CPM Model and CPM network components
PERT Model
So far, the analysis was focused on the determination of the critical path, event slacks, and activity floats. For this purpose we used single time estimates of activity duration though initially threetime estimates were developed for each activity. Now we consider the variability of project duration.
Measures of Variability
Variability in PERT analysis is measured by variance or its square root, standard deviation. Variance of a set of numbers is the average squared difference of the numbers in the set from their arithmetic average. A simple example may be given to illustrate the calculation of variance. Let a series consist of numbers 4, 6 and 8. The average of this series is 6. The differences of various numbers in the series from this average are 2, 0, and 2. Squaring them we get 4, 0 and 4. Hence, variance, the average
of squared difference, is 8/3 and standard deviation is 8/3 .
The steps involved in calculating the standard deviation of the duration of critical path are as follows:
1. Determine the standard deviation of the duration of each activity on the critical path.
2. Determine the standard deviation of the total duration of the critical path on the basis of information obtained in step 1.
For determining the standard deviation of the duration of an activity we require the entire probability distribution of the activity distribution. We, however, have only three values from this distribution: tp, tm, and to. In PERT analysis, a simplification is used in 2 calculating the standard deviation. It is estimated by the formula
Standard deviation = (tp – to) / 6
where, tp = pessimistic time
to = optimistic time
Variance is obtained by squaring standard deviation.
Activity

tp

to

O=tpt0/6

Variance=0^{2}

(12)

21

9

2

4.00






(2.5)

24

10

2.33

5.43






Assuming that the probability distribution of various activities on the critical path is independent, the variance of the critical path duration is obtained by adding variance of activities on the critical path.
For real life projects which have a large number of activities on the critical path we can reasonably assume that the critical path duration is approximately normally distributed, with mean and standard deviation obtained by the method described above.
A normal distribution looks like a bellshaped curve as shown in Figure 5.13. It is symmetric and single peaked and is fully described by its mean and standard deviation. The probability of values lying within certain ranges is as follows:
Range

Probability

Mean ± One standard deviation

0.682

Mean ± Two standard deviations

0.954

Mean ± Three standard deviations

0.998

CPM Model
The PERT model was developed for projects characterized by uncertainty and the CPM model was developed for projects which are relatively riskfree. While both the approaches begin with the development of the network and a focus on the critical path, the PERT approach is ‘probabilistic’ and the CPM approach is ‘deterministic’. This does not, however, mean that in CPM analysis we work with single time estimates. In fact, the principal focus of CPM analysis is on variations in activity times as a result of changes in resource assignments. These variations are planned and related to resource assignments and are not caused by random factors beyond the control of management as in the case of PERT analysis. The main thrust of CPM analysis is on time cost relationships and it seeks to determine the project schedule which minimizes total cost.
Assumptions
The Usual assumptions underlying CPM analysis are:
The costs associated with a project can be divided into two components: direct costs and indirect costs. Direct costs are incurred on direct material and direct labour. Indirect costs consist of overhead items like indirect supplies, rent, insurance, managerial services, etc.
Activities of the project can be expedited by crashing which involves employing more resources.
Crashing reduces time but enhances direct costs because of factors like overtime payments, extra payments, and wastage. The relationship between time and direct activity cost can be reasonably approximated by a downward sloping straight line.
Procedure
Given the above assumptions, CPM analysis seeks to examine the consequences of crashing on total cost (direct cost plus indirect cost). Since the behaviour of indirect project cost is well defined, the bulk of CPM analysis is concerned with the relationship between total direct cost and project duration. The procedure used in this respect is generally as follows:
1. Obtain the critical path in the normal network. Determine the project duration and direct cost.
2. Examine the costtime slope of activities on the critical path obtained and crash the activity which has the least slope.
3. Construct the new critical path after crashing as per step 2. Determine project duration and cost.
4. Repeat steps 2 and 3 till activities on the critical path (which may change every time) are crashed.