【正文】 this is termed a multivariate distribution. [4] Random number generations from other types of distributions are also possible. [5] Once a set of random variable distributions is obtained, then the process of applying a scheduling algorithm is required as described in previous sections. Example 112: A ThreeActivity Project Example Suppose that we wish to apply a Monte Carlo simulation procedure to a simple project involving three activities in series. As a result, the critical path for the project includes all three activities. We assume that the durations of the activities are normally distributed with the following parameters: Activity Mean (Days) Standard Deviation (Days) A B C To simulate the schedule effects, we generate the duration realizations shown in Table 113 and calculate the project duration for each set of three activity duration realizations. For the twelve sets of realizations shown in the table, the mean and standard deviation of the project duration can be estimated to be days and days respectively. In this simple case, we can also obtain an analytic solution for this duration, since it is only the sum of three independent normally distributed variables. The actual project duration has a mean of days, and a standard deviation of days. With only a limited number of simulations, the mean obtained from simulations is close to the actual mean, while the estimated standard deviation from the simulation differs significantly from the actual value. This latter difference can be attributed to the nature of the set of realizations used in the simulations。d|x = xk) and standard deviation ( 39。 where k is a constant which can be expressed in terms of and . Several beta distributions for different sets of values of and are shown in Figure 111. For a beta distribution in the interval having a modal value m, the mean is given by: () If + = 4, then Eq. () will result in Eq. (). Thus, the use of Eqs. () and () impose an additional condition on the beta distribution. In particular, the restriction that = (b a)/6 is imposed. Figure 111 Illustration of Several Beta Distributions Since absolute limits on the optimistic and pessimistic activity durations are extremely difficult to estimate from historical data, a mon practice is to use the niyfifth percentile of activity durations for these points. Thus, the optimistic time would be such that there is only a one in twenty (five percent) chance that the actual duration would be less than the estimated optimistic time. Similarly, the pessimistic time is chosen so that there is only a five percent chance of exceeding this duration. Thus, there is a niy percent chance of having the actual duration of an activity fall between the optimistic and pessimistic duration time estimates. With the use of niyfifth percentile values for the optimistic and pessimistic activity duration, the calculation of the expected duration according to Eq. () is unchanged but the formula for calculating the activity variance bees: () The difference between Eqs. () and () es only in the value of the divisor, with 36 used for absolute limits and 10 used for niyfive percentile limits. This difference might be expected since the difference between bi,j and ai,j would be larger for absolute limits than for the niyfifth percentile limits. While the PERT method has been made widely available, it suffers from three major problems. First, the procedure focuses upon a single critical path, when many paths might bee critical due to random fluctuations. For example, suppose that the critical path with longest expected time happened to be pleted early. Unfortunately, this does not necessarily mean that the project is pleted early since another path or sequence of activities might take longer. Similarly, a longer than expected duration for an activity not on the critical path might result in that activity suddenly being critical. As a result of the focus on only a single path, the PERT method typically underestimates the actual project duration. As a second problem with the PERT procedure, it is incorrect to assume that most construction activity durations are independent random variables. In practice, durations are correlated with one another. For example, if problems are encountered in the delivery of concrete for a project, this problem is likely to influence the expected duration of numerous activities involving concrete pours on a project. Positive correlations of this type between activity durations imply that the PERT method underestimates the variance of the critical path and thereby produces overoptimistic expectations of the probability of meeting a particular project pletion deadline. Finally, the PERT method requires three duration estimates for each activity rather than the single estimate developed for critical path scheduling. Thus, the difficulty and labor of estimating activity characteristics is multiplied threefold. As an alternative to the PERT procedure, a straightforward method of obtaining information about the distribution of project pletion times (as well as other schedule information) is through the use of Monte Carlo simulation. This technique calculates sets of artificial (but realistic) activity duration times and then applies a deterministic scheduling procedure to each set of durations. Numerous calculations are required in this process since simulated activity durations must be calculated and the scheduling procedure applied many times. For realistic project works, 40 to 1,000 separate sets of activity durations might be used in a single scheduling simulation. The calculations associated with Monte Carlo simulation are described in the following section. A number of different indicators of the project sched