## Evaluating risks to the schedule

We have seen that not all risks can be eliminated - even those that are classified as avoidable or manageable can, in the event, still cause problems affecting activity durations. By identifying and categorizing those risks, and in particular, their likely effects on the duration of planned activities, we can assess what impact they are likely to have on our activity plan.

We will now take a look at two methods for assessing the effects of these uncertainties on the project schedule.

### Using PERT to evaluate the effects of uncertainty

PERT was developed to take account of the uncertainty surrounding estimates of task durations. It was developed in an environment of expensive, high-risk and

PERT (program evaluation and review technique) was published in the same year as CPM. Developed for the Fleet Ballistic Missiles Program it is said to have saved considerable time in development of the Polaris missile.

state-of-the-art projects - not that dissimilar to many of today's large software projects.

The method is very similar to the CPM technique (indeed many practitioners use the terms PERT and CPM interchangeably) but, instead of using a single estimate for the duration of each task, PERT requires three estimates.

• Most likely time - the time we would expect the task to take under normal circumstances. We shall denote this by the letter m.

• Optimistic time - the shortest time in which we could expect to complete the activity, barring outright miracles. We shall use the letter a to denote this.

• Pessimistic time - the worst possible time allowing for all reasonable eventualities but excluding 'acts of God and warfare' (as they say in most insurance exclusion clauses). We shall denote this by b.

PERT then combines these three estimates to form a single expected duration, fe, using the formula

Exercise 7.4 Table 7.3 provides additional activity duration estimates for the network shown in

Figure 6.17. There are new estimates for a and b and the original activity duration estimates have been used as the most likely times, m. Calculate the expected duration, te, for each activity.

### Using expected durations

The expected durations are used to carry out a forward pass through a network; using the same method as the CPM technique. In this case, however, the calculated event dates are not the earliest possible dates but are the dates by which we expect to achieve those events.

Exercise 7.5 Before reading further, use your calculated expected activity durations to carry out a forward pass through the network (Figure 6.17) and verify that the project duration is 13.5 weeks.

What does an expected duration of 13.5 weeks mean in terms of the completion date for the project?

The PERT network illustrated in Figure 7.3 indicates that we expect the project to take 13.5 weeks - unlike CPM, this does not indicate the earliest date by which we could complete the project but the expected (or most likely) date. An advantage of this approach is that it places an emphasis on the uncertainty of the real world.

Table 7.3 PERT activity time estimates r

Activity durations (weeks) Activity Optimistic (a) Most likely (m) Pessimistic (b)

 A 5 6 8 B 3 4 5 C 2 3 3 D 3.5 4 5 E I 3 4 F 8 10 15 G 2 3 4 H 2 2 2.5

Rather than being tempted to say 'the completion date for the project is we are lead to say 'we expect to complete the project by

It also focuses attention on the uncertainty of the estimation of activity durations. Requesting three estimates for each activity emphasizes the fact that we are not certain what will happen - we are forced to take into account the fact that estimates are approximate.

 ( 2 >6.17
 f 1 t= 4.00 w r s I 0 J-^ 14 \„ 10.5 t =

Y 4

V

. 9

J

V *

 I 5 1^10.5 )
 ( 6 ^Jl 3.5 J
 f Event number Target^ date Expected V date Standard deviation/

The PERT event labelling convention adopted here indicates event number and its target date along with the calculated values for expected time and standard deviation.

Figure 7.3 The PERT network after the forward pass.

### Activity standard deviations

This standard deviation formula is based on the rationale that there are approximately six standard deviations between the extreme tails of many statistical distributions.

A quantitative measure of the degree of uncertainty of an activity duration estimate may be obtained by calculating the standard deviation s of an activity time, using the formula b-a

The activity standard deviation is proportional to the difference between the optimistic and pessimistic estimates, and can be used as a ranking measure of the degree of uncertainty or risk for each activity. The activity expected durations and standard deviations for our sample project are shown in Table 7.4.

Table 7.4 Expected times and standard deviations

Activity durations (weeks

 Activity Optimistic (a) Most likely (m) CJ Standard deviation (s) A 5 6 8 6.17 0.50 B 3 4 5 4.00 0.33 C 2 3 3 2.83 0.17 D 3.5 4 5 4.08 0.25 E 1 3 4 2.83 0.50 F 8 10 15 10.50 1.17 G 2 3 4 3.00 0.33 H 2 2 2.5 2.08 0.08

The likelihood of meeting targets

The main advantage of the PERT technique is that it provides a method for estimating the probability of meeting or missing target dates. There might be only a single target date - the project completion - but we might wish to set additional intermediate targets.

Suppose that we must complete the project within 15 weeks at the outside. We expect it will take 13.5 weeks but it could take more or, perhaps, less. In addition, suppose that activity C must be completed by week 10, as it is to be carried out by a member of staff who is scheduled to be working on another project and that event 5 represents the delivery of intermediate products to the customer. These three target dates are shown on the PERT network in Figure 7.4.

The PERT technique uses the following three-step method for calculating the probability of meeting or missing a target date:

• calculate the standard deviation of each project event;

• calculate the z value for each event that has a target date;

• convert z values to a probabilities.

Calculating the standard deviation of each project event

Standard deviations for the project events can be calculated by carrying out a forward pass using the activity standard deviations in a manner similar to that used with expected durations. There is, however, one small difference - to add two standard deviations we must add their squares and then find the square root of the sum. Exercise 7.5 illustrates the technique.

The square of the standard deviation is known as the variance. Standard deviations may not be added together but variances may.

The standard deviation for event 3 depends solely on that of activity B. The Exercise 7.6 standard deviation for event 3 is therefore 0.33.

For event 5 there are two possible paths, B + E or F. The total standard deviation for path B + E is V(0.332 + 0.502) = 0.6 and that for path F is 1.17; the standard deviation for event 5 is therefore the greater of the two, 1.17.

Verify that the standard deviations for each of the other events in the project are as shown in Figure 7.4.

 r 2 ^6.17 0.50j

f 1

t = 4.00

I 0

0 J

 f 0 "N L> 0 / - 4.08 \ 4 0.33 J s = 0.25
 f 4 10 > t=' I9 0.53y s =
 1 5 io->| 0.5 1.17J
 ( 6 15>| k13-5 1.22J

Figure 7.4 The PERT network with three target dates and calculated event standard deviations.