irwin/mcgraw-hill © the mcgraw-hill companies, inc., 2003 8.1 pert/cpm models for project...

© The McGraw-Hill Companies, Inc., 20038.1Irwin/McGraw-Hill

PERT/CPM Models for Project Management

© The McGraw-Hill Companies, Inc., 20038.2McGraw-Hill/Irwin

Project Management

• Characteristics of Projects– Unique, one-time operations

– Involve a large number of activities that must be planned and coordinated

– Long time-horizon

– Goals of meeting completion deadlines and budgets

• Examples– Building a house

– Planning a meeting

– Introducing a new product

• PERT—Project Evaluation and Review TechniqueCPM—Critical Path Method

– A graphical or network approach for planning and coordinating large-scale projects.


Example: Building a House

Activity Time (Days)ImmediatePredecessor

Foundation 4 —

Framing 10 Foundation

Plumbing 9 Framing

Electrical 6 Framing

Wall Board 8 Plumbing, Electrical

Siding 16 Framing

Paint Interior 5 Wall Board

Paint Exterior 9 Siding

Fixtures 6 Int. Paint, Ext. Paint


Gantt Chart

Foundation

Framing

Plumbing

Electrical

Siding

Wall Board

Paint Interior

Paint Exterior

Fixtures

Activity Start 5 10 15 20 25 30 35 40 45 50Days After Start

Days After StartStart 5 10 15 20 25 30 35 40 45 50


PERT and CPM

• Procedure1. Determine the sequence of activities.

2. Construct the network or precedence diagram.

3. Starting from the left, compute the Early Start (ES) and Early Finish (EF) time for each activity.

4. Starting from the right, compute the Late Finish (LF) and Late Start (LS) time for each activity.

5. Find the slack for each activity.

6. Identify the Critical Path.


Notation

t Duration of an activity

ES The earliest time an activity can start

EF The earliest time an activity can finish (EF = ES + t)

LS The latest time an activity can start and not delay the project

LF The latest time an activity can finish and not delay the project

Slack The extra time that could be made available to an activity without delaying the project (Slack = LS – ES)

Critical Path The sequence(s) of activities with no slack


PERT/CPM Project Network

START FINISHd

c

f

e

h

g

iba0 0

Foundation Framing

Siding PaintExterior

Plumbing

WallBoard Paint

Interior FixturesElectrical

4 10

9

6

16

8

5

9

6


Calculation of ES, EF, LF, LS, and Slack

GOING FORWARD

• ES = Maximum of EF’s for all predecessors

• EF = ES + t

GOING BACKWARD

• LF = Minimum of LS for all successors

• LS = LF – t

• Slack = LS – ES = LF – EF


Building a House: ES, EF, LS, LF, Slack

Activity ES EF LS LF Slack

(a) Foundation 0 4 0 4 0

(b) Framing 4 14 4 14 0

(c) Plumbing 14 23 17 26 3

(d) Electrical 14 20 20 26 6

(e) Wall Board 23 31 26 34 3

(f) Siding 14 30 14 30 0

(g) Paint Interior 31 36 34 39 3

(h) Paint Exterior 30 39 30 39 0

(i) Fixtures 39 45 39 45 0


PERT/CPM Project Network

START0

FINISH0

a

b c

d

e

f

h

g i

j

4

4

4 5 5

533

7

8


Example #2: ES, EF, LS, LF, Slack

Activity ES EF LS LF Slack

a 0 4 0 4 0

b 0 4 1 5 1

c 4 7 5 8 1

d 4 8 4 8 0

e 4 12 5 13 1

f 4 11 6 13 2

g 8 13 8 13 0

h 8 11 10 13 2

i 13 18 13 18 0

j 11 16 13 18 2


Reliable Construction Company Project

• The Reliable Construction Company has just made the winning bid of $5.4 million to construct a new plant for a major manufacturer.

• The contract includes the following provisions:– A penalty of $300,000 if Reliable has not completed construction within 47 weeks.

– A bonus of $150,000 if Reliable has completed the plant within 40 weeks.

Questions:1. How can the project be displayed graphically to better visualize the activities?

