Mathematical analysis is used in project schedule development to determine early and late start dates, as well as early and late finish dates for all project activities. The outcome indicates the time period in which the activity should be scheduled. Note that this analysis phase does not take into account any resource pool limitations or constraints.
The most widely known mathematical techniques used by project management teams are the: Critical Path Method (CPM), Program Evaluation and Review Technique (PERT), and Graphical Evaluation and Review Technique (GERT).
Critical Path Method (CPM)
The most common mathematical technique is the Critical Path Method (CPM). The CPM is used to predict project duration by analyzing which sequence of activities, or path, has the least amount of scheduling flexibility.
Once you have determined the early and late start and finish dates, you can determine float. Float is equal to the difference between the late finish and early finish dates, or the difference between the late start and early start dates.
The next step in the CPM is to determine the critical path (CP), which is the longest path for the project that has little or no float. To determine the critical path, you begin with the first activity in the network. Look at its successors, compare the successors' float values, and select the one with zero float. This is the second activity on the critical path.
Next, you would continue from the second activity on the critical path and compare float for its successors, selecting the activity that has zero float and including it in the critical path.
You continue this process to the final activity for a complete critical path. The project can finish no sooner than the time it takes to complete the activities on the critical path.
To calculate an activity's duration, you subtract the early start from the early finish or the late start from the late finish. In example that follows, the numbers indicate days.
- activity A - 1 day
- activity B - 2 days
- activity C - 3 days
- activity D - 2 days
Critical Path activities are, indeed, critical to a project's success. They need management's careful attention. The order and duration of these activities are important because any delays will result in the project going over the anticipated completion date. In addition, project improvements are most effective when made along the critical path.
Program Evaluation and Review Technique (PERT)
Have you ever performed activity duration estimates, then questioned your findings? There is a technique available for checking your findings.
Program Evaluation and Review Technique (PERT) is used when there is a high level of uncertainty about how long it will take to perform a given task.
PERT uses network logic—the collection of activity dependencies that make up a project network diagram—to determine duration. In PERT, network logic is used by applying the critical path method to a weighted average duration estimate.
Although very similar, there is one significant difference between PERT and CPM. CPM uses the most likely estimate instead of the expected value of the estimate that PERT uses.
PERT time estimating requires the following three estimates for each activity.
- TM = most likely time
- TO = optimistic time
- TP = pessimistic time
Once you have calculated the estimated times for your project you can plot those values on an s-curve. The s-curve allows you to easily see all three times—optimistic, most likely, and pessimistic.
Graphical Evaluation and Review Technique (GERT)
There is one additional mathematical analysis method that is rarely used today because it has been proven to be less accurate than PERT and CPM. This method is the graphical evaluation and review technique (GERT). GERT allows for probabilistic treatment of both network logic and activity duration estimates. GERT is mainly used on project activities that are only performed in part, as well as those activities that may be performed more than once (loop). The above graphic illustrates a GERT diagram with a simple loop.
For example, on a high-rise development project, the electrical outlets for each floor may be installed as each floor is completed instead of waiting for the completion of the entire building. Since this activity will be performed more than once, using GERT will enable you to calculate the entire duration of this activity.
Regardless of the type of mathematical analysis you apply to your projects, the ultimate objective is to produce a schedule with realistic start and finish dates.