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1 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Chapter 10Chapter 10Project Scheduling: Project Scheduling:
PERT/CPMPERT/CPM Project Scheduling with Known Project Scheduling with Known
Activity TimesActivity Times
Project Scheduling with Uncertain Project Scheduling with Uncertain Activity TimesActivity Times
Considering Time-Cost Trade-OffsConsidering Time-Cost Trade-Offs
2 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT/CPMPERT/CPM PERTPERT
• Program Evaluation and Review TechniqueProgram Evaluation and Review Technique•Developed by U.S. Navy for Polaris missile Developed by U.S. Navy for Polaris missile
projectproject•Developed to handle uncertain activity timesDeveloped to handle uncertain activity times
CPMCPM•Critical Path MethodCritical Path Method•Developed by Du Pont & Remington RandDeveloped by Du Pont & Remington Rand•Developed for industrial projects for which Developed for industrial projects for which
activity times generally were knownactivity times generally were known Today’s project management software packages Today’s project management software packages
have combined the best features of both have combined the best features of both approaches.approaches.
3 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT/CPMPERT/CPM
PERT and CPM have been used to PERT and CPM have been used to plan, schedule, and control a wide plan, schedule, and control a wide variety of projects:variety of projects:
•R&D of new products and processesR&D of new products and processes
•Construction of buildings and Construction of buildings and highwayshighways
•Maintenance of large and complex Maintenance of large and complex equipmentequipment
•Design and installation of new Design and installation of new systemssystems
4 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT/CPMPERT/CPM
PERT/CPM is used to plan the PERT/CPM is used to plan the scheduling of individual scheduling of individual activitiesactivities that make up a project.that make up a project.
Projects may have as many as Projects may have as many as several thousand activities.several thousand activities.
A complicating factor in carrying A complicating factor in carrying out the activities is that some out the activities is that some activities depend on the activities depend on the completion of other activities completion of other activities before they can be started.before they can be started.
5 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT/CPMPERT/CPM Project managers rely on PERT/CPM to Project managers rely on PERT/CPM to
help them answer questions such as:help them answer questions such as:
•What is the What is the total timetotal time to complete to complete the project?the project?
•What are the What are the scheduled start and scheduled start and finish datesfinish dates for each specific activity? for each specific activity?
•Which activities are Which activities are criticalcritical and must and must be completed exactly as scheduled to be completed exactly as scheduled to keep the project on schedule?keep the project on schedule?
•How long can How long can noncritical activitiesnoncritical activities be be delayed before they cause an delayed before they cause an increase in the project completion increase in the project completion time?time?
6 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Project NetworkProject Network
A A project networkproject network can be can be constructed to model the constructed to model the precedence of the activities. precedence of the activities.
The The nodesnodes of the network of the network represent the activities. represent the activities.
The The arcsarcs of the network reflect the of the network reflect the precedence relationships of the precedence relationships of the activities. activities.
A A critical pathcritical path for the network is a for the network is a path consisting of activities with path consisting of activities with zero slack.zero slack.
7 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Frank’s Fine Floats is in the business of Frank’s Fine Floats is in the business of building elaborate parade floats. Frank and building elaborate parade floats. Frank and his crew have a new float to build and want to his crew have a new float to build and want to use PERT/CPM to help them manage the use PERT/CPM to help them manage the projectproject . .
The table on the next slide shows the The table on the next slide shows the activities that comprise the project. Each activities that comprise the project. Each activity’s estimated completion time (in days) activity’s estimated completion time (in days) and immediate predecessors are listed as well.and immediate predecessors are listed as well.
Frank wants to know the total time to Frank wants to know the total time to complete the project, which activities are complete the project, which activities are critical, and the earliest and latest start and critical, and the earliest and latest start and finish dates for each activity.finish dates for each activity.
8 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Immediate CompletionImmediate Completion
ActivityActivity DescriptionDescription PredecessorsPredecessors Time (days)Time (days)
A Initial Paperwork A Initial Paperwork --- --- 3 3
B Build Body B Build Body A A 3 3
C Build Frame C Build Frame A A 2 2
D Finish Body D Finish Body B B 3 3
E Finish Frame E Finish Frame C C 7 7
F Final Paperwork F Final Paperwork B,C B,C 3 3
G Mount Body to Frame D,EG Mount Body to Frame D,E 6 6
H Install Skirt on Frame CH Install Skirt on Frame C 2 2
9 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Project NetworkProject Network
StartStart FinishFinish
BB33
DD33
AA33
CC22
GG66
FF33
HH22
EE77
10 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Earliest Start and Finish TimesEarliest Start and Finish Times
Step 1:Step 1: Make a forward pass through the Make a forward pass through the network as follows: For each activity network as follows: For each activity i i beginning at the Start nodebeginning at the Start node, , compute:compute:
• Earliest Start TimeEarliest Start Time = the maximum of = the maximum of the earliest finish times of all activities the earliest finish times of all activities immediately preceding activity immediately preceding activity ii. (This is . (This is 0 for an activity with no predecessors.)0 for an activity with no predecessors.)
