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PERT differs from CPM in that it bases the duration of an activity on three estimates:

**PERT ****Networks**

PERT
differs from CPM in that it bases the duration of an activity on three
estimates:

**a.
****Optimistic time, a,** which occurs when execution goes extremely welL

**b.
Most likely time, m,**

**c.
Pessimistic time, b,**

The range
*(a,* *b)* encloses all possible estimates
of the duration of an activity. The estimate *m* lies somewhere in the range *(a,* *b).* Based on
the estimates, the average duration time, Bar(*D),* and variance, *v,* are approximated as:

CPM
calculations given in Sections 6.5.2 and 6.5.3 may be applied directly, with Bar(D)
replacing the single estimate D.

It is
possible now to estimate the probability that a node *j* in the
network will occur by a prespecified scheduled time, *s _{j}.* Let ej
be the earliest occurrence time of node j. Because
the durations of the activities leading from the start node to node

Once the
mean and variance of the path to node *j,* *E{* *ej}* and var{*ej},* have been computed, the probability
that node *j* will be realized by a preset time *s _{j}* is calculated using the
following formula:

The
standard normal variable *z* has mean 0
and standard deviation 1 (see Section 12.4.4). Justification for the use of the
normal distribution is that *ej* is the
sum of independent random variables. According to the *central limit theorem* (see Section 12.4.4), *e _{j}* is approximately normally distributed.

**Example **6.5-6

Consider
the project of Example 6.5-2. To avoid repeating critical path calculations,
the values of *a, m, *and* b *in the table below are selected such
that* Bar(D*_{ij}* ) *=* D*_{ij}* *for all* i *and* *j* *in Example 6.5-2.

The mean Bar(*D** _{ij}*) and
variance

The next
table gives the longest path from node 1 to the different nodes, together with
their associated mean and standard deviation.

Finally,
the following table computes the probability that each node is realized by time
*s _{j}*
specified by the analyst.

**PROBLEM
SET 6.5E**

Consider Problem 2, Set 6.Sb. The estimates *(a, m,* *b)* are
listed below. Determine the probabilities that the different nodes of the
project will be realized without delay.

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