metodos cuantitativosm. en c. eduardo bustos farias1 project scheduling: pert/cpm
TRANSCRIPT
Metodos Cuantitativos M. En C. Eduardo Bustos Farias 1
Project Scheduling: PERT/CPM
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Characteristics of a Project 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 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: Program Evaluation and
Review Technique CPM: 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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Project Schedule Converts action plan into operating timetable Basis for monitoring & controlling project
activity More important for projects than for day-to-
day operations 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:
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:
Program Evaluation and Review Technique estimates probability of on-time completion
CPM: 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 Activity: task or set of tasks
uses resources and takes time Event: result of completing an activity:
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 EOT:
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
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 Activities - scheduled and actual Precedence relationships Milestones (identifiable points in
project) 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
MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
Locate newfacilities
Interview staff
Hire and train staff
Select and orderfurniture
Remodel and installphones
Move in/startup
Gantt Chart
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Gantt Basics 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 Step 1 – list the tasks in the project
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Making a Gantt chart Step 2 – add task durations
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Making a Gantt chart
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 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 before affecting the end date of 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 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
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 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
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 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 Probability Z Probability<3.0 0 -1.5 0.066813.0 0.00135 -1.4 0.08076-2.9 0.00187 -1.3 0.09680-2.8 0.00256 -1.2 0.11507-2.7 0.00347 -1.1 0.13566-2.6 0.00466 -1.0 0.15865-2.5 0.00621 -0.9 0.18406-2.4 0.00820 -0.8 0.21185-2.3 0.01072 -0.7 0.24196-2.2 0.01390 -0.6 0.27425-2.1 0.01786 -0.5 0.30853-2.0 0.02275 -0.4 0.34457-1.9 0.02872 -0.3 0.38209-1.8 0.03593 -0.2 0.42074-1.7 0.04456 -0.1 0.46017-1.6 0.05480 0.0 0.50000
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Normal Distribution Table for Positive Values of Z
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: 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:
Probability that project takes more than x days= 1-Probability that project takes x days or less
Probability that project takes more than 7 days= 1-Probability that project takes 7 days or less = 1-0.8413=0.1587
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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=0.9039Use 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
$34,000
$33,000
$32,000
$31,000
$30,000
ActivityCost
CrashCost
NormalCost
Crash Time Normal Time
Time (Weeks)
Crash Cost/Week =
1 2 3
Crash Cost - Normal CostNormal Time - Crash Time
$34,000 - $30,0003 - 1
=
= $4,0002 Weeks
= $2,000/Week
Normal
Crash
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CRASH COSTING 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
Cost
Time
Costs of Crashing
Indirect/Penalty Costs
Total Costs
10-9
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Beta Probability Distribution with Three Time Estimates
Pro
bab
ilit
y
OptimisticTime(a)
MostLikelyTime(m)
PessimisticTime(b)
Activity Time
Probability of1 in 100
(b) Occuring
Probability of1 in 100
(a) Occuring
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Time Estimates (in weeks) for project
t =a + 4m + b
6Variance =
b - a
6( )
2
ActivityOptimistic
aMost
Probable- mPessimistic
bExpected Time
t = [(a + 4m + b)/6]Variance[(b - a)/6]2
A
B
C
D
E
F
G
H
1
2
1
2
1
1
3
1
2
3
2
4
4
2
4
2
3
4
3
6
7
9
11
3
2
3
2
4
4
3
5
2
3 - 16( ) 4
36=2
4 - 26( ) 4
36=2
3 - 16( ) 4
36=2
6 - 26( ) 16
36=2
7 - 16( ) 36
36=2
9 - 16( ) 64
36=2
11 - 36( ) 64
36=2
3 - 16( ) 4
36=2
Total 25 weeks
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Probability of Project Meeting the Deadline
Project StandardDeviation, T
= Project Variance
Z = Due Date - Expected Completion DateT
= 16 - 151.76
= 0.57
.57 Standard Deviations
16Weeks
15Weeks
Time
Probability(T 16 Weeks)is 71.6%
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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 budget cost from the actual cost to date.
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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:
D
3
Start
A
9
I
4
H
5
E
4
B
8Finish
C
10
J
8
F
4
G
3
D
3
Start
A
9
I
4
H
5
E
4
B
8Finish
C
10
J
8
F
4
G
3
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Example: How Are We Doing?
Earliest/Latest Times Activity ES EF LS LF Slack A 0 9 0 9 0 B 0 8 5 13 5 C 0 10 7 17 7 D 8 11 22 25 14 E 8 12 13 17 5 F 9 13 13 17 4 G 9 12 9 12 0 H 12 17 12 17 0 I 12 16 21 25 9 J 17 25 17 25 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