Critical Path Method and Network Models

The CPM is a scheduling model, which like the Gantt chart, assumes that activ- ity durations and precedence relations are known. This method was developed in the late 1950s for the purpose of scheduling the construction of large chemi- cal plants. Unlike the Gantt model, the CPM does not assume that all activities should start as early as possible. Generally, this method is applied using a net- work project model, for example, in which each activity is represented by a net- work node and each precedence relation is represented by an arrow or an arc. Each arrow starts at the preceding activity and points to the succeeding activity.

CPM model assumptions are as follows: • All project activities are known.

• There are no resource or cost constraints.

• The duration of each activity is known and there is no uncertainty. Activity A B C D E F G 0 1 2 3 4 5 6 7 8 9 10 11 12 Duration in weeks FIGURE 3.1

Gantt chart for the example in Table 3.10.

The model focuses on the critical path of the project, which is the longest sequence of activities connecting the start of the project to its end. Using this method, the critical activities or activities that cannot be delayed without delaying the project are identified. Noncritical activities can be delayed with- out delaying the project.

Network CPM analysis is performed in two passes:

1. The forward pass: From the start of the project to its end, this pass calculates for each activity the earliest start time and the earliest fin- ish time, taking into account the precedence relations, which is a very similar analysis to the Gantt chart.

2. The backward pass: From the end of the project to its start, this pass calculates for each activity the latest finish time and the latest start time, taking into account the precedence relations.

The example used to illustrate the Gantt chart is used next to illustrate the CPM method. In Figure 3.2, each node (rectangle) represents an activity. The activity’s name and duration are presented inside the corresponding node.

Each arc or arrow represents the precedence relations between its starting node (the preceding activity) and its ending node (the succeeding activity). 3.3.3.1 CPM Analysis

For each activity, we calculate the earliest start time possible, without violat- ing the precedence constraints. This time is marked as the Early Start—ES. The earliest start time of all operations that have no predecessors is assumed to be zero. These activities, which may start at time zero, have a known duration and, their Earliest Finish—EF—is their duration. For each of the activities that have one or more predecessors, the largest EF of their prede- cessors becomes their early start (ES) and adding their duration to their ES, we get their Early Finish—EF. We continue in iterations of calculating ES and

A 2 C3 E1 G 1 F 3 D 5 B 3 FIGURE 3.2

Precedence diagram for the example of Table 3.10.

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Project Management

EF for the successor activities of the succeeding activities. The earliest start time of activities with predecessors is the latest earliest finish time of all its predecessors. The earliest finish time of every activity is equal to the earliest start time of the activity plus its duration.

If we assume that the project starts at time zero and the unit of time is a week of work (i.e., holidays or vacations are not present on the time line—the X axis), the resulting schedule is known as an ARO (after receiving an order) schedule. Using these simple rules, the ES and EF times of the ARO of each activity can be calculated as part of the forward pass:

ES (A) = 0, ES (B) = 0, ES (C) = 3, ES (D) = 3, ES (E) = 8, ES (F) = 8, ES (G) = 11

EF (A) = 2, EF (B) = 3, EF (C) = 6, EF (D) = 8, EF (E) = 9, EF (F) = 11, EF (G) = 12

Next, the backward pass from the end of the project to its beginning is performed. This time we first calculate the latest finish time of each activity, which is the latest time an activity can be finished without delaying the proj- ect, that is, the project still finishes in the earliest possible time. We denote the Late Finish as LF and Late Start as LS. The LS time of each activity is also calculated in the backward pass: the LS is the latest time an activity can start so that the project still finishes in the earliest possible time.

Based on the above assumptions, the LF of the last activity is equal to its EF. The late start LS of each activity is calculated by subtracting the duration of the activity from its late finish time. (This is the latest time that the activity can start without delaying the project.)

For all activities that have successor activities, the LF of each such activity is equal to the earliest LS of the succeeding activities.

Using these simple rules, the LS and LF times of the ARO of each activity can be calculated as part of the forward pass:

LS (A) = 1 LS (B) = 0 LS (C) = 5 LS (D) = 3 LS (E) = 10 LS (F) = 8 LS (G) = 11 LF (A) = 3 LF (B) = 3 LF (C) = 8 LF (D) = 8 LF (E) = 11 LF (F) = 11 LF (G) = 12 It is convenient to arrange the data of each activity as shown in Table 3.11. Using this convention for each activity and calculating the values yields a network model with all the data (Figure 3.3).

TABLE 3.11

The Convention of Depicting Values of ES, EF, LS, and LF

ES EF

LS LF