• Earliest Finish TimeEarliest Finish Time = (Earliest Start = (Earliest Start Time) + (Time to complete activity Time) + (Time to complete activity i i ).).
The project completion time is the The project completion time is the maximum of the Earliest Finish Times at maximum of the Earliest Finish Times at the Finish node.the Finish node.
11 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Earliest Start and Finish TimesEarliest Start and Finish Times
StartStart FinishFinish
BB33
DD33
AA33
CC22
GG66
FF33
HH22
EE77
0 30 3
3 63 6 6 96 9
3 53 5
12 12 1818
6 96 9
5 75 7
5 125 12
12 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Latest Start and Finish TimesLatest Start and Finish Times
Step 2:Step 2: Make a backwards pass through Make a backwards pass through the network as follows: Move the network as follows: Move sequentially backwards from the Finish sequentially backwards from the Finish node to the Start node. At a given node, node to the Start node. At a given node, jj, consider all activities ending at node, consider all activities ending at node j j. . For each of these activities, For each of these activities, ii, compute:, compute:
• Latest Finish TimeLatest Finish Time = the minimum of = the minimum of the latest start times beginning at the latest start times beginning at node node jj. (For node . (For node NN, this is the project , this is the project completion time.)completion time.)
• Latest Start TimeLatest Start Time = (Latest Finish = (Latest Finish Time) - (Time to complete activity Time) - (Time to complete activity i i ).).
13 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Latest Start and Finish TimesLatest Start and Finish Times
StartStart FinishFinish
BB33
DD33
AA33
CC22
GG66
FF33
HH22
EE77
0 30 3
3 63 6 6 96 9
3 53 5
12 12 1818
6 96 9
5 75 7
5 125 12
6 96 9 9 9 1212
0 30 3
3 53 5
12 12 1818
15 15 1818
16 16 1818
5 125 12
14 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Determining the Critical PathDetermining the Critical Path
Step 3:Step 3: Calculate the slack time for each Calculate the slack time for each activity by: activity by:
SlackSlack = (Latest Start) - (Earliest = (Latest Start) - (Earliest Start), or Start), or
= (Latest Finish) - = (Latest Finish) - (Earliest Finish).(Earliest Finish).
15 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Activity Slack TimeActivity Slack Time
ActivityActivity ESES EFEF LSLS LFLF SlackSlack A 0 3 0 3 0 A 0 3 0 3 0
(critical)(critical) B 3 6 6 9 3B 3 6 6 9 3 C 3 5 3 5 0 C 3 5 3 5 0
(critical)(critical) D 6 9 9 12 3D 6 9 9 12 3 E 5 12 5 12 0 E 5 12 5 12 0
(critical)(critical) F 6 9 15 18 9F 6 9 15 18 9 G 12 18 12 18 0 G 12 18 12 18 0
(critical)(critical) H 5 7 16 18 11H 5 7 16 18 11
16 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Determining the Critical PathDetermining the Critical Path
• A A critical pathcritical path is a path of activities, from the is a path of activities, from the Start node to the Finish node, with 0 slack Start node to the Finish node, with 0 slack times.times.
• Critical Path: A – C – E – GCritical Path: A – C – E – G
• The The project completion timeproject completion time equals the equals the maximum of the activities’ earliest finish maximum of the activities’ earliest finish times.times.
• Project Completion Time: 18 daysProject Completion Time: 18 days
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
17 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Critical PathCritical Path
StartStart FinishFinish
BB33
DD33
AA33
CC22
GG66
FF33
HH22
EE77
0 30 3
3 63 6 6 96 9
3 53 5
12 12 1818
6 96 9
5 75 7
5 125 12
6 96 9 9 9 1212
0 30 3
3 53 5
12 12 1818
15 15 1818
16 16 1818
5 125 12
18 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
In the In the three-time estimate approachthree-time estimate approach, the , the time to complete an activity is assumed to time to complete an activity is assumed to follow a Beta distribution. follow a Beta distribution.
