Transcript Project Scheduling: PERT/CPM Metodos Cuantitativos M. En C. Eduardo Bustos Farias 1

Project Scheduling: PERT/CPM
Metodos Cuantitativos
M. En C. Eduardo Bustos Farias
1
Characteristics of a Project

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A unique, one-time effort
Requires the completion of a large
number of interrelated activities
Resources, such as time and/or
money, are limited
Typically has its own management
structure
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Project Management
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A project manager is appointed to
head the project management team
The team members are drawn from
various departments and are
temporarily assigned to the project
The team is responsible for the
planning, scheduling and controlling
the project to its completion
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PERT and CPM

PERT:

CPM:

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Program Evaluation and
Review Technique
Critical Path Method
Graphically displays project activities
Estimates how long the project will take
Indicates most critical activities
Show where delays will not affect project
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Metodos Cuantitativos
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Project Schedule
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Converts action plan into operating timetable
Basis for monitoring & controlling project
activity
More important for projects than for day-today operations

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projects lack continuity of on-going functions
more complex coordination needed
One schedule for each major task level in
WBS
Maintain consistency among schedules
Final schedule reflects interdependencies,
departments.
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Network Model

Serves as a framework for:


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planning, scheduling, monitoring,
controlling
interdependencies and task coordination
when individuals need to be available
communication among departments and
functions needed on the project
Identifies critical activities and slack
time
Reduces interpersonal conflict
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PERT / CPM

PERT:
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CPM:
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Program Evaluation and Review Technique
estimates probability of on-time completion
Critical Path Method
deterministic time estimates
control both time and cost
Similar purposes, techniques, notation
Both identify critical path and slack time
Time vs. performance improvement
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PERT / CPM Definitions
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Activity: task or set of tasks

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Event: result of completing an activity:
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uses resources and takes time
has identifiable end state at a point in time
Network: combined activities & events in a
project
Path: series of connected activities
Critical: activities, events, or paths which, if
delayed, will delay project completion
Critical path: sequence of critical activities
from start to finish
Node / Arrow (Arc) - PERT / CPM notation
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The Basics of Using PERT/CPM
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The Project Network Model
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PERT / CPM Notations
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EOT:
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earliest occurrence time for event
time required for longest path leading to
event
LOT: latest occurrence time for event
EST: earliest starting time for activity
LST: latest starting time for activity
Critical time: shortest time in which
the project can be completed
Notation: AOA, AON, dummy activities
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Slack Time
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Project Network
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Example
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Partial Network
How should activity K be added?
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This works, but there is a better way.
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Earliest Time for an Event
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Earliest Time for Each Event
Expected time to complete the project is 44 days.
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Latest Time for an Event
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Latest Time for Each Event
Expected time to complete the project is 44 days.
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Slack Time
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Critical Activities
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Probabilistic Time Estimation
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Expected completion time:
Based on optimistic, pessimistic, most
likely
Take weighted average of the 3 times
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TE = (a + 4m + b)/6
Uncertainty = variance (range of
values)
Probability of completion of project in
desired time D
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Transforming Plan to Network
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Know activities which comprise project
Determine predecessor and successor
activities
Time and resources for activities
Interconnections depend on technical
interdependencies
Expected completion time

as soon as possible versus as late as
possible
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GANTT Chart
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Gantt
Charts
Henry Laurence Gantt (1861-1919)
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Planned and actual progress
for multiple tasks on horizontal time
scale
easy to read, easy to construct
effective monitoring and control of
progress
requires frequent updating
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Components of GANTT Chart
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Activities - scheduled and actual
Precedence relationships
Milestones (identifiable points in
project)
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usually represents reporting requirements
usually corresponds to critical events
Can add budget information
Does not show technical
interdependencies
Need PERT network to interpret,
control, and compensate for delays
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Planning and Scheduling
Gantt Chart
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Locate new
facilities
Interview staff
Hire and train staff
Select and order
furniture
Remodel and install
phones
Move in/startup
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Gantt Basics
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Basically, a timeline with tasks that
can be connected to each other
Note the spelling!
It is not all-capitals!
Can be created with simple tools like
Excel, but specialised tools like
Microsoft Project make life easier
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Making a Gantt chart
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Step 1 – list the tasks in the project
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Making a Gantt chart
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Step 2 – add task durations
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Making a Gantt chart
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Step 3 – add dependencies (which tasks
cannot start before another task
finishes)
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Notes
•The arrows indicate dependencies.
•Task 1 is a predecessor of task 2 – i.e. task 2 cannot start before task 1 ends.
•Task 3 is dependent on task 2. Task 7 is dependent on two other tasks
•Electrics, plumbing and landscaping are concurrent tasks and can happen at
the same time, so they overlap on the chart. All 3 can start after task 4 ends.
•Painting must wait for both electrics and plumbing to be finished.
•Task 9 has zero duration, and is a milestone
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Making a Gantt chart

