3 – 1Copyright © 2017 Pearson Education, Inc.

► CPM assumes we know a fixed time estimate for each activity and there is no variability in activity times

► PERT uses a probability distribution for activity times to allow for variability

Variability in Activity Times

3 – 2Copyright © 2017 Pearson Education, Inc.

► Three time estimates are required

► Optimistic time (a) – if everything goes according to plan

► Pessimistic time (b) – assuming very unfavorable conditions

► Most likely time (m) – most realistic estimate

Variability in Activity Times

3 – 3Copyright © 2017 Pearson Education, Inc.

Estimate follows beta distribution

Variability in Activity Times

Expected activity time:

Variance of activity completion times:

t = (a + 4m + b)/6

v = [(b – a)/6]2

3 – 4Copyright © 2017 Pearson Education, Inc.

Expected activity time:

Variance of activity completion times:

t = (a + 4m + b)/6

v = [(b – a)/6]2

Estimate follows beta distribution

Variability in Activity Times

t = (a + 4m + b)/6

v = [(b − a)/6]2Probability of 1 in 100 of > b occurring

Probability of 1 in 100 of < a occurring

P ro

b a

b ili

ty

Optimistic Time (a)

Most Likely Time (m)

Pessimistic Time (b)

Activity Time

Figure 3.11

3 – 5Copyright © 2017 Pearson Education, Inc.

Computing Variance

TABLE 3.4 Time Estimates (in weeks) for Milwaukee Paper's Project

ACTIVITY OPTIMISTIC

a

MOST

LIKELY m

PESSIMISTIC b

EXPECTED TIME t = (a + 4m + b)/6

VARIANCE

[(b – a)/6]2

A 1 2 3 2 .11

B 2 3 4 3 .11

C 1 2 3 2 .11

D 2 4 6 4 .44

E 1 4 7 4 1.00

F 1 2 9 3 1.78

G 3 4 11 5 1.78

H 1 2 3 2 .11

3 – 6Copyright © 2017 Pearson Education, Inc.

Probability of Project

Completion

Project variance is computed by summing the variances of critical activities

s2 = Project variance

= (variances of activities on critical path)

p

3 – 7Copyright © 2017 Pearson Education, Inc.

Probability of Project

Completion

Project variance is computed by summing the variances of critical activitiesProject variance

s2 = .11 + .11 + 1.00 + 1.78 + .11 = 3.11

Project standard deviation

sp = Project variance

= 3.11 = 1.76 weeks

p

3 – 8Copyright © 2017 Pearson Education, Inc.

Probability of Project

Completion

PERT makes two more assumptions:

► Total project completion times follow a normal probability distribution

► Activity times are statistically independent

3 – 9Copyright © 2017 Pearson Education, Inc.

Probability of Project

Completion

Standard deviation = 1.76 weeks

15 Weeks

(Expected Completion Time)

Figure 3.12

3 – 10Copyright © 2017 Pearson Education, Inc.

Probability of Project

Completion

What is the probability this project can be completed on or before the 16 week deadline?

Z = – /sp

= (16 weeks – 15 weeks)/1.76

= 0.57

Due Expected date date of completion

Where Z is the number of standard deviations the due

date or target date lies from the mean or expected date

3 – 11Copyright © 2017 Pearson Education, Inc.

Probability of Project

Completion

What is the probability this project can be completed on or before the 16 week deadline?

Z = − /sp

= (16 wks − 15 wks)/1.76

= 0.57

due expected date date of completion

Where Z is the number of standard deviations the due

date or target date lies from the mean or expected date

.00 .01 .07 .08

.1 .50000 .50399 .52790 .53188

.2 .53983 .54380 .56749 .57142

.5 .69146 .69497 .71566 .71904

.6 .72575 .72907 .74857 .75175

From Appendix I

3 – 12Copyright © 2017 Pearson Education, Inc.

Probability of Project

Completion

Time

Probability

(T ≤ 16 weeks)

is 71.57%

Figure 3.13

0.57 Standard deviations

15 16

Weeks Weeks

3 – 13Copyright © 2017 Pearson Education, Inc.

Determining Project

Completion Time

Probability of 0.01

Z

Figure 3.14

From Appendix I

Probability of 0.99

2.33 Standard deviations

0 2.33

3 – 14Copyright © 2017 Pearson Education, Inc.

Variability of Completion Time

for Noncritical Paths

► Variability of times for activities on noncritical paths must be considered when finding the probability of finishing in a specified time

► Variation in noncritical activity may cause change in critical path

3 – 15Copyright © 2017 Pearson Education, Inc.