An activity’s An activity’s mean completion timemean completion time is: is:
tt = ( = (aa + 4 + 4mm + + bb)/6)/6
• aa = the = the optimisticoptimistic completion time completion time estimateestimate
• bb = the = the pessimisticpessimistic completion time completion time estimateestimate
• mm = the = the most likelymost likely completion time completion time estimateestimate
Uncertain Activity TimesUncertain Activity Times
19 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
An activity’s An activity’s completion time variancecompletion time variance is: is:
22 = (( = ((bb--aa)/6))/6)22
• aa = the = the optimisticoptimistic completion time completion time estimateestimate
• bb = the = the pessimisticpessimistic completion time completion time estimateestimate
• mm = the = the most likelymost likely completion time completion time estimateestimate
Uncertain Activity TimesUncertain Activity Times
20 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Uncertain Activity TimesUncertain Activity Times
In the three-time estimate approach, In the three-time estimate approach, the critical path is determined as if the the critical path is determined as if the mean times for the activities were mean times for the activities were fixed times. fixed times.
The The overall project completion timeoverall project completion time is is assumed to have a normal distribution assumed to have a normal distribution with mean equal to the sum of the with mean equal to the sum of the means along the critical path and means along the critical path and variance equal to the sum of the variance equal to the sum of the variances along the critical path.variances along the critical path.
21 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC Associates Example: ABC Associates
Consider the following project:Consider the following project:
Immed. Optimistic Most Likely PessimisticImmed. Optimistic Most Likely Pessimistic
ActivityActivity Predec.Predec. Time (Hr.Time (Hr.) ) Time (Hr.)Time (Hr.) Time (Hr.)Time (Hr.) A A -- 4 -- 4 6 6 8 8 B B -- 1 -- 1 4.5 4.5
5 5 C C A A 3 3 3 3
3 3 D D A 4 5 A 4 5 6 6 E E A 0.5 1 A 0.5 1
1.51.5 F F B,C 3 4 5 B,C 3 4 5 G G B,C B,C 1 1.5 1 1.5
5 5 H H E,F E,F 5 6 5 6
7 7 I I E,F 2 5 8 E,F 2 5 8 J J D,H D,H 2.5 2.75 2.5 2.75
4.5 4.5 K K G,I 3 5 7 G,I 3 5 7
22 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC AssociatesExample: ABC Associates
Project NetworkProject Network
E
S tart
A
H
D
F
J
I
K
F in ish
B
C
G
66
44
33
55
55
22
44
1166
33
55
23 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC AssociatesExample: ABC Associates
Activity Expected Times and VariancesActivity Expected Times and Variances
tt = ( = (aa + 4 + 4mm + + bb)/6 )/6 22 = (( = ((bb--aa)/6))/6)22
ActivityActivity Expected TimeExpected Time VarianceVariance A A 6 6 4/9 4/9
B B 4 4 4/9 4/9 C C 3 3 0 0 D D 5 5 1/9 1/9 E E 1 1 1/36 1/36 F F 4 4 1/9 1/9 G G 2 2 4/9 4/9 H H 6 6 1/9 1/9 I I 5 5 1 1 J J 3 3 1/9 1/9 K K 5 5 4/9 4/9
24 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC AssociatesExample: ABC Associates
Earliest/Latest Times and SlackEarliest/Latest Times and Slack
ActivityActivity ESES EF EF LSLS LFLF SlackSlack A A 0 6 0 6 0 * 0 6 0 6 0 *
B B 0 4 5 9 5 0 4 5 9 5 C 6 9 6 9 0 *C 6 9 6 9 0 * D D 6 11 15 20 9 6 11 15 20 9 E E 6 7 12 13 6 6 7 12 13 6 F F 9 13 9 13 0 * 9 13 9 13 0 * G 9 11 16 18 7G 9 11 16 18 7 H H 13 19 14 20 1 13 19 14 20 1 I I 13 18 13 18 0 * 13 18 13 18 0 * J J 19 22 20 23 1 19 22 20 23 1 K K 18 23 18 23 0 * 18 23 18 23 0 *
25 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Determining the Critical PathDetermining the Critical Path
• A A critical pathcritical path is a path of activities, from the is a path of activities, from the Start node to the Finish node, with 0 slack Start node to the Finish node, with 0 slack times.times.
• Critical Path: A – C – F – I – KCritical Path: A – C – F – I – K
• The The project completion timeproject completion time equals the equals the maximum of the activities’ earliest finish maximum of the activities’ earliest finish times.times.
