0002-3 8 5 3 500 1,5002-4 4 3 0 3,000 03-4 0 0 0 0 04-5 4 1 0 200 04-6 12 9 3 7,000 21,0005-6 4 1 0 200 06-7 4 3 1 7,000 7,00012 $31,500Normal Crash Crash Crash CrashingTime Time Time Cost per CostActivity (wks) (wks) Used Week IncurredCh 17 - 41© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eCrashed Project1 2 4673512 78 504 312 9444 3Original time Crashed timesCh 17 - 42© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eTime-Cost Relationship •Crashing costs increase as project duration decreases•Indirect costs increase as project duration increases•Reduce project length as long as crashing costs are less than indirect costsCh 17 - 43© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/eTime-Cost TradeoffCost ($)Project DurationCrashingTotal costIndirect costDirect costTimeMinimum cost = optimal project
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