2. What is the total time required to complete the project if no delays occur?

3. When do the individual activities need to start and finish?

4. What are the critical bottleneck activities?

5. For other activities, how much delay can be tolerated?

6. What is the probability the project can be completed in 47 weeks?

7. What is the least expensive way to complete the project within 40 weeks?

8. How should ongoing costs be monitored to try to keep the project within budget?


Activity List for Reliable Construction

Activity Activity DescriptionImmediate

PredecessorsEstimated

Duration (Weeks)

A Excavate — 2

B Lay the foundation A 4

C Put up the rough wall B 10

D Put up the roof C 6

E Install the exterior plumbing C 4

F Install the interior plumbing E 5

G Put up the exterior siding D 7

H Do the exterior painting E, G 9

I Do the electrical work C 7

J Put up the wallboard F, I 8

K Install the flooring J 4

L Do the interior painting J 5

M Install the exterior fixtures H 2

N Install the interior fixtures K, L 6


Reliable Construction Project Network

A

START

G

H

M

F

J

K L

N

Activity Code

A. Excavate

B. Foundation

C. Rough wall

D. Roof

E. Exterior plumbing

F. Interior plumbing

G. Exterior siding

H. Exterior painting

I. Electrical work

J. Wallboard

K. Flooring

L. Interior painting

M. Exterior fixtures

N. Interior fixtures

2

4

10

746

7

9

5

8

4 5

6

2

0

0FINISH

D IE

C

B


The Critical Path

• A path through a network is one of the routes following the arrows (arcs) from the start node to the finish node.

• The length of a path is the sum of the (estimated) durations of the activities on the path.

• The (estimated) project duration equals the length of the longest path through the project network.

• This longest path is called the critical path. (If more than one path tie for the longest, they all are critical paths.)


The Paths for Reliable’s Project Network

Path Length (Weeks)

StartA B C D G H M Finish 2 + 4 + 10 + 6 + 7 + 9 + 2 = 40

Start A B C E H M Finish 2 + 4 + 10 + 4 + 9 + 2 = 31

Start A B C E F J K N Finish 2 + 4 + 10 + 4 + 5 + 8 + 4 + 6 = 43

Start A B C E F J L N Finish 2 + 4 + 10 + 4 + 5 + 8 + 5 + 6 = 44

Start A B C I J K N Finish 2 + 4 + 10 + 7 + 8 + 4 + 6 = 41

Start A B C I J L N Finish 2 + 4 + 10 + 7 + 8 + 5 + 6 = 42


ES and EF Values for Reliable Constructionfor Activities that have only a Single Predecessor

A

START

G

H

M

F

J

FINISH

K L

N

D IE

C

B

2

4

10

746

7

9

5

8

4 5

6

2

ES = 0 EF = 2

ES = 2 EF = 6

ES = 16 EF = 22

ES = 16 EF = 20

ES = 16 EF = 23

ES = 20 EF = 25

ES = 22 EF = 29

ES = 6 EF = 16

0

0


ES and EF Times for Reliable Construction

A

START

G

H

M

F

J

FINISH

K L

N

D IE

C

B

2

4

10

746

7

9

5

8

4 5

6

2

ES = 0 EF = 2

ES = 2 EF = 6

ES = 16 EF = 22

ES = 16 EF = 20

ES = 16 EF = 23

ES = 20 EF = 25

ES = 22 EF = 29

ES = 6 EF = 16

ES = 0 EF = 0

ES = 25 EF = 33

ES = 33 EF = 38

ES = 38 EF = 44

ES = 33 EF = 37

ES = 29 EF = 38

ES = 38 EF = 40

ES = 44 EF = 44

0

0


LS and LF Times for Reliable’s Project

A

START

G

H

M

F

J

FINISH

K L

N

D IE

C

B

2

4

10

746

7

9

5

8

4 5

6

2

LS = 0 LF = 2

LS = 2 LF = 6

LS = 20 LF = 26

LS = 16 LF = 20

LS = 18 LF = 25

LS = 20 LF = 25

LS = 26 LF = 33

LS = 6 LF = 16

LS = 0 LF = 0

LS = 25 LF = 33

LS = 33 LF = 38

LS = 38 LF = 44

LS = 34 LF = 38

LS = 33 LF = 42

LS = 42 LF = 44

LS = 44 LF = 44

0

0


The Complete Project Network

A

START

G

H

M

F

J

FINISH

K L

N

D IE

C

B

2

4

10

746

7

9

5

8

4 5

6

2

S = (0, 0) F = (2, 2)