Step 4 – find the critical path
The critical path is the sequence of tasks from beginning to end that takes
the longest time to complete.
It is also the shortest possible time that the project can be finished in.
Any task on the critical path is called a critical task.
No critical task can have its duration changed without affecting the end
Metodos Cuantitativos
date
of the project.
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MS Project can work out the critical path for
you!
 The length of the critical path is the sum of the
lengths of all critical tasks (the red tasks
1,2,3,4,5,7) which is 2+3+1+1.5+2+1 = 10.5
days.
 In other words, the minimum amount of time
required to get all tasks completed is 10.5 days
 The other tasks (6,8) can each run over-time
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before affecting the
date
the project
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The amount of time a task can be extended
before it affects other tasks is called slack (or
float).
Both tasks 6 and 8 can take one extra day
before they affects a following task, so each
has one day’s slack.
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Critical tasks, by definition, can have NO slack.
Tip:
If ever asked Can task X’s duration be changed
without affecting the end date of the
project?, if it is a critical task the answer is
always NO!
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Benefits of CPM/PERT
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Useful at many stages of project
management
Mathematically simple
Give critical path and slack time
Provide project documentation
Useful in monitoring costs
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Advantages of PERT/CPM
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useful at several stages of project management
straightforward in concept, and not mathematically
complex
uses graphical displays employing networks to help
user perceive relationships among project activities
critical path and slack time analyses help pinpoint
activities that need to be closely watched
networks generated provide valuable project
documentation and graphically point out who is
responsible for various project activities
applicable to a wide variety of projects and industries
useful in monitoring not only schedules, but costs as
well
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Limitations to CPM/PERT
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Clearly defined, independent and
stable activities
Specified precedence relationships
Subjective time estimates
Over emphasis on critical paths
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Limitations of PERT/CPM
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project activities must be clearly defined,
independent, and stable in their relationships
precedence relationships must be specified and
networked together
time activities in PERT are assumed to follow the
beta probability distribution -- this may be difficult
to verify
time estimates tend to be subjective, and are
subject to fudging by managers
there is inherent danger in too much emphasis
being placed on the critical path
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Probabilistic PERT/CPM
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Mean and Standard Deviation of Project
Duration
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Once the expected time te for all activities
has been computed, proceed to use te in
place of the single activity duration in CPM to
work out the critical path and the project
duration
The resulting project duration is the mean
project duration TE
We also need to work out the standard
deviation of the project duration  as
follows:

Project Duration = (Summation of
i2 f all the activities on the critical
path)
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Probability of Different Project Durations

From statistics, once we know the mean project duration,
TE, and the standard deviation of the project duration, 
we can work out the probability that the
project duration will be shorter than any
specific time, T (i.e. the project will take T
days or less) through the following formula:


Z=(T- TE )/  , where Z is the quantity called the
Normal variate
Knowing Z, we can read off the probability from
Normal Distribution Tables which are provided in
nest slides
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Normal Distribution Table for Negative
Values of Z
Z
<3.0
3.0
-2.9
-2.8
-2.7
-2.6
-2.5
-2.4
-2.3
-2.2
-2.1
-2.0
-1.9
-1.8
-1.7
-1.6
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Probability
0
0.00135
0.00187
0.00256
0.00347
0.00466
0.00621
0.00820
0.01072
0.01390
0.01786
0.02275
0.02872
0.03593
0.04456
0.05480
Z
-1.5
-1.4
-1.3
-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Probability
0.06681
0.08076
0.09680
0.11507
0.13566
0.15865
0.18406
0.21185
0.24196
0.27425
0.30853
0.34457
0.38209
0.42074
0.46017
0.50000
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Normal Distribution Table for Positive
Values of Z
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Z | Probability
--------------------0.0 | 0.5000
0.1 | 0.5398
0.2 | 0.5793
0.3 | 0.6179
0.4 | 0.6554
0.5 | 0.6915
0.6 | 0.7257
0.7 | 0.7580
0.8 | 0.7881
0.9 | 0.8159
1.0 | 0.8413
1.1 | 0.8643
1.2 | 0.8849
1.3 | 0.9032
1.4 | 0.9192
1.5 | 0.9332
Z | Probability
--------------------1.6 | 0.9452
1.7 | 0.9554
1.8 | 0.9641
1.9 | 0.9713
2.0 | 0.9772
2.1 | 0.9821
2.2 | 0.9861
2.3 | 0.9893
2.4 | 0.9918
2.5 | 0.9938
2.6 | 0.9953
2.7 | 0.9965
2.8 | 0.9974
2.9 | 0.9981
3.0 | 0.9987
>3.0| 1
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Example

Consider a project with TE = 5days and =2 days.If
we wish to find out the probability that the project
will take 7 days or less. Thus T=7days. First, work
out a value (calles the normal variate) Z, as follows:
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Z=(T- TE )/ =(7-5)/2=1
Read off the Normal Distribution Tables, the probability
for Z=1. We get the value 0.8413. Thus the probability
that the project will take 7 days or less is 0.8413
If we need to find the probability that the project takes
more than 7 days, we make use of the fact that:


