# PERT/CPM Calculations - Minnesota State University, Mankato URBS 609 PERT, Unit 2

### PERT/CPM CALCULATIONS

Basic Techniques Using MS Excel

And Manual Calculation

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This Unit of Instruction was crafted by Robert Hugg For Minnesota State University, Mankato Urban and Regional Studies Institute - 2004

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### Training Module Preview

• This module will provide:

– Introduction to manually calculating key Project

Management functions (both PERT and CPM)

– Introduction to using MS Excel to calculate key functions (PERT and Risk analysis)

– Step-by step instruction on building a PERT risk analysis calculator using MS Excel

– Use of PERT and CPM traditional techniques to manually lay out a project

• This module is constructed as the second block in a building block approach

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### PERT Calculations - Simplicity

• Simple steps in a logical order

– Step 2: Place Tasks in a logical order, find the critical path

• The longest time path through the task network.

(or even a single task) that dictates the calculated finish date

– Step 3: Generate estimates

• Optimistic, pessimistic, likely and PERT- expected

• Standard Deviation and variance

– Step 4: Determine earliest and latest dates

– Step 5:Determine probability of meeting expected date

• Steps 1 and 2 are logic and legwork, not calculation – these require a clear goal

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### PERT Calculations – Step 3

• Assuming steps 1 and 2 have been completed begin calculations – use a table to organize your calculations

• Simple calculations to estimate project durations

• Based on input of 3 estimated durations per task

Most Optimistic (T

O

) – best case scenario

Most Likely (T

L

) “normal” scenario

Most Pessimistic (T

P

) Worst case scenario

• Formula derives a probability-based expected duration (T

E

( T

O x 1 + T

L x 4 + T

P x 1 ) / 6 = T

E

– Read this formula as the sum of (optimistic x 1 + likely x 4 + pessimistic x 1) divided by 6 = expected task duration

• Complete this calculation for all tasks

)

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### PERT Calculations – Step 3

• Standard deviation and variance

– Standard deviation (SD) is the average deviation from the estimated time

• SD=(T

P

-T

0

• As a general rule, the higher the standard deviation the greater the amount of uncertainty

– Variance (V) reflects the spread of a value over a normal distribution

• V=SD 2 (Standard deviation squared)

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### PERT Calculations – Step 3

• When doing manual PERT Calculations it is helpful to construct a table to stay organized

• Consider the sample project in Unit 1 – planting trees and flowers, set up using a list

– Rough estimates and no risk analysis

• No Range, simply rough estimates - unreliable?

– PERT Analysis will better refine estimates

• Start by setting up a table to organize data

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Start

### Our Project – A Refresher

Set up in tabular form, it might look like this…

7

1

Description

Mark Utilities

2

3

Dig Holes

4

5

Plant Trees

6

7

8

Plant Flowers

Install Edging

Duration (Days)

?

?

?

?

?

?

?

?

3

1 2

Mark Utilities Dig Holes

5

Plant Trees

6

Plant Flowers

8

Install Edging Finish

4

Set up in visual form it might look like this…

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### PERT Step 3 – First Get Organized

In considering all tasks on the previous slide, a table might look like this

T

O

T

L

T

P

T

E

1

2

5

6

8

TOTAL

T

O

T

L

T

P

T

E

3

4

7

TOTAL

T

O

-Optimistic T

M

-Likely T

P

-Pessimistic T

E

-Expected (Derived by PERT)

Remember – tasks 3, 4 and 7 are concurrent and do not add to the timeline

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### PERT Step 3 – Durations

After generating estimates using the formula, the table might look like this

1

2

5

6

8

TOTAL

3

4

7

TOTAL

1

1

7

T

O

1

2

1

T

O

.5

.5

.5

1.5

T

L

1

1

1

3

T

L

3

4

3

3

2

15

T

P

5

T

E

3

7 4.17

6 3.17

5 3

4 2.17

28 15.6

T

P

T

E

3 1.25

3 1.25

3 1.25

9 3.75

SD

.67

.83

.83

.67

.5

3.5

SD

.42

.42

.42

1.26

V

.44

.69

.69

.44

.25

2.51

V

.17

.17

.17

.51

T

O

-Optimistic T

M

-Likely T

P

-Pessimistic T

E

-Expected (Derived by PERT)

