Scheduling is one of the areas that received considerable attention from researchers as well as practitioners in all types of applications including operations scheduling and project scheduling. Techniques are developed to develop optimum or near optimal schedules with respect to different possible performance measures. This chapter highlights some of these techniques and their application in maintenance scheduling.
Program Evaluation Review Techniques (PERT)
Maintenance activities are usually unique and commonly involve unexpected needs that make their time duration highly uncertain. CPM uses a single estimate of the time duration based on the judgment of a person. PERT, on the other hand, incorporates the uncertainty by three time estimates of the same activity to form a probabilistic description of their time requirement. Even though the three time estimates are judgmental they provide more information about the activity that can be used for probabilistic modeling. The three values are represented as follows:
Oi = optimistic time, which is the time required if execution goes extremely well;
Pi = pessimistic time, which is the time required under the worst conditions;
mi = most likely time, which is the time required under normal condition.
The activity duration is modeled using a beta distribution with mean ( ) and variance ( ) for each activity i estimated from the three points as follows:
Estimated means are then used to find the critical path in the same way of the CPM method. In PERT, the total time of the critical path is a random variable with a value that is unknown in advance. However, additional probabilistic analysis can be conducted regarding possible project durations based on the assumption that the total time of the project may be approximated by a normal probability
distribution with mean and variance 2 estimated as
Using the above approximation we can calculate the probability with which a project can be completed in any time duration, T, using the normal distribution as follows:
Where Φ is the distribution function of the standard normal distribution.
Tables exist for evaluating any probability under the standard normal distribution. To illustrate the PERT analysis, consider the previous example with additional time estimates shown in Table 11.7 below.
Table 11.7. The PERT calculation for the bearing overhaul example
The critical path calculations lead to the same critical path obtained in the previous CPM calculations. The total project time is expected to be 467 min. The estimated variance is 213.37 min. The probability that the project will complete in
467 min can be calculated from the standard normal distribution to be
0.5, or the project has a 50% chance of completing in 467 min.
The probability that the project may finish in 500 min can be calculated as: