1. 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
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Russell/Taylor Oper Mgt 3/e Ch

2. Example Beta Distributions
P (time)
P (time) b a m t b
P (time) a b
m = t
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Russell/Taylor Oper Mgt 3/e Ch

3. 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
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Russell/Taylor Oper Mgt 3/e Ch

4. Russell/Taylor Oper Mgt 3/e
Activity Information
Time estimates (wks) Mean Time Variance
Activity a b c t 2
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Russell/Taylor Oper Mgt 3/e Ch

5. Russell/Taylor Oper Mgt 3/e
Early And Late Times
Activity t 2 ES EF LS LF S
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Russell/Taylor Oper Mgt 3/e Ch

6. 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
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Russell/Taylor Oper Mgt 3/e Ch

7. Russell/Taylor Oper Mgt 3/e
Project Variance
Project variance is the sum of variances on the critical path
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Russell/Taylor Oper Mgt 3/e Ch

8. 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

9. Normal Distribution Of Project Time
Probability Z
 = tp x
Time
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Russell/Taylor Oper Mgt 3/e Ch

10. Probabilistic Analysis Example
What is the probability that the project is completed within 30 weeks?
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Russell/Taylor Oper Mgt 3/e Ch

11. Determining Probability From Z Value… . . . .
P( x<= 30 weeks) =
 = 25
x = 30
Time (weeks)
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Russell/Taylor Oper Mgt 3/e Ch

12. 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)
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Russell/Taylor Oper Mgt 3/e Ch

13. 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
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Russell/Taylor Oper Mgt 3/e Ch

14. House-building Network
Activity times in weeks 3 8 12 4 1 2 4 6 7 12 4 4 4 5
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Russell/Taylor Oper Mgt 3/e Ch

15. 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
$3,000 $5,000 5 $400
,000 3,
,000 7, ,000 ,
,000 71, ,000 ,
,000 22, ,000
$75,000 $110,700
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Russell/Taylor Oper Mgt 3/e Ch

16. 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
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Russell/Taylor Oper Mgt 3/e Ch

17. 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
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Russell/Taylor Oper Mgt 3/e Ch

18. 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
$400 $2,000
,500
,000 0
,000 21,000
,000 7,000
12 $31,500
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Russell/Taylor Oper Mgt 3/e Ch

19. Russell/Taylor Oper Mgt 3/e
Crashed Project 3 1 2 4 6 7 4 4 5
Original time Crashed times
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Russell/Taylor Oper Mgt 3/e Ch

20. 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
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Russell/Taylor Oper Mgt 3/e Ch

21. 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
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Russell/Taylor Oper Mgt 3/e Ch