What Project Management Has

Provided So Far

1. The project’s expected completion time is 15 weeks

2. There is a 71.57% chance the equipment will be in place by the 16 week deadline

3. Five activities (A, C, E, G, and H) are on the critical path

4. Three activities (B, D, F) are not on the critical path and have slack time

5. A detailed schedule is available

Sheet1

Sheet2

Sheet4

Sheet3

1 PERT Tutorial:

Suppose that you are a given the responsibility to manage a project and need to develop a budget forecast for the project, with the intention of submitting a budget request to your supervisor for approval. You have many line items in your forecast. These include deterministic items such as building rent, insurance, etc., and many probabilistic line items such as payroll, bonus payments, travel expenses, etc. Therefore, you must develop a budget based on a probabilistic approach and use statistics. Suppose your budget forecast is normally distributed, with a mean of $300,000, and a standard deviation of $10,000.

Figure 4-21

If you wanted to submit a budget for which you were 95% confident you would be able to make that budget, your calculations would be something like below.

zσ = x – μ

zσ + μ = x (1.645) ($10,000) + $300,000 = x

$16,450 + $300,000 = x $316,450 = x

Therefore, the budget submission would be for $316,450 – because there is a 95% probability that the project team will spend no more than that amount.

The question now becomes, “how did we get a normal distribution with a mean of $300,000 and a standard deviation of $10,000 in the first place?

Refer to the following spreadsheet. The first four columns show eight-line items of expenses (“Team

2

salaries,’ “Office rent,” “Travel expenses,” etc.). For each item, three forecasts are made for those expenses. The first, the “Optimistic” is the best-case scenario in terms of favorability of expenses (i.e., the least cost forecast if that situation arises). The second, “Most Likely,” is the realistic case scenario. The third, “Pessimistic,” represents the worst-case cost scenario. For “Team salaries,” the project manager believes that the size of the project team is probabilistic as some people may quit, the team may remain intact for the duration of the project, or unanticipated needs may arise and additional people may be hired. So, the least possible salary expense would be $222 thousand, the most probable expense would be $242 thousand, and the worst case situation would be $247 thousand.

Office rent is determined by contract, so it is deterministic. Each case of the three scenarios will be forecast at $9 thousand – the contracted amount.

All other line items are probabilistic, and their forecasts are so entered (in $thousands).

3

1 2 3 4 5 6 7

Expense Optimistic Most Likely Pessimistic (a + 4m + b)/6 (b-a)/6 Variance

Team salaries 222 242 247 239.50 4.17 17.36

Office rent 9 9 9 9.00 0.00 0.00

Travel expenses 8.4 18 17.9 16.38 1.58 2.51

Training expenses 0 6 9 5.50 1.50 2.25

IT maintenance share 3 4 4.7 3.95 0.28 0.08

Performance awards 0 19.6 49 21.23 8.17 66.69

Office supplies 0.1 1.27 1.4 1.10 0.22 0.05

Unplanned software 0 0 20 3.33 3.33 11.11

300.00

100.05 sum of the variances 10.00 square root of (sum of the variances)

Figure 4-22

PERT says that if the number of forecast line items is large (probably at least 30), then we are in the process of building a normal distribution curve of forecast costs. For this academic example, simulate that with just eight line items.

The next step is to calculate the mean of this curve. PERT says that the mean of each line item is the weighted average of the sum of the Optimistic value, 4 times the Most Likely value, and the Pessimistic value. This represents six weights, so divide that sum by 6 to obtain the weighted average. The formula is below.

(1*Optimistic + 4* Most Likely + 1 * Pessimistic) 6

This is often abbreviated as follows. (a + 4m + b)

6

For the first line item, “Team salaries,” the formula calculates as follows.

(1*222 + 4* 242 + 1 * 247) 6

or, 239.5. This is shown in the fifth column of Figure 4-22. All the rest of the calculations for the line items are in the fifth column. The total of these means is the mean of the budget normal distribution curve, or 300.00 – shown at the bottom of the fifth column.

To calculate the standard deviation of this curve, several steps are necessary. When forecasting the Optimistic and Pessimistic values for each line item, ensure that you are at least 99% sure that the final value will be within this range. In the “Team salaries” line item example, the forecaster is over 99%

4

certain that the actual amount spent on “Team salaries” will be within the range of $222 and $247 thous

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