| Chapter Contents | Previous | Next |
The CPM Procedure
This example illustrates the PERT statistical approach. Throughoutthis chapter, it has been assumed that the activity duration timesare precise values determined uniquely. In practice, however,each activity is subject to a number of chance sources ofvariation and it is impossible to know, apriori,the duration of the activity. The PERTstatistical approach is used to include uncertainty aboutdurations in scheduling. For a detailed discussion about variousassumptions, techniques, and cautions related to the PERT approach,refer to Moder, Phillips, andDavis (1983) and Elmaghraby (1977).A simple model is used here to illustrate howPROC CPM can incorporate some of these ideas. A more detailedexample can be found in SAS/OR Software: Project Management Examples.
Consider the widget manufacturing example. To perform PERTanalysis, 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 usedto obtain a weighted average that is assumed to be a reasonableestimate of the activity duration.Note that the time estimates for theactivities must beindependent for the analysis to be considered valid.Furthermore, the distribution of activity duration timesis purely hypothetical,as no statistical sampling is likely to be feasible on projectsof a unique nature to be accomplished at some indeterminate timein the future. Often, the time estimates used are based on pastexperience with similar projects.
To derive the formula for the mean, you must assume some functional form for the unknown distribution. Thewell-known Beta distribution is commonly used, as it hasthe desirable properties of being contained inside a finiteinterval and can be symmetric or skewed, depending on thelocation of the mode relative to the optimistic andpessimistic estimates. A linear approximation of the exactformula for the mean of the beta distribution weights thethree time estimates as follows:
(tp + (4*tm) + to) / 6
The following program saves the network (AOA format) from Example2.2 with three estimates of activity durations in a SAS data set. TheDATA step also calculates the weighted average duration for eachactivity. Following the DATA step, PROC CPM is invokedto produce the schedule plotted on a Gantt chart inOutput 2.21.1. The E_FINISH time for the final activity in theprojectcontains the mean project completion time based on the durationestimates that are used.
title 'PERT Assumptions and Calculations'; /* Activity-on-Arc representation of the project with three duration estimates */ data widgpert; input task $ 1-12 tail head tm tp to; 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; headnode head; duration dur; id task; run; proc sort; by e_start; run; goptions vpos=50 hpos=80 border; proc gantt graphics data=sched; chart / compress tailnode=tail headnode=head font=swiss height=1.5 nojobnum skip=2 dur=dur increment=7 nolegend cframe=ligr; id task; run;
Some words of caution are worth mentioning with regard to thetraditional PERT approach. The estimate of the mean project durationobtained in this instance always underestimates the true value sincethe length of acritical path is a convex function of the activity durations.The original PERT model developed by Malcolm et al. (1959)provides a way toestimate the variance of the project duration as well as calculatingthe probabilities of meeting certain target dates and so forth. Their analysisrelies on an implicit assumption that you may ignore all activitiesthat are not on the critical path in the deterministic problem that isderived by setting the activity durations equal to the mean value oftheir distributions. It then applies the Central Limit Theorem to theduration of this critical path and interprets the result as pertainingto the project duration.
Output 2.21.1: PERT Statistical Estimates: Gantt Chart
However, when the activity durations are random variables, each pathof the project network is a likely candidate to be the critical path.Every outcome of the activity durations could result in a differentlongest path. Furthermore, there could be several dependent paths inthe network in the sense that they share at least one common arc.Thus, in the most general case, the length of a longest path would bethe maximum of a set of, possibly dependent, random variables.Evaluating or approximating the distribution of the longest path,even under very specific distributional assumptions on the activitydurations is not a very easy problem. It is not surprising that thistopic is the subject of much research.
In view of the inaccuracies that can stem from the original PERTassumptions, many people prefer to resort to the use of Monte CarloSimulation. Van Slyke (1963) made the first attempt at straightforwardsimulation to analyze the distribution of the critical path.Refer to Elmaghraby (1977) for a detailed synopsis of the pitfalls of makingtraditional PERT assumptions and for an introduction to simulationtechniques for activity networks.
| Chapter Contents | Previous | Next | Top |
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
