## Toward Realistic Time Estimates

The calculations of expected network times, and the uncertainty associated withjj those time estimates performed in the preceding sections are based, as we notedA on estimating optimistic and pessimistic times at the .99 level. That is, a is esti-s'~ mated such that the actual time required for an activity will be a or higher 99 per-sf cent of the time and will be b or lower 99 percent of the time. We then noted, paren-T thetically, that no project managers of our acquaintance are comfortable making.;:-estimates at that level of precision. rM

Fortunately, in practice it is not necessary to make estimates at the one-in-a-*-hundred level. Unless the underlying distribution is very asymmetric, no great errorjs is introduced in finding TE if the pessimistic and optimistic estimates are made agsr the 95 percent, or even at the 90 percent levels; that is to say, only once in 20 times^'t (or ten times for the 90 percent level) will the actual activity time be greater than oJ|| less than the pessimistic or optimistic estimates, respectively. The formula for cal*» culating the variance of an activity, however, must be modified. -|j|'

Recall that the calculation of variance is based on the assumption that thep standard deviation of a beta distribution is approximately one-sixth of its range||L Another way of putting this assumption is that a and b are estimated at the and +3o- limits respectively—roughly at the 99+ percent levels. Let the 95 perceng| estimates be represented by a' and b' and the 90 percent estimates by a" and b" we use a 95 or 90 percent estimation level, we are actually moving both a and b t|j|[, from the distribution's tails so that the range will no longer represent ±3<r. Seffig.., Figure 8-17. V

It is simple to correct the calculation of variance for this error. Consider the percent estimates. Referring to Table 8-4 we can find the Z associated with .95 of th^. area under the curve from a' to For .95, Z is approximately -1.65. (Of course, thl& applies to the normal distribution rather than to the beta distribution, but thi,fi| huristic appears to work quite well in practice.) Similarly, Z = 1.65 for the area u"'. der the curve from to V.

8.2 NETWORK TECHNIQUES: PERT AND CPM 353

8.2 NETWORK TECHNIQUES: PERT AND CPM 353 The range between b' and a' represents 2(1.65)cr = 33a, rather than the 6cr used in the traditional estimation of the variance. Therefore when estimating a' and b' at the 95 percent level, we should change the variance calculation formula to read

For estimations at the 90 percent level (a" and b" in Figure 8-17), Z is approximately 1.3 and the variance calculation becomes a2 = ((Ir - a")/2.6)2

In order to verify that this modification of the traditional estimator for the variance of a beta distribution gave good estimates of the true variance, we ran a series of trials using Statistical Analysis Systems (SAS) PROC IML for beta distributions of different shapes and estimated a and b at the 95 and 90 percent levels. We then compared these estimates of a and b with the true variance of the distribution and found the differences to be quite small, consistently under five percent.

An alternate method for approximating the mean and variance of a beta distribution when a and b are estimated at the 95 percent level is given in the last section of Appendix E. For the full exposition of the method, see reference 5 of Appendix E.

toother Lotus 1-2-3® File

Just as we did in Chapter 7 on budgeting, we can construct a Lotus 1-2-3 template to do the calculations for finding the expected times, variances, and standard deviations associated with a series of three-time estimates for PERT/CPM networks. Figure 8-18 shows the file itself and Figure 8-19 shows the formulas used. Remember that the formula for calculation of variance must be modified according to the previous section unless a and b are estimated at the 99 + percent levels.

Most of the widely used project management software will not accept three-time estimates or do the necessary calculations to use such estimates, but a large majority of such software packages will routinely exchange information with Lotus 1-2-3,® Excel,® and similar spreadsheet software. It Is therefore quite simple to enter the three-time estimates into a Lotus 1-2-3® file and enter the expected activity times, TEs, into a project management scheduling package where they can be used as if they were deterministic times in finding a project's critical path and time. Calculations of the probability of completing a project on or before some elapsed time can easily be done by hand. 