S = (2, 2) F = (6, 6)

S = (16, 20) F = (22, 26)

S = (16, 16) F = (20, 20)

S = (16, 18) F = (23, 25)

S = (20, 20) F = (25, 25)

S = (22, 26) F = (29, 33)

S = (6, 6) F = (16, 16)

S = (0, 0) F = (0, 0)

S = (25, 25) F = (33, 33)

S = (33, 33) F = (38, 38)

S = (38, 38) F = (44, 44)

S = (33, 34) F = (37, 38)

S = (29, 33) F = (38, 42)

S = (38, 42) F = (40, 44)

S = (44, 44) F = (44, 44)

0

0


Slack for Reliable’s Activities

Activity Slack (LF–EF) On Critical Path?

A 0 Yes

B 0 Yes

C 0 Yes

D 4 No

E 0 Yes

F 0 Yes

G 4 No

H 4 No

I 2 No

J 0 Yes

K 1 No

L 0 Yes

M 4 No

N 0 Yes


Spreadsheet to Calculate ES, EF, LS, LF, Slack

3456789

10111213141516171819

B C D E F G H I JActivity Description Time ES EF LS LF Slack Critical?

A Excavate 2 0 2 0 2 0 YesB Foundation 4 2 6 2 6 0 YesC Rough Wall 10 6 16 6 16 0 YesD Roof 6 16 22 20 26 4 NoE Exterior Plumbing 4 16 20 16 20 0 YesF Interior Plumbing 5 20 25 20 25 0 YesG Exterior Siding 7 22 29 26 33 4 NoH Exterior Painting 9 29 38 33 42 4 NoI Electrical Work 7 16 23 18 25 2 NoJ Wallboard 8 25 33 25 33 0 YesK Flooring 4 33 37 34 38 1 NoL Interior Painting 5 33 38 33 38 0 YesM Exterior Fixtures 2 38 40 42 44 4 NoN Interior Fixtures 6 38 44 38 44 0 Yes

Project Duration 44


PERT with Uncertain Activity Durations

• If the activity times are not known with certainty, PERT/CPM can be used to calculate the probability that the project will complete by time t.

• For each activity, make three time estimates:– Optimistic time: o

– Pessimistic time: p

– Most-likely time: m


Beta Distribution

Assumption: The variability of the time estimates follows the beta distribution.

Elapsed time

p0

Beta distribution

mo


PERT with Uncertain Activity Durations

Goal: Calculate the probability that the project is completed by time t.

Procedure:

1. Calculate the expected duration and variance for each activity.

2. Calculate the expected length of each path. Determine which path is the mean critical path.

3. Calculate the standard deviation of the mean critical path.

4. Find the probability that the mean critical path completes by time t.


Expected Duration and Variance for Activities (Step #1)

• The expected duration of each activity can be approximated as follows:

• The variance of the duration for each activity can be approximated as follows:

2

p o6

2

o 4m p6


Expected Length of Each Path (Step #2)

• The expected length of each path is equal to the sum of the expected durations of all the activities on each path.

• The mean critical path is the path with the longest expected length.


Standard Deviation of Mean Critical Path (Step #3)

• The variance of the length of the path is the sum of the variances of all the activities on the path.

2path = ∑ all activities on path 2

• The standard deviation of the length of the path is the square root of the variance.

path path2


Probability Mean-Critical Path Completes by t (Step #4)

• What is the probability that the mean critical path (with expected length tpath and standard deviation path) has duration ≤ t?

• Use Normal Tables (Appendix A)

z

t (tpath)

path

+ +2 +3Ğ2Ğ3

Path duration

ProbabilityDensityFunction

tpath

t


Example

START0

FINISH

0a

b

c

d

e

2 - 3 - 4

2 - 4 - 5 3 - 4 - 6

1 - 3 - 7 2 - 3 - 8

Question: What is the probability that the project will be finished by day 12?


