# Calculation of Time Estimates in CPM

In the project network given in figure below, activities and their durations are specified at the activities. Find the critical path and the project duration.

Calculations in Network Analysis

The following calculations are required in network analysis in order to prepare a schedule of the project.

1. Total completion time of the project
2. Earliest time when each activity can start (i.e. earlist start time)
3. Earliest time when each activity can finish (i.e. earlist finished time)
4. Latest time when each activity can be started without delaying the project (i.e. latest start time)
5. Latest time when each activity can be finished without delaying the project (i.e. latest finish time)
6. Float on each activity (i.e. time by which the completion of an activity can be delayed without delaying the project)
7. Critical activity and critical path

The symbols used in the calculations are shown in table below.

 Symbol Description Ei Earliest occurance time of event i Lj Latest allowable occurance time of event j tEi-j Estimated completion time of activity (i,j) (EST)ij Earliest starting time of activity (i,j) (EFT)ij Earliest finishing time of activity (i,j) (LST)ij Latest starting time of activity (i,j) (LFT)ij Latest finishing time of activity (i,j)

The computations are made in following steps.

(a) Forward Pass Computations :

(b) Backward Pass Computations :

(c) Calculation of Slack:

Event slack is defined as the difference between the latest event and earlist event times.

Slack for head event = Lj - Ej

Slack for tail event = Li - Ei

The calculations for the above taken example network are summarised below in the table.

 Predecessor Event i Successor    Event j tEi-j (EST)ij (EFT)ij (LST)ij (LFT)ij S(i) Slack 5 10 7 0 7 0 7 0 5 15 12 0 12 7 19 - 5 20 17 0 17 5 22 - 10 20 15 7 22 7 22 0 10 25 9 7 16 21 30 - 15 30 11 12 23 19 30 7 20 25 5 22 27 25 30 - 20 30 8 22 30 22 30 0 25 35 10 27 37 30 40 3 25 45 15 27 42 35 50 - 30 35 10 30 40 30 40 0 30 40 8 30 38 35 43 - 35 45 10 40 50 40 50 0 40 45 7 38 45 43 50 5

(d) Determination of Critical Path:

The sequance of critical activities in a network is called the critical path. The activities with zero slack of head event and zero slack for tail event, are called as crititcal activities. In the taken network, the following activities are critical activities: 5 - 10, 10 - 20, 20 - 30, 30 - 35, 35 - 45.
Thus the critical path is A - E - G - K - M.
Critical path duration is 7 + 15 + 8 + 10 + 10 = 50.

# Calculation of Expected Time and Variance of a Path in PERT

The Expected Time of a chain of activities in series, is the sum of their expected times. Similarly the variance of the path, is the sum of variances of activities on the path. In Figure below, three activities A,B and C are connected in series, (i.e. form a path). Their time estimates to-tm-tpare given along the activity arrows. The expected time of the path 1-2-3-4 is calculated as:

As the length of the path ,that is the number of activies connected in series increases,the variance of the path and hence the uncertainty of meeting the expected time also increases.

An Example

In the network of figure below, the PERT time estimates of the activities are written along the activity arrows in the order to-tm-tp. Compute the expected time and variance for each activity. Also compute the expected duration and standard deviation for the following paths of the network.

(a) 10-20-50-80-90

(b) 10-30-50-70-90

(c) 10-40-60-80-90

The computation of expected times and variances for different activities are carried in a table given below.

 Activity i              j Time Estimates to           tm          tp Expected Time tE Variance σ2 10           20 6             9           12 9.00 1.00 10           30 3             5            9 5.33 1.00 10           40 10         14           18 14.00 1.78 20           50 7           10           13 10.00 1.00 20           70 3             4             8 4.5 0.69 30           50 4           10           12 9.33 1.78 40           50 8           11           14 11.00 1.00 40           60 5          10           15 10.00 2.78 50           70 3            4            5 4.00 0.11 50           80 11           15          17 14.67 1.00 60           80 7             9           12 9.17 0.69 70           90 4             8            10 7.67 1.00 80           90 6             7           9 7.17 0.25

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