Probabilistic Time Estimates Reflect uncertainty of activity times Beta distribution is used in PERT Mean (expected time): a + 4m + b 6 t = 2 b - a 6 ( ) Variance: 2 = Where, a = optimistic estimate m = most likely time estimate b = pessimistic time estimate © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 23

Example Beta Distributions P (time) P (time) b a m t b P (time) a b m = t © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 24

Russell/Taylor Oper Mgt 3/e PERT Example Equipment testing and modification 2 6 Final debugging Equipment installation Dummy System Training 1 3 5 7 9 System development Manual Testing System changeover System Testing Job training Dummy Position recruiting Orientation 4 8 © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 25

Russell/Taylor Oper Mgt 3/e Activity Information Time estimates (wks) Mean Time Variance Activity a b c t 2 1 - 2 6 8 10 8 .44 1 - 3 3 6 9 6 1.00 1 - 4 1 3 5 3 .44 2 - 5 0 0 0 0 .00 2 - 6 2 4 12 5 2.78 3 - 5 2 3 4 3 .11 4 - 5 3 4 5 4 .11 4 - 8 2 2 2 2 .00 5 - 7 3 7 11 7 1.78 5 - 8 2 4 6 4 .44 7 - 8 0 0 0 0 .00 6 - 9 1 4 7 4 1.00 7 - 9 1 10 13 9 4.00 © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 26

Russell/Taylor Oper Mgt 3/e Early And Late Times Activity t 2 ES EF LS LF S 1 - 2 8 0.44 0 8 1 9 1 1 - 3 6 1.00 0 6 0 6 0 1 - 4 3 0.44 0 3 2 5 2 2 - 5 0 0.00 8 8 9 9 1 2 - 6 5 2.78 8 13 16 21 8 3 - 5 3 0.11 6 9 6 9 0 4 - 5 4 0.11 3 7 5 9 2 4 - 8 2 0.00 3 5 14 16 11 5 - 7 7 1.78 9 16 9 16 0 5 - 8 4 0.44 9 13 12 16 3 7 - 8 0 0.00 13 13 16 16 3 6 - 9 4 1.00 13 17 21 25 8 7 - 9 9 4.00 16 25 16 25 0 © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 27

Russell/Taylor Oper Mgt 3/e Network With Times ( ) ES=8, EF=13 LS=16 LF=21 2 6 ( ) ES=0, EF=8 LS=1, LF=9 5 ( ) ES=13, EF=25 LS=16 LF=25 ( ) ES=8, EF=8 LS=9, LF=9 8 4 ( ) ES=0, EF=6 LS=0, LF=6 ( ) ES=9, EF=13 LS=9, LF=16 3 9 1 3 5 7 9 6 ( ) ES=6, EF=9 LS=6, LF=9 7 ( ) ES=13, EF=25 LS=16 LF=25 ( ) ES=9, EF=13 LS=12, LF=16 ( ) ES=0, EF=3 LS=2, LF=5 4 ( ) ES=13, EF=13 LS=16 LF=16 3 ( ) ES=3, EF=7 LS=5, LF=9 4 2 4 8 ( ) ES=3, EF=5 LS=14, LF=16 © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 28

Russell/Taylor Oper Mgt 3/e Project Variance Project variance is the sum of variances on the critical path © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 29

Probabilistic Network Analysis Determine probability that project is completed within specified time where  = tp = project mean time  = project standard deviation x = proposed project time Z = number of standard deviations x is from mean Z = x -   © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 30

Normal Distribution Of Project Time Probability Z  = tp x Time © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 31

Probabilistic Analysis Example What is the probability that the project is completed within 30 weeks? © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 32

Determining Probability From Z Value 1.9 0.4713 0.4719 … 0.4767 . . . . P( x<= 30 weeks) = 0.9719  = 25 x = 30 Time (weeks) © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 33

Russell/Taylor Oper Mgt 3/e What is the probability that the project is completed within 22 weeks? P( x<= 22 weeks) =0.1271 x = 22  = 25 Time (weeks) © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 34

Russell/Taylor Oper Mgt 3/e Project 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 cost © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 35

House-building Network Activity times in weeks 3 8 12 4 1 2 4 6 7 12 4 4 4 5 © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 36

Normal Activity And Crash Data Total Normal Crash Allowable Crash Time Time Normal Crash Crash Time Cost per Activity (wks) (wks) Cost Cost (wks) Week 1-2 12 7 $3,000 $5,000 5 $400 2-3 8 5 2,000 3,500 3 500 2-4 4 3 4,000 7,000 1 3,000 3-4 0 0 0 0 0 0 4-5 4 1 500 1,100 3 200 4-6 12 9 50,000 71,000 3 7,000 5-6 4 1 500 1,100 3 200 6-7 4 3 15,000 22,000 1 7,000 $75,000 $110,700 © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 37

Network With Crashing Costs Activity 1-2 can be crashed a total of 5 weeks for $2000 Crash cost per week = Total crash cost/Total crash time = $2,000/5 = $400 per week 3 $500 8 $7,000 12 4 $7,000 1 2 4 6 7 $400 $3,000 12 4 4 4 $200 $200 5 © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 38

Normal And Crash Relationships $ 12 4 2 6 8 10 14 1,000 3,000 4,000 5,000 7,000 2,000 6,000 Crash cost Crashed activity Slope = crash cost per week Normal cost Normal activity Crash time Normal time Weeks © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 39

Russell/Taylor Oper Mgt 3/e Crashing Solution Normal Crash Crash Crash Crashing Time Time Time Cost per Cost Activity (wks) (wks) Used Week Incurred 1-2 12 7 5 $400 $2,000 2-3 8 5 3 500 1,500 2-4 4 3 0 3,000 0 3-4 0 0 0 0 0 4-5 4 1 0 200 0 4-6 12 9 3 7,000 21,000 5-6 4 1 0 200 0 6-7 4 3 1 7,000 7,000 12 $31,500 © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 40

Russell/Taylor Oper Mgt 3/e Crashed Project 3 8 5 4 3 12 7 12 9 1 2 4 6 7 4 3 4 4 5 Original time Crashed times © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 41

Time-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 costs © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 42

Russell/Taylor Oper Mgt 3/e Time-Cost Tradeoff Minimum cost = optimal project time Total cost Cost ($) Indirect cost Direct cost Crashing Time Project Duration © 2000 by Prentice-Hall Inc Russell/Taylor Oper Mgt 3/e Ch 17 - 43