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• Program Evaluation and Review TechniqueProgram Evaluation and Review Technique•Developed by U.S. Navy for Polaris missile projectDeveloped by U.S. Navy for Polaris missile project•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.
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
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.
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 the to complete the project?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 increase delayed before they cause an increase in the project completion time?in the project completion time?
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.
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.
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.
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 ).).
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
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
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.
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
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.
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.
In the In the Critical Path Method (CPM)Critical Path Method (CPM) approach to approach to project scheduling, it is assumed that the project scheduling, it is assumed that the normal time to complete an activity, normal time to complete an activity, ttj j , which , which can be met at a normal cost, can be met at a normal cost, ccj j , can be crashed , can be crashed to a reduced time, to a reduced time, ttjj’, under maximum crashing ’, under maximum crashing for an increased cost, for an increased cost, ccjj’.’.
Using CPM, activity Using CPM, activity jj's maximum time 's maximum time reduction, reduction, MMj j , may be calculated by: , may be calculated by: MMj j = = ttjj - - ttjj'. '. It is assumed that its cost per unit reduction, It is assumed that its cost per unit reduction, KKj j , is linear and can be calculated by: , is linear and can be calculated by: KKjj = ( = (ccjj' - ' - ccjj)/)/MMjj..
YYHH << 1 1 XXFF >> XXBB + (10 - + (10 - YYFF) ) XXII << 26 26 YYII << 4 4 XXFF >> XXCC + (10 - + (10 - YYFF) ) XXGG >> XXEE + (2 - + (2 - YYGG) ) XXii, , YYjj >> 0 0 for all i for all i
Example: EarthMover, Inc.Example: EarthMover, Inc.
Linear Program for Minimum-Cost CrashingLinear Program for Minimum-Cost Crashing
Let: Let: XXii = earliest finish time for activity = earliest finish time for activity ii YYii = the amount of time activity = the amount of time activity ii is crashed is crashed
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?
Ch. 10 – 18The manager of the Oak Hills Swimming Club is planning the club’s swimming team program. The first team practice is scheduled for May 1. The activities, their immediate predecessors, and the activity time estimates (in weeks) are as follows.
ActivityImmediatePredecessor
ABCDEFGHI
-AA
B,CBADG
E,H,F
Description
Meet with boardHire coachesReserve poolAnnounce programMeet with coachesOrder team suitsRegister swimmersCollect feesPlan first practice
a. Draw a project network.b. Develop an activity schedule.c. What are the critical activities, and what is the expected
project completion time?d. If the club manager plans to start the project on
February1, what is the probability the swimming program will be ready by the scheduled May 1 date (13 weeks)? Should the manager begin planning the swimming program before February 1?