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Chapter 10
Project Scheduling: PERT/CPM
Critical Path SchedulingProject Scheduling with Uncertain Activity Times
PERT and CPMPERT (program evaluation and review technique)
– U.S. Navy Special Projects Office (1958) – Polaris missile project
CPM (critical path method)– J. E. Kelly of Remington-Rand and M. R. Walker
of Du Pont (1957)– Scheduling maintenance shutdowns of chemical
processing plants
Precedence RelationshipsPredecessor and Successor Activities• Activity A is called a predecessor of activity B if
activity A needs to be completed before activity B can be started.
• In this case, activity B is called a successor of activity A.
Note: An activity may have more than one predecessor and/or more than one successor.
Example: Development of a Compact Recording Device
A : Design deviceB : Build prototypeC : Evaluate assembly equipmentD : Test prototypeE : Write methods specification reportF : Write assembly equipment evaluation report G : Write final report
Precedence Relationships: Recording Device Development
ImmediateActivity Predecessor(s)
A -B AC AD BE C, DF C, DG E, F
Precedence Relationships: Recording Device Development
Activities C and D are the immediate predecessors of activity E. What are all of the predecessors of activity E? We call activity E an immediate successor of both activities C and D.How many immediate successors does activity C have? What are all of the successors of activity C?
Precedence Relationships: Recording Device Development (cont.)
Does activity A have an immediate predecessor?Does activity G have an immediate successor?
Estimating Activity DurationEstimated duration for each activity stated in terms of specified time unit– months, weeks, or days
Usually difficult in practiceIn certain instances, multiple time estimates are used– optimistic– most likely– pessimistic
Activity Duration Estimates: Recording Device Development
Immediate EstimatedActivity Predecessor(s) Duration (weeks)
A - 12B A 5C A 7D B 2E C, D 4F C, D 7G E, F 2
Prerequisites for Critical Path Analysis
A project must have:
well-defined jobs or tasks whose completion marks the end of the project;
independent jobs or tasks; and
tasks that follow a given sequence.
Steps in the CPM with Single Activity Time Estimate
1. Activity Identification2. Activity Sequencing and Network Construction3. Determination of the Critical Path
– From the critical path all of the project and activity timing information can be obtained
Activity on Node (AON)Network RepresentationNetwork consists of nodes & arcs
A CB
Arc(precedencerelationshipbet. activities)
Node(activity)
2 4 3
Activity time estimate
Activity on Arrow (AOA)Network Representation
1 32Construct forms Pour concrete
Arcs represent activities
AON vs. AOA Network Representation
In AON, nodes represent activities & arcs show precedence relationships.In AOA, arcs represent activities & nodes are ‘events’ (completion or beginning of an activity).Difference between AOA and AON only in form. AOA and AON are conceptually equivalent.We use AON.
AON Network Diagram: Necessary Elements
Nodes representing activitiesArcs representing precedence relationships between activitiesActivity time estimates (single or multiple)
Constructing the Network Diagram: Our Conventions
Predecessor node must be drawn to the left of its successor node/sSuccessor node must be drawn to the right of its predecessor node/sNo vertical arcs No arcs moving from right to leftThus, network clearly flows from left to rightDraw arrowheads on arcs
Network Diagram : Recording Device Development
A,12
B,5
C,7
D,2
F,7
E,4
G,2
‘Forward Pass’: Earliest Times
ES = earliest time activity can startForward pass starts at beginning of CPM/PERT network to determine ES timesUsually set ES = 0 for beginning activityEF = ES + t
• ES = largest EF of immediate predecessors• Why?
‘Backward Pass’: Late Times
LS = latest time activity can start & not delay projectBackward pass starts at end of CPM/PERT network to determine LS timesUsually set LF = EF for last activityLS = LF – t
• LF = smallest LS of immediate successors• Why?
Activity Slack, S
SlackS = LS - ES = LF - EF
Activity is a critical activity if ES=LS & EF=LFS = 0
Activity is not critical if it has positive slackS > 0
Critical Path
Path: sequence of connected activities running from start to end node in network.Critical path: path consisting entirely of critical activities.Critical path has longest duration among all paths on network.
Critical Path
Duration of critical path = earliest project completion time. Project cannot be completed in less than length (duration) of critical path.Activity slack: meaning?Critical path in project control: significance?Is critical path unique?
Critical Path Scheduling with 3 Activity Time Estimates
Reflect uncertainty of activity timesBeta distribution is used in PERT
a = optimistic time estimatem = most likely time estimateb = pessimistic time estimate
where
64 : time(expected)Mean bmat j
++=
22
6 :Variance ⎟
⎠⎞
⎜⎝⎛ −
=ab
jσ
Example
2.52.01.5F, HI
5.04.03.0GH
7.03.53.0B, DG
3.02.01.0EF
1.51.00.5B, DE
2.52.01.5CD
3.02.01.0-C
6.02.52.0AB
2.52.01.5-A
PessimisticMost LikelyOptimisticPredecessor/sActivity
Time (weeks)Immediate