SD=Standard Deviation V=Variance

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### PERT Step 4 – Dates

For each task, determine the latest allowable time for moving to the next task

The difference between latest time and expected time is called slack time

Tasks with zero slack time are on the critical path

1

2

5

6

8

TOTAL

T

O

1

2

1

1

1

7

T

L

3

4

3

3

2

15

T

P

5

7

6

5

T

3

E

4.17

3.17

3

ES

0

3

7

10

EF

3

7.17

10.17

13

LS

0

3

7

10

LF

3

7.17

10.17

13

4 2.17

13 15.17 13 15.17

28 15.51

Slack SD

0

0

0

0

0

.67

.83

.83

.67

.5

3.5

V

.444

.694

.694

.444

.254

2.530

3

4

7

TOTAL

T

O

.5

.5

.5

1.5

T

L

1

1

1

3

T

P

T

E

3 1.25

3 1.25

ES

0

0

EF

1.25

1.25

LS

3

3

LF

4.25

4.25

3 1.25

1.25

2.50 4.25 5.50

9 3.75

FLOAT

3

3

3

SD

.42

.42

.42

1.26

V

.17

.17

.17

.51

ES=Earliest Start EF= Earliest Finish LS=Latest Start LF=Latest Finish

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### PERT Step 5 – Probabilities

Manually computing probability using data compiled in your table

• Determine probability of meeting a date by using the table data

– Denote the sum of all expected durations on the critical path as S

– Denote the sum of all variances on the critical path as V

– Select a desired completion time, denote this as D

COMPUTE : (D-S)/square root (V) = Z ( the number of std. deviations that the due date is away from the expected date))

• Enter a standard normal table to find a probability that corresponds with Z or go online to:

– http://math.uc.edu/statistics/statbook/tables.html

) to enter a z number the application will retrieve the probability from the lengthy table

• For our project, figure a probability based on the most likely time, 15 days: (15-15.51)/square root(2.53) = (15-15.51)/1.59=-.3207 (Z)

• A corresponding probability is 37.7% (Rounded)

• This process can be repeated for any date desired

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### PERT Step 5 – Probabilities

Computing probability in Excel using data compiled in your table

• Excel has normal distribution functions built in and can compute PERT probabilities

• By creating a table as a spreadsheet, the addition of a few simple formulae will do the rest of the work

• Create a table as a template that can be used over and over again – simply change the input

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### PERT Step 5 – Probabilities

Computing probability in Excel using data compiled in your table

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Step 1 - Create a spreadsheet that resembles the table used earlier

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Step 2 – Use formulae as shown to calculate PERT Expectations

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Cell Formulae used for PERT Analysis- expected durations

• Computing PERT Expected duration

– For each task cell: (Optimistic + 4x Typical +

Pessimistic)/6

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Step 3 – Use formulae as shown to calculate variances

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Cell Formulae used for PERT Analysis – Variances

• Computing Variances

• ((Pessimistic-Optimistic)/6) 2

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Step 4 – Use formulae as shown to calculate STD. Deviations

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Cell Formulae used for PERT Analysis – Standard Deviations

• Computing Standard Deviations

• Square root of the variance for that task

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Step 5 – Use formula as shown to sum PERT expectations

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Cell Formula used for PERT Analysis – Summing PERT Expectations

• Sum Pert Expectations using either autosum feature or sum formula

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Step 6 – Use formula as shown to sum variances

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Cell Formula used for PERT Analysis – Summing Variances

• Sum Variances using either auto-sum feature or sum formula

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Step 6 – Use formula as shown to compute probability

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Cell Formula used for PERT Analysis – Completion Probability

• Excel uses a formula designed to compute the probability of placement of a combination of elements in a normal distribution – very accurate

NORMDIST(x,mean,standard_dev,cumulative)

X is the value for which you want the distribution

(desired date)

Mean is the arithmetic mean of the distribution (summed

PERT expected durations)

Standard_dev is the standard deviation of the distribution (square root of the summed variances)

Cumulative is a logical value that determines the form of the function. If cumulative is TRUE , NORMDIST returns the cumulative distribution function (probability of completion on the date entered)

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Cell Formula used for PERT Analysis – Hints and Tips

– If a error message shows up check cell addresses in the formulae first – formulae must reflect intent

• This set of formulae mirrors the manual calculations but takes less time for the user

• Because PERT is a probabilistic approach, these formulae can deliver a 100% probability – but no plan is perfect – these are always estimates

• Never feel there is a 100% probability of a project completing on the estimated date

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### PERT Analysis

Thoughts, Philosophy and Lessons Learned

• All Plans are estimates and are only as good as the task estimates – unrealistic estimates equal unrealistic plans

• If the scope of a plan changes, all estimates must change – adding tasks equals added time and cost

• PERT Analysis is a good way to “what if” before a project is launched – helps determine if it is needed at all

– What tasks will it take to do the project?

