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Project Management

Once a problem statement is finalized, it is broken into deliverables that are further broken down to work tasks. Work tasks are grouped into higher level project milestones. Milestones are control points for the project. An objective of a project team is to time phase the milestones and work tasks using the work breakdown structure method. This method creates a list of sequential and well-defined work tasks that are aggregated hierarchically into milestones for project management and reporting. Each work task or activity has a starting point and an ending point, a measurable output, and a defined resource requirement and time duration.

A Gantt chart and Project Evaluation and Review Technique (PERT) model of a project’s work tasks can be represented as a network that describes activity sequence and relationships. Figure 12.2 shows an example. This network has six activities spatially arranged into two parallel paths. For simplicity, we will not go deeper into the work task level of detail, but instead demonstrate how to build a network model using the work breakdown structure and the Gantt chart, calculating the critical path, and estimating the probability of completing this project example on schedule.

The analysis starts by estimating the time to complete each activity within the network. In the absence of historical or current lead time data


Developing a work breakdown structure.

for each activity, the team estimates the “most optimistic,” “expected,” and “most pessimistic” times to complete each activity based on the team’s collective knowledge. Depending on the required analytical detail, activities can be disaggregated into work tasks. Each activity in Figure 12.2


Gantt chart.

has optimistic, average and pessimistic estimates to complete it. These are used to calculate expected completion times and their variance. The expected completion time of the first activity is 5.17 days and its variance is 2.25 days. Calculations for the other five activities are summarized in Figure 12.3 with related statistics.

The work breakdown structure is shown in the Gantt chart in Figure 12.4, where the parallel paths of the network are clearly shown. The network’s critical path is through operations A,B,C, and D, and the calculated expected project completion time for these four operations is 23.5 days. The total time required to complete all six operations is 44.0 (rounded) days. This implies that operations 5 and 6 have slack or extra time available on their parallel path.


PERT and critical path summary statistics by activity.

Table 12.2 calculates the critical path by estimating the earliest starting time (ES), earliest finishing time (EF), latest starting time (LS), and latest finishing time (LF) of an activity. In complicated networks, these statistics are usually calculated using software. First, take the expected completion time of each activity and make a forward pass through the network to estimate the ES and EF statistics for each activity. This is shown in Step 2 of Table 12.2. Once the earliest finish time (EF) has been calculated for the last operation or activity of a project network (in this example, 23.5 days), a backward pass is made through the network. This backward pass is used to help calculate the LS and LF statistics as described in Step 3 of Table 12.2. The slack time of each activity is calculated as described in Step 4 of Table 12.2. An activity with zero slack must be started and finished on time and is on the critical path. The statistics described in Table 12.2 are summarized in Figure 12.4, where they are used to find the probability of completing the project in 20 days or less. Other completion (or lead) time targets can also be calculated. In Table 12.3, a completion probability of 34% is calculated using a hypothetical lead time of 20 days or less and assuming the distribution of completion times is normal. An interval can also be calculated assuming the expected time is 23.5 days

TABLE 12.2

Calculating a Critical Path


1. A “path” is a linked sequence of work tasks within a network beginning with a “start” node and ending with an “stop” node. The critical path is the longest sequence of network work tasks.

2. Earliest Start time (ES) and Earliest Finish time (EF) for a work task is defined by “EF = ES +1,” where “t” is the duration of the work task.

3. Latest Start time (LS) and Earliest Start time (ES) for a work task is defined by the relation “LS = LF-t.”

4. Slack time for a work task is designed by the relation “LS-ES = LF-EF.”

5. Make a forward pass through the network calculating the EF and LF times for every work task.

6. Make a backward pass through the network using the total calculated completion time as the LF time.

7. The critical path is defined as the sequence of work tasks having zero slack time as defined in Step 4 above.

TABLE 12.3

Statistics for Critical Path Operations


1. Expected completion time (t) = 5.17 + 5.17 + 8.00 + 5.17 = 23.5 days

2. Variance of completion time = 2.25 + 0.69 + 2.78 + 2.25 = 7.97 days

3. Probability of completing project in 20 days = probability Z < (20-3.5) / 7.97) = -0.41 = 34%

with a 7.97 day variance. The 95% interval based on a Normal distribution assumption is 23.5 ± (1.96) (2.83 days) = 23.5± 5.5 days = 18 to 29 days.

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