【正文】 deadline is equal to the probability that the standard normal distribution is less than or equal to (PD D)| D where PD is the project deadline, D is the expected duration and D is the standard deviation of project duration. For example, the probability of project pletion within 35 days is: where z is the standard normal distribution tabulated value of the cumulative standard distribution appears in Table of Appendix B. Monte Carlo simulation results provide slightly different estimates of the project duration characteristics. Assuming that activity durations are independent and approximately normally distributed random variables with the mean and variances shown in Table 111, a simulation can be performed by obtaining simulated duration realization for each of the nine activities and applying critical path scheduling to the resulting work. Applying this procedure 500 times, the average project duration is found to be days with a standard deviation of days. The PERT result is less than this estimate by days or three percent. Also, the critical path considered in the PERT procedure (consisting of activities A, C, F and I) is found to be the critical path in the simulated works less than half the time. TABLE 111 Activity Duration Estimates for a Nine Activity Project Activity Optimistic Duration Most Likely Duration Pessimistic Duration Mean Variance A B C D E F G H I 3 2 6 5 6 10 2 4 4 4 3 8 7 9 12 2 5 6 5 5 10 8 14 14 4 8 8 If there are correlations among the activity durations, then significantly different results can be obtained. For example, suppose that activities C, E, G and H are all positively correlated random variables with a correlation of for each pair of variables. Applying Monte Carlo simulation using 500 activity work simulations results in an average project duration of days and a standard deviation of days. This estimated average duration is days or 20 percent longer than the PERT estimate or the estimate obtained ignoring uncertainty in durations. If correlations like this exist, these methods can seriously underestimate the actual project duration. Finally, the project durations obtained by assuming all optimistic and all pessimistic activity durations are 23 and 41 days respectively. Other whatif simulations might be conducted for cases in which peculiar soil characteristics might make excavation difficult。d|x = xk) of a normally distributed variable given a realization of the second variable is: () where dx is the correlation coefficient between d and x. Once xk is known, the conditional mean and standard deviation can be calculated from Eq. () and then a realization of d obtained by applying Equation (). Correlation coefficients indicate the extent to which two random variables will tend to vary together. Positive correlation coefficients indicate one random variable will tend to exceed its mean when the other random variable does the same. From a set of n historical observations of two random variables, x and y, the correlation coefficient can be estimated as: () The value of xy can range from one to minus one, with values near one indicating a positive, near linear relationship between the two random variables. It is also possible to develop formulas for the conditional distribution of a random variable correlated with numerous other variables。 a numerical example of this optimism appears below. Second, the use of single activity durations often produces a rigid, inflexible mindset on the part of schedulers. As field managers appreciate, activity durations vary considerable and can be influenced by good leadership and close attention. As a result, field managers may loose confidence in the realism of a schedule based upon fixed activity durations. Clearly, the use of fixed activity durations in setting up a schedule makes a continual process of monitoring and updating the schedule in light of actual experience imperative. Otherwise, the project schedule is rapidly outdated. A second simple approach to incorporation uncertainty also deserves mention. Many managers recognize that the use of expected durations may result in overly optimistic schedules, so they include a contingency allowance in their estimate of activity durations. For example, an activity with an expected duration of two days might be scheduled for a period of days, including a ten percent contingency. Systematic application of this contingency would result in a ten percent increase in the expected time to plete the project. While the use of this ruleofthumb or heuristic contingency factor can result in more accurate schedules, it is likely that formal scheduling methods that incorporate uncertainty more formally are useful as a means of obtaining greater accuracy or in understanding the effects of activity delays. The most mon formal approach to incorporate uncertainty in the scheduling process is to apply the critical path scheduling process (as described in Section ) and then analyze the results from a probabilistic perspective. This process is usually referred to as the PERT scheduling or evaluation method. [1] As noted earlier, the duration of the critical path represents the minimum time required to plete the project. Using expected activity durations and critical path scheduling, a critical path of activities can be identified. This critical path is then used to analyze the duration of the project incorporating the uncertainty of the activity durations along the critical path. The expected project duration is equal to the sum of the expected durations of the activities along the critical path. Assuming that activity durations are independent rand