– What is the optimum order of the project tasks?

– How long will it take to do the project?

– How likely is the project to succeed?

– What if “ The Boss ” wants it earlier, what is the likelihood then?

• A great way to get organized and stay organized

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### CPM Analysis

• In comparison to PERT, CPM analysis is simple

• CPM Analysis is a series of easy steps

1. Develop time and cost data ("normal" and "crashed") for all tasks

2. Develop cost-per-week for crashing ( crashed costs divided by time saved)

3. Develop project network (PERT)

4. Crash the activity crashing on the critical path with the lowest cost-for-

5. Recalculate the project network (the critical path might change!)

• Repeat steps 4 & 5 until all the paths have been crashed.

• Ease up on all non-critical paths, just to the point that all paths are critical.

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CPM Analysis

• A typical CPM table might have the following structure:

Activity Begin End

Time

(Crashed)

Time

(Normal)

Cost

(Crashed)

Cost

(Normal)

Time

Saved

Cost

Increase

Cost /

Week

Foundation

Frame

1

2

2

3

1

1

2

4

4000

8000

3000

4000

1

3

1000

4000

1000

1333 cost-per-week for crashing = crashed costs divided by time saved

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### CPM Analysis

Thoughts, Philosophy and Lessons Learned

• All Plans are estimates and are only as good as the task estimates – unrealistic estimates equal unrealistic plans

• If the scope of a plan changes, all estimates must change – adding tasks equals added time and cost

• CPM Analysis is a good way to “what if” before a project is launched – helps control expectations

– How much will it cost?

– How long will it take?

– How long will it take if it needs to be done sooner?

– How much will it cost if it needs to be done sooner?

• A great way to get organized and stay organized

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### Use PERT and CPM Together

• PERT & CPM are totally complementary both require the same preparation:

1. Define the Project and all of its significant activities or tasks. The

Project should have only a single start activity and a single finish activity.

2. Develop the relationships among the activities; decide which activities must precede and which must follow others.

3. Draw a Network Diagram connecting all the activities (each activity should have a unique number).

4. Assign time and/or cost estimates to each activity.

5. Compute the longest time path through the network. (The critical path)

6. Use the Network to help plan, schedule, monitor & control the project.

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### PM Calculations Overview

• PERT and CPM can be used together

• Calculations are based on a few simple formulae:

– PERT Derived duration estimates

– Standard Deviation

– Variance

– Probability of meeting expectation

– Crash costs and time & normal costs and time

• Calculations can be done manually or using

Excel – same formulae, different tools

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### Resources Used in This Unit

• Bonini, Charles, et al, Quantitative

Analysis for Management, Columbus:

McGraw Hill, 1997

• Dr. Anthony Filipovitch

• Goldratt, Eli, Dr., The Goal: A Process of

Ongoing Improvement, Great

Barrington: New River Press, 1996

• Mednick, Barry, PERT-CPM on

Excel,Fullerton: Cal State, 2000

• MS Project, by Microsoft Corporation

• MS Excel, by Microsoft Corporation

• PM Body of Knowledge (PMBOK),

• Project Management Institute (PMI)

Resource Center

– Project Management Institute Website

• ProjeX, by WAA, Inc

• Systema, Sid, Probabilistic Solutions to

Project Scheduling, Ferris State, 1999

• US National Performance Survey, The

Standish Group, 1998

• Verma, Vijay K., Managing the Project

Team: The Human Aspects of Project

• Wiest, Jerome D., and Levy, Ferdinand

K., A Management Guide to

PERT/CPM, New Delhi: Prentice-Hall of

India Private Limited, 1974

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You have completed

URBS 609 PERT Unit 2