• Project Completion Time: 23 hoursProject Completion Time: 23 hours
Example: ABC AssociatesExample: ABC Associates
26 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC AssociatesExample: ABC Associates
Critical Path (A-C-F-I-K)Critical Path (A-C-F-I-K)
E
S tart
A
H
D
F
J
I
K
F in ish
B
C
G
66
44
33
55
55
22
44
1166
33
55
0 60 60 60 6
9 139 139 139 13
13 1813 1813 1813 18
9 119 1116 1816 18
13 1913 1914 2014 20
19 2219 2220 2320 23
18 2318 2318 2318 23
6 76 712 1312 13
6 96 96 96 9
0 40 45 95 9
6 116 1115 2015 20
27 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Probability the project will be completed within 24 Probability the project will be completed within 24 hrshrs
22 = = 22AA + + 22
CC + + 22FF + + 22
HH + + 22KK
= 4/9 + 0 + 1/9 + 1 + 4/9 = 4/9 + 0 + 1/9 + 1 + 4/9
= 2= 2
= 1.414= 1.414
zz = (24 - 23)/ = (24 - 23)/(24-23)/1.414 (24-23)/1.414 = .71= .71
From the Standard Normal Distribution table: From the Standard Normal Distribution table:
P(z P(z << .71) = .5 + .2612 = .7612 .71) = .5 + .2612 = .7612
Example: ABC AssociatesExample: ABC Associates
28 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
EarthMover is a manufacturer of road EarthMover is a manufacturer of road constructionconstruction
equipment including pavers, rollers, and graders. equipment including pavers, rollers, and graders. TheThe
company is faced with a newcompany is faced with a new
project, introducing a newproject, introducing a new
line of loaders. Managementline of loaders. Management
is concerned that the project mightis concerned that the project might
take longer than 26 weeks totake longer than 26 weeks to
complete without crashing somecomplete without crashing some
activities.activities.
Example: EarthMover, Inc.Example: EarthMover, Inc.
29 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Immediate Immediate CompletionCompletion ActivityActivity DescriptionDescription PredecessorsPredecessors Time (wks)Time (wks)
A Study Feasibility A Study Feasibility --- --- 6 6 B Purchase Building B Purchase Building A A
4 4 C Hire Project Leader C Hire Project Leader A A
3 3 D Select Advertising StaffD Select Advertising Staff B B
6 6 E Purchase Materials E Purchase Materials B B
3 3 F Hire Manufacturing Staff B,CF Hire Manufacturing Staff B,C
10 10 G Manufacture Prototype E,FG Manufacture Prototype E,F
2 2 H Produce First 50 Units GH Produce First 50 Units G
6 6 II Advertise Product D,G Advertise Product D,G
8 8
Example: EarthMover, Inc.Example: EarthMover, Inc.
30 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT NetworkPERT Network
Example: EarthMover, Inc.Example: EarthMover, Inc.
C
S tart
D
E
I
A
F in ish
H
G
B
F
C
S tart
D
E
I
A
F in ish
H
G
B
F
6644
331010
33
66
22 66
88
31 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Earliest/Latest TimesEarliest/Latest Times
ActivityActivity ESES EFEF LSLS LFLF SlackSlack A A 0 6 0 6 0 * 0 6 0 6 0 * B B 6 10 6 10 0 * 6 10 6 10 0 * C C 6 9 7 10 1 6 9 7 10 1 D 10 16 16 22 6D 10 16 16 22 6 E E 10 13 17 20 7 10 13 17 20 7 F F 10 20 10 20 0 * 10 20 10 20 0 * G G 20 22 20 22 0 * 20 22 20 22 0 * H H 22 28 24 30 2 22 28 24 30 2 I I 22 30 22 30 0 * 22 30 22 30 0 *
Example: EarthMover, Inc.Example: EarthMover, Inc.
32 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: EarthMover, Inc.Example: EarthMover, Inc.
Critical ActivitiesCritical Activities
C
S tart
D
E
I
A
F in ish
H
G
B
F
C
S tart
D
E
I
A
F in ish
H
G
B
F
6644
331010
33
66
22 66
880 60 60 60 6
10 2010 20 10 2010 20
20 2220 2220 2220 22
10 1610 1616 2216 22 22 3022 30
22 3022 30
22 2822 2824 3024 30
6 96 9 7 107 10
10 1310 1317 2017 20
6 106 10 6 106 10
33 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Ch. 10 – 7A project involving the installation of a computer system comprises eight activities. The following table lists immediate predecessors and activity times (in weeks).
ActivityImmediate
Predecessor Time
ABCDEFGH
--A
B,CDE
B,CF,G
36254393
a. Draw a project network.b. What are the critical activities?c. What is the expected project completion time?
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