Expected Duration and Variance of Activities (Step #1)

Activity o m p

a 2 3 4 3.00 1/9

b 2 4 5 3.83 1/4

c 1 3 7 3.33 1

d 3 4 6 4.17 1/4

e 2 3 8 3.67 1

o 4m p6

2 p o6

2


Expected Length of Each Path (Step #2)

Path Expected Length of Path

a - b - d 3.00 + 3.83 + 4.17 = 11

a - c - e 3.00 + 3.33 + 3.67 = 10

The mean-critical path is a - b - d.


Standard Deviation of Mean-Critical Path (Step #3)

• The variance of the length of the path is the sum of the variances of all the activities on the path.

2path = ∑ all activities on path 2 = 1/9 + 1/4 + 1/4 = 0.61

• The standard deviation of the length of the path is the square root of the variance.

path path2 0.610.78


Probability Mean-Critical Path Completes by t=12 (Step #4)

• The probability that the mean critical path (with expected length 11 and standard deviation has duration ≤ 12?

• Then, from Normal Table: Prob(Project ≤ 12) = Prob(z ≤ 1.41) = 0.92

z

t (tpath)

path

12 110.71

1.41


Reliable Construction Project Network

A

START

G

H

M

F

J

K L

N

Activity Code

A. Excavate

B. Foundation

C. Rough wall

D. Roof

E. Exterior plumbing

F. Interior plumbing

G. Exterior siding

H. Exterior painting

I. Electrical work

J. Wallboard

K. Flooring

L. Interior painting

M. Exterior fixtures

N. Interior fixtures

2

4

10

746

7

9

5

8

4 5

6

2

0

0FINISH

D IE

C

B


Reliable Problem: Time Estimates for Reliable’s Project

Activity o m p Mean Variance

A 1 2 3 2 1/9

B 2 3.5 8 4 1

C 6 9 18 10 4

D 4 5.5 10 6 1

E 1 4.5 5 4 4/9

F 4 4 10 5 1

G 5 6.5 11 7 1

H 5 8 17 9 4

I 3 7.5 9 7 1

J 3 9 9 8 1

K 4 4 4 4 0

L 1 5.5 7 5 1

M 1 2 3 2 1/9

N 5 5.5 9 6 4/9


Pessimistic Path Lengths for Reliable’s Project

Path Pessimistic Length (Weeks)

StartA B C D G H M Finish 3 + 8 + 18 + 10 + 11 + 17 + 3 = 70

Start A B C E H M Finish 3 + 8 + 18 + 5 + 17 + 3 = 54

Start A B C E F J K N Finish 3 + 8 + 18 + 5 + 10 + 9 + 4 + 9 = 66

Start A B C E F J L N Finish 3 + 8 + 18 + 5 + 10 + 9 + 7 + 9 = 69

Start A B C I J K N Finish 3 + 8 + 18 + 9 + 9 + 4 + 9 = 60

Start A B C I J L N Finish 3 + 8 + 18 + 9 + 9 + 7 + 9 = 63


Three Simplifying Approximations of PERT/CPM

1. The mean critical path will turn out to be the longest path through the project network.

2. The durations of the activities on the mean critical path are statistically independent. Thus, the three estimates of the duration of an activity would never change after learning the durations of some of the other activities.

3. The form of the probability distribution of project duration is the normal distribution. By using simplifying approximations 1 and 2, there is some statistical theory (one version of the central limit theorem) that justifies this as being a reasonable approximation if the number of activities on the mean critical path is not too small.


Calculation of Project Mean and Variance

Activities on Mean Critical Path Mean Variance

A 2 1/9

B 4 1

C 10 4

E 4 4/9

F 5 1

J 8 1

L 5 1

N 6 4/9

Project duration p = 44 2p = 9


Spreadsheet for PERT Three-Estimate Approach

3

456789

101112131415161718

B C D E F G H I J KTime Estimates On Mean

Activity o m p Critical Path

A 1 2 3 * 2 0.1111 Mean CriticalB 2 3.5 8 * 4 1 PathC 6 9 18 * 10 4 44D 4 5.5 10 6 1 9E 1 4.5 5 * 4 0.4444F 4 4 10 * 5 1 P(T<=d) = 0.8413G 5 6.5 11 7 1 whereH 5 8 17 9 4 d = 47I 3 7.5 9 7 1J 3 9 9 * 8 1K 4 4 4 4 0L 1 5.5 7 * 5 1M 1 2 3 2 0.1111N 5 5.5 9 * 6 0.4444

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