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 The CPM Procedure

## Example 2.21: PERT Assumptions and Calculations

This example illustrates the PERT statistical approach. Throughout this chapter, it has been assumed that the activity duration times are precise values determined uniquely. In practice, however, each activity is subject to a number of chance sources of variation and it is impossible to know, apriori, the duration of the activity. The PERT statistical approach is used to include uncertainty about durations in scheduling. For a detailed discussion about various assumptions, techniques, and cautions related to the PERT approach, refer to Moder, Phillips, and Davis (1983) and Elmaghraby (1977). A simple model is used here to illustrate how PROC CPM can incorporate some of these ideas. A more detailed example can be found in SAS/OR Software: Project Management Examples.

Consider the widget manufacturing example. To perform PERT analysis, you need to provide three estimates of activity duration: a pessimistic estimate (tp), an optimistic estimate (to), and a modal estimate (tm). These three estimates are used to obtain a weighted average that is assumed to be a reasonable estimate of the activity duration. Note that the time estimates for the activities must be independent for the analysis to be considered valid. Furthermore, the distribution of activity duration times is purely hypothetical, as no statistical sampling is likely to be feasible on projects of a unique nature to be accomplished at some indeterminate time in the future. Often, the time estimates used are based on past experience with similar projects.

To derive the formula for the mean, you must assume some functional form for the unknown distribution. The well-known Beta distribution is commonly used, as it has the desirable properties of being contained inside a finite interval and can be symmetric or skewed, depending on the location of the mode relative to the optimistic and pessimistic estimates. A linear approximation of the exact formula for the mean of the beta distribution weights the three time estimates as follows:

(tp + (4*tm) + to) / 6

The following program saves the network (AOA format) from Example 2.2 with three estimates of activity durations in a SAS data set. The DATA step also calculates the weighted average duration for each activity. Following the DATA step, PROC CPM is invoked to produce the schedule plotted on a Gantt chart in Output 2.21.1. The E_FINISH time for the final activity in the project contains the mean project completion time based on the duration estimates that are used.

title 'PERT Assumptions and Calculations';
/* Activity-on-Arc representation of the project
with three duration estimates */
data widgpert;
dur = (tp + 4*tm + to) / 6;
datalines;
Approve Plan    1   2    5   7   3
Drawings        2   3   10  11   6
Anal. Market    2   4    5   7   3
Write Specs     2   3    5   7   3
Prototype       3   5   15  12   9
Mkt. Strat.     4   6   10  11   9
Materials       5   7   10  12   8
Facility        5   7   10  11   9
Init. Prod.     7   8   10  12   8
Evaluate        8   9    9  13   8
Test Market     6   9   14  15  13
Changes         9  10    5   6   4
Production     10  11    0   0   0
Marketing       6  12    0   0   0
Dummy           8   6    0   0   0
;

proc cpm data=widgpert out=sched
date='2dec91'd;
tailnode tail;
duration dur;
run;

proc sort;
by e_start;
run;

goptions vpos=50 hpos=80 border;
proc gantt graphics data=sched;
font=swiss height=1.5 nojobnum skip=2
dur=dur increment=7 nolegend
cframe=ligr;