Metodos Cuantitativos
Probability that project
Probability that project
Probability that project
Probability that project
0.8413=0.1587
takes
takes
takes
takes
more than x days= 1x days or less
more than 7 days= 17 days or less = 1-
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Interpolating from the Normal Distribution
Table
In the previous example, the ‘Z’ value was 1.0 and could
be read off directly. If you had a value like 1.01, you could
still round it off to 1.0
 However there will be instances when you will get a value
like 1.275, in which case you will need to interpolate from
the table
 From the table Z1=1.2, P1=0.8849
Z2=1.3, P2=0.9039
Use the interpolation formula:
P=P1+Z-Z1 *(P2-P1)
Z2-Z1
Therefore at Z=1.275,
P=0.8849 + 1.275 -1.2 * (0.9039-0.8849) = 0.8992
1.3-1.2

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Crash and Normal Times and Costs
Activity
Cost
Crash
Crash Cost - Normal Cost
Normal Time - Crash Time
$34,000 - $30,000
=
3-1
= $4,000 = $2,000/Week
2 Weeks
Normal
$34,000
Crash
Cost
Crash Cost/Week =
$33,000
$32,000
$31,000
$30,000
Normal
Cost
1
Crash Time
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2
3
Time (Weeks)
Normal Time
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CRASH COSTING
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1. Find critical path.
2. Find cheapest act. in critical path
3. Reduce time until:




a. Can’t be reduced
b. Another path becomes critical
c. Increase in direct costs exceeds savings
from shortening project
4. Return to Step 1, as long as savings.
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Time-Cost Trade-Off
Total Costs
Indirect/Penalty Costs
Cost
Costs of Crashing
Time
10-9 Metodos Cuantitativos
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Probability
Beta Probability Distribution with
Three Time Estimates
Probability of
1 in 100
(a) Occuring
Optimistic
Time
(a)
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Probability of
1 in 100
(b) Occuring
Most
Likely
Time
(m)
Pessimistic Activity Time
Time
(b)
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Time Estimates (in weeks) for project
a + 4m + b
Variance =
t=
6
Optimistic
Most
Pessimistic Expected Time
a
Probable- m
b
t = [(a + 4m + b)/6]
Activity
A
1
2
B
2
3
C
1
2
D
2
4
E
1
4
F
1
2
G
3
4
Metodos Cuantitativos
H
1
2
3
2
(
2
b-a
)
(
6
Variance
[(b - a)/6]2
3-1 2
4
) =
6
4-2 2
4
3
( 6 )
3-1 2
( 6 )
3
2
6-2 2
( 6 )
6
4
7-1 2
7
4
( 6 )
9-1 2
9
3
( 6 )
11 - 3 2
11
5
( 6 )
2
3
1
3
2
( 6 )
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Total 25 weeks
36
4
= 36
4
= 36
16
= 36
36
= 36
64
= 36
64
= 36
4
=58 36
Probability of Project Meeting the Deadline
Project Standard
=
Deviation, T
Project Variance
Z = Due Date - Expected Completion Date
T
= 16 - 15 = 0.57
1.76
.57 Standard Deviations
Probability
(T  16 Weeks)
is 71.6%
Metodos Cuantitativos
15 16
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Weeks Weeks
Time
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PERT/Cost
PERT/Cost is a technique for monitoring costs
during a project.
 Work packages (groups of related activities)
with estimated budgets and completion times
are evaluated.
 A cost status report may be calculated by
determining the cost overrun or underrun for
each work package.
 Cost overrun or underrun is calculated by
subtracting the budgeted cost from the actual
cost of the work package.
 For work in progress, overrun or underrun may
be determined by subtracting the prorated
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budget cost from M.the
actual
cost

PERT/Cost

The overall project cost overrun or
underrun at a particular time during a
project is determined by summing the
individual cost overruns and
underruns to date of the work
packages.
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Example: How Are We Doing?

Consider the following PERT network:
A
9
Start
B
8
G
3
I
4
F
4
H
5
Finish
D
3
J
8
E
4
C
10
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Example: How Are We Doing?

Earliest/Latest Times
Activity ES EF
A
0
B
0
C
0
D
8
E
8
F
9
G
9
H
12
I
12
J
17
Metodos Cuantitativos
LS LF Slack
9
0
9
8
5 13
10
7 17
11
22 25
12
13 17
13
13 17
12
9 12
17
12 17
16
21 25
25
17 25
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0
5
7
14
5
4
0
0
9
0
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Example: How Are We Doing?

Activity Status (end of eleventh week)
Activity
Actual Cost
% Complete
A
$6,200
100
B
5,700
100
C
5,600
90
D
0
0
E
1,000
25
F
5,000
75
G
2,000
50
H
0
0
I
0
0
J
0
0
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Example: How Are We Doing?

Cost Status Report
(Assuming a budgeted cost of $6000 for each activity)
Activity Actual Cost
Value
Difference
A
$6,200
(1.00)x6000 = 6000
$200
B
5,700
(1.00)x6000 = 6000
- 300
C
5,600
(.90)x6000 = 5400
200
D
0
0
0
E
1,000
(.25)x6000 = 1500
- 500
F
5,000
(.75)x6000 = 4500
500
G
2,000
(.50)x6000 = 3000
-1000
H
0
0
0
I
0
0
0
J
0
0
0
Totals
$25,500
$26,400
$- 900
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