## Info

.5359 ;» .5753 J' ■6141 k, ■6517 .6879*4'. .7224sfc

We turn now to Table 8-4, which shows the probabilities associated with vat ous levels of Z. (Table 8-4 also appears as Appendix C. It is shown here for the rea.. er's convenience.) We go down the left column until we find Z = 1.2, and then across to column .02 to find Z = 1.22. The probability value of Z = 1.22 shown in the table is .8888, which is the likelihood that we will complete the critical path our sample project within 50 days of the time it is started. Figure 8-16 shows the resulting probability distribution of the project completion times.*

'Our use of the normal distribution is allowed by the Central Limit Theorem which attests tot fact that the sum of independent activity times is normally distributed if the number of activities is lar,„

43 50

Time (Days)

43 50

Time (Days)

Figure 8-16: Probability distribution of project completion times.

We can work the problem backward, too. What deadline is consistent with a .95 probability of on-time completion? First, we go to Table 8-4 and look through the table until we find .95. The Z value associated with .95 is 1.645. (The values in the table are not strictly linear, so our interpolation is only approximate.) We know that p. is 43 days, and that Verbis 5.745. Solving the equation for D, we have

Thus, there is a 95 percent chance of finishing the project by 52.45 days.

Note that as D approaches p., Z gets smaller, approaching zero. Table 8-4 shows that for Z = 0, the chance of on-time completion is 50-50. The managerial implications are all too clear. If the PM wants a reasonable chance of meeting a project deadline, there must be some slack in the project schedule. When preparing a project budget, it is quite proper to include some allowance for contingencies. The same principle holds for preparing a project schedule. The allowance for contingencies in a schedule is network slack, and the wise PM will insist on some.

Finally, to illustrate an interesting point, let's examine a noncritical path, activities b-g-i. The variance of this path (from Figure 8-15) is 0 + 0 + 28.4 = 28.4, which is slightly less than the variance of the critical path. The path time is 42 days. The numerator of the fraction (D —p,)/ Vo^2 is larger, and in this case the denominator is smaller. Therefore, Z will be larger, and the probability of this path delaying project completion is less than for the critical path. But consider the noncritical path c-h-i with a time of 10+11 + 18 = 39 days, and a total variance of 37,8. (Remember, we are trying to find the probability that this noncritical path with its higher variance but shorter completion time will make us late, given that the critical path is 43 days.)

The result is that we have a 96 percent chance for this noncritical path to allow the project to be on time.

If the desired time for the network equaled the critical time, 43 days, we have seen that the critical path has a 50-50 chance of being late. What are the chances that the noncritical path c-h-1, will make the project late? D is now 43 days, so we have

Z = .65 is associated with a probability of .74 of being on time, or 1 — .74 = .26 of being late.

Assuming that these two paths (a-d-j and c-h-i) are independent, the probabil- 5 ity that both paths will be completed on time is the product of the individual probabilities, (,50)(.74) = .37, which is considerably less than the 50-50 we thought the chances were. (If the paths are not independent, the calculations become more complicated.) Therefore, it is a good idea to consider noncritical paths that have activities with large variances and/or path times that are close to critical in duration * (i.e., those with little slack). \

Simulation is an obvious way to check the nature and impacts of interactions» between probabilistic paths in a network. While this used to be difficult and time'*'.' consuming, software has now been developed which simplifies matters greatly. Two»], excellent software packages have been developed which link to widely available" J-spreadsheets: Crystal Ball ® which runs as a part of Excell®, and At Risk® which 4a runs as a part of Lotus 1-2-3®. Both allow easy simulation of network interactions It, 