and 
represent the number
of days we decrease the time needed for excavation, foundation, walling,
and siding, respectively, then we get the L.P.:
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PERT/CPM
If we let new variables
and represent the number
of days we decrease the time needed for excavation, foundation, walling,
and siding, respectively, then we get the L.P.:
So far we have assumed that reasonably accurate estimates can be made of the time required for each activity in the project network. In reality, ther is frequently some uncertainty about the time an activity can take.
In a PERT network, this uncertainty is summed up in three numbers about each activity: the most likely value for the duration (m), a pessimistic value (b) and an optimistic value (a). PERT then fits a particular type of probability distribution to these values. This distribution (the beta distribution) assumes that the range from a and b encompasses 6 standard deviations (3 on either side of the mean). The mean itself is calculated as
and the variance:
Based on these values, PERT will use an activity network to calculate a mean finishing time along with a variance about that finishing time. There are two critical assumptions: the times for the activities are independent of each other, and the critical path identified is always the longest path in the network, no matter how the activity lengths turn out.
With these assumptions, you can solve a PERT network as follows. Find a critical path using the CPM method with the mean activity times on the arcs. This gives the mean finishing time. The variance of the finishing time is simply the sum of the variances of the activities on the critical path. The overall finishing value is assumed to be normally distributed, so quantiles are based on the normal distribution.
PERT allows you to answer such questions as:
