Metodos CuantitativosM. En C. Eduardo Bustos Farias1 Project Scheduling: PERT/CPM

Metodos Cuantitativos M. En C. Eduardo Bustos Farias 1

Project Scheduling: PERT/CPM

M. En C. Eduardo Bustos Farias 2 Metodos Cuantitativos

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


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


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



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.


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


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


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


The Basics of Using PERT/CPM


The Project Network Model


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


Slack Time


Project Network


Example


Partial Network

How should activity K be added?


This works, but there is a better way.



Earliest Time for an Event


Earliest Time for Each Event

Expected time to complete the project is 44 days.


Latest Time for an Event


Latest Time for Each Event

Expected time to complete the project is 44 days.


Slack Time


Critical Activities


Probabilistic Time Estimation


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


Transforming Plan to Network


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


GANTT Chart


Gantt Charts

Henry Laurence Gantt (1861-1919)


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


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


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


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


Making a Gantt chart Step 1 – list the tasks in the project


Making a Gantt chart Step 2 – add task durations


Making a Gantt chart

Step 3 – add dependencies (which tasks cannot start before another task finishes)


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


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.


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


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.


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!


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


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


Limitations to CPM/PERT Clearly defined, independent and

stable activities Specified precedence relationships Subjective time estimates Over emphasis on critical paths


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


Probabilistic PERT/CPM


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)


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


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


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


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


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


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


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.


Time-Cost Trade-Off

Cost

Time

Costs of Crashing

Indirect/Penalty Costs

Total Costs

10-9


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


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


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%


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.


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.


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


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


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


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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Metodos CuantitativosM. En C. Eduardo Bustos Farias1 Project Scheduling: PERT/CPM