Critical Path Method (CPM)

CPM is a networking methodology, which is employed to find the minimum time required to complete the project and at the same time identify the sequence of tasks to be performed. CPM is used in identifying the precedence relationship among various tasks of a project. CPM helps project managers track various activities of the project and it enables the identification of the critical activities of the project.

Table: Activities, their Preceding Activities, and Durations  

Problem 

Based on the information given in Table, identify the critical path and the total duration of the project.

Solution

To identify the critical path, we have to find the earliest and latest times. The network diagram is shown in the Figure. The steps involved in identifying the critical path are described below:

Figure: Network Diagram for Problem  

Computation of earliest time (Forward Pass)

Forward pass computation is employed to find the total project time.

Using forward pass computation, the earliest start and earliest finish time for each activity is identified.

The earliest start time for an activity (i,j) is given by the earliest expected occurrence time of the event i. For instance, earliest start time for the activity 1-2 would be earliest expected occurrence time for event 1.   

The earliest event time for the starting event is taken as zero. The earliest start time for an event is given by the earliest starting time plus activity duration of the preceding activity.

Therefore the earliest event time for event 1 in the network be 0 or E1 = 0

    E2 = E1+D1-2 = 0+2 = 2

    E3 = E1+D1-3= 0+2 = 2

    E4 = E1+D1-4= 0+2 = 2

    E5 = E2+D2-5= 2+4 = 6

Note, two activities converge at node 6 i.e., activity 6 can begin only when both 3-6 and 4-6 activities are completed. In this case, the earliest event time of 6 is given by E6 = max [E3+D3-6, E4+D4-6] = 10

Similarly,

    E7 = max [E5+D5-7, E6+D6-7] = 15

    E8 = E7+D7-8 = 15+4 = 19

Next, we will find the EFT or earliest finish time, which is determined by adding activity duration and EST (Refer Table). 

Table 20.2: Calculation of Earliest and Latest Time for an Activity

Computation of latest times: backward pass

With forward pass, we can identify the duration of the project. Backward pass is used to find the float for an activity, which helps in determining the critical path. Backward pass begins from the end node, and proceeds towards the first node. The latest finish time for the last activity is equal to the earliest finish time of the last activity.

So,

    L8 = E8 = 19

    L7 = L8-D7-8 = 19-4 = 15

    L5 = L7-D5-7 = 15-2 = 13

    L6 = L7-D6-7 = 15-5 = 10

    L4 = L6-D4-6 = 10-4 = 6

    L3 = L6-D3-6 = 10-8 = 2

    L2 = L5-D2-5 = 13-4 = 9

As three activities are originating from A, LFT is the minimum of previous LSTs. 

L1 = min [L2-D1-2, L3-D1-3, L4-D1-4]

= min [7, 0, 4] = 0

If the L1 value is not equal to 0, then there is an error in the forward or backward pass calculation.

Critical paths can be identified by calculating the float or slack values, as shown in Table. The float value of each activity can be calculated by using either starting time or finishing time. The activities with zero float are part of the critical path. In this network, the critical path includes activities 1-3, 3-6, 6-7, 7-8 and the total duration of the project is 19 days. 

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